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Addressing Alternative ExplanationsExplanations Multiple Regression
17871 Spring 2012
1
t t
Did Cli h G lDid Clinton hurt Gore example
Did Clinton hurt Gore in the 2000 election Treatment is not liking Bill Clinton Treatment is not liking Bill Clinton
2
Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer
Clinton thermometer
3
t t
Did Cli h G lDid Clinton hurt Gore example
What alternative explanations would you need to address
Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources
Letrsquos address one source party identification How could we do this
Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents
Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive
4
t tDDemocratiic piicture
Clinton thermometer
5
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t t
Did Cli h G lDid Clinton hurt Gore example
Did Clinton hurt Gore in the 2000 election Treatment is not liking Bill Clinton Treatment is not liking Bill Clinton
2
Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer
Clinton thermometer
3
t t
Did Cli h G lDid Clinton hurt Gore example
What alternative explanations would you need to address
Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources
Letrsquos address one source party identification How could we do this
Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents
Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive
4
t tDDemocratiic piicture
Clinton thermometer
5
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Bivariate regression of Gore thermometer onBivariate regression of Gore thermometer on Clinton thermometer
Clinton thermometer
3
t t
Did Cli h G lDid Clinton hurt Gore example
What alternative explanations would you need to address
Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources
Letrsquos address one source party identification How could we do this
Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents
Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive
4
t tDDemocratiic piicture
Clinton thermometer
5
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t t
Did Cli h G lDid Clinton hurt Gore example
What alternative explanations would you need to address
Nonrandom selection into the treatment group (disliking Nonrandom selection into the treatment group (disliking Clinton) from many sources
Letrsquos address one source party identification How could we do this
Matching compare Democrats who like or donrsquot like Clinton do the same for Republicans and independents
Multivariate regression control for partisanship statistically Also called multiple regression Ordinary Least Squares (OLS) Presentation below is intuitive
4
t tDDemocratiic piicture
Clinton thermometer
5
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t tDDemocratiic piicture
Clinton thermometer
5
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t tI dIndependdent piicture
Clinton thermometer
6
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
tR bli Republican piicture
Clinton thermometer
7
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t t C bi d d i Combined data picture
Clinton thermometer
8
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Combined data picture withCombined data picture with regression bias
Clinton thermometer
9
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Combined data picture withCombined data picture with ldquotruerdquo regression lines overlaid
Clinton thermometer
10
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Tempting yet wronggp g y normalizations
Subtract the Gore therm from the avg Gore therm avg Gore therm score
Clinton thermometer
Subtract the ClintonSubtract the Clinton therm from the avg Clinton therm score
Clinton thermometer
11
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
12
3D R l hi3D Relatitionship
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
13
3D Linear Relationship3D Linear Relationship
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
14
t3D R 3D Rellationshihip Clintonti Cli
100 50 0
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
15
t3D R l hi3D Relatitionship party
Rep Ind Dem
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
The Linear Relationship between ThreeThe Linear Relationship between Three Variables
ClintonGore
XXY
Clinton thermometer
Gore thermometer Party ID
iiii XXY 22110
16
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
The method of least squaresThe method of least squares (again)
Pick and to minimizePick and to minimize 0 1 2
n
((YY YY ))2 oror ii ii i1
n
(Y X X )2 i 0 1 i 2 2
i1
17
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
nn
Th Sl C ffiThe Slope Coefficiients n n
(Y Y )(X X ) (X X )(X Xi 1 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - and 11 nn 22 nn
(X X )2 (X X1 1i 1 1 )2
i i1 i1
n n
(Y Y )(X X ) (X X )(X Xi 2 1i 1 1i 2 2 )i ˆ i1 ˆ i1 - 2 n 1 n
2 ((XX 2 XX 2i )) ((XX 2 XX 2i )) 2 22
i1 i1 0 1 X1 2 XY 2
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
18
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
The Slope Coefficients MoreThe Slope Coefficients More Simply
ˆ cov(X1Y ) ˆ cov(X1 X 2 ) and - and 1 var(X1) 2 var(X1)
ˆ cov(X 2 Y ) ˆ cov(X1 X 2 )2 - 1var(X 2 ) var(X 2 ))var(X ) var(X
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer
19
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
Th M fThe Matriix form
y1
y2
hellip
yyn
1 x11 x21 hellip xk1
1 x12 x22 hellip xk2
1 hellip hellip hellip hellip
11 xx1n xx2n hellip xxkn
(( X X )) 1 X yy 20
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Multivariate slope coefficientsMultivariate slope coefficients
cov((X1Y )) Bivariate estimate B 1 vs Bivariate estimate vs1 var( X1)
ˆ M cov(X1Y ) ˆ M
Are Gore and Party ID related
cov(X1 X 2 ) Multivariate estimate - Multivariate estimate 1 22
Clinton effect ( G ) i (on Gore) in
multivariate (M) regression
1 var( X1) var( X1)
Clinton effect (on Gore) in bivariate (B)bivariate (B) regression
Are Clinton and Party IDy related
cov( X X )cov( X X )B M 1 2B MM MWhen does ˆ ˆ Obviously when 021 1 var( X 1 )
X1 is Clinton thermometer X2 is PID and Y is Gore thermometer 21
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t t
-
------------------------------------------------------------------------------
------------------------------------------------------------------------------
Th OThe Output reg gore clinton party3
Source | SS df MS Number of obs = 1745 -------------+------------------------------ F( 2 1742) = 104804
Model | 6292619162926191 22 314630955 Prob
Prob
gt F =
Model | 00000 Residual | 522964934 1742 300209492 R-squared = 05461
-------------+------------------------------ Adj R-squared = 05456 Total | 115222684 1744 66068053 Root MSE = 17327
gore | Coef Std Err t Pgt|t| [95 Conf Interval] -------------+----------------------------------------------------------------
clinton | 5122875 0175952 2912 0000
4777776 5467975 party3 || 5770523 5594846 1031
0000 4673191 6867856
_cons | 286299 1025472 2792 0000 2661862 3064119
Interpretation of clinton effect Holding constant party identification a one-Interpretation of clinton effect Holding constant party identification a one point increase in the Clinton feeling thermometer is associated with a 51 increase in the Gore thermometer
22
2 314630955 Prob Prob F 00000
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
tS iSeparate regressions
(1) (2) (3) Interceptp 231 559 286 Clinton 062 -- 051 PartyParty -- 157157 5858
ˆ cov(X11Y ) ˆ cov(X11 X 22 )1 - 2 and and var(X1) var(X1)
cov(X Y ) ˆ cov(X X )ˆ cov(X 22 Y ) cov(X 11 X 22 )2 - 1var(X 2 ) var(X 2 ) 23
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
Why did the Clinton CoefficientWhy did the Clinton Coefficient change from 062 to 051
corr gore clinton party cov (obs=1745)
| gore clinton party3 -------------++---------------------------
gore | 660681 clinton | 549993 883182
3 | party3 | 13 7008 137008 16 905 16905 87358735
24
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
ThThe CCallcullatitions B cov(gore )cov(goreclintonclinton) 549549993993 B 062271 var(clinton) 883182
M cov(goreclinton) M cov(clinton party)ˆ ˆ 1 221 var((clilintton)) var((clilintton)) 549993 16905
57705 883182 883182
corr gore clinton partycov (obs=1745) 06227 01105
| gore clinton party3 -------------+---------------------------
gore | 660681 05122 clinton | 549993 883182 party3 | 137008 16905 8735
25
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Another way of thinking aboutAnother way of thinking about this
R it Rewrite cov(goreclinton) cov(clinton party)ˆ M ˆ M 1 221 var(clinton) var(clinton)
as cov(goreclinton) cov(clinton party)
2 1 M ˆ M
var(clinton) var(clinton)
Total effect = Direct effect + indirect effect
The Total Effect of the Clinton thermometer on the Gore thermometer (61) can be Broken down into a direct effect of 51 plus an indirect effect (though party) of 11
26
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
D i ki d G k Lif E lDrinking and Greek Life Example
Why is there a correlation between living in a fraternityysororityy house and drinkingg Greek organizations often emphasize social
gatheringgs that have alcohol The effect is g being in the Greek organization itself not the house
Therersquos something about the House environment itself
27
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
Dependent variable TimesDependent variable Times Drinking in Past 30 Days
HFKVOHU+HQUampROOHJH$OFRKRO6WXG+DUYDUG6FKRRORI3XEOLF+HDOWK +DUYDUG6FKRRORI3XEOLF+HDOWK$OOULJKWVUHVHUYHG7KLVFRQWHQWLVH[FOXGHGIURP RXUampUHDWLYHampRPPRQVOLFHQVH)RUPRUHLQIRUPDWLRQVHHKWWSRFZPLWHGXIDLUXVH
28
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
infix age 10-11 residence 16 greek 24 screen 102 timespast30 103 howmuchpast30 104 gpa 278-279 studying 281 i h 325 h hh 326 i li i 283 99 475 493timeshs 325 howmuchhs 326 socializing 283 stwgt_99 475-493 weight99 494-512 using da3818datclear (14138 observations read)
recode timespast30 timeshs (1=0) (2=15) (3=4) (4=75) (5=145) (6=295) (7=45) (timespast30 6571 changes made) (timeshs 10272 changes made)(timeshs 10272 changes made)
replace timespast30=0 if screenlt=3 (4631 real changes made)
29
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
tab timespast30
timespast30 | Freq Percent Cum ------------++-----------------------------------
0 | 4652 3337 3337 15 | 2737 1964 5301 4 | 2653 1903 7204
75 | 1854 1330 8534 145 | 1648 1182 9717 295 | 350 251 9968 45 |45 | 4545 0 32 032 100 00 10000
------------+-----------------------------------Total | 13939 10000
30
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
K l bl Key explanatory variiables
Live in fraternitysorority house Indicator variable (dummy variable) Indicator variable (dummy variable) Coded 1 if live in 0 otherwise
Member of fraternitysororityMember of fraternitysorority Indicator variable (dummy variable) Coded 1 if member 0 otherwise Coded 1 if member 0 otherwise
31
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
Th RThree Regressiions Dependent variable number of times drinking in past 30 days
Live in fratsor house (indicator variable)
M b f f Member of fratsor (indicator variable)
Intercept
SER
R2R2
N
444 (035)
---
454 (056)
649
011 011
13876
---
2 88 288 (016) 427
(0059)
644
023023
13876
226 (038)
2 44 244 (018) 427
(0059)
644
025025
13876 What is the substantive interpretation of the coefficients N t St d d i th C B t li i iNote Standard errors in parentheses Corr Between living in fratsor house and being a member of a Greek organization is 42
32
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
Th PiThe Picture
i k Living in frat house
=226
X
M 2
Drinks per 30 days
house
=01921 Y
X2
Member of fraternity =244
X1
M 1 y 1
33
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t t t
A ti f th l ffAccounting for the total effect
ˆ B ˆ M ˆ M 1 1 2 21
Total effect = Direct effect + indirect effect
M =226
Drinks per Living in frat house X2
2
Drinks per 30 days
=01921 Y
Member of fraternity =244
X1
M 1
34
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
t
Live in frat
Accounting for the effects of fratg house living and Greek membership on drinkingmembership on drinking From
accounting identity T=D+I
Effect Total Direct Indirect Member of 288 244 044 Greek org (85) (15)
444 226 218 sor house (51) (49)
From bi i
From multiple regressions
bivariate regressions
35
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms
MIT OpenCourseWarehttpocwmitedu
17871 Political Science LaboratorySpring 2012
For information about citing these materials or our Terms of Use visit httpocwmiteduterms