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General Linear Models -- #1. things to remember b weight interpretations 1 quantitative predictor 1 2-group predictor 1 k-group predictor 1 quantitative & a 2-group predictors 1 quantitative & a k-group predictors 2 quantitative predictors 2x2 – main effects 2x2 with interactions - PowerPoint PPT Presentation
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General Linear Models -- #1
• things to remember• b weight interpretations• 1 quantitative predictor• 1 2-group predictor• 1 k-group predictor• 1 quantitative & a 2-group predictors • 1 quantitative & a k-group predictors• 2 quantitative predictors• 2x2 – main effects• 2x2 with interactions• 2x3 – main effects• 2x3 with interactions
A few important things to remember…
• we plot and interpret the model of the data, not the data
• if the model fits the data poorly, then we’re carefully describing and interpreting nonsense
• the interpretation of regression weights in a main effects model (without interactions) is different than in a model including interactions
• regression weights reflect “main effects” in a maineffects model
• regression weights reflect “simple effects” in a modelincluding interactions
b weight interpretations
Constant
the expected value of y when the value of all predictors = 0
Centered quantitative variable
the direction and extent of the expected change in the value of y for a 1-unit increase in that predictor, holding the value of all other predictors constant at 0
Dummy Coded binary variable
the direction and extent of expected mean difference of the Target group from the Comparison group, holding the value of all other predictors constant
Dummy Coded k-group variable
the direction and extent of the expected mean difference of the Target group for that dummy code from the Comparison group, holding the value of all other predictors constant.
b weight interpretations
Interaction between quantitative variables
the direction and extent of the expected change in the slope of the linear relationship between y and one predictor for each 1-unit change in the other predictor, holding the value or all other predictors constant at 0
Interaction between quantitative & Dummy Coded binary variablesthe direction and extent of expected change in the slope of the linear relationship between y and the quantitative variable of the Target group from the slope of the Comparison group, holding the value of all other predictors constant at 0
Interaction between quantitative & Dummy Coded k-group variables
the direction and extent of expected change in the slope of the linear relationship between y and the quantitative variable of the Target group for that dummy code from the slope of the Comparison group, holding the value of all other predictors constant at 0
Interaction between Dummy Coded Variablesthe direction and extend of expected change in the mean difference between the IVx Target & Comparison groups of the IVz Target group from mean difference between the IVx Target & Comparison groups of the IVz Comparison group, holding the value of all other predictors constant at 0.
0
10
20
30
4
0
50
60
y’ = b0 + b1 X
b0
b1
-20 -10 0 10 20 X
b0 = ht of line
b1 = slp of line
X = X – Xmean
Single quantitative predictor (X) Bivariate Regression
0
10
20
30
4
0
50
60
Cx
Tx
2-group predictor (Tx Cx) 2-grp ANOVA
b0 = ht Cx
b1 = htdif Cx & Tx
X Tx = 1 Cx = 0
X = Tx vs. Cx
b0
b1
y’ = b0 + b1X
0
10
20
30
4
0
50
60
b2
Cx
Tx2
Tx1
b0 = ht Cx
b1 = htdif Cx & Tx1
b2 = htdif Cx & Tx2
3-group predictor (Tx1 Tx2 Cx) k-grp ANOVA
y’ = b0 + b1X1 + b2X2
X1 Tx1=1 Tx2=0 Cx=0 X2 Tx1=0 Tx2=1 Cx=0
X1 = Tx1 vs. Cx X2 = Tx2 vs. Cx
b0
b1
0
10
20
30
4
0
50
60
y’ = b0 + b1X + b2Z
b0
b1b2
Cz
Tz
quantitative (X) & 2-group (Tz Cz) predictors 2-grp ANCOVA
-20 -10 0 10 20 X
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
Z-lines all have same slp(no interaction)
0
10
20
30
4
0
50
60
b0
b1b2
Cz
Tz
-20 -10 0 10 20 Xcen
XZ = Xcen * Z
b3
b0 = ht of Cz line
b1 = slp of Cz line
b2 = htdif Cz & Tz
b3 = slpdif Cz & Tz
quantitative (X) & 2-group (Tz Cz) predictors w/ interaction
y’ = b0 + b1X + b2Z + b3XZ
X = X – Xmean Z Tz = 1 Cz = 0
Z = Tz vs. Cz
0
10
20
30
4
0
50
60
b0
b1
b2
Cz
Tz2
Tz1
b3
-20 -10 0 10 20 X
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
y’ = b0 +b1X + b2Z1 + b3Z2
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
quantitative (X) & 3-group (Tz1 Tz2 Cz) predictors 3-grp ANCOVA
Z-lines all have same slp(no interaction)
0
10
20
30
4
0
50
60
y’ = b0 + b1Xcen + b2Z1 + b3Z2 + b4XZ1 + b5XZ2
b0
b1b2
Cx
Tx1
Tx2 b3
-20 -10 0 10 20 Xcen
b0 = ht of Cz line
b2 = htdif Cz & Tz1
b3 = htdif Cz & Tz2
b1 = slp of Cz line
b4 = slpdif Cz & Tz1
b5 = slpdif Cz & Tz2
XZ1 = Xcen * Z1
XZ2 = Xcen * Z2
b4
b5
Models with quant (X) & 3-group (Tz1 Tz2 Cz) predictors w/ interaction
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
X = X – Xmean Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
0
10
20
30
4
0
50
60
y’ = b0 + b1X + b2Z
b0
b1 b2Z=0
+1std Z
-1std Z
b2
Z = Z – Zmean
2 quantitative predictors multiple regression
-20 -10 0 10 20 X
b0 = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
X = X – Xmean
Z-lines all have same slp(no interaction)
0
10
20
30
4
0
50
60
y’ = b0 + b1Xcen + b2Zcen + b3XZ
b0
b1
-b2Z=0
+1std Z
-1std Z
b2
Zcen = Z – Xmean
2 quantitative predictors w/ interaction
-20 -10 0 10 20 Xcen
a = ht of Zmean line
b1 = slope of Zmean line
b2 = htdifs among Z-lines
Xcen = X – Xmean ZX = Xcen * Zcen
b3
b3b3 = slpdifs among Z-lines
0
10
20
30
4
0
50
60
b0
b1
b2
2-group (Tx Cx) & 2-group (Tz Cz) predictors Main Effects Model
0 1 X
b0 = mean of CxCz
b1 = htdif of CxCz & TxCz
b2 = htdifs of CxCz
& CxTz
= simple effects (no interaction)
X = Tx vs. Cx
y’ = b0 + b1X + b2Z
Z = Tz vs. Cz
XC T
TZ
C CxCz
CxTz
TxCz
TxCz
CxCz
CxTz
TxTz
TxTz
Z Tz = 1 Cz = 0X Tx = 1 Cx = 0
0
10
20
30
4
0
50
60
b0
b1
b2
Models with 2-group (Tx Cx) & 2-group (Tz Cz) predictors 2x2 ANOVA
0 1 X
b0 = mean of CxCz
b1 = htdif of CxCz & TxCz
b2 = htdifs of CxCz
& CxTz
X = Tx vs. Cx
y’ = b0 + b1X + b2Z + b3XZ
Z = Tz vs. Cz
TxCz
CxCz
CxTzTxTz
Z Tz = 1 Cz = 0X Tx = 1 Cx = 0 XZ = X * Z
b3 = dif htdifs of
CxCz - TxCz & CxTz - TxTz
b3
vs.
0
10
20
30
4
0
50
60
b0
b1b2
Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors ME model
0 1 X
b0 = mean of CxCz
b1 = htdif of CxCz & TxCz
b2 = htdifs of CxCz & CxTz1
= simple effects (no interaction)
y’ = b0 + b1X + b2Z1 + b3Z2
b3
CxCz
CxTz1
CxTz2
TxCz
TxTz1
TxTz2
b3 = htdifs of CxCz & CxTz2
Z C T1 T2
CX
T
CxCzTxCz TxTz1
CxTz1CxTz2
TxTz2
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
X = Tx vs. Cx
X Tx = 1 Cx = 0
0
10
20
30
4
0
50
60
b0
b1
b2
Models with 2-group (Tx Cx) & 3-group (Tz1 Tz2 Cz) predictors 2x3 ANOVA
0 1 X
b0 = mean of CxCz
b1 = htdif of CxCz & TxCz
b2 = htdifs of CxCz & CxTz1
y’ = b0 + b1X + b2Z1 + b3Z2 +b4XZ1 + b5XZ2
b3
CxCz
CxTz1
CxTz2
TxCzTxTz1
TxTz2
b3 = htdifs of CxCz & CxTz2
Z1 = Tz1 vs. Cz Z2 = Tz2 vs. Cz
Z1 Tz1=1 Tz2=0 Cx=0 Z2 Tz1=0 Tx2=1 Cx=0
X = Tx vs. Cx
X Tx = 1 Cx = 0 XZ1 = X * Z1
XZ2 = X * Z2
b4 = dif htdifs of
CxCz - TxCz & CxTz1 – TxTz1
b5 = dif htdifs of
CxCz - TxCz & CxTz2 – TxTz2
b4
vs.
b5
vs.
