2002 Prentice-Hall, Inc.Chap 12-1 Statistics for Managers Using
Microsoft Excel 3 rd Edition Chapter 12 Multiple Regression
Slide 2
2002 Prentice-Hall, Inc. Chap 12-2 Chapter Topics The multiple
regression model Residual analysis Testing for the significance of
the regression model Inferences on the population regression
coefficients Testing portions of the multiple regression model
Slide 3
2002 Prentice-Hall, Inc. Chap 12-3 Chapter Topics The quadratic
regression model Dummy variables Using transformation in regression
models Collinearity Model building Pitfalls in multiple regression
and ethical considerations (continued)
Slide 4
2002 Prentice-Hall, Inc. Chap 12-4 Population Y-intercept
Population slopesRandom Error The Multiple Regression Model
Relationship between 1 dependent & 2 or more independent
variables is a linear function Dependent (Response) variable for
sample Independent (Explanatory) variables for sample model
Residual
Slide 5
2002 Prentice-Hall, Inc. Chap 12-5 Population Multiple
Regression Model Bivariate model
Slide 6
2002 Prentice-Hall, Inc. Chap 12-6 Sample Multiple Regression
Model Bivariate model Sample Regression Plane
Slide 7
2002 Prentice-Hall, Inc. Chap 12-7 Simple and Multiple
Regression Compared Coefficients in a simple regression pick up the
impact of that variable plus the impacts of other variables that
are correlated with it and the dependent variable. Coefficients in
a multiple regression net out the impacts of other variables in the
equation.
Slide 8
2002 Prentice-Hall, Inc. Chap 12-8 Simple and Multiple
Regression Compared:Example Two simple regressions: Multiple
regression:
Slide 9
2002 Prentice-Hall, Inc. Chap 12-9 Multiple Linear Regression
Equation Too complicated by hand! Ouch!
Slide 10
2002 Prentice-Hall, Inc. Chap 12-10 Interpretation of Estimated
Coefficients Slope (b i ) Estimated that the average value of Y
changes by b i for each 1 unit increase in X i holding all other
variables constant (ceterus paribus) Example: if b 1 = -2, then
fuel oil usage (Y) is expected to decrease by an estimated 2
gallons for each 1 degree increase in temperature (X 1 ) given the
inches of insulation (X 2 ) Y-intercept (b 0 ) The estimated
average value of Y when all X i = 0
Slide 11
2002 Prentice-Hall, Inc. Chap 12-11 Multiple Regression Model:
Example ( 0 F) Develop a model for estimating heating oil used for
a single family home in the month of January based on average
temperature and amount of insulation in inches.
Slide 12
2002 Prentice-Hall, Inc. Chap 12-12 Sample Multiple Regression
Equation: Example Excel Output For each degree increase in
temperature, the estimated average amount of heating oil used is
decreased by 5.437 gallons, holding insulation constant. For each
increase in one inch of insulation, the estimated average use of
heating oil is decreased by 20.012 gallons, holding temperature
constant.
Slide 13
2002 Prentice-Hall, Inc. Chap 12-13 Multiple Regression in
PHStat PHStat | regression | multiple regression EXCEL spreadsheet
for the heating oil example.
Slide 14
2002 Prentice-Hall, Inc. Chap 12-14 Venn Diagrams and
Explanatory Power of Regression Oil Temp Variations in Oil
explained by Temp or variations in Temp used in explaining
variation in Oil Variations in Oil explained by the error term
Variations in Temp not used in explaining variation in Oil
Slide 15
2002 Prentice-Hall, Inc. Chap 12-15 Venn Diagrams and
Explanatory Power of Regression Oil Temp (continued)
Slide 16
2002 Prentice-Hall, Inc. Chap 12-16 Venn Diagrams and
Explanatory Power of Regression Oil Temp Insulation Overlapping
variation NOT estimation Overlapping variation in both Temp and
Insulation are used in explaining the variation in Oil but NOT in
the estimation of nor NOT Variation NOT explained by Temp nor
Insulation
Slide 17
2002 Prentice-Hall, Inc. Chap 12-17 Coefficient of Multiple
Determination Proportion of total variation in Y explained by all X
variables taken together Never decreases when a new X variable is
added to model Disadvantage when comparing models
Slide 18
2002 Prentice-Hall, Inc. Chap 12-18 Venn Diagrams and
Explanatory Power of Regression Oil Temp Insulation
Slide 19
2002 Prentice-Hall, Inc. Chap 12-19 Adjusted Coefficient of
Multiple Determination Proportion of variation in Y explained by
all X variables adjusted for the number of X variables used
Penalize excessive use of independent variables Smaller than Useful
in comparing among models
Slide 20
2002 Prentice-Hall, Inc. Chap 12-20 Coefficient of Multiple
Determination Excel Output Adjusted r 2 reflects the number of
explanatory variables and sample size is smaller than r 2
Slide 21
2002 Prentice-Hall, Inc. Chap 12-21 Interpretation of
Coefficient of Multiple Determination 96.56% of the total variation
in heating oil can be explained by different temperature and amount
of insulation 95.99% of the total fluctuation in heating oil can be
explained by different temperature and amount of insulation after
adjusting for the number of explanatory variables and sample
size
Slide 22
2002 Prentice-Hall, Inc. Chap 12-22 Using The Model to Make
Predictions Predict the amount of heating oil used for a home if
the average temperature is 30 0 and the insulation is six inches.
The predicted heating oil used is 278.97 gallons
Slide 23
2002 Prentice-Hall, Inc. Chap 12-23 Predictions in PHStat
PHStat | regression | multiple regression Check the confidence and
prediction interval estimate box EXCEL spreadsheet for the heating
oil example.
Slide 24
2002 Prentice-Hall, Inc. Chap 12-24 Residual Plots Residuals
vs. May need to transform Y variable Residuals vs. May need to
transform variable Residuals vs. May need to transform variable
Residuals vs. time May have autocorrelation
Slide 25
2002 Prentice-Hall, Inc. Chap 12-25 Residual Plots: Example No
Discernable Pattern Maybe some non- linear relationship
Slide 26
2002 Prentice-Hall, Inc. Chap 12-26 Influence Analysis To
determine observations that have influential effect on the fitted
model Potentially influential points become candidate for removal
from the model Criteria used are The hat matrix elements h i The
Studentized deleted residuals t i * Cooks distance statistic D i
All three criteria are complementary Only when all three criteria
provide consistent result should an observation be removed
Slide 27
2002 Prentice-Hall, Inc. Chap 12-27 The Hat Matrix Element h i
If, X i is an influential point X i may be considered a candidate
for removal from the model
Slide 28
2002 Prentice-Hall, Inc. Chap 12-28 The Hat Matrix Element h i
: Heating Oil Example No h i > 0.4 No observation appears to be
candidate for removal from the model
Slide 29
2002 Prentice-Hall, Inc. Chap 12-29 The Studentized Deleted
Residuals t i * : difference between the observed and predicted
based on a model that includes all observations except observation
i : standard error of the estimate for a model that includes all
observations except observation i An observation is considered
influential if is the critical value of a two-tail test at 10%
level of significance
Slide 30
2002 Prentice-Hall, Inc. Chap 12-30 The Studentized Deleted
Residuals t i * :Example t 10 * and t 13 * are influential points
for potential removal from the model
Slide 31
2002 Prentice-Hall, Inc. Chap 12-31 Cooks Distance Statistic D
i is the Studentized residual If, an observation is considered
influential is the critical value of the F distribution at a 50%
level of significance
Slide 32
2002 Prentice-Hall, Inc. Chap 12-32 Cooks Distance Statistic D
i : Heating Oil Example No D i > 0.835 No observation appears to
be candidate for removal from the model Using the three criteria,
there is insufficient evidence for the removal of any observation
from the model
Slide 33
2002 Prentice-Hall, Inc. Chap 12-33 Testing for Overall
Significance Shows if there is a linear relationship between all of
the X variables together and Y Use F test statistic Hypotheses: H 0
: k = 0 (no linear relationship) H 1 : at least one i ( at least
one independent variable affects Y ) The null hypothesis is a very
strong statement Almost always reject the null hypothesis
Slide 34
2002 Prentice-Hall, Inc. Chap 12-34 Testing for Overall
Significance Test statistic: Where F has p numerator and (n-p-1)
denominator degrees of freedom (continued)
Slide 35
2002 Prentice-Hall, Inc. Chap 12-35 Test for Overall
Significance Excel Output: Example p = 2, the number of explanatory
variables n - 1 p value
Slide 36
2002 Prentice-Hall, Inc. Chap 12-36 Test for Overall
Significance Example Solution F 03.89 H 0 : 1 = 2 = = p = 0 H 1 :
At least one i 0 =.05 df = 2 and 12 Critical Value(s) : Test
Statistic: Decision: Conclusion: Reject at = 0.05 There is evidence
that at least one independent variable affects Y = 0.05 F 168.47
(Excel Output)
Slide 37
2002 Prentice-Hall, Inc. Chap 12-37 Test for Significance:
Individual Variables Shows if there is a linear relationship
between the variable X i and Y Use t test statistic Hypotheses: H 0
: i 0 (no linear relationship) H 1 : i 0 (linear relationship
between X i and Y)
Slide 38
2002 Prentice-Hall, Inc. Chap 12-38 t Test Statistic Excel
Output: Example t Test Statistic for X 1 (Temperature) t Test
Statistic for X 2 (Insulation)
Slide 39
2002 Prentice-Hall, Inc. Chap 12-39 t Test : Example Solution H
0 : 1 = 0 H 1 : 1 0 df = 12 Critical Value(s): Test Statistic:
Decision: Conclusion: Reject H 0 at = 0.05 There is evidence of a
significant effect of temperature on oil consumption. t 0 2.1788
-2.1788.025 Reject H 0 0.025 Does temperature have a significant
effect on monthly consumption of heating oil? Test at = 0.05. t
Test Statistic = -16.1699
Slide 40
2002 Prentice-Hall, Inc. Chap 12-40 Venn Diagrams and
Estimation of Regression Model Oil Temp Insulation Only this
information is used in the estimation of This information is NOT
used in the estimation of nor
Slide 41
2002 Prentice-Hall, Inc. Chap 12-41 Confidence Interval
Estimate for the Slope Provide the 95% confidence interval for the
population slope 1 (the effect of temperature on oil consumption).
-6.169 1 -4.704 The estimated average consumption of oil is reduced
by between 4.7 gallons to 6.17 gallons per each increase of 1 0
F.
Slide 42
2002 Prentice-Hall, Inc. Chap 12-42 Contribution of a Single
Independent Variable Let X k be the independent variable of
interest Measures the contribution of X k in explaining the total
variation in Y (SST)
Slide 43
2002 Prentice-Hall, Inc. Chap 12-43 Contribution of a Single
Independent Variable Measures the contribution of in explaining SST
From ANOVA section of regression for
Slide 44
2002 Prentice-Hall, Inc. Chap 12-44 Coefficient of Partial
Determination of Measures the proportion of variation in the
dependent variable that is explained by X k while controlling for
(holding constant) the other independent variables
Slide 45
2002 Prentice-Hall, Inc. Chap 12-45 Coefficient of Partial
Determination for (continued) Example: Two Independent Variable
Model
Slide 46
2002 Prentice-Hall, Inc. Chap 12-46 Venn Diagrams and
Coefficient of Partial Determination for Oil Temp Insulation =
Slide 47
2002 Prentice-Hall, Inc. Chap 12-47 Coefficient of Partial
Determination in PHStat PHStat | regression | multiple regression
Check the coefficient of partial determination box EXCEL
spreadsheet for the heating oil example
Slide 48
2002 Prentice-Hall, Inc. Chap 12-48 Contribution of a Subset of
Independent Variables Let X s be the subset of independent
variables of interest Measures the contribution of the subset x s
in explaining SST
Slide 49
2002 Prentice-Hall, Inc. Chap 12-49 Contribution of a Subset of
Independent Variables: Example Let X s be X 1 and X 3 From ANOVA
section of regression for
Slide 50
2002 Prentice-Hall, Inc. Chap 12-50 Testing Portions of Model
Examines the contribution of a subset X s of explanatory variables
to the relationship with Y Null hypothesis: Variables in the subset
do not improve significantly the model when all other variables are
included Alternative hypothesis: At least one variable is
significant
Slide 51
2002 Prentice-Hall, Inc. Chap 12-51 Testing Portions of Model
Always one-tailed rejection region Requires comparison of two
regressions One regression includes everything Another regression
includes everything except the portion to be tested
(continued)
Slide 52
2002 Prentice-Hall, Inc. Chap 12-52 Partial F Test For
Contribution of Subset of X variables Hypotheses: H 0 : Variables X
s do not significantly improve the model given all others variables
included H 1 : Variables X s significantly improve the model given
all others included Test Statistic: with df = m and (n-p-1) m = #
of variables in the subset X s
Slide 53
2002 Prentice-Hall, Inc. Chap 12-53 Partial F Test For
Contribution of A Single Hypotheses: H 0 : Variable X j does not
significantly improve the model given all others included H 1 :
Variable X j significantly improves the model given all others
included Test Statistic: With df = 1 and (n-p-1) m = 1 here
Slide 54
2002 Prentice-Hall, Inc. Chap 12-54 Testing Portions of Model:
Example Test at the =.05 level to determine whether the variable of
average temperature significantly improves the model given that
insulation is included.
Slide 55
2002 Prentice-Hall, Inc. Chap 12-55 Testing Portions of Model:
Example H 0 : X 1 (temperature) does not improve model with X 2
(insulation) included H 1 : X 1 does improve model =.05, df = 1 and
12 Critical Value = 4.75 (For X 1 and X 2 )(For X 2 ) Conclusion:
Reject H 0 ; X 1 does improve model
Slide 56
2002 Prentice-Hall, Inc. Chap 12-56 Testing Portions of Model
in PHStat PHStat | regression | multiple regression Check the
coefficient of partial determination box EXCEL spreadsheet for the
heating oil example.
Slide 57
2002 Prentice-Hall, Inc. Chap 12-57 Do We Need to Do this for
One Variable? The F test for the inclusion of a single variable
after all other variables are included in the model is IDENTICAL to
the t test of the slope for that variable The only reason to do an
F test is to test several variables together
Slide 58
2002 Prentice-Hall, Inc. Chap 12-58 The Quadratic Regression
Model Relationship between one response variable and two or more
explanatory variables is a quadratic polynomial function Useful
when scatter diagram indicates non- linear relationship Quadratic
model : The second explanatory variable is the square of the first
variable
Slide 59
2002 Prentice-Hall, Inc. Chap 12-59 Quadratic Regression Model
(continued) Quadratic models may be considered when scatter diagram
takes on the following shapes: X1X1 Y X1X1 X1X1 YYY 2 > 0 2 <
0 2 = the coefficient of the quadratic term X1X1
Slide 60
2002 Prentice-Hall, Inc. Chap 12-60 Testing for Significance:
Quadratic Model Testing for Overall Relationship Similar to test
for linear model F test statistic = Testing the Quadratic Effect
Compare quadratic model with the linear model Hypotheses (No 2 nd
order polynomial term) (2 nd order polynomial term is needed)
Slide 61
2002 Prentice-Hall, Inc. Chap 12-61 Heating Oil Example ( 0 F)
Determine whether a quadratic model is needed for estimating
heating oil used for a single family home in the month of January
based on average temperature and amount of insulation in
inches.
Slide 62
2002 Prentice-Hall, Inc. Chap 12-62 Heating Oil Example:
Residual Analysis No Discernable Pattern Maybe some non- linear
relationship (continued)
Slide 63
2002 Prentice-Hall, Inc. Chap 12-63 Heating Oil Example: t Test
for Quadratic Model Testing the quadratic effect Compare quadratic
model in insulation With the linear model Hypotheses (No quadratic
term in insulation) (Quadratic term is needed in insulation)
(continued)
Slide 64
2002 Prentice-Hall, Inc. Chap 12-64 Example Solution H 0 : 3 =
0 H 1 : 3 0 df = 11 Critical Value(s): Test Statistic: Decision:
Conclusion: Do not reject H 0 at = 0.05 There is not sufficient
evidence for the need to include quadratic effect of insulation on
oil consumption. Z 0 2.2010 -2.2010.025 Reject H 0 0.025 Is
quadratic model in insulation needed on monthly consumption of
heating oil? Test at = 0.05. t Test Statistic = 1.6611
Slide 65
2002 Prentice-Hall, Inc. Chap 12-65 Example Solution in PHStat
PHStat | regression | multiple regression EXCEL spreadsheet for the
heating oil example.
Slide 66
2002 Prentice-Hall, Inc. Chap 12-66 Dummy Variable Models
Categorical explanatory variable (dummy variable) with two or more
levels: Yes or no, on or off, male or female, Coded as 0 or 1 Only
intercepts are different Assumes equal slopes across categories The
number of dummy variables needed is (number of levels - 1)
Regression model has same form:
Slide 67
2002 Prentice-Hall, Inc. Chap 12-67 Dummy-Variable Models (with
2 Levels) Given: Y = Assessed Value of House X 1 = Square footage
of House X 2 = Desirability of Neighborhood = Desirable (X 2 = 1)
Undesirable (X 2 = 0) 0 if undesirable 1 if desirable Same
slopes
Slide 68
2002 Prentice-Hall, Inc. Chap 12-68 Undesirable Desirable
Location Dummy-Variable Models (with 2 Levels) (continued) X 1
(Square footage) Y (Assessed Value) b 0 + b 2 b0b0 Same slopes
Intercepts different
Slide 69
2002 Prentice-Hall, Inc. Chap 12-69 Interpretation of the Dummy
Variable Coefficient (with 2 Levels) Example: : GPA 0 Female 1 Male
: Annual salary of college graduate in thousand $ On average, male
college graduates are making an estimated six thousand dollars more
than female college graduates with the same GPA. :
Slide 70
2002 Prentice-Hall, Inc. Chap 12-70 Dummy-Variable Models (with
3 Levels)
Slide 71
2002 Prentice-Hall, Inc. Chap 12-71 Interpretation of the Dummy
Variable Coefficients (with 3 Levels) With the same footage, a
Split- level will have an estimated average assessed value of 18.84
thousand dollars more than a Condo. With the same footage, a Ranch
will have an estimated average assessed value of 23.53 thousand
dollars more than a Condo.
Slide 72
2002 Prentice-Hall, Inc. Chap 12-72 Interaction Regression
Model Hypothesizes interaction between pairs of X variables
Response to one X variable varies at different levels of another X
variable Contains two-way cross product terms Can be combined with
other models E.G., Dummy variable model
Slide 73
2002 Prentice-Hall, Inc. Chap 12-73 Effect of Interaction
Given: Without interaction term, effect of X 1 on Y is measured by
1 With interaction term, effect of X 1 on Y is measured by 1 + 3 X
2 Effect changes as X 2 increases
Slide 74
2002 Prentice-Hall, Inc. Chap 12-74 Y = 1 + 2X 1 + 3(1) + 4X 1
(1) = 4 + 6X 1 Y = 1 + 2X 1 + 3(0) + 4X 1 (0) = 1 + 2X 1
Interaction Example Effect (slope) of X 1 on Y does depend on X 2
value X1X1 4 8 12 0 010.51.5 Y Y = 1 + 2X 1 + 3X 2 + 4X 1 X 2
Slide 75
2002 Prentice-Hall, Inc. Chap 12-75 Interaction Regression
Model Worksheet Multiply X 1 by X 2 to get X 1 X 2. Run regression
with Y, X 1, X 2, X 1 X 2
Slide 76
2002 Prentice-Hall, Inc. Chap 12-76 Interpretation when there
are more than Three Levels MALE = 0 if female and 1 if male MARRIED
= 1 if married; 0 if not DIVORCED = 1 if divorced; 0 if not
MALEMARRIED = 1 if male married; 0 otherwise = (MALE times MARRIED)
MALEDIVORCED = 1 if male divorced; 0 otherwise = (MALE times
DIVORCED)
Slide 77
2002 Prentice-Hall, Inc. Chap 12-77 Interpretation when there
are more than Three Levels (continued)
Slide 78
2002 Prentice-Hall, Inc. Chap 12-78 Interpreting Results FEMALE
Single: Married: Divorced: MALE Single: Married: Divorced: Main
Effects : MALE, MARRIED and DIVORCED Interaction Effects :
MALEMARRIED and MALEDIVORCED Difference
Slide 79
2002 Prentice-Hall, Inc. Chap 12-79 Hypothesize interaction
between pairs of independent variables Contains 2-way product terms
Hypotheses: H 0 : 3 = 0 (no interaction between X 1 and X 2 ) H 1 :
3 0 (X 1 interacts with X 2 ) Evaluating Presence of
Interaction
Slide 80
2002 Prentice-Hall, Inc. Chap 12-80 Using Transformations
Requires data transformation Either or both independent and
dependent variables may be transformed Can be based on theory,
logic or scatter diagrams
Slide 81
2002 Prentice-Hall, Inc. Chap 12-81 Inherently Linear Models
Non-linear models that can be expressed in linear form Can be
estimated by least squares in linear form Require data
transformation
Slide 82
2002 Prentice-Hall, Inc. Chap 12-82 Transformed Multiplicative
Model (Log-Log) Similarly for X 2
Slide 83
2002 Prentice-Hall, Inc. Chap 12-83 Square Root Transformation
1 > 0 1 < 0 Similarly for X 2 Transforms one of above model
to one that appears linear. Often used to overcome
heteroscedasticity.
Slide 84
2002 Prentice-Hall, Inc. Chap 12-84 Linear-Logarithmic
Transformation 1 > 0 1 < 0 Similarly for X 2 Transformed from
an original multiplicative model
Slide 85
2002 Prentice-Hall, Inc. Chap 12-85 Exponential Transformation
(Log-Linear) Original Model 1 > 0 1 < 0 Transformed
Into:
Slide 86
2002 Prentice-Hall, Inc. Chap 12-86 Interpretation of
Coefficients The dependent variable is logged The coefficient of
the independent variable can be approximately interpreted as: a 1
unit change in leads to an estimated percentage change in the
average of Y The independent variable is logged The coefficient of
the independent variable can be approximately interpreted as: a 100
percent change in leads to an estimated unit change in the average
of Y
Slide 87
2002 Prentice-Hall, Inc. Chap 12-87 Interpretation of
coefficients Both dependent and independent variables are logged
The coefficient of the independent variable can be approximately
interpreted as : a 1 percent change in leads to an estimated
percentage change in the average of Y. Therefore is the elasticity
of Y with respect to a change in (continued)
Slide 88
2002 Prentice-Hall, Inc. Chap 12-88 Interpretation of
Coefficients If both Y and are measured in standardized form: And
The are called standardized coefficients They indicate the
estimated number of average standard deviations Y will change when
changes by one standard deviation (continued)
Slide 89
2002 Prentice-Hall, Inc. Chap 12-89 Collinearity
(Multicollinearity) High correlation between explanatory variables
Coefficient of multiple determination measures combined effect of
the correlated explanatory variables No new information provided
Leads to unstable coefficients (large standard error) Depending on
the explanatory variables
Slide 90
2002 Prentice-Hall, Inc. Chap 12-90 Venn Diagrams and
Collinearity Oil Temp Insulation Overlap NOT Large Overlap in
variation of Temp and Insulation is used in explaining the
variation in Oil but NOT in estimating and Overlap Large Overlap
reflects collinearity between Temp and Insulation
Slide 91
2002 Prentice-Hall, Inc. Chap 12-91 Detect Collinearity
(Variance Inflationary Factor) Used to Measure Collinearity If is
Highly Correlated with the Other Explanatory Variables.
Slide 92
2002 Prentice-Hall, Inc. Chap 12-92 Detect Collinearity in
PHStat PHStat | regression | multiple regression Check the variance
inflationary factor (VIF) box EXCEL spreadsheet for the heating oil
example Since there are only two explanatory variables, only one
VIF is reported in the excel spreadsheet No VIF is > 5 There is
no evidence of collinearity
Slide 93
2002 Prentice-Hall, Inc. Chap 12-93 Model Building Goal is to
develop a good model with the fewest explanatory variables Easier
to interpret Lower probability of collinearity Stepwise regression
procedure Provide limited evaluation of alternative models
Best-subset approach Uses the c p statistic Selects model with
small c p near p+1
Slide 94
2002 Prentice-Hall, Inc. Chap 12-94 Model Building Flowchart
Choose X 1,X 2,X p Run Regression to find VIFs Remove Variable with
Highest VIF Any VIF>5? Run Subsets Regression to Obtain best
models in terms of C p Do Complete Analysis Add Curvilinear Term
and/or Transform Variables as Indicated Perform Predictions No More
than One? Remove this X Yes No Yes
Slide 95
2002 Prentice-Hall, Inc. Chap 12-95 Pitfalls and Ethical
Considerations Understand that interpretation of the estimated
regression coefficients are performed holding all other independent
variables constant Evaluate residual plots for each independent
variable Evaluate interaction terms To avoid pitfalls and address
ethical considerations:
Slide 96
2002 Prentice-Hall, Inc. Chap 12-96 Additional Pitfalls and
Ethical Considerations Obtain VIF for each independent variable and
remove variables that exhibit a high collinearity with other
independent variables before performing significance test on each
independent variable Examine several alternative models using best-
subsets regression Use other methods when the assumptions necessary
for least-squares regression have been seriously violated
(continued) To avoid pitfalls and address ethical
considerations:
Slide 97
2002 Prentice-Hall, Inc. Chap 12-97 Chapter Summary Developed
the multiple regression model Discussed residual plots Addressed
testing the significance of the multiple regression model Discussed
inferences on population regression coefficients Addressed testing
portion of the multiple regression model
Slide 98
2002 Prentice-Hall, Inc. Chap 12-98 Chapter Summary Described
the quadratic regression model Addressed dummy variables Discussed
using transformation in regression models Described collinearity
Discussed model building Addressed pitfalls in multiple regression
and ethical considerations (continued)