Homework 5 - 11 Questions
Covers mainly Chapter 9 but also Categorical Coding (8) and VIF, Cook's Distance (10)
Use the Project Talent data set.
1. A lack-of-fit test regressing Math on Gender, SES, Sociability, Reading, and
Mechanical Reasoning indicates that a possible curvature exists for the variables Reading
and Mechanical Reasoning. Without centering, create new, second-order variables for
Reading and for Mechanical Reasoning. Repeat the multiple regression analysis
including these second order terms (will result in a model with seven (7) predictors).
Select Options and select Variance Inflation Factor AND Lack-of-Fit > Data Subsetting
and answer the following questions.
a. What are the VIF values for all of the predictors?
Predictor
Gender
Reading
Mech
Social
SES
r2
me2
Coef
-1.238
-1.6071
-2.040
0.5179
0.22559
0.03581
0.14575
SE Coef
1.895
0.6963
1.159
0.2971
0.09012
0.01264
0.06323
T
-0.65
-2.31
-1.76
1.74
2.50
2.83
2.30
P
0.522
0.034
0.096
0.099
0.023
0.011
0.034
VIF
1.542
63.366
43.422
1.365
1.543
69.894
52.568
b. Does this indicate the presence of Multicollinearity? Why or why not?
Yes, since the VIF values for Reading, Mechanical and their second order
terms exceed 10
c. What is the p-value of the lack-of-fit tests and what does this indicate about model fit?
No evidence of lack of fit (P >= 0.1) indicating that there is no
lack of model fit
2. Create new second order terms for Reading and Mechanical Reasoning after centering
both the first order terms. Repeat the multiple regression analysis including these
centered first and second order terms. Select Options and select Variance Inflation Factor
AND Lack-of-Fit > Data Subsetting and answer the following questions.
a. What are the VIF values for all of the predictors?
Predictor
Gender
Social
SES
cR
cM
cR2
cM2
Coef
-1.238
0.5179
0.22559
0.5960
1.2242
0.03581
0.14575
SE Coef
1.895
0.2971
0.09012
0.1435
0.4320
0.01264
0.06323
T
-0.65
1.74
2.50
4.15
2.83
2.83
2.30
P
0.522
0.099
0.023
0.001
0.011
0.011
0.034
VIF
1.542
1.365
1.543
2.692
6.033
2.651
2.963
1
b. Does this indicate the presence of Multicollinearity and support your answer?
With all VIF values being less than 10 there is no indication of multicollinearity.
The centering reduced the correlation between the first and second order terms for
Reading and Mechanical Reasoning
c. What is the p-value of the lack-of-fit test and what does this indicate about model fit?
No evidence of lack of fit (P >= 0.1) indicating that there is no
lack of model fit
3. The variable School Size is interpreted as follows:
1 = number of students is less than 100 (call this "small")
2 = number of students is from 100 to 399 (call this "medium")
3 = number of students is 400 or more (call this "large")
The data set includes dummy coding, effect coding (large school as reference group for
both), and orthogonal coding for Small - Medium, and Small/Medium - Large.
Regressing Math on just these coded variables (i.e. regress Math separately on the two
DV, then the two effect variables, then the two orthogonal variables) and answer the
following questions:
a: Provide an interpretation of the t-tests for regressing Math on the DV.
Predictor
Constant
Small_DV
Medium_DV
Coef
31.571
-13.429
-8.481
SE Coef
3.495
4.943
4.471
T
9.03
-2.72
-1.90
P
0.000
0.013
0.071
Since dummy coded, the T-tests are comparing the means for the categorical level in
the model to the mean for the category not included in the model. In this case, the
mean Math score of the Small schools and mean Math score of the Medium schools
are being compared to the mean Math score of the Large schools. Using alpha of
0.05, we would conclude a significant difference between the mean math scores for
the Small and Large schools. The negative slope for Small indicates that the mean
math score for Small schools is less than the mean math score for Large schools.
Since the p-value for Medium exceeds 0.05 we cannot conclude a difference in mean
math scores between the Medium and Large schools. Note that in this model the
intercept is the mean math score for the Large schools.
b: Provide an interpretation of the t-tests for regressing Math on the effect variables.
Predictor
Constant
Small_eff
Med_eff
Coef
24.268
-6.126
-1.177
SE Coef
1.892
2.766
2.484
T
12.83
-2.21
-0.47
P
0.000
0.037
0.640
2
Since effect coded AND unequal sample sizes, the T-tests are comparing the means
for the categorical level in the model to the average of the three group means. That
is, if you find the mean math score for the three school sizes (18.14, 23.09, and 31.57)
this average is 24.3 – the intercept of the model. In this model, the mean Math
score of the Small schools and mean Math score of the Medium schools are being
compared to the this average mean. Using alpha of 0.05, we would conclude a
significant difference between the mean math scores for the Small and average of
the mean Math scores for the three sizes. The negative slope for Small indicates that
the mean math score for Small schools is less than this average. Since the p-value
for Medium exceeds 0.05 we cannot conclude a difference between the mean math
scores for the Medium and average of the mean Math scores for the three sizes.
c: Provide an interpretation of the t-tests for regressing Math on the orthogonal variables.
Predictor
Constant
Small-Med
SM-Large
Coef
24.080
-0.2749
-0.4162
SE Coef
1.850
0.2484
0.1648
T
13.02
-1.11
-2.53
P
0.000
0.280
0.019
Since orthogonal coded, the T-tests are comparing the means for the categorical
levels being compared in that specific contrast. For Small-Med this compares the
mean Math score between these two school sizes; for SM-Large we are comparing
the mean Math score of the Small/Medium schools to the mean math score of the
Large schools. Again using alpha of 0.05, we only reject the hypothesis for the SMLarge contrast and conclude a difference in mean math scores for the
Small/Medium schools compared to the Large schools. With a negative slope and
the contrast difference being calculated by “SM minus Large” we can add that the
Large school mean math scores are higher than the mean math scores of the
Small/Medium sized schools. Note that the intercept in this model is the overall
mean of the math scores, i.e. the grand mean for math.
4. Regress Math on each orthogonal coded variable separately (i.e. conduct two simple
linear regressions) and answer the following:
a. What is the R-squared value for regressing Math on Small-Med?
R-sq = 4.1%
b. What is the R-squared value for regressing Math on SM-Large?
R-sq = 21.6%
c. What is the R-squared value for the multiple regression model when regressing Math
on both orthogonal variables (i.e. from the model in 3c)?
R-sq = 25.7%
d. Did the sum of the R-squared values in parts a and b equal the R-squared value in c?
3
Yes they did. NOTE: this is the result of having orthogonal (i.e. correlation between
predictors is zero). In such instances when all the model predictors are orthogonal,
the R-squared values are additive.
Use the High School and Beyond (HSB) data set.
The data is explained in the HSB Read Me file.
USE MATH AS RESPONSE
1. With Math as the response and the remaining variables as predictors (excluding ID as
that serves only as an identifier), how many models are possible (assume an intercept for
all models)?
213 – 1 = 8191
2. Using Math as response, analyze the data using Minitab Backward, Forward, and
Stepwise Regression (keep default settings). Specify the “best” regression equation
identified by these three methods. How many steps did it take for each method? Do they
agree?
Backward: Math = 9.28 – 1.42SEX + 0.83SES +0.26RDG + 0.28WRTG + 0.22SCI
0.07CIV
Steps: 8
Forward: Math = 8.79 + 0.25RDG + 0.28WRTG + 0.22SCI - 1.41SEX + 0.78SES +
0.07CIV + 0.09CAR
Steps: 7
Stepwise: Math = 8.79 + 0.25RDG + 0.28WRTG + 0.22SCI - 1.41SEX + 0.78SES +
0.07CIV + 0.09CAR
Steps: 7
Agree? No, they do not all agree; the forward and stepwise methods do but not
backward.
3. In the Backward Elimination analysis, which variable was removed first and why?
After the full model is run, School Type (SCTYP) is removed first since it has the
largest p-value (0.981) exceeding the alpha to remove of 0.1
4. In the Forward and Stepwise analyses which variable entered first and why?
After the full model is run, Reading (RDG) is entered first since it has the smallest
p-value (0.000) below the alpha to enter of 0.25 (Note: with rounding there could be
4
other variables with three digit p-values of 0.000. In such cases the distinction as to
which would be added first could be determined by the variable with the largest T
statistic.)
5. In the Backward Elimination analysis how much of a change in R2 is there between the
model in Step 4 and the final model?
After step 4, the R-sq is 57.33% and in the final step this is 58.20% - no a great
increase.
6. Now regress Math on all of the predictors and use the Best Subsets in Minitab to
determine the variables that comprise the best model using R-squared, adjusted Rsquared, and lowest Cp. What are the variables, criterion values, and are the models the
same? [Remember that goal is reduce number of variables from full model.]
Lowest Cp: SEX, SES, CAR, RDG, WRTG, SCI, CIV
Value: 4.0
R-squared: SEX, SES, LOCUS, CAR, RDG, WRTG, SCI, CIV
Value: 58.3%
Adj. R-Squared: SEX, SES, CAR, RDG, WRTG, SCI, CIV
Value: 57.7%
All models the same? No, the two models based on lowest Cp and adjusted Rsquared are the same, but not when using R-squared.
7. Regress Math on Reading, Writing and Science. Click Storage and select Cook’s
Distance (Di). Determine if any of these Di value(s) indicate if any observation(s) as
influential by seeing if any of these Di values exceed 0.5 of the F-distribution with p and
n-p degrees of freedom. That is, find the cumulative F probability for this column of Di
values. If any cumulative probabilities exceed 0.5 then that observation would be
considered and outlier. Also, in the output under Unusual Observations any observation
marked with an “X” indicates and influential outlier. Do any exist in this regression
analysis?
DF: 4, 596
Number of Di values greater than 0.5: None
Observations that are considered influential outliers: Using Cook’s Distance there
is no observations that would be classified as influential outliers. NOTE however in
the output under Unusual Observations that there are three observations designated
with an ‘X’ indicating they are influential using the high leverage method. If you
stored the leverages and compared these to 0.02 (from min[0.99, 3p/n]) you would
find that the leverages for these three observations exceed 0.02
5