MS4225:
Business Research Methods
Mid-session test
Time allowed: 75 minutes
An educationalist used a LOGIT model to analyze the effectiveness of a new method of teaching
mathematics. The dependent variable for the study was GRADE, an indicator of the student’s
grade improved on an examination (=1 if there was an improvement, 0 otherwise). The
explanatory variables were GPA, the grade point average; TUCE, the score on a pretest that
indicates entering knowledge of the material; and PSI, which equals 1 if the student was exposed
to the new teaching method, 0 otherwise.
1)
Write down the LOGIT model to be estimated and derive an expression (in terms of the
explanatory variables) for the odds of the event of an improvement.
(5)
2)
The SAS program and estimation results are given on the next page. What is the purpose
of the DESCENDING option in PROC LOGISTIC? What will happen if this option is
omitted?
(5)
3)
Compute the estimates of the odds ratio for the variables GPA, TUCE and PSI. Interpret
these odds estimates.
(10)
4)
Interpret the LOGIT coefficient estimate of 2.3785 for PSI. Would the interpretation be
different if the model estimated were a Linear Probability Model instead? Explain. (10)
5)
Consider a student with GPA = 2.45, TUCE = 20 and PSI = 1. Predict the probability
that this student’s grade will improve after exposing to the new teaching method.
(5)
6)
Calculate the effect of the new teaching method on the probability of an improvement for
a student with GPA = 2.44, TUCE = 19.
(15)
7)
In the portion of the output corresponding to the LACKFIT option, compute the observed
and expected frequencies for GRADE = 0, and discuss how the Hosmer-Lemeshow test
statistic is calculated.
(10)
8)
Use the Hosmer-Lemeshow test to evaluate the model’s goodness of fit. Carefully set up
the null and alternative hypotheses. Use a 10% significance level.
(5)
9)
Compute the Likelihood Ratio test statistic for the overall significance of the model. Use
this test and the Wald and Lagrange Multipler tests to test the model’s overall
significance. Carefully set up the null and alternative hypotheses. Use a 5% significance
level.
(10)
10)
What is a discordant? How many discordant pairs are there in the sample?
(10)
11)
Comment on the model’s goodness of fit using the Somer’s D, Gamma and Tau-a
statistics.
(5)
12)
Use the Likelihood Ratio test to test the hypothesis that the new teaching method has no
impact on the students’ mathematics performance. Compare the results with the
corresponding Wald test. Set up the null and alternative hypotheses in terms of the
unknown coefficients. Use a 10% significance level.
(10)
data test06;
infile 'd:\teaching\ms4225\TABLE19.txt';
input obs gpa tuce psi grade;
proc logistic data=test06 descending ;
model grade=psi gpa tuce/lackfit;
run;
proc logistic data=test06 descending ;
model grade=gpa tuce;
run;
Output for models with PSI, GPA and TUCE as explanatory variables
The LOGISTIC Procedure
Model Information
Data Set
Response Variable
Number of Response Levels
Model
Optimization Technique
WORK.TEST06
grade
2
binary logit
Fisher's scoring
Number of Observations Read
Number of Observations Used
32
32
Response Profile
Ordered
Value
grade
Total
Frequency
1
2
1
0
11
21
Probability modeled is grade=1.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Criterion
AIC
SC
-2 Log L
Intercept
Only
Intercept
and
Covariates
43.183
44.649
41.183
33.779
39.642
25.779
Testing Global Null Hypothesis: BETA=0
Test
Chi-Square
Likelihood Ratio
Score
Wald
???
13.3088
8.3762
DF
Pr > ChiSq
3
3
???
0.0040
0.0388
?
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept
psi
gpa
tuce
1
1
1
1
-13.0204
2.3785
2.8259
0.0951
4.9310
1.0645
1.2629
0.1415
6.9723
4.9925
5.0072
0.4518
0.0083
0.0255
0.0252
0.5015
Odds Ratio Estimates
Effect
psi
gpa
tuce
Point
Estimate
95% Wald
Confidence Limits
???
???
???
1.339
1.420
0.833
86.917
200.567
1.451
Association of Predicted Probabilities and Observed Responses
Percent Concordant
Percent Discordant
Percent Tied
Pairs
88.3
11.3
0.4
231
Somers' D
Gamma
Tau-a
c
0.771
0.774
0.359
0.885
Partition for the Hosmer and Lemeshow Test
Group
Total
1
2
3
4
5
6
7
8
9
10
11
3
3
3
3
3
3
3
3
3
3
2
grade = 1
Observed
Expected
0
0
0
1
0
1
1
1
3
2
2
grade = 0
Observed
Expected
0.08
0.09
0.14
0.23
0.49
0.74
1.16
1.69
1.99
2.53
1.85
?
?
?
?
?
?
?
?
?
?
?
???
???
???
???
???
???
???
???
???
???
???
The LOGISTIC Procedure
Hosmer and Lemeshow Goodness-of-Fit Test
Chi-Square
DF
Pr > ChiSq
6.8661
9
0.6511
Output for models with only GPA and TUCE as explanatory variables
The LOGISTIC Procedure
Model Information
Data Set
Response Variable
Number of Response Levels
Model
Optimization Technique
WORK.TEST06
grade
2
binary logit
Fisher's scoring
Number of Observations Read
Number of Observations Used
Response Profile
32
32
Ordered
Value
grade
Total
Frequency
1
2
1
0
11
21
Probability modeled is grade=1.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Criterion
Intercept
Only
Intercept
and
Covariates
43.183
44.649
41.183
37.983
42.380
31.983
AIC
SC
-2 Log L
Testing Global Null Hypothesis: BETA=0
Test
Chi-Square
DF
Pr > ChiSq
9.2005
8.3699
6.6259
2
2
2
0.0100
0.0152
0.0364
Likelihood Ratio
Score
Wald
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept
gpa
tuce
1
1
1
-10.6560
2.5383
0.0856
4.0572
1.1819
0.1332
6.8981
4.6125
0.4126
0.0086
0.0317
0.5207
Odds Ratio Estimates
Effect
gpa
tuce
Point
Estimate
95% Wald
Confidence Limits
12.658
1.089
1.248
0.839
128.344
1.414
Association of Predicted Probabilities and Observed Responses
Percent Concordant
Percent Discordant
Percent Tied
Pairs
80.5
19.5
0.0
231
Somers' D
Gamma
Tau-a
c
0.610
0.610
0.284
0.805