MS4225: Business Research Modeling
Mid-session test
8 March 2011
Question 1
Answer True/False to each of the following questions:
1)
In a binary Logit model, the dependent variable is assumed to follow a logistic
distribution.
2)
The Hosmer-Lemeshow test compares the fitted model with the model that
contains no explanatory variable but the intercept.
3)
For the binary Logit model, the predicted value of the dependent variable cannot
be a probability unless an explicit constraint is imposed on the model.
4)
Fisher Scoring used under PROC LOGISTIC in SAS is another name for
maximum likelihood.
5)
Given models A and B, the log-likelihood for A is always no smaller than the loglikelihood for B provided that B is nested within A.
6)
Imposing restrictions on the unknown parameters can never result in an increase
in the model’s likelihood value even if the restrictions are correct. It is for this
reason that the likelihood ratio test statistic cannot take on a negative value.
7)
The term “asymptotic” is a statistical jargon referring to small samples.
8)
For the binary Logit model, the change in the log-odds in favour of the target
event due to a change in an explanatory variable is not a constant; instead, it
varies across the population observations.
9)
In the binary Logit model, if an estimated coefficient has a negative sign, then an
increase in the corresponding explanatory variable always results in a decrease in
the probability being modeled.
10)
An estimated Logit model cannot be very good if the number of discordants
exceeds the number of concordants.
Question 2
A binary logit model has been fitted to explain the choice between a cat (Y = 1) and a
dog (Y = 0) as the family pet for n = 23 households, each having either a cat or a dog.
The following explanatory variables are used:
X1:
X2:
X3:
X4:
= 1 if the family lives in a house, 0 if the family lives in an apartment;
= the number of males minus the number of females in the household;
= the average age of the household;
= the gross annual income of the household (in thousands of U.S. dollars)
The SAS results based on PROC LOGISTIC are given in the Appendix.
1)
Write down the estimated Logit model.
2)
Compute the likelihood-ratio test statistic and use it to test for the overall
significance of the model. Use a 5% level of significance.
3)
In terms of AIC and SBC, is the estimated model any better than the model that
contains an intercept only? Explain.
4)
Compute the Tau-a, Gamma, Somer’s D and c statistics. What conclusion can
you draw from these statistics?
5)
Compute the Generalized R2 for the above regression and interpret the result.
6)
What are the values of X1, X2, X3 and X4 for the following household:
A 35 year old female earning $U.S. 70,000 per year and living alone in an
apartment.
7)
Predict the probability of the above household having a dog as a pet.
8)
Verify the odds ratio estimate of 0.804 for X3 as shown in the SAS output and
interpret this estimate.
Appendix
Model Information
Response Variable
Number of Response Levels
Model
Optimization Technique
y
2
binary logit
Fisher's scoring
Number of Observations Read
Number of Observations Used
23
23
Response Profile
Ordered
Value
y
Total
Frequency
1
2
1
0
9
14
Probability modeled is y=1.
Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.
Model Fit Statistics
Criterion
AIC
SC
-2 Log L
Intercept
Only
Intercept
and
Covariates
32.789
33.925
30.789
25.712
31.389
15.712
Testing Global Null Hypothesis: BETA=0
Test
Chi-Square
DF
Pr > ChiSq
?????
11.9009
4.5367
?
?
?
0.0045
0.0181
0.3382
Likelihood Ratio
Score
Wald
The LOGISTIC Procedure
Analysis of Maximum Likelihood Estimates
Parameter
DF
Estimate
Standard
Error
Wald
Chi-Square
Pr > ChiSq
Intercept
x1
x2
x3
x4
1
1
1
1
1
7.9627
-8.9105
1.7876
-0.2181
0.0266
6.6899
4.9690
1.3850
0.1374
0.0587
1.4167
3.2156
1.6660
2.5202
0.2051
0.2339
0.0729
0.1968
0.1124
0.6506
Odds Ratio Estimates
Effect
x1
x2
x3
x4
Point
Estimate
95% Wald
Confidence Limits
<0.001
5.975
0.804
1.027
<0.001
0.396
0.614
0.915
2.290
90.205
1.052
1.152
Association of Predicted Probabilities and Observed Responses
Percent Concordant
Percent Discordant
Percent Tied
Pairs
91.3
8.7
0.0
126
Somers' D
Gamma
Tau-a
c
???
???
???
???