Download Test 2 - City University of Hong Kong

MS4225: Business Research Modeling
Test 2
Time Allowed: 55 minutes
The year 2012 marks the 15th anniversary of the establishment of the HKSAR. A
university research centre conducted a survey about Hong Kong residents’ perception of
post-British Hong Kong. Each interviewee was given the following question:
“Do you think Hong Kong has been generally better, worse, or much the same after
British rule ended in 1997?”
The response, represented by the variable REP, takes on one of the following values:
1 = Worse, 2 = Much the Same, 3 = Better
In addition, the following information was collected from each interviewee:
AGE: In which age group do you belong? (1 = 25 – 40, 0 = above 40)
DEM: Would you consider yourself a supporter of the Democratic Alliance? (1 = Yes, 0
= No)
INC: What is your monthly level? (1 = below $20,000, 2 = between $20,000 and
$50,000, 3 = over $50,000)
Over six thousand respondents aged between 25 and 65, and all had Hong Kong as their
place of birth, took part in the survey. The data collected were subsequently analyzed
using a multinomial logit model with REP as dependent variable, and AGE, DEM and
INC as explanatory variables.
The SAS output based on the PROC CATMOD is given in the Appendix.
1)
What are the estimated log-odds equations for “Worse” versus “Better”, “Much
the Same” versus “Better”, and “Worse” versus “Much the Same”?
(8)
2)
What are “Populations”, “Total Frequency” and “Observations” in the Data
Summary section of the output? Why are these three numbers different?
(3)
3)
Interpret the coefficient estimates of 4.1355 and 3.4272 for DEM in terms of
odds. Other things being equal, is it likely for supporters of the Democratic
Alliance to have a favourable opinion of post-British Hong Kong? Carefully
explain your answer.
(6)
1
4)
Interpret the coefficient estimates of -0.6095 and -0.7651 for INC in terms of
odds. How do residents with different income levels perceive post-British Hong
Kong? Carefully explain your answer.
(6)
5)
Interpret the Likelihood Ratio test statistic of 63.33. What are the null and
alternative hypotheses? Why are there sixteen degrees of freedom, and what do
you conclude from the test?
(4)
6)
Perform a joint test to test the null hypothesis that the coefficients of AGE across
the two equations are zero.
(4)
7)
What is the underlying probability distribution of REP?
8)
The number of respondents answering “Worse” and “Better” to the main question
of the survey are 2490 and 2611 respectively. If 15 interviewees are selected at
random, what is probability that 3 answer “Worse”, 5 answer “Much the Same”,
and 7 answer “Better”? (Hint: see Formula below)
(4)
9)
Suppose that the response to the main question is represented by 7 instead of 3
different answers. How many log-odds equations do these 7 responses yield?
How many log-odds equations require estimation in order to obtain coefficient
estimates for all log-odds equations?
(4)
10)
(Bonus question) Explain the meaning of the following English slangs:
i)
ii)
iii)
iv)
(2)
Rings a bell
Blimey
Bring a plate
Keep your nose to the grind stone
(4)
Formula
Suppose that there are n independent trials, and the probability that the outcome of a
k
single trial falling in class i is pi , with
p
i 1
i
 1 and k being the total number of classes.
Further, let Si be the number of trials in which the outcome falls in class i. Note that
k
S
i 1
i
 n. Now, the joint p.d.f. for S1,…,Sk is given by
Pr( S1 , S2 ,....Sk ) 
n!
p1s1 p2s2 ...... pksk
S1 ! S2 !....S k !
2
Appendix
The CATMOD Procedure
Data Summary
Response
Weight Variable
Data Set
Frequency Missing
rep
freq
TEST2
0
Response Levels
Populations
Total Frequency
Observations
3
12
6227
36
Population Profiles
Sample
age
dem
inc
Sample Size
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1
0
0
1
443
2
0
0
2
475
3
0
0
3
834
4
0
1
1
669
5
0
1
2
577
6
0
1
3
515
7
1
0
1
385
8
1
0
2
476
9
1
0
3
529
10
1
1
1
627
11
1
1
2
391
12
1
1
3
306
Response Profiles
Response
rep
ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ
1
1
2
2
3
3
Maximum Likelihood Analysis
Maximum likelihood computations converged.
Maximum Likelihood Analysis of Variance
Source
DF
Chi-Square
Pr > ChiSq
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Intercept
2
22.53
<.0001
age
2
104.66
<.0001
dem
2
2002.86
<.0001
inc
2
219.97
<.0001
Likelihood Ratio
16
63.33
<.0001
The CATMOD Procedure
Analysis of Maximum Likelihood Estimates
Function
Standard
ChiParameter Number
Estimate
Error
Square
Pr > ChiSq
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Intercept
1
-0.3124
0.1177
7.04
0.0080
2
-0.6103
0.1286
22.51
<.0001
age
1
-0.4884
0.0825
35.08
<.0001
2
0.2444
0.0882
7.67
0.0056
3
dem
inc
1
2
1
2
4.1355
3.4272
-0.6095
-0.7651
0.0941
0.1023
0.0496
0.0543
1933.04
1122.41
151.14
198.46
<.0001
<.0001
<.0001
<.0001
4