## Final Exam

### Outline of solutions

Notes:

A series of numbers is given in brackets for each problem; these are the point values assigned to the different parts of the problem. Thus, 1. [8/4/8] would mean that problem 1 was assigned 20 points: 8 for part (a), 4 for part (b) and 8 for part (c).

This is not meant to be a full solution set, but rather an indication of some of the important aspects of the problems, or an indication of the direction for a solution.

The exams were graded with students' names concealed.

```Problem I.
1-7. [2 pts each]   7612435 or cdfegba

8. 
(ICD) -> (ID,IC,DC) -> (IC,DC) -> (IC,D) -> (I,D,C)
|
(IC)

[Also accepted: ... (IC,D) -> (IC) -> (I,D,C), although technically
this would be true only for the model (IC) fit to uncollapsed data.]

9.   loglin count irritate conc drug, fit(irritate conc, drug)

10. [3/3/3/3]
a) 5.046/5/ p>0.25
b) drug is jointly independent of concentration and score
c) [1.6] fits as well and is simpler
d) Irritation level is affected similarly by the two drugs, although
higher concentrations are strongly associated with higher
irritation.
Note: this ignores the information in the order of irritation scores.
Note: points were deducted for serious deviations from the "one
brief sentence" requirement.

11. [3/3/3/3]
a) 0.045 / 2 / approx 1
b) At each fixed concentration, drug and irritation exhibit uniform
association. (The conditional statement is important here.)
c) [1.13] fits as well and is simpler
d) At each concentration, the association of irritation and drug (as
measured by the local odds ratio) is equal at high, medium, and
low levels of irritation.
Note: points were deducted for serious deviations from
the "one brief sentence" requirement.

12. [4/3/3]
a) Models [1.16], [1.18], and [1.19] all show moderate
evidence for an effect (p=0.035).  The estimated log-odds
parameter is fairly constant in these models (about 0.82), Model
[1.14] does not involve drug.  The drug coefficient in model
[1.20] cannot be directly interpreted, as the model also contains
a higher-order interaction involving drug.
b) The positive coefficient indicates that Drug 2 causes higher
levels of irritation.
c) Given concentration level, the odds ratio for for being above any
given irritation level is about 2.3 times larger for drug 2 than
for drug 1.  [2.3 = exp(0.82)]

```

```Problem II.
1. 
indedpendence, row effects, column effects, uniform association,
saturated

2. 
Many residuals > 2 in magnitude; patterns of signs on residuals;
strong positive residuals in upper left, lower right corners with
strong negative residuals in the other corners.

3. 
Chi-squared should be about 47 with 15 df, p less than 0.001.  [LRT
and Pearson chi-squared are nearly equivalent, especially in
moderately sized samples.]  This gives strong evidence against independence.

4. 
The local log odds of being in a higher category of SES (or mental
health) is approximately 0.0907.  This means the odds ratio in any
2x2 subtable is approximately exp(0.0907) = 1.095.

5. 
Given uniform spacing of PSES, these represent optimal scores for
mental health status.  The scores in model [2.8] are proportional
to (0, 1, 1, 2).  This suggests that mild and moderate symptoms can't
be easily distinguished for these purposes.

```

```Problem III.
1. 
Each factor (location, gender, seatbelt) contributes separately to
risk.  The effect on the odds of injury is multiplicative.

2. 
Using seatbelt decreases odds of injury, by factor of exp(-0.817) = 0.442
Rural accidents have increased odds of injury, by factor of exp(0.758) = 2.134
Males have decreased odds of injury, by factor of exp(-0.545) = 0.580

3. 
[3.13]:  (SLG, IS, IL, IG)
[3.14]:  ( SL, SG, IS, IL, IG, LG)

4.  (a)

5. 
[3.13] contains a three-factor interaction missing from [3.14]

6. 
[3.14] exhibits statistically significant lack of fit (see
G-squared), while [3.13] does not.

7. 
2.264  2.255   =    exp(0.817)   exp(0.813)
2.134  2.128        exp(0.758)   exp(0.755)
0.580  0.580        exp(-0.544)  exp(-0.545)

8. 
The effect of the three-factor interaction affects parameters very
little, but the differences attain statistical significance because
of the large sample size (n=68694).

```