## Midterm 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] means 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.

```1. [8/4/8]

a) Several answers.  The most appropriate might be the chi-squared test for
independence and logistic regression, although one could also compare mean
Karnofsky scores between groups, or conduct a linear trend test.  Measures
of association do not directly test the hypothesis, although they may still
be useful descriptors.

b) Tests which take the ordering of the Karnofsky scores into account will
be more powerful, and also may lead to interpretible parameters.

c) Under the hypothesis of 1:3 randomization, the expected number
randomized to Placebo is 53/4 = 13.25 and to MGDF 3*(53/4) = 39.75.  Using
the chi-squared goodness-of-fit test, we obtain

chi-sq = (13-13.25)^2/13.25 +  (40-39.75)^/39.75 = 0.005

with one degree of freedom.  The p-value is nearly one; the actual
allocations are about as close to the 1:3 ratio as they could be.
```
```2. [4/4/4/4/4]

a) 179/648 = 0.296

b)  72/119 = 0.605
110/306 = 0.359
44/223 = 0.197

c) 226/648 = 0.349

d) Independence of males' education level and their opinions about the
role of women.

e) The chi-squared statistic of 57.09 on 2 df (calculated by Stata)
indicates a strong dependence between the variables.  The answer to
(b) above indicates that more inclusive views on women's roles are
associated with higher levels of education among men.

```
```3. [6/6/6]  (+5 bonus)
Note: A general description of the different sampling schemes does
not related to Table 1-m, and consequently did not get full credit.

a) Men were sampled for a fixed period of time; this resulted in
completed surveys from 648 men, who were then cross-clasified by
education and opinion.

b) Men are sampled until 119 people with grade-school education, 306
with high-school education, and 223 with some college are
obtained.  (These numbers are specified in advance by the survey
director.)  They are then classified by opinion within each education
level.

c) Men are sampled until exactly 648 men are selected (this was the
number which the survey budget would support).  They were then
cross-clasified by education and opinion.

d) Men are sampled until 119 people with grade-school education, 306
with high-school education, and 223 with some college are
obtained.  (These numbers are specified in advance by the survey
director.)  These men are placed in a room with 226 cards that say
"Agree" and 422 cards that say "Disagree".  The group is instructed
that each man must take one card, and that so far as possible, the
cards should reflect the individuals' opinions about the question.
The men are allowed to talk among themselves and to trade cards.  The
number of men from each education level who selected each type of
card is recorded in the table.

```
```4. [6/6/9]

a) The male-female odds ratio for white victims for a "firearm" attack is
1.08, which represents only a small association between sex and type
of attack.  For those who prefer statistical guidance, the
chi-squared statistic is 0.85 on 1 df (p=0.4).  The relationship, if
any, is very weak.

b) The male-female odds ratio for black victims for a "firearm" attack is
1.21, which represents a modest association between sex and type of
attack.  The chi-squared statistic is 6.72 on 1 df (p=0.01).  This
suggests that there is a real relationship (with men being more likely
than women to be assaulted using firearms/explosives), even though it is
not a large effect.

c) After adjusting for racial differences, female victims are less likely
than male victims to be assualted with firearms/explosives (coeff =
-0.13).  After adjusting for any differences associated with the sex
of the victim, black victims are less likely than white victims to
be assaulted with firearms/explosives (coeff = -0.27).  There is
strong evidence that both of these effects are real (small
p-values).  A numerical interpretation would be that relative to
black women, assualts on males have odds increased by 15% and
assaults on whites have odds increased by an (additional) 31% for a
firearm attack.   [Note exp(0.273)=1.31 and exp(0.13)=1.15]
```
```5. [8/7/6]

a) 2 x [ -147.674 - ( -336.886 ) ] = 2 x 189.2 = 378.42
df = 2 (1 df each for A, B).

b) 147.674 - 145.212 = 2.46
2 x 2.46 = 4.92.  This is a 1 df comparison.  The p-value is somewhat
less than 0.05, which suggests that C is helpful in any model which
contains A and B.

c) If the regression coefficients in the logistic model are 6, 2, and
1.5, then then odds factors corresponding to the conditions are 403
for being in a single-parent household, 7.4 for overcrowded
households, and 4.5 for foreign birth.  The amount by which the odds
are increased for any individual child is the product of the odds
factors which pertain to the child's classification.

```