STAT 557 Assignment #2
FALL 2002
Reading Assignment: Lloyd, Categorical Data Analysis: You should have already read Chapter 1 and Sections 3.1-3.3. Read Chapter 2 and finish reading Chapter 3.
Also read Sections 7.1 and 7.2.
Written Assignment: Due Monday, September 23, in class.
1. In a study of obesity, samples of white noninstitutionalized women between the ages of 20 and 24 inclusive were obtained from Great Britain, Canada, and the United States (Lloyd, page 114). Each woman was classified into one of four categories according to the Quetelet index which divides weight by height squared. The results are shown in the following table and posted in the file obesity.dat on the course web page.
Great Britain
Canada
United States
Underweight Normal Overweight Obese
63
148
78
153
249
174
44
30
37
13
8
22
(a) Fit a model that assumes that the distribution across the obesity categories is identical for the three countries, assuming a multinomial distribution for each country. Give the value of the X
2 and
(b) Examine residuals, or deviations from expected counts. Are there any patterns?
2. Births of twins at a hospital in Melbourne were recorded and classified according to sex
(Andersen, 1990, p. 84). The observed counts were:
Sex
Two Boys
Two Girls
One Boy/One Girl
Count
29
36
33
Total 98
(a) Is it reasonable to use a multinomial distribution with sample size n=98 as a model for the counts? Explain.
(b) Let Y
1
, Y
2
, Y
3
denote the counts for two boys, two girls, one boy/one girl respectively. Assume that Y
~
(
π
1
,
π
2
,
π
3
)
T
.
=
( Y
1
, Y
2
, Y
3
)
T
~
π
~
where
π
=
~
Call this model A and find formulas and numerical values for the maximum likelihood estimates of
π
1
,
π
2
,
π
3
.

(c) Consider a second model, call it model B, that further assumes that sex of one twin is determined independently of the sex of the other twin in each pair and the probability of a boy is
τ
for any birth. Derive a formula for the m.
l .e. for
τ
. Obtain a numerical value for the m.
l .e. for
τ
and use it to obtain m.
l .e.’s for
π
1
,
π
2
,
π
3
.
2
(d) Compute the values of the G
2
and X
2 statistics for comparing how well model B fits the observed data, relative to model A. Report degrees of freedom and p-values for these two tests and state your conclusion.
(e) Twins can be classified as either monozygotes or dizygotes. Monozygotes develop from the same egg, so the sex of the twins must be the same. Dizygotes develop from different eggs and dizygotic twins are not necessarily of the same sex. Assume that for dizygotic twins, the sex of one twin is determined independently of the sex of the other twin. Further assume the boys and girls are equally likely to occur for both monozygotic and dizygotic pairs of twins. Call this model C. Let
θ
denote the probability of monozygotic twins. Find the m.
l .e. for
θ
and use it to find the m.
l .e.’s for
π
1
,
π
2
,
π
3
in this model.
(f) Compute the value of G
2
and X
2
for comparing how well models A and C fit the data. Report degrees of freedom and p-values and state your conclusions.
(g) Are models A, B, and C a set of nested models? If so, report an analysis of deviance table.
(h) Suppose you wanted to assess the fit of another model, called model D, that is model
C without the restriction that girls and boys are equally likely. Instead, model D makes the assumption that the proportion of male births is
τ
, 0 <
τ
< 1, for both monozygotic and dizygotic pairs of twins. (In model C, it is assumed that
τ
= 0.5)
Can you assess the fit of this model with the data provided in this problem? If not, describe what could be done to obtain a set of counts that could be used to assess the fit of model D, derive formulas (if possible) for the m.
l .e.’s of
τ
and
θ
for the data you propose to collect, and determine the degrees of freedom associated with the G
2 and X
2 test statistics.
3. The following table, (Bergman, et. al., 1972), presents the numbers of births and the numbers of SIDS (Sudden Infant Death Syndrome) cases for each of 9 contiguous birth weight categories. Birth weight increases as the category number increases.
Birth Weight Category SIDS Cases Total Births
1 1 279
2
3
4
5
3
12
12
37
413
904
2.838
11,509
6
7
8
9
52
34
13
2
24,941
21,832
8,408
2,061

3
(a) Assuming a multinomial distribution for the counts, use the Pearson chi-square statistic to test the null hypothesis that the incidence of SIDS is the same for all 9 birth weight categories. Report degrees of freedom and a p-value and state your conclusion. Is there a trend in SIDS incidence across birth weight categories?
Explain.
(b) The computation of the Pearson statistic in part (a) involved some small expected counts. One way to avoid small counts is to combine some categories. Compute a new value of the Pearson statistic after combining the three lowest birth weight categories into a single category and combining the two highest birth weight categories into a single category. Report values for reach the same conclusion as part (a)?
(c) Use the normal approximation to the binomial distribution to compute a 95% confidence interval for the probability of a SIDS case for infants in the highest birth weight category.
(d) Use the binomial likelihood to compute an exact 95% confidence interval for the probability of a SIDS case for infants in the highest birth weight category.
(e) Use large sample normal approximation to the distribution of multinomial counts to construct an approximate 95% confidence interval for the difference in probabilities of SIDS cases in weight groups 5 and 7. Is the incidence of SIDS cases significantly higher in weight group 5 than in weight group 7?
4. Edwards and Gurland (1961) present the following data on the numbers of accidents sustained by 166 London bus drivers during a two year period.
Number of
Accidents
0
1
2
3
Number of
Drivers
15
32
26
29
Expected Counts
Poisson Model Neg. Binomial Model
8
9
10
11
12
13
14
15
4
5
6
7
0
1
0
3
1
0
0
1
22
19
9
8

4
(a) Assuming that an i.i.d. Poisson (m) model is correct for London bus drivers, compute the m.
l .e. for m, and compute expected counts for the accident categories shown in the table.
(b) Use the Pearson chi-square test to assess the fit of the i.i.d. Poisson (m) model.
Combine 8 or more accidents into a single category. Report values for
2
X , d.f., and p-value. State your conclusion.
(c) An alternative test that is often used to assess the fit of the Poisson model is Fisher’s dispersion index. Let X
1
, .
.
.
, X k
denote the counts for k bus drivers, and let
X
= k
1 k
i
=
1
X i
and S
2 = k
1 i k
=
1
( X i
−
X )
2
.
Since the mean is equal to the variance of Poisson distribution, both X and consistent estimates of the variance for the i.i.d. Poisson (m) model. The i.i.d.
Poisson(m) model is declared inadequate when
k S
2
/ X >
χ
2 k 1,
α
This test generally has more power against mixed Poisson alternatives, like the negative binomial model, than the test in part (b) . Report a value for n S
2
/ X and state your conclusion.
(d) Obtain maximum likelihood estimates of expected counts for the negative binomial model for the bus driver accidents.
(e) Use the Pearson chi-square statistic to assess the fit of the negative binomial model to the bus driver accident data. Combine 10 or more accidents into a single category.
Report values for
(f) Construct a 95% confidence interval for the mean number of accidents per bus driver in a two year period. Indicate how your confidence interval was constructed.
5. Vianna, et al. (1971, Lancet, 1, 431-432) considered a sample of 109 patients with Hodgkin's disease. Some of those patients had undergone a tonsillectomy prior to the development of
Hodgkin’s disease, and the purpose of the study was to determine if tonsillectomy is associated with increased incidence of Hodgkin’s disease. A second sample of 109 "control" patients was selected from hospital records of patients with no history of Hodgkin's disease or any other malignant disease or chronic illness. The control patients were selected from a set of hospital records that generally matched the composition of the group of patients with Hodgkin's disease with respect to age, sex, race, county of residence, and date of hospital admission. Eight of the patients with Hodgkin's disease and two control patients were deleted from the analysis because their tonsillectomy history could not be obtained. The remaining 208 patients were cross-classified into the following 2
×
2 contingency table.

5
Hodgkin’s
Disease
Controls
Had
Tonsillectomy
67
43
Did not have
Tonsillectomy
34
64
Totals
101
107
The Pearson Chi-square test for independence is 14.26 on 1 d.f. with p-value < .001.
Vianna, et al. used the odds ratio ˆ
=
2 .
93 as an approximate measure of relative risk and concluded that tonsillectomy increases the risk of contracting Hodgkin's disease by a factor of nearly 3. They concluded that tonsillectomy removes a protective barrier against
Hodgkin's disease.
(a) Compute a 95% confidence interval for the odds ratio.
(b) A year later, Johnson and Johnson (1972, New England Journal of Medicine, 287,
1122-1125) reported results from a different study of 175 patients treated for
Hodgkin's disease at the Radiation Branch of the National Cancer Institute.
Information was available on 472 siblings of the 172 patients. The authors chose the closest sibling of the same sex within five years of age of each patient. This matching reduced the data to 85 patient-sibling pairs and the following table was reported.
Hodgkin’s
Disease
Controls
Had
Tonsillectomy
41
33
Did not have
Tonsillectomy
44
52
Totals
85
85
The Pearson chi-square test for independence is computed as 1.53 with p-value = .22, and the estimated odds ratio is ˆ
=
1 .
47 with 95% confidence bounds (0.80, 2.70).
On the basis of this, Johnson and Johnson claim to have refuted the contention of
Vienna, et al., that tonsils constitute a lymphoid barrier to Hodgkin's disease.
Which authors, if any, do you agree with? State your reasons. If you think any mistakes were made in either of the analyses, describe the mistakes and explain how the data should be analyzed, including formulas for test statistics, relative risk, and a confidence interval for relative risk. (Note: You may only be able to formulas for an improved analysis because you may not have all the information needed to complete the analysis you would like to do.)

6
6. Forty women at high risk for pregnancy induced high blood pressure participated in a randomized clinical trial to examine the effects of a 100 mg daily dose of aspirin on controlling blood pressure during the last three months of pregnancy. The result for each woman was classified as either “high blood pressure” or “satisfactory blood pressure”. The results are shown in the following table.
Aspirin Treatment
Placebo Treatment
High Blood
Pressure
1
5
Satisfactory Blood
Pressure
19
15
Use Fisher’s exact test to test the null hypothesis that aspirin has the same effect as the placebo on incidence of high blood pressure against the alternative that a daily dose of 100 mg of aspirin reduces the incidence of high blood pressure among high risk pregnant women. Report a p-value and state your conclusion.
7. In a study of the effects of treating multiple sclerosis patients with human fibroblast interferon (IFN-B) (reported by Jacobs, O'Malley, Freeman, and Ekes (1981), Science, 214, pp. 1026-1028), 20 multiple sclerosis patients were randomly divided into a group of 10
IFN-B recipients and a group of 10 controls. At the beginning of the study the severity of each patient's symptoms was evaluated and at the end of the study each patient was reevaluated and classified as either improved, unchanged, or worsened. The data are given in the following table.
Treated with IFN-B
Controls
Result of Treatment
Improved
5
1
Unchanged
4
4
Worsened
1
5
TOTALS
10
10
Perform an "exact" randomization test of the null hypothesis that the IFN-B treatment produces the same results as the treatment given to the controls against the null hypothesis that the IFN-B treatment gives better results. Using whatever criterion you think is best to order the tables, report the possible tables that are less consistent with the null hypothesis than the observed table. Compute the p-value for your test and state your conclusion. Be sure to state the criterion you used to order the tables.
8. (a) A newspaper article reports that in a survey of 1100 adults in the U.S., 63% think the
President is going a good job. The article claims that “this estimate is accurate to within 3 percentage points.” What does this mean with respect to concepts of statistical inference? (Assume the survey was done using simple random sampling from the adult population of the United States.)
(b) How large must the sample size be to guarantee accuracy within 1 percentage point?

7
9. (a) A researcher is planning a study to compare two programs for helping smokers to quit smoking. Program A is a program that is endorsed by the American Cancer Society and has been widely used. Based on extensive experience with program A, the researchers expect that about 15% of the smokers who try program A will be successful. Success is defined as never smoking during the first 12 months after completing the program. The researchers believe they will achieve a 22% success rate with program B, a new stop smoking program that has not been widely used.
Smokers volunteering for this study will be randomly divided into two treatment groups with n subjects in each group. How many smokers are needed for this study if the researchers want to have power = .85 of demonstrating a significantly higher success rate for program B using a .05 type I error level?
(b) Suppose the researchers simply want to construct a 95% confidence interval for the difference in the success rates for the two programs. How many subjects are needed in each group to produce a 95% confidence interval of width .06?
10. The data in the following table were obtained by classifying 156 dairy calves as to whether or not they developed pneumonia within the first two months of life. Those who developed an infection were also classified by whether or not they developed a second infection within two weeks of the first infection.
Bovine Pneumonia Infections
Secondary Infection
Yes No
Primary Infection
No Primary Infection
30
0
63
63
Note that one cell of the table must contain a zero count as a calf cannot get a secondary infection without a primary infection. A zero count of this type is called a “structural zero” because it is not a random observation. (Lloyd, page 117)
(a) Let
π ij denote the probability of a randomly selected calf being classified into the
(i, j) cell of the table. Consider the null hypothesis that the conditional probability of a secondary infection, given a primary infection, is equal to the probability of a primary infection. Express this null hypothesis in terms of the
π ij
’s.
(b) Assuming the null hypothesis in part (a) is true and using cell (i, j), write down the log-likelihood as a function of
Y ij
to denote the count in
τ
, the probability of a primary infection.
(c) Use the log-likelihood from part (b) to find the m.
l .e. for
τ
and its standard error.

8
(d) Use the deviance and Pearson chi-square statistics to test the fit of the model in part
(a) against the general multinomial alternative. Report values of p-values, and state your conclusion.
G 2 and X 2
, d.f.,
(e) Compute the m.
l .e. for error.
τ
under the general alternative and also report its standard
(f) Using the large sample normality of the m.
l .e., construct 95% confidence intervals for
τ
from the results from parts (c) and (e). Which is the better confidence interval to use? Explain.