Description
1. The following table shows the results of a study comparing radiation
therapy with surgery in treating cancer of the larynx. The response
indicates the cancer was controlled for at least two years following
treatment. Let θ be the odds ratio in the 2 × 2 table. Significance
level 0.05 is used for testing.
Controlled Not controlled
Surgery 21 2
Radiation therapy 8 9
(a) Calculate and interpret the difference of proportions (D), relative risk (RR), and odds ratio (OR), including their confidence
intervals.
(b) Calculate and interpret the Pearson and likelihood-ratio (LR)
chi-squared statistics for testing H0 : θ = 1 against H1 : θ 6= 1.
(c) Calculate and interpret the p-value for Fisher’s exact test with
H0 : θ = 1 and H1 : θ 6= 1.
2. A study aspirations of high school students measured aspirations with
the scale (some high school, high school graduate, some college, college
graduate). The data are shown in the following table. The table also
contains the estimated expected frequencies under independence and
the standardized Pearson residuals in the parentheses.
1
Family Income
Study Aspirations low middle high Total
some high school 9 11 9 29
(?, ?) (11.47, -0.19) (9.45, -0.19)
high school graduate 44 52 41 137
(38.14, 1.58) (54.20, -0.54) (44.66, -0.95)
some college 13 23 12 48
(13.36, -0.13) (18.99, 1.30) (15.65, -1.24)
college graduate 10 22 27 59
(16.43, -2.11) (23.34, -0.40) (?, ?)
Total 76 108 89 273
(a) Test independence of educational aspirations and family income
using the Pearson χ
2
statistic (X2
), and the likelihood ratio χ
2
statistic (G2
).
(b) Complete the above table. Do the residuals suggest any association pattern?
(c) Calculate and interpret the odds ratio (OR) from the cells at the
four corners.
3. Consider the previous data for investigating the association between
educational aspirations and family income. Assign the scores to study
aspiration and family income in the following manner: 1=some high
school, 2=high school graduate, 3=some college, 4=college graduate
and 1=low, 2=middle, 3=high.
(a) Calculate the sample correlation between educational aspirations
and family income, and test whether there is an increasing or
descreasing trend.
(b) Let 10=some high school, 20=high school graduate, 30=some
college, 40=college graduate and -1=low, 0=middle, 1=high. Do
the results change?
(c) Let 1=some high school, 3=high school graduate, 5=some college,
10=college graduate and 1=low, 5=middle, 10=high. Do the
results change?
4. The following table refers to applicants to graduate school at the University of California at Berkeley, for fall 1973. It presents admissions
decisions by gender of applicant for the size largest graduate departments. Denote the three variables by A = whether admitted, G =
gender, and D = department.
2
Whether Admitted
Male Female
Department Yes No Yes No
1 512 313 89 19
2 353 207 17 8
3 120 205 202 391
4 138 279 131 244
5 53 138 94 299
6 22 351 24 317
Total 1198 1493 557 1278
(a) Compute and interpret the sample AG conditional odds ratios.
(b) Compute and interpret the sample AG marginal odds ratios.
(c) Explain why they give such different indications of the AG association.
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