Introduction to Categorical Data Analysis

PROBLEMS 63
Table 2.17. Data for Problem 2.30
Cancer Cancer Not
Controlled Controlled
Surgery 21 2
Radiation therapy 15 3
Source: W. Mendenhall et al., Int. J. Radiat. Oncol. Biol. Phys., 10:
357–363, 1984. Reprinted with permission from Elsevier Science Ltd.
2.32 Of the six candidates for three managerial positions, three are female and three
are male. Denote the females by F1, F2, F3 and the males by M1, M2, M3.
The result of choosing the managers is (F2, M1, M3).
a. Identify the 20 possible samples that could have been selected, and construct
the contingency table for the sample actually obtained.
b. Let p1 denote the sample proportion of males selected and p2 the sample
proportion of females. For the observed table, p1 − p2 = 1/3. Of the 20
possible samples, show that 10 have p1 − p2 ≥ 1/3. Thus, if the three
managers were randomly selected, the probability would equal 10/20 =
0.50 of obtaining p1 − p2 ≥ 1/3. This reasoning provides the P -value for
Fisher’s exact test with Ha: π1 >π2.
2.33 In murder trials in 20 Florida counties during 1976 and 1977, the death penalty
was given in 19 out of 151 cases in which a white killed a white, in 0 out of
9 cases in which a white killed a black, in 11 out of 63 cases in which a
black killed a white, and in 6 out of 103 cases in which a black killed a black
(M. Radelet, Am. Sociol. Rev., 46: 918–927, 1981).
a. Exhibit the data as a three-way contingency table.
b. Construct the partial tables needed to study the conditional association
between defendant’s race and the death penalty verdict. Find and interpret
the sample conditional odds ratios, adding 0.5 to each cell to reduce the
impact of the 0 cell count.
c. Compute and interpret the sample marginal odds ratio between defendant’s
race and the death penalty verdict. Do these data exhibit Simpson’s paradox?
Explain.
2.34 Smith and Jones are baseball players. Smith had a higher batting average than
Jones in 2005 and 2006. Is it possible that, for the combined data for these two
years, Jones had the higher batting average? Explain, and illustrate using data.
2.35 At each age level, the death rate is higher in South Carolina than in Maine, but
overall the death rate is higher in Maine. Explain how this could be possible.
(For data, see H. Wainer, Chance, 12: 44, 1999.)