Topics to be covered this week. STA 138 Winter Quarter, 2020 Monday, Jan 27 Two-way contingency tables (chaps 2.1-2.3 in the text and Handout 5). Wednesday, Jan 29 Tests for independence in a two-way table (chap 2.4 in the text, and Handout 6). Friday, Jan 31 Tests for independence in a two-way table (chap 2.4 in the text, and Handout 6), Test of independence in a two-way table and partitioning of chi-square (chap 2.4 in the text and Handout 7), Test for independence for ordinal data (chap 2.5 in the text, and Handout 7). Homework 3: Due on Wednesday, Feb 5. You may form a group of 3 students registered in this course and submit one completed homework for the group. The front page should display only the names of the students in the group. The actual work should start from the second page. 1. Consider the following contingency table, in which two species of mice were tested for a speci c parasite. Infected Not Infected Species 1 40 16 Species 2 24 34 (a) If 1 and 2 are the proportions of the infected in species 1 and species 2 respectively, then nd an approximate con dence interval for 1 2. Interpret your result. (b) Estimate the odds of infection for the two species (two numbers). Also nd the odds ratio comparing species 1 to species 2. (c) Obtain an approximate 95% con dence interval for the odds ratio. Interpret your result. 2. Consider the following contingency table (based on the past record of 100,000 individuals), in which the true diagnosis (by an invasive test) and the predicted diagnosis (by a mouth swab) are shown. Tested Positive Tested Negative Condition True 475 25 Condition False 4970 94530 (a) Estimate the overall risk (probability) of having the condition. 1 (b) Estimate the sensitivity and speci city. (c) Estimate the false negative (having the condition given that the test is negative), and false positive (do not have the condition given that the test is positive). (d) Would you say that this test does well? Explain your answer. 3. The following table is obtained from a General Social Survey in 2002. Party Identi cation Democrat Independent Republican White 875 441 871 Black 302 80 43 (a) Carry out a test for the null hypothesis of independence between party identi cation and race. Use = 0:05. Find the p-value of your test and state your conclusion. (b) Obtain the standardized residuals and summarize your ndings. (c) Partition the chi-squared into two components, and explain your results. 4. In a large country, smoking behavior of a random sample of 1000 lung cancer patients were recorded. Independently, a random sample of 1000 lung cancer free patients was taken and their smoking behavior were recorded. The data is from 1950. The counts are given below. Have smoked Cancer Cancer-free Yes 970 917 No 30 83 Total 1000 1000 (a) Identify the response variable and the explanatory variable. Describe the sampling scheme. Is it joint multinomial or independent multinomials? If it a joint multinomial, are the rows independent multinomials or are the columns independent multinomials. Explain (b) Can you obtain the proportion who have lung cancer in the country? Ex- plain. (c) Estimate the odds ratios of smoking among the cancer, and among the cancer free patients. Estimate the odds ratio which compares the odds of cancer to cancer-free patients. Summarize your ndings. (d) Test the hypothesis H0 : 1 and against H1 : > 1 at level = 0:05. Find the p-value of your test and state your conclusion. 5. In an I J contingence table, let ^ij = ni+n+j=n be the estimated expected counts under H0, the hypothesis of independence. (a) Show that the row and column totals of f^ijg are the same as those of the observed counts fnijg. (b) When the rows are independent multinomials, then show that E(^ij) = E(nij) under H0. 2 (c) Show that E(n+j=n) = +j under H0, irrespective of whether fnijg are joint multinomial or the rows of fnijg are independent multinomials. (d) If the rows of fnijg are independent multinomials, and if fi+g were known, then show that ~j = PI i=1 niji+=ni+ is unbiased for +j (even when H0 is false). 6. In order to investigate how the educational levels of adults are related to their race, two surveys were done in a major metropolitan area. In survey 1, 225 adults were randomly taken, and their highest educa- tional level (1=none, 2=high school, 3=college) as well as their race (1=black, 2=white) were recorded. In survey 2, 100 blacks were randomly selected and independently, 100 whites were selected. Educational level of each of the sample subjects was recorded. Denote by ij be the proportion of individuals who are of the ith race with jth educational level in the metropolitan area. The data are given below. Survey 1 Educational Level None High School College Row Total Race Black 25 62 37 124 White 20 40 41 101 Column Total 45 102 78 225 Survey 2 Educational Level None High School College Row Total Race Black 21 49 30 100 White 22 29 49 100 Column Total 43 78 79 225 The questions below are conceptual in nature, and numerical calculations are not necessary. Please provide adequate explanation for each answer. (a) For each of the two surveys, describe if the distribution of the counts fnijg is jointly multinomial or independent multinomials. Write down the parameters of the distributions. (b) The Welfare Department of the state is interested in estimating ijs, i+s, +js. Can you estimate these from the data in Survey 1? Survey 2? (c) The state is interested in obtaining estimates of the proportion of None, High School and College graduates among blacks as well as among whites. Can these be estimated from the data in Survey 1 and/or Survey 2? (d) The state would like to have estimates of the percentage of blacks and whites among those who are college graduates. Can you obtain these estimates from Survey 1 data? How about Survey 2 data? 3 (e) It is of interest to test if educational level is independent of race. Can this done using the data from Survey 1? How about Survey 2 data? 4
