Reference no: EM132239435
Assignment Questions -
Q1. For a 2 by 2 contingency table, let πij and nij be the multinomial cell probability and observed cell count for the (i, j)th cell, respectively, for i=1, 2; j=1, 2. Let n= Σij nij. Show that the maximum likelihood estimate (MLE) of πij is nij/n.
Q2. For adults who sailed on the Titanic on its fateful voyage, the odds ratio between gender (female, male) and survival (yes, no) was 11.4.
(a) What is wrong with the interpretation, "The probability of survival for females was 11.4 times that for males."
(b) When would the quoted interpretation be approximately correct? Why?
(c) The odds of survival for females equaled 2.9. For each gender, find the proportion who survived.
Q3. A study on educational aspirations of high school students (S. Crysdale, Int. J. Compar. Sociol., 16:19 - 36 (1975)) measured aspirations using the scale (some high school, high school graduate, some college, college graduate). For students whose family income was low, the counts in these categories were (9; 44; 13; 10); when the family income was middle, the counts were (11; 52; 23; 22); when family income was high, the counts were (9; 41; 12; 27).
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Family Income
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Education Aspiration
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|
Some School
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High School
|
Some College
|
College
|
|
Low
|
9
|
44
|
13
|
10
|
|
Middle
|
11
|
52
|
23
|
22
|
|
High
|
9
|
41
|
12
|
27
|
(a) Test independence of educational aspirations and family income using Pearson's χ2 (chisquare) test and likelihood ratio test. Interpret, and explain the deficiency of this test for these data.
(b) Calculate the adjusted residuals. Do they suggest any association pattern?
(c) Conduct a test for linear trend alternative which is more powerful in the case of ordinal data.
For that purpose, you may assign the equally spaced scores for educational aspirations as 1, 2, 3 and 4 and for income categories as 1, 2 and 3. Interpret your test results.
Q4. The following data consist of a random sample of 793 individuals who were involved in bicycle accidents during a specified one-year period.
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Head Injury
|
Wearing Helmet
|
|
Yes
|
No
|
Total
|
|
Yes
|
17
|
218
|
235
|
|
No
|
130
|
428
|
558
|
|
Total
|
147
|
646
|
793
|
a) Construct a 90% confidence interval for the difference of proportions. Test hypothesis of equal proportions at α = 0.10 and interpret your test results.
b) Construct a 90% confidence interval for odds ratio. Test hypothesis of independence at α = 0.10 and interpret your test results.
c) Conduct Pearson's χ2 (chi-square) test and likelihood ratio test of statistical independence. Interpret your test results.
d) Are the conclusions in a), b) and c) consistent?
Q5. In a small survey, a researcher asked 23 individuals if they received a flu shot this year, and whether they caught the flu this winter. The results indicate that, of the ten people who did not receive a flu shot, four got the flu and six did not. Of the thirteen people who received a flu shot, one got the flu and twelve did not.
|
|
Flu
|
No Flu
|
|
No shot
|
4
|
6
|
|
Shot
|
1
|
12
|
Test null hypothesis that flu shot is ineffective.
a) Perform one sided Fisher's exact test manually. Would you reject null hypothesis of independence at significance level of α = 0.10.
b) Perform Fisher's exact test using Minitab which is two sided. Would you reject null hypothesis of independence at significance level of α = 0.10.
c) Fisher's exact test using Minitab is actually a two sided. Explain how to perform this two sided Fisher's exact test manually.
d) For this particular application, would you prefer one or two sided Fisher's exact test? Justify your answer.
Q6. Let's go back to the example of Kidney Stones (Charig et. al., British Medical Journal (Clinical Research Ed), March 1986, 292 (6524): 879-882).
|
|
Treatment
|
|
A
|
B
|
|
kidney stones
|
success
|
failure
|
success
|
failure
|
|
Small
|
81
|
6
|
234
|
36
|
|
Large
|
192
|
71
|
55
|
25
|
|
Total:
|
273
|
77
|
289
|
61
|
To study the association between treatment type and treatment outcome given types of stones, do the following analyses.
a) Write down the 2 by 2 marginal table and calculate the relevant odds ratio. Conduct Pearson Chi-square and likelihood ratio tests for independence.
b) Write down the 2 by 2 partial tables and calculate the relevant odds ratios. Conduct Pearson Chi-square and likelihood ratio tests for independence for each of the partial tables.
c) Using the Cochran-Mantel-Haenszel test in Minitab, test conditional independence between treatment type and treatment outcome given all types of stones.
d) Summarize your conclusions.