Question

In an epidemiologic study, 147 incident bladder cancer cases were compared to 179 controls selected from the population that gave rise to the cases with respect to smoking history. The data were as follows:

Exposure status	Disease Status
	Case	Control	Total
Smoker	90	82	172
Nonsmoker	57	97	154
Total	147	179	326

             Perform the appropriate statistical test of the relationship between bladder cancer and smoking history using the attached SAS file. Assume α = 0.05 (two-tailed) and 80% power.

1. State the null and alternative hypotheses.

2. List the critical value.

3. List the estimated odds ratio.

4. Using the chi-square, P-value, and odds ratio output from SAS, report your decision based on the critical value (reject Ho and accept H_A OR fail to reject Ho), P-value, and 95% confidence interval

5. If the decision was to fail to reject Ho, can Ho be accepted?

Answer 1

Expected Values Column 1 Column 2 Total smoker 147x172 326 77.558 179 x 172 326 94.442 172 non smoker 147 x 154 326 69.442 179 x 154 326 = 84.558 154 Total 147 179 326 Based on the observed and expected values, the squared distances can be computed according to the following formula: (E – 0)2 /E. The table with squared distances is shown below: Squared Distances Column 1 Column 2 smoker (90–77.558) 77.558 = 1.996 (82–94.442) 94.442 = 1.639 non smoker (57–69.442) 69.442 2.229 (97–84.558) 84.558 = 1.831 Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: The two variables are independent H: The two variables are dependent This corresponds to a Chi-Square test of independence. Rejection Region Based on the information provided, the significance level is a = 0.05, the number of degrees of freedom is df = (2 – 1) (2 – 1) = 1, so then the rejection region for this test is R= {x? : x² > 3.841}. Test Statistics The Chi-Squared statistic is computed as follows: (Oij – Eij)? Eij = 1.996 + 2.229 +1.639 +1.831 = 7.695 i=1 Decision about the null hypothesis Since it is observed that x2 = 7.695 > x = 3.841, it is then concluded that the null hypothesis is rejected. Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the two variables are dependent, at the 0.05 significance level. The corresponding p-value for the test is p = Pr(xỉ > 7.695) = 0.0055.

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