Question

A 1970s case-control study on cerebrovascular disease (thrombotic stroke) and oral contraceptive use in young women matched cases to controls according to neighborhood, age, sex, and race. The table below displays data from this study for thrombotic stroke. Perform an appropriate test at alpha level 5%.

	Case exposed	Case Not exposed
Control exposed	2	5
Control Not exposed	44	55

1. Which test is appropriate to test the association between cerebrovascular disease (thrombotic stroke) and oral contraceptive use? Why?

2. State null and alternative hypotheses.

3. Compute a test statistics.

4. Compute a p-value using an appropriate table.

5. Can you reject the null hypothesis at alpha level 5%?

6. State your conclusion.

Answer 1

We will use a Chi-Square Independence test. This is because we have categorical and count data and want to find whether there is ascoocation between the two vairables.

Chi-Square Independence test

(1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: H0​: The two variables - cerebrovascular disease and oral contraceptive use are independent Ha​: The two variables - cerebrovascular disease and oral contraceptive use are dependent This corresponds to a Chi-Square test of independence. (2) Degrees of Freedom The number of degrees of freedom is df = (2 - 1) * (2 - 1) = 1 (3) Critical value and Rejection Region Based on the information provided, the significance level is α=0.05, the number of degrees of freedom is df = (2 - 1) * (2 - 1) = 1, so the critical value is 3.8415. Then the rejection region for this test becomes R={χ2:χ2>3.8415}. (4)Test Statistics The Chi-Squared statistic is computed as follows: (5)P-value The corresponding p-value for the test is p=Pr(χ2​>0.6706)=0.4129 (6)The decision about the null hypothesis Since it is observed that χ2=0.6706<χ2_c​rit=3.8415, it is then concluded that the null hypothesis is NOT rejected. (7)Conclusion It is concluded that the null hypothesis Ho is NOT rejected. Therefore, there is NOT enough evidence to claim that the two variables - cerebrovascular disease and oral contraceptive use are dependent, at the 0.05 significance level. Conditions: a. The sampling method is simple random sampling. b. The data in the cells should be counts/frequencies c. The levels (or categories) of the variables are mutually exclusive.

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