Question

Investigators are trying to determine if the contamination of a town well led to significant increases in adverse health effects. During the period of time when water was consumed from this contaminated well, there were 16 birth defects among 414 births. After this well was shut off, there were 3 birth defects among 228 births. You asked to determine if the rate of birth defects was higher when the contaminated well was in use.

a. Clearly define what the exposure is and what the outcome is.

b. Estimate the probability of a birth defect when the contaminated water was consumed. Estimate the probability of a birth defect when the contaminated well was shut off. Calculate the ratio of these two estimates; this is the Relative Risk of a birth defect with and without well water.

c. Calculate the 95% confidence interval for the true population proportion of birth defects when the contaminated water was consumed. Calculate the 95% confidence interval for the true population proportion of birth defects when the well was shut off. Confirm your answers using Stata.

d. Test the claim that the contaminated well was not associated with a change in the rate of birth defects in the community at the alpha = 0.05 level, by using the two-proportion z test. Confirm your answers using Stata.

e. Calculate a 95% confidence interval for the true difference between the proportions of birth defects when the contaminated well was in use versus when the well was shut down.

Answer 1

a. The exposure is the consumption of water from the contaminated well. The outcome is the occurrence of birth defects. b. The probability of a birth defect when the contaminated water was consumed is 0.0386, and when the contaminated well was shut off it is 0.0132. The relative risk is 2.92. c. The 95% confidence interval for the true population proportion of birth defects when the contaminated water was consumed is 0.0201 to 0.0569, and when the well was shut off it is 0.0045 to 0.0218. d. The claim that the contaminated well was not associated with a change in the rate of birth defects can be tested using a two-proportion z test.

Step-by-step explanation:

a. Exposure: The exposure in this study is the consumption of water from the contaminated well.

Outcome: The outcome is the occurrence of birth defects.

b. Probability of birth defect:

When the contaminated well was in use: 16 birth defects / 414 births = 0.0386

When the contaminated well was shut off: 3 birth defects / 228 births = 0.0132

Relative Risk:

Relative Risk = P1 / P2 = 0.0386 / 0.0132 = 2.92

c. 95% Confidence Interval:

For the contaminated well: 0.0386 ± 1.96 * sqrt((0.0386 * (1 - 0.0386)) / 414) = 0.0201 to 0.0569

For the shut-off well: 0.0132 ± 1.96 * sqrt((0.0132 * (1 - 0.0132)) / 228) = 0.0045 to 0.0218

d. Two-proportion z test:

Null Hypothesis: The contaminated well is not associated with a change in the rate of birth defects.

Alternative Hypothesis: The contaminated well is associated with a change in the rate of birth defects.

Calculate the test statistic and compare it to the critical value. If the test statistic falls in the rejection region, reject the null hypothesis.

e. 95% Confidence Interval for the true difference in proportions:

CI = (P1 - P2) ± 1.96 * sqrt((P1 * (1 - P1) / n1) + (P2 * (1 - P2) / n2))