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.

Exposure is "Consumption of water from contaminated well".

Outcome is "Birth defects".

b.

The probability of a birth defect when the contaminated water was consumed =p1 =x1/n1 =16/414 =0.039

The probability of a birth defect when the contaminated water was shut off =p2 =x2/n2 =3/228 =0.013

Ratio of these estimates =Relative Risk, RR =p1/p2 =0.039/0.013 =3

c.

95% confidence interval for the true population proportion of birth defects when the contaminated water was consumed =P1 = =0.039 (1.96* ) =(0.020, 0.576)

95% confidence interval for the true population proportion of birth defects when the well was shut off =P2 = =0.013 (1.96* ​​​​​​) =(0.002, 0.028)

d.

H0: The difference between the population proportion is not significant. P1 - P2 =0

H1: The deference between the population proportions is significant. P1 - P2 0

Overall proportion, p =(x1+x2)/(n1+n2) =(16+3)/(414+228) =19/642 =0.03

Test statistic, Z =(p1 - p2)/ =(0.039 - 0.013)/ =0.026/0.01407 =1.85

Critical Z-value at 0.05 significance level for two-tailed test is Zcrit =1.96

Conclusion: Z < Zcrit (1.85 < 1.96) - No sufficient evidence to reject the null hypothesis(H0) and so, we failed to reject H0. Thus, we cannot claim that the difference between two population proportions (P1 - P2) is significant.

e.

Difference in sample proportions =p1 - p2 =0.039 - 0.013 =0.026

Standard error, SE =0.01407 (since from d. above)

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 is:

P1 - P2 =(p1 - p2) (Z*SE) =0.026 (1.96*0.01407) =(-0.0016, 0.0536)

The above interval includes 0. So, the difference between true (population) proportions is not significant (same conclusion as in d. above).