Question

In 1954, an experiment was conducted to test the effectiveness of the Salk Vaccine as protection against the devastating effects of polio. Approximately 200,000 children were injected with an ineffective salt solution, and 200,000 other children were injected with the vaccine. The experiment was "double blind". Only 30 of the 200,000 vaccinated children later developed paralytic polio, whereas 120 of the 200,000 injected with the salt solution later developed paralytic polio.

a) Estimate the 95% confidence interval for the relative risk of the children in two groups.

b) Summarize the conclusion in one or two sentences. Use level of significance at 0.05 to make your conclusion by stating whether p-value is greater or less than cut off. Report the point estimate and 95% confidence interval of relative risk but describe it in plain words how much risk of one group is different than other group.

Answer 1

Relative Risk(RR) is defined as the ratio of Probability of event in exposed group to Probability of event in not exposed group. Here exposed group is one injected with salt solution.

There fore relative risk of developing paralytic polio in children injected with salt solution compared to that vaccinated is 120/30

ie children injected with salt solution is 4 times likely to get paralytic polio than vaccinated.

Now to calculate the confidence interval for relative risk we first estimate the S.E for Ln(RR) and then calculate the 95% CI for Ln(RR)

ie. where x₁ and x₂ are number of paralytic polio patients in the two groups. and and are the total population in each group.Substituting the values and now computing S.E=0.2041

Now Ln(RR)=Ln(4)=1.3863

Hence the 95% confidence interval is calculated using the formula and here z=1.96.

ie (0.9862,1.7863) This is the 95% CI for Ln(RR)

Now the confidence interval fro RR is

The above interval is expected to contain the true RR .if repeated samples are taken then 95% of the samples would have its true RR within the calculated confidence interval.

Null hypoyhesis in this case is

z statistic is calculated using fisher's exact test and i used a calculator to find the z statistic =6.792

The p value which corresponds to this statistic is 0 <0.05. Hence we conclude that the relative risk is greater than 1