Goals: In this lab, you will

• Consider the relative risk as an alternative statistic in a two-way table
• Learn how to construct a confidence interval for relative risk
• Practice interpreting a confidence interval for relative risk

Peanut allergies have increased in prevalence in the last decade, but can they be prevented? Even among infants with a high risk of allergy? Is it better to avoid the problematic food or to encourage early introduction? Du Toit et al. (New England Journal of Medicine, Feb. 2015) randomly assigned U.K. infants (4-11 months old) with pre-existing sensitivity to peanut extract to either consume 6 g of peanut protein per week or to avoid peanuts until 60 months of age. The table below shows the results for infants who were not initially sensitized to peanuts and whether or not the child had developed a peanut allergy at 60 months.

	Peanut avoidance	Peanut consumption	Total
Peanut allergy	11	2	13
No allergy	172	193	365
Total	183	195	378
Proportion	0.0601	0.0103

These data provide strong evidence that avoiding peanuts increases the probability of developoing a penaut allergy (Fisher's Exact Test p-value = P(X ≥ 11) = 0.0074 < 0.05).  But telling the public that the probability increases by 0.0498 might not be very compelling...

Research question: How much does peanut avoidance increase infants' likelihood of developing a peanut allergy?

 

Changing the statistic to the relative risk does not change how we find the p-value (1-to-1 correspondence with the other statistics).

However, now we would also like a confidence interval for the corresponding parameter, the ratio of the underlying probabilities of allergy between these two treatments. When we produced confidence intervals for other parameters, we examined the sampling distribution of the corresponding statistic to see how values of that statistic varied under repeated random sampling. If the sampling distribution of the statistic was approximately normal and centered around the parameter, we said a 95% confidence interval could be something like statistic + z* x SE(statistic).

So now let’s examine the behavior of the relative risk of conditional proportions using the Analyzing Two-Way Tables applet to simulate the random assignment process (as opposed to simulating the random sampling from a binomial process) under the (null) assumption that there’s no difference between the two treatments probabilities of success.

Generate a null distribution for Relative Risks: Enter the two-way table into the applet (including naming the rows and columns) and press Use Table. Use the Statistic pull-down menu to select Rel Risk and confirm the Relative risk value. Generate 10,000 random shuffles. Note: If the number of successes in either group equals zero, the applet adds 0.5 to each cell of the table before calculating the relative risk in order to avoid dividing by zero. Include a screen capture of your null distribution of simulated relative risk values.

In fact, this distribution is usually well modeled by a log normal distribution (meaning the logged values follow a normal distribution). To verify this, check the ln relative risk box (in the lower left corner, or choose from pull-down menu) to take the natural log of each relative risk value and display a new histogram of these transformed values.

While the log transformation does not impact the p-value, it does impact the confidence interval. So far you have seen the standard deviation of the null distribution, but that assumes the probability of success is the same for both treatments. Instead, we want to find a standard deviation of the ln rel risk values without making that assumption.

But our parameter is the long-run relative risk, not the long-run ln relative risk. Luckily there is an easy fix.

Confirm your calculations match the applet output:

If not, check your work!

For this lab, after you submit Jotform will send you an email with a pdf attachment. Upload this pdf file into Canvas.

If you worked with a partner on this lab, both of you need to join a Lab 6 group before submitting.