Question

In December 2001, the Department of Veterans Affairs announced that it would begin paying benefits to soldiers suffering from Lou Gehrig's disease who had served in the Gulf War (The New York Times, December 11,2001 ). This decision was based on an analysis in which the Lou Gehrig's disease incidence rate (the proportion developing the disease) for the approximately 700,000 soldiers sent to the Gulf between August 1990 and July 1991 was compared to the incidence rate for the approximately \(1.8\) million other soldiers who were not in the Gulf during this time period. Based on these data, explain why it is not appropriate to perform a formal inference procedure (such as the two-sample \(z\) test) and yet it is still reasonable to conclude that the incidence rate is higher for Gulf War veterans than for those who did not serve in the Gulf War.

Short answer

Formal inference procedures like the two-sample z test are not appropriate in this case because they are designed to estimate parameters of a population from a sample. Here, we are not dealing with samples but with entire populations of soldiers. Therefore, we are not estimating but comparing known parameters. Further, the incidence rate is a ratio, so a higher rate among Gulf War veterans would directly indicate a higher risk of developing the disease in this group.

Step-by-step solution

Understanding Data

First, let's understand the exercise. It deals with two groups - soldiers who served in the Gulf War (700,000) and other soldiers who did not (1.8 million). The incidence rate of Lou Gehrig's disease was compared between these groups.

Why not Formal Inference

The reason it's not appropriate to perform a formal inference here is that formal tests like two-sample z test are designed to infer from a sample to a population. Here, we are not dealing with samples. The incidence rates are for the entire populations of soldiers who were, and were not, in the Gulf War during the specified time period. We are not estimating an unknown parameter based on a sample but rather comparing known parameters derived from entire populations.

How to Conclude Higher Incidence Rate

Despite this, we can still conclude that the incidence rate is higher for Gulf War veterans because we are comparing the incidence rates (a ratio of disease frequency to the group size) for the entire groups. If the ratio is higher for the Gulf War veterans, then it is reasonable to conclude that these veterans had a higher risk of developing the disease.