Chapter 9Comparing Two Populations: Binomial andPoisson9.1 Four Types of StudiesWe will focus on the binomial distribution in this chapter. In the last (optional) section we extendthese ideas to the Poisson distribution.When we have a dichotomous response we have focused on BT. The idea of finite populationswas introduced in Chapter 2 and presented as a special case of BT. In this section it is convenientto begin with finite populations.The four in the title of this section is obtained by multiplying 2 by 2. When we compare twopopulations both populations can be trials or both can be finite populations. In addition, as we shalldiscuss soon, a study can be observational or experimental. Combining these two dichotomies,we get four types of study, for example an observational study on finite populations.It turns out that the math results are (more or less) identical for the four types of studies, butthe interpretation of the math results depends on the type of study.We begin with an observational study on two finite populations. This was a real study per-formed over 20 years ago; it was published in 1988. The first finite population is undergraduatemen at at the University of Wisconsin-Madison and the second population is undergraduate menat Texas A&M University. Each man’s response is his answer to the following question:If a woman is interested in dating you, do you generally prefer for her: to ask you out;to hint that she wants to go out with you; or to wait for you to act.The response ‘ask’ is labeled a success and either of the other responses is labeled a failure. Thepurpose of the study is to compare the proportion of successes at Wisconsin with the proportion ofsuccesses at Texas A&M.The two populations obviously fit our definition of finite populations. Why is it called ob-servational? The dichotomy of observational/experimental refers to the control available to theresearcher. Suppose that Matt is a member of one of these populations. As a researcher, I havecontrol overwhether I have Matt in my study, but I do not have control over the population to which91Table 9.1: Responses to the Dating Study.Observed Frequencies Row ProportionsPrefer Women to: Prefer Women to:Population Ask Other Total Ask Other TotalWisconsin 60 47 107 0.56 0.44 1.00Texas A&M 31 69 100 0.31 0.69 1.00Total 91 116 207he belongs. The variable that determines to which population a subject belongs, is often called thestudy factor. Thus, in the current study, the study factor is school attended and it has two levels:Wisconsin and Texas A&M. This is an observational factor, sometimes called, for obvious reasons,a classification factor, because each subject is classified according to his school.Table 9.1 presents the data for this Dating Study.Next, we have an example of comparing finite populations in an experimental study. Medicalresearchers were searching for an improved treatment for persons with Crohn’s Disease. Theywanted to compare a new drug therapy, cyclosporine, to an inert drug, called a placebo.Now we are at a hugely important distinction from the Dating Study. Below we are going totalk about comparing the ‘cyclosporine population’ to the ‘placebo population.’ But, as we shallsee, and perhaps is already obvious, there is, in reality, neither a ‘cyclosporine population’ nor a‘placebo population.’ Certainly not in the physical sense of there being a UW and Texas A&M.Indeed, as I formulate a ‘population approach’ to this medical study, the only population Ican imagine is one superpopulation of all persons, say in the US, who have Crohn’s Disease.This superpopulation gives rise to two imaginary populations: first, imagine that everybody in thesuperpopulation is given cyclosporine and, second, imagine that everybody in the superpopulationis given the placebo.To summarize the differences between observational and experimental:1. For observational, there exists two distinct finite populations.For experimental, there exists two ‘treatments’ of interest and one superpopulation of sub-jects. The two populations are generated by imagining what would happen if each memberof the superpopulation was assigned each treatment.2. Here is a very important consequence of 1: For an observational study, the two populationsconsist of different subjects whereas for an experimental study, the two populations consistof the same subjects.For the Dating study, the two populations are comprised of different men (Bubba, Bobby Lee,Tex, etc. for one; and Matt, Eric, Brian, etc. for the other). For the Crohn’s study, both populationsconsist of the same persons, namely the persons in the superpopulation.An experimental study also requires something called randomization. I will discuss it inthe next section. Also, these ideas can and will be extended to BT that are for trials, not finitepopulations.929.2 Assumptions and ResultsWe begin with an observational study on finite populations. Assume that we have a random sampleof subjects from each population and that the samples are independent of each other. Independencehere is much the same idea as it was for trials. For our Dating study, independence means that themethod of selecting subjects from Texas was totally unrelated to the method used in Wisconsin.‘Totally unrelated’ is, of course, rather vague, but bear with me for now. Additionally and sadly, Iwill not at this time give you an example where independence fails to be true in a major way. Laterwhen we consider paired data we will revisit this issue.All this talk of independence should not make us forget that, just like for a single finite popu-lation, the biggest challenge is to actually get a random sample. Usually the sample is clearly notrandom and the researcher simply pretends that it is. This is too big of a topic for typing; I willdiscuss it in lecture.The sample sizes are n1from the first population and n2from the second population. We defineX to be the total number of successes in the sample from the first population and Y to be the totalnumber of successes in the sample from the second population. Given our assumptions, X ∼Bin(n1, p1) and Y ∼ Bin(n2, p2), where piis the proportion of successes in population i, i = 1, 2.Always remember that you can study the populations separately using the estimation methodsof Chapter 3. The purpose of this chapter is to compare the populations, or, more precisely, tocompare the two p’s. We will consider both estimation and testing.For
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