Chapter 9 Comparing Two Populations Binomial and Poisson 9 1 Four Types of Studies We will focus on the binomial distribution in this chapter In the last optional section we extend these ideas to the Poisson distribution When we have a dichotomous response we have focused on BT The idea of finite populations was introduced in Chapter 2 and presented as a special case of BT In this section it is convenient to begin with finite populations The four in the title of this section is obtained by multiplying 2 by 2 When we compare two populations both populations can be trials or both can be finite populations In addition as we shall discuss 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 but the 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 performed over 20 years ago it was published in 1988 The first finite population is undergraduate men at at the University of Wisconsin Madison and the second population is undergraduate men at 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 The purpose of the study is to compare the proportion of successes at Wisconsin with the proportion of successes at Texas A M The two populations obviously fit our definition of finite populations Why is it called observational The dichotomy of observational experimental refers to the control available to the researcher Suppose that Matt is a member of one of these populations As a researcher I have control over whether I have Matt in my study but I do not have control over the population to which 91 Table 9 1 Responses to the Dating Study Observed Frequencies Row Proportions Prefer Women to Prefer Women to Population Ask Other Total Ask Other Total Wisconsin 60 47 107 0 56 0 44 1 00 Texas A M 31 69 100 0 31 0 69 1 00 Total 91 116 207 he belongs The variable that determines to which population a subject belongs is often called the study 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 Medical researchers were searching for an improved treatment for persons with Crohn s Disease They wanted 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 to talk about comparing the cyclosporine population to the placebo population But as we shall see 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 I can 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 the superpopulation is given cyclosporine and second imagine that everybody in the superpopulation is 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 subjects The two populations are generated by imagining what would happen if each member of the superpopulation was assigned each treatment 2 Here is a very important consequence of 1 For an observational study the two populations consist of different subjects whereas for an experimental study the two populations consist of 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 populations consist of the same persons namely the persons in the superpopulation An experimental study also requires something called randomization I will discuss it in the next section Also these ideas can and will be extended to BT that are for trials not finite populations 92 9 2 Assumptions and Results We begin with an observational study on finite populations Assume that we have a random sample of subjects from each population and that the samples are independent of each other Independence here is much the same idea as it was for trials For our Dating study independence means that the method 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 I will not at this time give you an example where independence fails to be true in a major way Later when 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 population the biggest challenge is to actually get a random sample Usually the sample is clearly not random and the researcher simply pretends that it is This is too big of a topic for typing I will discuss it in lecture The sample sizes are n1 from the first population and n2 from the second population We define X to be the total number of successes in the sample from the first population and Y to be the total number of successes in the sample from the second population Given our assumptions X Bin n1 p1 and Y Bin n2 p2 where pi is the proportion of successes in population i i 1 2 Always remember that you can study the populations separately using the estimation methods of Chapter 3 The purpose of this chapter is to compare the populations or more precisely to compare the two p s We will consider both estimation and testing For estimation our goal is to estimate p1 p2 Define p 1 X n1 and p 2 Y n2 these are out point estimators of the p s The obvious and
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