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UW-Madison STAT 371 - Ch. 15

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Comparing Two Binomial PopulationsThe Ubiquitous 2 2 TableComparing Two Populations; the Four Types of StudiesAssumptions and Results`Blind' Studies and the Placebo EffectAssumptions, RevisitedBernoulli TrialsSimpson's ParadoxSimpson's Paradox and BasketballSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 15Comparing Two Binomial PopulationsIn this chapter and the next, our data are presented in a 2 × 2 contingency table. As you willlearn, however, not all 2 × 2 contingency t ables are analyzed the same way. Thus, I b egin with anintroductory section.15.1 The Ubiquitous 2 × 2 TableI have always liked the word ubiquitous; and who can argue taste? According to dictionary.comthe definiti on of ubiquit ous is:Existing or being everywhere, especially at the same time; omnipresent.Table 15.1 is a partial recreation of Table 8.5 on page 170 in Chapter 8 of these notes.Let me explain why I refer to this table as ubiquitous. You will learn of many different scientificscenarios that yield data of the form presented in Table 15.1. Depending on the scenario, you willlearn the different appropriate ways to summarize and analyze the data in this table.I say that Table 15.1 is only a partial recreation o f Chapter 8’s table because of the followingchanges:1. In Chapter 8, the rows were treatments 1 and 2; in Table 15.1, they are simply called rows 1and 2. In some scenarios, these rows will represent treatments and in some scenarios theywon’t.2. In Chapter 8, the col umns were the two poss ible responses, success and failure; in Table 15.1,they are simply called columns 1 and 2. In some scenarios, these columns will represent theresponse and in some scenarios they won’t.3. The table in Chapter 8 also includ ed row proportions that served two purposes: they de-scribed/summarized the d at a; and they were the basis (via the computation of x = ˆp1− ˆp2)for findi ng the observed value of the test statistic, X, for Fisher’s test. Again, in some sce-narios in this and the next chapter we will compute row proportions, and in some scenarioswe won’t.353Table 15.1: The general notation for the ubiquitous 2 × 2 contingency table of d at a.ColumnRow 1 2 Total1 a b n12c d n2Total m1m2nHere are the main features to note about the ubi quitous 2 × 2 contingency table of data.• The values a, b, c and d are called the cell counts. They are necessarily nonnegative integers.• The values n1and n2are the row totals of the cell counts.• The values m1and m2are the column totals of the cell counts.• The value n is the sum of the four cell counts; alternatively, it is the sum of the row [column]totals.Thus, there are nine counts in the 2 × 2 contingency table, all of which are determined by the fourcell counts.15.2 Compari n g Two Populations; th e Four Typ es of StudiesThe first appearance of the ubi quitous 2 × 2 contingency t able was in Chapter 8 for a CRD witha dichotomous response. Recall that Fisher’s Test is used to evaluate the Skeptic’s Argument. Asstated at the b eginnin g of this Part II of these notes, a limitation of the Skeptic’s Argument is thatit is concerned only with the units under study. In this section, you will learn how to extend theresults of Chapter 8 to populations. In add ition, we will extend results to observational studies that,as you may recall from Chapter 1, do not involve randomization.In Chapter 8 the un its can be trials or subjects. The list ing below summaries the studies ofChapter 8.• Units are subjects: The infidelity study; the prison er study; and the artificial HeadacheStudy-2.• Units are trials: The go lf putting study.The idea of a population depends on the type of unit. In particular,• When units are subjects, we have a finite population. The mem b ers of the finite populationcomprise all potential subjects of interest t o the researcher.• When unit s are trials, we assume that they are Bernoulli Trials.354The number four in the ti tle of this sectio n is obtained by multiplying 2 by 2. When wecompare two populations both populations can be Bernoulli trial s or both can be finite populations.In additi on, as we shall discuss soon, a study can be observational or experimental. Combiningthese two dichotomies, we get four types of study; for example an observational study on finitepopulations.It turns o ut that the mathematical formulas are identical for the four types of studies, but theinterpretat ion of our analysis depends o n the type of stud y.We begin with an observational study o n two finite populations. Thi s is a real study that waspublished in 1988; [1].Example 15.1 (The Dating study.) The first finite popul ation is undergraduate men at at the Uni-versity of Wis consin–Madison and the second population is undergraduate men at Texas A&MUniversity. Each man’s response is his answer to the following question:If a woman is int erested 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 t he study is to compare the proportion of successes at Wisconsin with the proportion ofsuccesses at Texas A&M.These two populati ons 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 M at t is a member of one of these p opulations. As a researcher, I havecontrol over whether I have Matt in my study, but I do not have control over the population to whichhe belongs. Consistent with our usage in Chapter 1, the variable that determines a subj ect’s popu-lation, is called the study factor. In the current example, the study factor is school attended and ithas two levels: Wisconsi n and Texas A&M. Thi s is an observati o nal factor, sometimes called, forobvious reasons, a classification factor, because each subject is classified according to his school.Table 15.2 present s the data from this Dating Study. Please no te the following decisions that Imade in creating this table.1. Similar to our tables in Chapter 8, the columns are for the response and the rows are forthe levels of the st u dy factor; i.e., the p opulations. Note that because the researcher did notassign men to university by randomization , we do not refer to the rows as treatments.2. As in Chapter 8, I find the row


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UW-Madison STAT 371 - Ch. 15

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