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UW-Madison STAT 371 - Ch. 16

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One Population with Two Dichotomous ResponsesPopulations: Structure, Notation and ResultsFinite PopulationsConditional ProbabilityHow Many Probabilities are There?Screening Test for a DiseaseTrials and Bayes' FormulaRandom Samples from a Finite PopulationRelative Risk and the Odds RatioComparing P(A) to P(B)The Computer Simulation of PowerPaired Data; Randomization-based InferenceMaslow's Hammer RevisitedSummaryPractice ProblemsSolutions to Practice ProblemsHomeworkChapter 16One Population with Two DichotomousResponsesThis chapter focuses on a new idea. Thus far in these notes, a unit (subject, trial) has yielded oneresponse. In this chapter, w e cons ider situations in which each unit yields two responses, bothdichotomies. Later in these Course Notes we will examine situations in which both responsesare numbers and the mixed situation of one response being a number and the other a dicho tomy.Multi-category responses can be added to the mix, but—with one exception—we won’t have timefor that topic.Sometimes the examples of this chapter will look very much like our examples of Chapter 15.Other times, it will be natural to view our two responses as paired data. As a result, you need tobe extra careful as y ou read through this material.16.1 Populations: Structure, Notation and Res u l tsA population model for two dichotomous responses can arise for a collection of individuals—a finite population—or as a mathematical model for a process that generates two dichotomousresponses per trial.Here are two examples.1. Consider the pop ulation of students at a small college. The two responses are sex withpossible values female and male; and t he answer to the following quest ion, with possiblevalues yes and no .Do y ou usually wear corrective lenses when you attend lectures?2. Recall the data on Larry Bird in Chapter 15, presented in Table 15.16 on page 37 4. We viewhis shooting a pair of free throws as a trial with two responses: the outcome of the first shotand t he outcome o f the s econd shot.Recall that I treated the Larry Bird data as Chapter 15 d ata; i.e., independent random samples fromtwo Bernoulli trials processes. Later in this chapter we will view his results as paired data. Both387perspectives are valid, but it will require so me care for yo u to be comfortable with such movingbetween models. Also, my example o f sex and lenses can be viewed as Chapter 15 data, but Iwould find it awkward to refer to it as paired data.We begin with some notation. Wi th two responses per unit, sometimes it would be confusingto speak of successes and failures. Instead, we proceed as follows.• The first response has possible values A and Ac. Note that Acis read A-complement o r not-A.• The second response has possible values B and Bc. Note that Bcis read B-complement ornot-B.In the above example of a finite population, A could denote female; Accould d eno te male; B cou lddenote the answer ‘yes;’ and Bccould deno te the answer ‘no.’ In the above example of trials, Acould denote that t he first s hot is made; Accould denote that the first shot is missed; B coulddenote that the second sho t is made; and Bccould deno te that the second shot is missed. In fact,with data naturally viewed as paired, such as Larry Bird’s sh o ts, it is natural to view A [B] as asuccess on the first [second] respon se and Ac[Bc] as a failure on the first [second] response.It will b e easier if we consider finite populations and trials separately. We will begin with finitepopulations.16.1.1 Finite PopulationsTable 16.1 presents our notation for population counts for a finite popu lation. Remember t h at ,in practice, only Nature would know these num bers. This notation is fairly si mple to rem ember:all counts are represented by N, with o r witho u t subscripts. The symbol N without subscriptsrepresents the t otal number of members of the popu lation. An N with sub scripts counts the numberof m embers of the population with t he feature(s) given by the subscripts. For example, NABis thenumber of popu lation members with response values A and B; NAcis the number of populati onmembers with value Acon the first response; i.e., for this, we don’t care about the second respo n se.Note also that these guys sum in the obvious way:NA= NAB+ NABc.In words, if you take the number of population members whose response values are A and B; andadd to it the numb er of population members whose response values are A and Bc, then you get thenumber of population members whose value on t he first response is A.It might help if we have some hypothet ical values for the population counts. I put some inTable 16.2.If we take the table of population counts and divide each entry by N, we get the tabl e ofpopulation p roportions or probabilities—see the discussi on in the next paragraph. I d o this inTables 16.3 and 16.4, for the general notation and our particular hy pothetical data.Now we must face a n otational annoyance. Consider the number 0.36 in Table 16.4, derivedfrom our hypothetical population counts for th e sex and lenses study. There are two ways tointerpret th is number. First, it is the proportion of the popul at ion who have value A (female) o n the388Table 16.1: The table of population counts.B BcTotalA NABNABcNAAcNAcBNAcBcNAcTotal NBNBcNTable 16.2: Hypothetical population counts for the study o f sex and corrective lenses.Yes (B) No (Bc) TotalFemale (A) 360 240 600Male (Ac) 140 260 400Total 500 500 1000Table 16.3: The table of population proportions—lower case p’s with subscripts—orprobabilities—upper case P ’s followed by parentheses.B BcTotalA pAB= P (AB) pABc= P (ABc) pA= P (A)AcpAcB= P (AcB) pAcBc= P (AcBc) pAc= P (Ac)Total pB= P (B) pBc= P (Bc) 1Table 16.4: Hy pothetical popul at ion prop ortions or probabilities for the study of sex and correctivelenses.Yes (B) No (Bc) TotalFemale (A) 0.36 0 . 24 0.60Male (Ac) 0.14 0.26 0.40Total 0.50 0.50 1.00389first response and value B (yes) o n the s econd response. From this perspective, it is natural to view0.36 as pABbecause we use lower case p’s for population proportions—with a s u bscript, if needed,to clarify which one. But consider our most commonly used chance mechanism when s tudyinga finite population: Select a member of the p opulation at random. For this chance mechanism itis natural to view 0.36 as the probability of selecting a person who is female and woul d answer‘yes.’ We use upper case ‘P’ to denote the word probability.


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UW-Madison STAT 371 - Ch. 16

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