Dichotomous Responses; Critical Regions Introduction and NotationThe Test of Hypotheses: Fisher's TestThe Critical Region of a TestMotivation: Comparing P-valuesFrom P-values to Critical RegionsTwo Types of ErrorsThe Significance Level of a TestTwo Final RemarksChoosing the Alternative after Looking at the Data: Is it Really Cheating?The Two-Sided Alternative RevisitedSummaryPractice ProblemsSolutions to Practice ProblemHomework ProblemsChapter 8Dichotomous Responses; Critical Regions8.1 Introduction and NotationIn all previous studies in these notes, the response has been either a numerical variable or anordered categorical variable with at least three categories. For a numerical response we comparedthe treatments by comparing the means or the ranks of their responses. For an ordered categoricalresponse we compared the treatments by comparing the ranks of their responses.In this chapter we consider stu dies that have a dichotomous response—a categorical responsewith two categories. We begin with four examples.Example 8.1 (Therese’s infidelity study.) Therese studied 20 of her adult female friends. Thewomen were divided into two treatment groups, both of size 10, by randomization. Women as-signed to the first treatment group read the following question:• You are friend s with a married couple and are equally fond of the man and the woman. Youdiscover that the husband is having an affair. The wife susp ects that something is going onand asks you if y ou know anything about her husband having an affair. Do you tell?Women assigned to the second treatment group read the following question:• You are friend s with a married couple and are equally fond of the man and the woman. Youdiscover that the wife is h aving an affair. The husband suspects that something is going onand asks you if y ou know anything about his wife having an affair. Do you tell?Each s ubject was instructed to respond either yes or no.Example 8.2 (Ruth’s prisoner study.) Ruth’s subjects were 50 male inmates at a minimum secu-rity federal prison camp in Wisconsin. All of the men were first-t ime nonviolent criminal offendersserving two or more years of prison time. The men were divided into two treatment groups of 25each by randomization. M en assigned to the first treatment group were given the following ques-tion:167• The prison is b eginnin g a p rogram in which inmates have the opportunity to volunteer forcommunity service with developmentally disabled adults. Inmates who volunteer will re-ceive a sentence reduction. Would you participate?Men assigned to the second treatment group were given the following question:• The prison is b eginnin g a p rogram in which inmates have the opportunity to volunteer forcommunity service with developmentally disabled adults. Would you participate?Each s ubject responded either yes or no.Example 8.3 (Thomas’s golf putting study.) Thomas wanted t o investigate the difference in dif-ficulty between fou r and eight foot pu tts in golf. He performed a balanced st udy with randomiza-tion and a total of 50 putts . The first treatment was pu tting from four feet o n a level surface andthe second treatment was putt ing from eight feet o n a level su rface. Each putt was either made ormissed.Example 8.4 (The artificial Headache Study-2 (HS-2).) A researcher has 100 persons availablefor study. Each person routin el y suffers mild tension headaches (not migraines). Th e researcherwants t o compare two active drugs, call them A and B, for the t reatm ent of mild headaches. The100 subjects are divided into t wo groups of size 50 each by randomization. Each subject is giventhe following instructions:The next time you have a mild headache take t he drug we have given you. Fifteenminutes later answer the following qu estion with a response of either yes or no: Hasyour headache pain diminished?When the response is a dichotomy, there are technical names for the two poss ible responses:one is called a success and the other is called a failure. The methods we l earn will focus oncounting successes. We use the fol lowing method for deciding which possible outcome gets thedistinctio n of being called a success.1. If one of the possi b le responses is very rare (admittedly vague), then it is labeled the success.2. If neither possible response is very rare, then the more desirable response is labeled thesuccess.3. If neither of the previous two scenarios applies, then the researcher arbitrarily assigns thelabel success to one of the possible responses.Here is the idea behind the first rule. Every ti me I drive a car I have the potential to be involvedin a traffic accident. Fortunately, in my 47 y ears of driving I have been in only one accident and itwas very minor. When something occurs only rarely, it i s much easier to keep track of how manytimes it happens rather than how many times it fails to happen.In our examples above, the researchers labeled as successes: telling, agreeing to volunteer,making a putt and reporting that the pain has diminished. Tables 8.1–8.4 present and summarizethe data for each of our four studies.168Table 8.1: The 2 × 2 contingency table of observed counts for Therese’s infidelity study.RowTell? ProportionsCheater was: Yes No Total Yes NoThe Husband 7 3 10 0.70 0.30The Wife 4 6 10 0.40 0.60Total 11 9 20Table 8.2: The 2 × 2 contingency table of observed counts for Ruth’s prisoner study.Volunteer? Row Prop.Version Read: Yes No Total Yes NoSentence Reduction 18 7 25 0.72 0.28No Sentence Reduction 23 2 25 0.92 0.08Total 41 9 50Table 8.3: 2 × 2 Contingency table of observed counts for Thomas’s golf putting study.Putt was Row Prop.Distance:Made Missed Tot al Made Mi ssedFour feet 18 7 25 0.72 0.28Eight feet10 15 25 0.40 0.60Total 28 22 50Table 8.4: 2 × 2 contingency table of ob served counts for the artificial Headache Study-2 (HS-2).Pain relieved? Row Prop.Drug : Yes No Tot al Yes NoA 29 21 50 0.58 0.42B 21 29 50 0.42 0.58Total 50 50 100169Table 8.5 : General notation for a 2 × 2 contingency table of observed counts for a CRD with adichotomous response.RowResponse ProportionsTreatment : S F Total S F1 a b n1ˆp1= a/n1ˆq1= b/n12 c d n2ˆp2= c/n2ˆq2= d/ n2Total m1m2nTable 8.5 presents our general notation for a CRD with a dichoto mous response. When Idevelop ideas below it will be convenient to use the general n o tation.First, a few comm ents:1. The orientation for these tables in these notes will follow the four examples above; namely,the rows
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