Chapter 8 Dichotomous Responses Critical Regions 8 1 Introduction and Notation In all previous studies in these notes the response has been either a numerical variable or an ordered categorical variable with at least three categories For a numerical response we compared the treatments by comparing the means or the ranks of their responses For an ordered categorical response we compared the treatments by comparing the ranks of their responses In this chapter we consider studies that have a dichotomous response a categorical response with two categories We begin with four examples Example 8 1 Therese s infidelity study Therese studied 20 of her adult female friends The women were divided into two treatment groups both of size 10 by randomization Women assigned to the first treatment group read the following question You are friends with a married couple and are equally fond of the man and the woman You discover that the husband is having an affair The wife suspects that something is going on and asks you if you know anything about her husband having an affair Do you tell Women assigned to the second treatment group read the following question You are friends with a married couple and are equally fond of the man and the woman You discover that the wife is having an affair The husband suspects that something is going on and asks you if you know anything about his wife having an affair Do you tell Each subject 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 security federal prison camp in Wisconsin All of the men were first time nonviolent criminal offenders serving two or more years of prison time The men were divided into two treatment groups of 25 each by randomization Men assigned to the first treatment group were given the following question 167 The prison is beginning a program in which inmates have the opportunity to volunteer for community service with developmentally disabled adults Inmates who volunteer will receive a sentence reduction Would you participate Men assigned to the second treatment group were given the following question The prison is beginning a program in which inmates have the opportunity to volunteer for community service with developmentally disabled adults Would you participate Each subject responded either yes or no Example 8 3 Thomas s golf putting study Thomas wanted to investigate the difference in difficulty between four and eight foot putts in golf He performed a balanced study with randomization and a total of 50 putts The first treatment was putting from four feet on a level surface and the second treatment was putting from eight feet on a level surface Each putt was either made or missed Example 8 4 The artificial Headache Study 2 HS 2 A researcher has 100 persons available for study Each person routinely suffers mild tension headaches not migraines The researcher wants to compare two active drugs call them A and B for the treatment of mild headaches The 100 subjects are divided into two groups of size 50 each by randomization Each subject is given the following instructions The next time you have a mild headache take the drug we have given you Fifteen minutes later answer the following question with a response of either yes or no Has your headache pain diminished When the response is a dichotomy there are technical names for the two possible responses one is called a success and the other is called a failure The methods we learn will focus on counting successes We use the following method for deciding which possible outcome gets the distinction of being called a success 1 If one of the possible 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 the success 3 If neither of the previous two scenarios applies then the researcher arbitrarily assigns the label success to one of the possible responses Here is the idea behind the first rule Every time I drive a car I have the potential to be involved in a traffic accident Fortunately in my 47 years of driving I have been in only one accident and it was very minor When something occurs only rarely it is much easier to keep track of how many times 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 summarize the data for each of our four studies 168 Table 8 1 The 2 2 contingency table of observed counts for Therese s infidelity study Row Tell Proportions Cheater was Yes No Total Yes No The Husband 7 3 10 0 70 0 30 The Wife 4 6 10 0 40 0 60 Total 11 9 20 Table 8 2 The 2 2 contingency table of observed counts for Ruth s prisoner study Volunteer Version Read Yes No Sentence Reduction 18 7 No Sentence Reduction 23 2 Total 41 9 Row Prop Total Yes No 25 0 72 0 28 25 0 92 0 08 50 Table 8 3 2 2 Contingency table of observed counts for Thomas s golf putting study Distance Four feet Eight feet Total Putt was Row Prop Made Missed Total Made Missed 18 7 25 0 72 0 28 10 15 25 0 40 0 60 28 22 50 Table 8 4 2 2 contingency table of observed counts for the artificial Headache Study 2 HS 2 Drug A B Total Pain relieved Row Prop Yes No Total Yes No 29 21 50 0 58 0 42 21 29 50 0 42 0 58 50 50 100 169 Table 8 5 General notation for a 2 2 contingency table of observed counts for a CRD with a dichotomous response Response Treatment S F Total 1 a b n1 2 c d n2 Total m1 m2 n Row Proportions S F p 1 a n1 q 1 b n1 p 2 c n2 q 2 d n2 Table 8 5 presents our general notation for a CRD with a dichotomous response When I develop ideas below it will be convenient to use the general notation First a few comments 1 The orientation for these tables in these notes will follow the four examples above namely the rows distinguish between treatments and the columns identify the possible responses Many materials texts research papers etc reverse this orientation Thus when reading other materials be careful to identify which orientation is being used 2 We summarize the table of counts by computing the row proportions the p s and q s given above There is a great deal of redundancy in these namely in each row the sum of its p and q is always one Thus after you get more familiar with these ideas I usually will suppress the q s 3 In these tables I do not calculate the row proportions for the
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