+ +Lecture NotesStatistics 301Professor WardropGo to ‘Wardrop’s webpage:’www.stat.wisc.edu/~wardrop/Scroll down to ‘Courses;’ and click on Stat is-tics 301 for the current semester (don’t clickon an earlier semester !).The resultant location is henceforth referredto as the ‘course website .’These notes and essential documents are onthe course website.+ 9+ +Chapter 1: Comparative Studies (CS)Example: The Infidelity Study (IS) on page11; read it.There are four components to a CS.1. The subjects (usually called units). Theseare the people, objects, t ri als, whatever, fromwhich we obtain information.In the IS, the subjects are Therese’s 20 fem alefriends.2. The response is obtained from each sub-ject. In the IS the r e sponse is the answer yesor no. It is a dichotomy giving us a dichoto-mous response.A response could have more than two cate-gories or it could be a number. Multiple re-sponses are also possible.+ 10+ +We are interested in studies in which the re-sponse varies over subjects. We invent the no-tion of factors. A factor is a characteristic ofa subject that might influence (a strong wor d)or be associated with (weaker) the value ofthe response.In any CS there are many possible fac tors;a fact or may be specific and relatively easyto determine (e.g. marital status of subje c t)or vague and difficult to determine (e.g. sub-ject’s attitude towards marriage).Of all possible factors, the researcher selectsone factor to be the study factor. The studyfactor is the third component of a compara-tive study.+ 11+ +In the IS, Therese chose t he study factor tobe the gender of the cheater (discuss otherpossible wordings).The possible values of the study factor arecalled its levels.In the IS, the levels are: husband and wife.Very roughly speaking, the purpose of a CS isto investigate whether the lev e l of the studyfactor influences (that strong word again) oris associat e d with (weaker, again) t he valueof the response.In this cour se, we re st r ic t attention to studyfactors that have exactly two levels. In Chap-ters 1–3 and 5–7 we restrict attention to re-sponses t ha t are dichotomous. In Chapters12, 15 and 16 we consider numerical responses.In Chapte rs 8 and 13 we consider studies withtwo responses.+ 12+ +Consider the IS a gai n. The following questionis very import ant.Should each subject read and answer bothversions of the question, or only one?In other words, should we obtain a responsefor both levels of the study factor or for onlyone?Discuss.In Chapters 1–3, we restri c t attention to stud-ies in which eac h subject is ‘exposed’ to ex-actly one level of the study factor .This raises an obvious question:How do we assign subjects to levels?+ 13+ +The widely accepted answer is: by random-ization.Randomization dates back at le ast to Ben-jamin Franklin and studies of anim al mag-netism, but statisticia ns tend to identify itspopularity with the efforts of Sir Ronald F i sher(1890–1962), a statistician and geneticist.Randomization in the IS:• Assign the numbers 1, 2, . . . , 20 to thesubjects in any manner.• Place 20 identi c al cards in a box; the cardsbeing numbered 1, 2, . . . , 20.• Select 10 cards at random from the box.• The subjects corresponding to the selecte dcards are assigned to the first lev e l (hus-band cheats) and the remaining subjectsare assigned to the second level.+ 14+ +Whenever a researcher assigns subjects to lev-els by ra ndomization, we cal l the levels treat-ments.Thus, in the IS, the treatments are husbandcheating and wife cheating.The basic question becomes: Does the tre at-ment influence (not e the stronger word) theresponse.There are (up to) three reasons statistici ansadvocat e randomization.• It is fair to the subjects• It is fair to the treatments• It is a basis for inference (se e Chapter 2).+ 15+ +Data for the IS (p. 16):Tell?Cheater Yes No TotalHusband 7 3 10Wife 4 6 10Total 11 9 20Can you describe this?Table of row proportions (p. 16; divide eachrow entry by its total):Tell?Cheater Yes No TotalHusband 0.70 0.30 1.00Wife 0.40 0.60 1.00+ 16+ +General notation:Table of counts (p. 17; contingency table):ResponseTreat. S F Total1 a b n12 c d n2Total m1m2nTable of row proportions (p. 23)ResponseTreat. S F Total1 ˆp1ˆq112 ˆp2ˆq21+ 17+ +Medical StudiesA researcher wants to investiga te the proper-ties of a new therapy for a certain disease.The new therapy is treatment 1. The 2ndtreatment represents a control group.Question: Is there an existing therapy for thedisease?If ye s, the control group receives the existingtherapy.If no, the control group receives a place bo.Example: Chronic Crohn’s Disea se (CCD;p. 2 0)+ 18+ +New therapy: cyclosporine (immunosuppres-sant)Control group: placeboSubjects: 71 persons with CCD for whom t heexisting (standard) therapy was ineffective .Response: Im provement (S) or not (F) afterthree months of treat ment.Improved?Treatment Yes No Total ˆpCyclosporine 22 15 37 0.59Placebo 11 23 34 0.32Total 33 38 71‘Blind’ studies: (S hould be called ‘ignorant.’)Subject is ignorant of the t r e atment he/shereceives.Evaluator (of response) is ignorant of the as-signment of treatme nts to subjects.+ 19+ +Revisit ‘fairness’ of randomization.Subjects: The advantage/risk of the new ther-apy is assigned w/o favoritism.Treatments: See Table 1.11 on p. 21.Background factors:Gender: 24/37 (65%) of Sbjts on cycl osporineand 22/34 (65%) of Sbjts on placebo are fe-male.Thus, if females are part ic ularly good (or bad)subjects, neither trea tment is given an advan-tage.Remember: There is no guarantee thatrandomization will ‘work’ this well. Random-ization is about the process, not the outcome.+ 20+ +Disease site: Smal l bowel, colon, or both.Clearly (?) both is most serious. 20/37 (54%)of Sbjts on cyc l osporine and 13/34 (38%) ofSbjts on placebo have the disease at bothsites.Discuss.Section 3: A sequence of trials.So far, subjects = people; more generally, dis-tinct individuals.But also can have subjects = trials.3 point basket study (3PBS; p. 28 .)Playe r : Clyde Gaines.Trials: 100 shots in basketball.Response: Shot is ma de (S) or missed (F).Study Factor: Location of shot; 1 is behindthe 3-point line in front of the basket and 2 isbehind the 3-point line from the left corner.+ 21+ +Question: Does the location of the shot in-fluence the outcome?Read: Alternate way of randomization in lastparagraph on page 10.ResponseLocation Basket Miss Total ˆpFront 21 29 50 0.42Left