Populations: Getting StartedThe Population BoxAn Extended Example on a Very Small NHorseshoes …Meaning of ProbabilityThe Law of Large NumbersIndependent and Identically Distributed TrialsAn Application to GeneticsMatryoshka (Matrushka) Dolls, Onions and ProbabilitiesIn Praise of Dumb SamplingSome Practical IssuesThe Issue of NonresponseDrinking and Driving in WisconsinPresidents and BirthdaysComputingSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 10Populations: Getting StartedYou have now completed Part 1 of these notes, consisting of nine chapters. What have you learned?On the one hand, you could say that you have learned many thing s about the discipline of Stati stics.I am quite sure that you have expended a great deal of time and effort to learn, perhaps master, thematerial in the first nin e chapters. On the other hand, however, you could say, “I have learned morethan I ever wanted t o know about the Skeptic’s Argument and not much else.” I hope t h at you feeldifferently, but I cannot say this comment is totally lacking in merit.So, why have we spent so much time on the Skeptic’s Argument? First, b ecause the idea ofOccam’s Razor is very impo rtant in science. It is important to be skeptical and not just jump on thebandwagon of the newest i dea. For data-based conclusi ons, we shoul d give the benefit of the doubtto the notion that nothing is h appening and only conclude that , indeed, something is happening ifthe data tell us that the n othing is happening hypot h es is is inadequate. The Skeptic’s Argument is,in my opinion, the purest way to introduce you t o how to use Statisti cs in science.The analyses you have learned in the first nine chapters require you t o make decisions: thechoice of the components of a CRD; the choice of the alternative for a test of hypotheses; fornumerical data, the choice of test statistic; for a power stu dy, the choi ce of an alternative of interest.The analyses require you to take an action: you must rando mize. But, and this is the key point,the analyses make no assumptions. The remainder of these notes will focus on population-basedinference. Assumptions are always necessary in order to reach a conclusion on a population-basedinference. The two most basic of t hese assumptions involve:1. How do the units actually studi ed relate to the entire popu lation of units?2. What structure is assumed for the population?By the way, if either (or both) of these questions makes no s ense to you that is fine. We will learnabout th ese question s and more later in these notes.As we will see, in population-based inference, we never (some might say rarely; I don’t wantto quibble about this) know with certainty whether our assumptions are true. Indeed, we us uallyknow that they are not true; in this sit uation, we spend time investigating how much it matters thatour assumptions are no t true. (In m y experience, t h e reason why many—certainly not all, perhapsnot even most—math teachers have so much trou ble teaching Statistics is because they just don’t215get the idea that an assumpti on can be wrong. If a mathematician says, “Assume we have a triangleor a rectangle or a continuous function” and I say, “How do you know the assumption is true,” themathematician will look at me and say, “Bob, you are hopeless!”)The above di scussion raises an obvious qu estion: If population-based inference techniques relyon assumptions that are not true, why learn them? Why not limit ourselves to studies for whichwe can examine the Skeptic’s Arg ument? Well, as much as I l ove the Skeptic’s Argument, I mus tacknowledge its fundamental weakness: It is concerned only with the u nits in the study; it has noopinion on the units that are not in the study. Here is an example o f what I mean.Suppose that a bal anced CRD is performed on n = 200 persons suffering from colon cancer.There are two competing treatments, 1 and 2, and the data give a P-val ue of 0.0100 for the alter-native 6= with the data supporting the notion that treatment 1 is better. The Skeptic’s Argu ment is,literally, concerned only with the n = 200 persons in the study. The Skeptic’s Argument makesno claim as to how the treatments would work on any of the thous ands of people with colon can-cer who are no t in the study. If you are a physician caring for one of these thousands you willneed to decide which treatment you recommend. Th e Skeptic cannot tell you what to do. Bycontrast, with popul at ion-based inference a P-value equal to 0.01 00 allows one to conclude thatoverall treatment 1 is bet ter than treatment 2 for the entire population. By m aking more assump-tions, populati on-based inference obtains a stronger conclusion. The difficulty, of course, is thatthe assum ptions of the popu lation-based inference might not be t rue and, if not true, might give amisleading conclu sion.Of course, there i s another difficulty in my colon cancer example. As we saw in Case 3 inTable 5.3 on page 90 in Chapter 5, even if we conclude that treatm ent 1 is bett er than treatment 2overall, this does not imply that treatment 1 is better t h an treatment 2 for every subject; this istrue for the Skeptic’s argument and it’s true for pop ulation-based inference.There is, of course, a second weakness of the methods we covered in Part 1 of these notes:They require the assignment of units to study factor levels by randomization. For many s tudiesin science, randomization is either impossible or, if possible, highly unethical. For an exampleof the former, consider any study that compares the responses given by m en and women. For anexample of the latter, imagi ne a study that assigns p ersons, by randomization, to the smokes threepacks of cigarettes per d ay treatment. As we will discuss often in the remainder of these notes,studies with randomization yield greater scientific validity—in a carefully explained way—thanstudies without randomization. This does not mean, however, that studies without randomizationare inherently bad or are to be avoided.One of the g reatest strengths o f population-based inference is that it allow s a scientist to makepredictions about future u ncertain outcomes. The Skeptic’s A rgument cannot be made to do this.Predictions are impo rt ant in real life and they give us a real-world measure of whether the answerswe get from a statistical analysi s have any validity.Anyways, I have gotten very far ahead of myself.
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