Chapter 20 Comparing Two Numerical Response Populations Paired Data This chapter is an extension of Chapter 16 In Chapter 16 we considered populations in which each population member or trial yields two dichotomous responses In the current chapter each population member or trial yields two numbers In other ways however this chapter also extends the work we did in Chapters 17 19 20 1 Subject Reuse I will introduce you to the idea of subject reuse with an artificial study of drug therapy for tension headaches We will compare two different ways to design a study I cannot use a real scientific study to make my comparisons because to my knowledge medical researchers select a design and use it They do not investigate a medical issue twice with two different designs just to make me happy I am interested in studying drug therapies for a fairly mild health ailment tension headaches As you will see shortly it is important that I have chosen an ailment that is both nonlethal and recurrent I want to compare two drug therapies for the treatment of a tension headache I will refer to the two therapies as drug A treatment 1 and population 1 and drug B treatment 2 and population 2 We need a response that is a number Each subject is given the following instructions The next time you experience a tension headache take the drug we have given to you Wait 20 minutes Write down your assessment of your pain on a scale from 0 no pain to 10 worst pain ever How can I study this Going all the way back to Chapter 1 I can use a completely randomized design Following Chapter 19 I can perform population based inference on the data I obtain from my completely randomized design In particular I can compare the mean of population 1 drug A 1 to the mean of population 2 drug B 2 I can estimate 1 2 with confidence and test the 513 Table 20 1 Artificial data from a CRD on headache pain sorted within each treatment Position 1 2 Drug A 2 2 Drug B 0 1 3 4 3 3 2 2 5 6 4 4 3 3 7 8 9 5 5 6 3 4 4 10 11 12 6 7 7 4 5 5 13 14 15 16 8 8 9 9 6 7 7 8 null hypothesis that 1 2 In order to choose an alternative we need more information about the drugs Three scenarios come to mind listed below Drug A is a placebo and drug B supposedly is beneficial In this situation remembering that smaller responses are preferred to larger responses my alternative would be Drug A is the extra strength version of drug B In this situation my alternative would be Drugs A and B are different active drugs In this situation my alternative would be 6 Suppose now that I have 32 subjects available for study and I am willing to pretend that they are a random sample from my superpopulation of interest I decide to use a balanced design Thus I will use the online randomizer to assign 16 subjects to each treatment The artificial data for my CRD on the 32 subjects is given in Table 20 1 The data have been separated by treatments and sorted within each treatment You can verify the following values of summary statistics or trust me if you don t need additional practice on these computations x 5 500 s1 2 366 y 4 000 s2 2 251 and n1 n2 16 Next I calculate s2p 2 366 2 2 251 2 5 3325 and sp 5 3325 2 309 2 The 95 confidence interval estimate of 1 2 is see Formula 19 9 on page 501 q 5 50 4 00 2 042 2 309 2 16 1 50 2 042 0 8164 1 50 1 67 0 17 3 17 This interval is inconclusive because it contains both positive and negative numbers For future reference note that the half width of this interval is 1 67 For a test of hypotheses from Equation 19 11 on page 502 the observed value of the test statistic is t 1 50 0 8164 1 837 With the help of our website calculator http stattrek com online calculator t distribution aspx 514 we find that the area under the t curve with df 16 16 2 30 to the right of 1 837 is equal to 0 0381 Thus the approximate P value for the alternative is 0 0381 and the approximate P value for the alternative 6 is 2 0 0381 0 0762 Let s look at the data in Table 20 1 again In the drug A row two subjects gave a response of 2 not much pain and two gave a response of 9 a great deal of pain In words for drug A there is a large amount of subject to subject variation The same is true for drug B The idea behind the randomized pairs design RPD is to attempt to reduce this subject to subject variation I mentioned above that it is important that tension headaches are nonlethal and recurrent Recurrence is important because if each subject has a headache which is necessary in the CRD for us to obtain a response from each subject then the subject will have a second headache The RPD we learn about below will use responses from two headaches per subject compared to the CRD which looked at one headache per subject Nonlethal is important because and I don t mean to be insensitive in order to have a second headache the subject must survive the first one Admittedly I am ignoring studies that would involve looking at 3 4 5 or more headaches per subject I must draw the line somewhere You can now see the reason for the term subject reuse We reuse each subject and thus obtain two responses per subject And somewhat obviously because our goal is to compare the two treatments for each subject we obtain a response from both treatments Thus for example subject Sally gives us two numbers her pain with drug A and her pain with drug B My next step is to provide you with artificial headache pain data from an RPD My goal is to compare my RPD to my CRD for the artificial headache pain study What is a fair way to do this Well my CRD had 32 subjects with one response per subject yielding a total of 32 observations I could have 32 subjects in my RPD but that would yield 32 2 64 observations This strikes me as an unfair comparison Thus instead my RPD below has only 16 subjects with each subject giving two responses I will have a total of 16 2 32 observations the same as I had in my CRD In fact my RPD has exactly the same 32 observations as my CRD did The data for my RPD is given in Table 20 2 Let s take a moment to make sure we can read this table correctly I have 16 subjects …
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