1 Models for Matched Pairs
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1 1 Models for Matched Pairs Matched pairs occur whe we aalyse samples such that for each measuremet i oe of the samples there is a measuremet i the other sample that directly relates to the measuremet i the first sample. Matched pairs typically occur whe for example two measuremets are take for the same idividual, e.g. respose before ad after itervetio. If the measuremets are categorical, a atural way to display such data would be a cotigecy table. Example 1 For istace (this is a example from the olie otes by Jaso Newsom at Portlad State Uiversity), we might examie the favorability of voters for gu cotrol legislatio i April ad i Jue. Jue Yes No Total April Yes No Total We ow could ask if the opiios have chaged from April to Jue. The we are ot iterested to test for idepedece, because if every participat kept their opiio all measuremets would be i the cells o the diagoal. We also might wat to compare the proportio of voters who agree with gu cotrol i April ad Jue. For the distributio this would mea comparig π yes+ with π +yes. I geeral: Let π ij be the joit distributio of a 2 2 table. The the differece betwee π 1+ ad π +1 is of iterest. It is π 1+ π +1 = π 11 + π 12 (π 11 + π 21 ) = π 12 π 21 Therefore i fact this becomes a questio about the off diagoal etries. To aswer if the margial distributios are equal we ca check if π 12 = π McNemar Test To test if π 12 = π 21 oly the cell couts for those two cells are eeded, ad the relevat sample size is = With this data we ow test if the probability (to fall i cell 12, give it is either i cell 12 or 21) of a biomial distributio is 0.5. Applyig the Z-score for biomial distributios to this settig, we get Z = 12 (0.5) (0.5)(0.5) = ( ) = = which is approximately stadard ormally distributed, if the true probability is 0.5. The McNemar statistic χ 2 = Z 2 is therefore χ 2 -distributed with df = 1. 1
2 To test if the margial distributios are the same for paired samples with biary respose: McNemar Test 1. H 0 : π 12 π 21 = 0 or H 0 : π 1+ π +1 = 0 versus H 0 is ot true. Choose α. 2. Radom sample for the table, Test Statistic χ 2 0 = ( ) , df = 1 4. P-value = P (χ 2 > χ 2 0) Example 2 Test if the proportio of people who support gu cotrol has chaged from April to Jue. 1. H 0 : π 12 π 21 = 0 versus H 0 is ot true. α = The data represets a radom sample, ad = Test Statistic 4. P-value = P (χ 2 > 73.6) < Reject H 0. χ 2 0 = (90)2 110 = 73.6, df = 1 6. At sigificace level of 5% the data provide sufficiet evidece that the proportio of voters who support gu cotrol has chaged. 1.2 Estimatig the differece betwee two proportios based o paired samples I geeral a cofidece iterval is preferable over a test. The cofidece iterval ca be based o the differece i the sample proportios: ˆπ i+ ˆπ +i which is a ubiased estimator for the true differece i the probabilities, s.t. µˆπi+ ˆπ +i = π i+ π +i ad has estimated stadard deviatio s ˆπi+ (1 ˆπ i+ ) + ˆπ +i (1 ˆπ +i ) 2(ˆπ 11ˆπ 22 ˆπ 12ˆπ 21 ) SE = Rewritig the SE i terms of observed cell couts gives SE = ( ) 2( ) 2 / 2
3 McNemar Cofidece Iterval for π i+ π +i ( ) 2( ) 2 / (ˆπ i+ ˆπ +i ) ± z α/2 Example 3 A 95% comfidece iterval for the differece i the proportio of voters supportig gu cotrol: ( ) 2(10 100)2 / ± 1.96 (110 54) ( ) ± ± [ , ] We are 95% cofidet that the percetage of voters who support gu cotrol icreased betwee 25% ad 35% (based o a sample where the same people were asked i April ad Jue). 1.3 Logistic Regressio Models for Matched Pairs Margial Models I a first step we will model the situatio above. Let Y 1, Y 2 be the radom variables represetig success/failure for observatio 1 ad observatio 2, respectively, ad x t is a dummy variable which is oe for observatio 1 (t = 1) ad zero for observatio 2 (t = 2). Now aalyse the followig model logit(p (Y t = 1)) = α + βx t The for t = 1 : for t = 2 : logit(p (Y 1 = 1)) = α + β logit(p (Y 2 = 1)) = α Which meas that e β is the odds ratio for success i observatio 1 versus observatio 2. Example 4 From the data Θ = 120/ /90 = The odds for support of gu cotrol i April are times the odds i Jue, or the odds for support of gu cotrol i Jue are 1/0.2856=3.49 times the odds i April. Fittig the model gives (iputtig the data as if it would be idepedet, therefore the tests ca ot be used) logit(p(y t =Yes)) = x t with e =
4 The model logit(p (Y t = 1)) = α + βx t is called the margial model because it models the margial distributio of the respose for the paired observatios Coditioal models These models are based o the perceptio that matched pairs data ca be viewed as three way cotigecy tables, oe table for each idividual i the sample. Example 5 Assume the first idividual i the sample was agaist gu cotrol i April, but for it i Jue, the the table for this idividuals respose would be: How may of these tables would we eed? Respose Time Yes No April 0 1 Jue 1 0 The kth table shows the resposes Y 1, Y 2 for idividual k, ad the sample data is represeted as a 2 2 table. Models based o this perceptio of matched pairs are called coditioal models, the tables are coditioed o the idividual, ad permit that the probability distributio for the respose is differet for each idividual. Notatio: Let Y it be observatio t for idividual i (1 = success). The coditioal model: or logit(p (Y i1 = 1)) = α i + β, logit(p (Y i2 = 1)) = α i, logit(p (Y it = 1)) = α i + βx it with x i1 = 1, x i2 = 0. The model implies a homogeeous relatioship because the odds ratio for success for comparig the two observatios is e β for all idividuals. I the case β = 0, for each idividual the odds (probabilities) for success are the same for the two observatios. The odds (probabilities) for success ca be differet for idividuals depedig o α i. To aswer if the probabilities for success are differet is equivalet to test withi the coditioal model if β = 0. The may parameters (+1) i the model ca cause difficulties with the estimatio. Oe solutio is to cosider coditioal maximum likelihoods, where the coditioal likelihood fuctio is maximized, by coditioig out the idividual parameters. For this simple model this leads to the coditioal maximum likelihood estimator of exp( ˆβ) = 12 / 21 from the origial 2-way table. 4
5 Example 6 The coditioal maximum likelihood estimate for the odds ratio comparig the support for gu cotrol i April ad Jue equals 10/100=0.10. The odds for support of gu cotrol is i April 10% of the odds i Jue. This is very differet from the we foud usig the margial model, reflectig the differece betwee margial ad coditioal odds ratios. 5
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