Confidence Intervals for Association Parameters Testing Independence in Two-Way Contingency Tables Following-Up Chi-Squared Tests

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1 Outlie Cofidece Itervals for Associatio Parameters Testig Idepedece i Two-Way Cotigecy Tables Refereces : Ala Agresti,, Wiley Itersciece, New Jersey, 00 Iterval Estimatio of Odds Ratios The sample odds ratio θ = for a x table equals 0 or if ay =0 ad it is udefied if both etries i a row or colum are zero. Sice these outcomes have positive probabilities, the expected value ad variace of θˆ ad log θˆ do ot exist. Aother ameded estimators ~ ( + 0.5)( θ = ( + 0.5)( Ad log ~ θ behave well ˆ + 0.5) + 0.5) Iterval Estimatio of Odds Ratios A estimated stadard error for log θˆ is ˆ σ ( log ˆ θ ) = + + By the large sample ormality of log log ˆ θ ± zα / ˆ σ / is a Wald cofidece iterval for log θ. Expoetiatig (takig atilogs of) its edpoits provides a cofidece iterval for θ + ( log ˆ θ ) ^ θ 3

2 Aspiri ad Myocardial Ifarctio Example Iterval Estimatio of Odds Ratios Swedish Study o Aspiri Use ad Myocardial Ifarctio Myocardial Ifarctio Yes No Placebo Aspiri The sample odds ratioθ Sice o cell cout is especially small. The stadard error of ˆ ~ =.56 is close to θ =.55 ( log ˆ) log ˆ θ = 0.5 is ˆ σ θ = 5 A 95% cofidece iterval for log θ is 0.5±.96 (0.307) or (-0.57,.07). The correspodig iterval for θ is [exp(-0.57), exp(.07)] or (0.85,.85) Sice the cofidece iterval for θ cotais.0, it is plausible that the true odds of death due to myocardial ifarctio are equal for aspiri ad placebo. If there truly is a beeficial effect of aspiri but the odds ratio is ot large, it may require a large sample size to show that beefit because of the relatively, small umber of myocardial ifarctio cases. 6 Iterval Estimatio of Differece of Proportios The differece of proportios ad the relative risk compare coditioal distributios of a respose variable for two groups. For these measures, we treat the samples as idepedet biomials. For group i, y i ~ bi ( i, π i ). The sample proportio πˆ i = yi i has expectatio π I ad variace π I ( - π I )/ i. Sice ˆ ad πˆ are idepedet, their differece has E ( ˆ π ˆ π ) = π π π Iterval Estimatio of Differece of Proportios Ad stadard error The estimate by πˆ i. The σ ( ˆ π ˆ π ) ˆ σ ( π π ) ˆ ˆ π( π) π ( π ) = + / Is Wald cofidece iterval for π - π. uses formula () with π I replaced ( ˆ π ˆ π ) ± zα ˆ σ ( ˆ π π ) / ˆ 7 8

3 Iterval Estimatio of Relative Risk Iterval Estimatio of Relative Risk The sample relative risk is r = πˆ ˆ π. Like the odds ratio, it coverges to ormality faster o log scale. The asymptotic stadard error for log r is σ ( log r) π π = + π π The Wald iterval expoetiates edpoits of log r ± zα ˆ(log σ / / r ) 9 πˆ sample proportio of myocardial ifarctio deaths for subjects takig placebo πˆ sample proportio of myocardial ifarctio deaths for subjects takig Aspiri Sample risk is / % Cofidece iterval log relative risk of log (.5)±.96 (0.97) (0.86,.75) : We ifer that the death rate for those takig placebo was betwee 0.86 ad.75 times that for those takig aspiri The Wald 95% cofidece iterval for (-0.005,0.033) π - π is 0.0 ±.96 (0.0098) Accordig to either measure, substatial public health beefit could result from takig aspiri, but o effect or a slight egative effect are also plausible. 0 Pearso Chi Squared Tests For multiomial samplig with probabilities {π } i a I x J cotigecy table, the ull hypothesis of statistical idepedece is H o : π = π i+ π +j for all i ad j The Pearso X test statistic for testig H o : idepedece is ( ) = ˆ μ ˆ π X i+ = i+, ˆ π + j = + j ˆ μ i j ˆ μ = ˆ π i+ ˆ π + j = i+ + j X is asymptotically chi-squared with df = (I-)(J-) Likelihood-Ratio Chi Squared Tests The ratio of the likelihoods equals The likelihood-ratio Chi squared statistic is - log Λ. Deoted by G, it equals G = log Λ = log ˆ μ, ˆ μ = i j The larger the values of G ad X, the more evidece exist agaist idepedece. G has a chi-squared ull distributio with df = (I-)(J-). The approximatio is usually poor for G whe /IJ < 5. i j i+ Λ = ( ) i + j j ( ) i+ + j

4 Educatio ad Religious Fudametalist Example Educatio ad Religious Fudametalist Example Highest Degree Less tha high school High school or juior college Bachelor or graduate Fudametalist 78 (37.8) (.5) 570 (539.5) (.6) 38 (08.7) (-6.8) 886 Religious Beliefs Moderate 38 (6.5) (-.6) 68 (63.) (.3) 5 (.5) (0.7) 038 Estimated expected frequecies for testig idepedece Liberal 08 (.7) (-.9) (88.) (-.0) 5 (88.9) (6.3) The table above cross classifies the degree of fudametalism of subjects religious beliefs by their highest degree of educatio. The table also cotais the estimated expected frequecies for H o : idepedece. For istace, ˆ μ = + + = (x886) / 76 = The chi-squared statitics are X = 69., G =69.8, with df = (3-)(3-) =. The P- values are < These statistics provide extremely strog evidece of a associatio Stadardized Pearso residuals 3 Pearso ad Stadardized Residuals A small P-value idicates strog evidece of associatio but provides little iformatio about the ature or stregth of the associatio. Statisticias have log wared about dagers of relyig solely o results of chi-squared tests rather tha studyig the ature of the associatio. A cell-by-cell compariso of observed ad estimated expected frequecies helps show the ature of the depedece. Uder H o, larger differeces μˆ ted to occur i cell with larger μ Cofidece Itervals for Associatio Ptarameters Pearso ad Stadardized Residuals The Pearso residual, defied for a cell by Pearso residual relate to the Pearso statistic by A stadardized Pearso residual is e = ˆ μ ˆ/ μ i j X e = ˆ μ [ ˆ μ ( p )( p + j) ] / A stadardized Pearso residual that exceeds about or 3 i absolute value idicates lack of fit of H o i that cell. Larger values are more relevat whe df is larger ad it becomes more likely that at least oe is large simply by chace i+ 5 6

5 Educatio ad Religious Fudametalism Revisited Partitioig Chi - Squared Table i p.3 also shows stadardized Pearso residuals for testig idepedece. For istace, = 78, μˆ = 37.8 The relevat margial proportios equal p + = / 76 = 0.56 ad p + = 886/76 = The stadardized Pearso residual for this cell equals ( )/ [ (37.8)( 0.56) ( 0.35)] / =.5 This cell shows a much greater discrepacy betwee ad ˆμ tha expected if the variables were truly idepedet. 7 We begi with a partitioig for the test of idepedece i x J tables. We partitio G, which has df = (J-), ito J- compoets. The jth compoet is G for a x table where The first colum is colum combies colums through j of the full table ad The secod colum is colum j+ G for testig idepedece i a x J tables equals a statistic that : Compares the first two colums, plus a statistic that combies the first two colums ad compares them to the third colum, ad so o up to a statistic that combies the first J- colums ad ad compares them to the last colum. Each compoet statistic has df = 8 Partitioig Chi - Squared Origi of Schizophreia Example For a I x J table, idepedet chi squared compoets result from comparig colum ad ad the combiig them ad comparig them to colum 3, ad so o. Each of the J statistics has df = I -. More refied partitios cotai (I )(J ) statistics, each havig df =. Oe such partitio ( Lacaster, 99) applies to the (I - )(J-) separate x tables a< ib< j ib b< j ab aj a<i for i =,..,I ad j =,...,J. School of Pyschiatric Though Eclectic ical Psychoaalytic Ecl Bio 90 3 Ev Ecl Biogeic Bio + Ev 0 Com 78 6 Origi of Schizophreia Evirometal Ecl+ Psy 3 Bio 03 9 Ev 3 3 Ecl + Psy Combiatio Bio+ 6 3 Com

6 Rules for Partitioig Rules for Partitioig Goodma ad Lacaster gave rules for determiig idepedet compoets of chi-squared. For formig subtables, amog the ecessary coditios are the followig. The df for the subtables must sum to the df for the full table.. Each cell cout i the full table must be a cell cout i oe ad oly oe subtable. 3. Each margial total of the full table must be a margial total for oe ad oly oe subtable. For a certai partitioig, whe the subtable df values sum proporly, but the G values do ot, the compoets are ot idepedet. For the G statistic, exact partitioogs occur. The Pearso X eed ot equal the sum of the X for the full table. Limitatios of Chi Squared Test Small Sample Test of Idepedece Chi squared tests of idepedece merely idicate the degree of evidece of associatio. They are rarely adequate for aswerig all questios about a data set. Rather tha relyig solely o results of these test, ivestigate the ature of the associatio : Studi residuals Decompose chi squared ito compoets Estimate parameters such as odds ratios that describe the stregth of associatio. The chi squared tests also have limitatios i the types of data to which they apply. For istace, they require large Whe is small, alterative methods use exact small-sample distributios rather tha large-sample approximatios. Fisher s Exact Test for x Tables We kow that, for Poisso samplig othig is fixed, for multiomial samplig oly is fixed, ad for idepedet biomial samplig i the two rows oly the row of margial totals are fixed. I ay of these cases, uder H 0 : idepedece, coditioig o both sets of margial totals yields the hypergeometric distributio p( t) = p( + + t + t = + samples. 3 = t ) This formula expresses the distributio of { } i terms of oly. Give the margial totals, determies the other three cell couts.

7 Small Sample Test of Idepedece Small Sample Test of Idepedece Fisher s Tea Driker For x tables, idepedece is equivalet to the odds ratio θ =. To test H 0 : θ =, the P-value is the sum of certai hypergeometric probabilities. To illustrate, cosider H a : θ >. For the give margial totals, tables havig larger have larger odds ratios ad hece stroger evidece i favor of H a. Thus, the P-value equals P( t 0 ), where t 0 deotes the observed value of. This test for x tables is called Fisher s exact test Muriel Bristol, a colleague of Fisher s, claimed that whe drikig tea she could distiguish whether milk or tea was added to the cup first (she preferred milk added first) Poured First Milk Tea Milk 3 Guess Poured First Tea Small Sample Test of Idepedece Fisher s Tea Driker Distiguishig the order of pourig better tha with pure guessig correspods to θ >, reflectig a positive associatio betwee order of pourig ad the predictio. We coduct Fisher s exact test of H 0 : θ = agaist H a : θ > The observed table, t 0 = 3 correct choices of the cups havig milk added first, has ull probability 3 8 = 0.9 The P-value is P( 3) = 0.3. This result does ot establish a associatio betwee the actual order of pourig ad her predictios. It is difficult to do so with such a small sample. Accordig to Fisher s daughter (Box, 978,p.3), i reality Bristol did covice Fisher of her ability. 7 Each of 00 multiple-choice questios o a exam has four possible aswers, oe of which is correct. For each questio, a studet guesses by selectig a aswer radomly. Specify the distributio of the studet s umber of correct aswers. Fid the mea ad stadard deviatio of that distributio. Would it be surprisig if the studet made at least 50 correct resposes? Why? Specify the distributio of (,, 3, ), where j is the umber of times the studet picked choice j. Fid E( j ), var( j ). 8

8 A ame-brad household washig machie is sold i five differet colors, ad a market researcher wishes to study the popularity of the various colors. The frequecies give i Table. are observed from a radom sample of 300 recet sales. As a iitial step, the researcher may wat to test the hypothesis that all five colors are equally popular. Table. Sales data of five colors of a Name-brad washer. Avocado 88 Ta 65 Red 5 Blue 0 White I the Uited States, the estimated aual probability that a woma over the age of 35 dies of lug cacer equals for curret smokers ad for osmokers. Fid ad iterpret the differece of proportios ad the relative risk. Which measure is more iformative for these data? Why? Fid ad iterpret the odds ratio. Explai why the relative risk ad odds ratio take similar values A article i the New York Times about the PSA blood test for detectig prostate cacer stated: The test fails to detect prostate cacer i i me who have the disease (false-egative results), ad as may as two-thirds of the me tested receive false-positive results. Let C(C*) deote the evet of havig (ot havig) prostate cacer, ad let + (-) deote a positive (egative) test result. Which is true: P(- C) = ¼ or P(C -) = ¼? P(C* +)=/3 or P(+ C*) Determie the sesitivity ad specificity. A diagostic test has sesitivity = specificity = fid the odds ratio betwee true disease status ad the diagostic test result. 3 3