Two Sample TestsBios 662Michael G. Hudgens, [email protected]://www.bios.unc.edu/∼mhudgens2008-09-08 13:47BIOS 662 1 Two Sample TestsTwo Sample Test Settings• Single cross-sectional sample, comparing two sub-samples• Compare samples from two different populations (2 cross-sectional samples, case control study)• Single sample; s ubjects randomly allocated to differentinterventions (experiment, clinical trials)BIOS 662 2 Two Sample TestsFundamentals• Def 5.2 Two rvs Y1and Y2are independent if for all y1and y2Pr[Y1≤ y1, Y2≤ y2] = Pr[Y1≤ y1] Pr[Y2≤ y2]• Result 5.1 If Y1and Y2are independent rvs, then forany two constants a1and a2the rv W = a1Y1+ a2Y2has mean and varianceE(W ) = a1E(Y1) + a2E(Y2)V (W ) = a21V (Y1) + a22V (Y2)BIOS 662 3 Two Sample TestsFundamentals• Result 5.2 If Y1and Y2are independent rvs that arenormally distributed, then W = a1Y1+ a2Y2is nor-mally distributed with mean and var given by Result5.1• Corrollary: If¯Y1and¯Y2are based on two independentrandom samples of size n1and n2from two normal dis-tributions with means µ1and µ2and variances σ21andσ22, then¯Y1−¯Y2∼ N µ1− µ2,σ21n1+σ22n2!BIOS 662 4 Two Sample TestsFundamentals• Result 5.3 If¯Y1and¯Y2are based on two independentrandom samples of size n1and n2from two normal dis-tributions with means µ1and µ2and the same variancesσ21= σ22= σ2, then(¯Y1−¯Y2) − (µ1− µ2)spp1/n1+ 1/n2∼ tn1+n2−2wheres2p=(n1− 1)s21+ (n2− 1)s22n1+ n2− 2• Note n1= n2impliess2p=12(s21+ s22)BIOS 662 5 Two Sample TestsTwo Sample t-test Example• An experiment was conducted to see if a drug couldprevent prem ature birth• 30 women at risk of premature birth were randomlyassigned to take drug or placebo (15 in each group)• Endpoint: birthweightBIOS 662 6 Two Sample TestsTwo Sample t-test Example• Let 1=drug, 2=placebo• H0: µ1= µ2vs HA: µ1> µ2• Cα= {t : t > t1−α;28}• C.05= {t : t > 1.7}BIOS 662 7 Two Sample TestsTwo Sample t-test ExampleDrug Placebo6.9 6.47.6 6.77.3 5.47.6 8.26.8 5.37.2 6.68.0 5.85.5 5.75.8 6.27.3 7.18.2 7.06.9 6.96.8 5.65.7 4.28.6 6.8BIOS 662 8 Two Sample TestsTwo Sample t-test ExampleHistogram of drugdrugDensity4 5 6 7 8 90.0 0.1 0.2 0.3 0.4 0.5Histogram of placeboplaceboDensity4 5 6 7 8 90.0 0.1 0.2 0.3 0.4 0.5 0.6●drug placebo5 6 7 8BIOS 662 9 Two Sample TestsTwo Sample t-test Example•¯y1= 7.08 s1= 0.899•¯y2= 6.26 s2= 0.961• Thuss2p=14(.899)2+ 14(.961)228= 0.8695t =7.08 − 6.26.931p2/15= 2.41• Since t ∈ C.05, reject H0p = 1 − Ft28(2.41) = .011BIOS 662 10 Two Sample TestsTwo Sample t-test: BW example• R> t.test(bw$drug,bw$placebo,var.equal=TRUE,alternative="greater")Two Sample t-testdata: bw$drug and bw$placebot = 2.4136, df = 28, p-value = 0.01129alternative hypothesis: true difference in means is greater than 0BIOS 662 11 Two Sample TestsTwo Sample t-test: BW example• SASproc ttest; class trt; var bw;The TTEST ProcedureT-TestsVariable Method Variances DF t Value Pr > |t|bw Pooled Equal 28 2.41 0.0226bw Satterthwaite Unequal 27.9 2.41 0.0226BIOS 662 12 Two Sample TestsHomogeneity of Variance• Want to testH0: σ21= σ22versus HA: σ216= σ22• We know that (assuming normality)(nk− 1)s2kσ2k∼ χ2nk−1for k = 1, 2• If X1and X2are independe nt rvs with X1∼ χ2v1andX2∼ χ2v2, thenX1/v1X2/v2∼ Fv1,v2BIOS 662 13 Two Sample TestsF Distribution0 1 2 3 4 5 60.0 0.2 0.4 0.6 0.8fdensity1,1 df4,4 df4,25 dfBIOS 662 14 Two Sample TestsHomogeneity of Variance• LetXk=(nk− 1)s2kσ2kfor k = 1, 2• It follows thatY =X1/(n1− 1)X2/(n2− 1)∼ Fn1−1,n2−1• ThusY =s21/σ21s22/σ22∼ Fn1−1,n2−1BIOS 662 15 Two Sample TestsHomogeneity of Variance• Under H0: σ21= σ22, such thats21s22∼ Fn1−1,n2−1• For HA: σ216= σ22, reject null if s21/s22is v large or vsmall (i.e., near ze ro)• Formally,Cα= {f : f < Fn1−1,n2−1,α/2or f > Fn1−1,n2−1,1−α/2}where f = s21/s22BIOS 662 16 Two Sample TestsHomogeneity of Variance• Note: Fv1,v2,α= 1/Fv2,v1,1−α• Table A.5 and A .6 of text for two-sided α = .10 andα = .02; see errata• R> qf(.975,14,14)[1] 2.978588• SASdata; y = finv(.975,14,14);BIOS 662 17 Two Sample TestsHomogeneity of Variance: BW example• H0: σ21= σ22; HA: σ216= σ22• For α = .05,C.05= {f : f < F14,14,.025or f > F14,14,.975}= {f : f < 0.34 or f > 2.98}• Observed test statisticf =.89942.96052= 0.8768• Therefore, do not reject H0p = 2 ∗ F14,14(.8768) = 0.809BIOS 662 18 Two Sample TestsHomogeneity of Variance: BW example• SASproc ttest; class trt; var bw;The TTEST ProcedureT-TestsVariable Method Variances DF t Value Pr > |t|bw Pooled Equal 28 2.41 0.0226bw Satterthwaite Unequal 27.9 2.41 0.0226Equality of VariancesVariable Method Num DF Den DF F Value Pr > Fbw Folded F 14 14 1.14 0.8090BIOS 662 19 Two Sample TestsHomogeneity of Variance: BW example• R> var.test(bw$drug,bw$placebo)F test to compare two variancesdata: bw$drug and bw$placeboF = 0.8767, num df = 14, denom df = 14, p-value = 0.809alternative hypothesis: true ratio of variances is not equal to 1BIOS 662 20 Two Sample TestsHomogeneity of Variance• Cf page 133 of text• Genuine interes t in whether vars equal• WRT testing H0: µ1= µ2– For small samples, potential for type II error– For large samples, CLT/Slutsky– Adjustment for sequential testing• For additional reading, see Moser and Stevens (TAS1992)BIOS 662 21 Two Sample TestsTesting µ1= µ2• What if σ216= σ22and unknown?• Solutions1. Large sample approximation2. Normality: Welch-Satterthwaite approximation(Behrens-Fisher problem)3. Transformation4. Nonparametric methods: Wilcoxon RanksumBIOS 662 22 Two Sample TestsLarge Sample Approximation• If n1and n2are large, homoegeneity of variance as-sumption is not important• Recall CLT plus Slutsky implies¯Y ∼ N(µ,s2n)• Thus¯Y1−¯Y2∼ N(µ1− µ2,s21n1+s22n2)BIOS 662 23 Two Sample TestsLarge Sample Approximation• Therefore, to test H0: µ1− µ2= δ, we can useZ =(¯Y1−¯Y2) − δrs21n1+s22n2• Under H0, Z ∼ N(0, 1)• Approximation gets better as n1, n2→ ∞• Generally, require nj≥ 25 for j = 1, 2• Note assumption that Y ’s normally distributed no longerneeded either (CLT)BIOS 662 24 Two Sample TestsLarge Sample Approximation: Example• A st udy was done to compare the percent body fat of 3rdgraders at schools on 2 Native American reservations:Tohona and Apache• H0: µT= µAvs HA: µT6= µA• nT= 63, nA= 35• C.05= {z : |z| >