last observation analysis in anova and ancova - Institute of Statistical ...

860 BIN CHENG, JUN SHAO AND BOB ZHONG2. One-Way ANCOVAConsider a clinical trial consisting of I treatments, n i patients randomized totreatment group i, and T scheduled post-baseline visits for each patient. Supposethat under treatment i, n it patients drop out after visit t. Hence, n i1 +· · ·+n iT =n i and (n i1 , . . . , n iT ) has the multinomial(n i , p i1 , . . . , p iT ) distribution, where p itis the population proportion of patients dropping out after visit t under treatmenti. We assume a pattern-mixture one-way ANCOVA model, i.e., for patients whodrop out after visit t, their last observed responses y itk ’s are independent withmeans µ it + b ′ z itk , where k = 1, . . . , n it , t = 1, . . . , T , i = 1, . . . , I, the µ it ’sare unknown fixed treatment effects, z itk is a q-vector of covariates observed foreach patient, and b is a q-vector of unknown parameters. Unlike the y-responsevariable, the z-covariate for a patient does not vary with t, although we usethe notation z itk to specify it. No other condition is imposed on the dropoutmechanism (i.e., dropout may be nonignorable).When there is no dropout, testing for treatment effect may be carried outby using responses from the end of the study and the method of ANCOVA.When there are dropouts, as we discussed in Section 1, the LOAN considers thehypothesisH 0 : µ 1 = · · · = µ I , (1)where µ i = p i1 µ i1 + · · · + p iT µ iT .The LOCF test is the ANCOVA test that treats y itk as the observation atthe end of the trial. Since µ it usually changes with t, one wonders what theLOCF tests for. If (1) is the hypothesis of interest, there is the question ofvalidity of the LOCF test. The following result shows when the LOCF test isasymptotically valid for (1). Since Shao and Zhong (2003) showed that whenI ≥ 3, the LOCF test is asymptotically wrong for testing (1) in a one-wayANOVA without covariates, we only consider the case of I = 2 treatments.In this paper, χ 2 ddenotes the chi-square distribution with d degree of freedom,while χ 2 d,α and F l,m,α denote, respectively, the 1 − α quantiles of χ 2 dand the F-distribution with degrees of freedom l and m, where α is a given nominal level.Let[MSTR = (ȳ 1.. − ˆb ′¯z 1.. ) − (ȳ 2.. − ˆb ′¯z ] / 2 2∑ T∑ ∑n it2.. )a 2 itk ,i=1 t=1 k=112∑ T∑ ∑n it [MSE =(y itk − ȳ i.. ) −n 1 + n 2 − q − 2ˆb ′ 2(z itk − ¯z i.. )],i=1 t=1 k=1where z itk is the covariate value associated with y itk , ȳ i.. and ¯z i.. are, respectively,the averages of y itk and z itk over the indexes t and k,2∑ T∑ˆb = (˜Z ′ ∑n it−1 ˜Z) (y itk − ȳ i.. )(z itk − ¯z i.. ),i=1 t=1 k=1

862 BIN CHENG, JUN SHAO AND BOB ZHONGThat is, the LOCF test is asymptotically valid for testing (1) when (6) or (7)holds. Note that Shao and Zhong (2003) showed that the condition lim(n 1 /n 2 ) =1 ensures the asymptotic validity of the LOCF test in one-way ANOVA. Whenthere are covariates, our result shows that in addition to the balance conditionlim n 1 /n 2 = 1, the validity of the LOCF test requires the covariates to be balancedin the sense that either the means of the covariates under two treatmentsare asymptotically the same (condition (6)) or the covariance matrices of thecovariates under two treatments are asymptotically the same (condition (7)).When there are I ≥ 3 treatments or the design is not balanced, the LOCFtest has the wrong asymptotic size for testing (1). An asymptotically valid test of(1) is derived as follows. For the ith treatment, let ˆb i =(˜Z ′ ˜Z i i ) −1∑ T∑ nitt=1 k=1 z itky itk ,where ˜Z i is the n i ×q matrix whose n i rows are z ′ i11 −¯z′ i.. , . . . , z′ i1n i1−¯z ′ i.. , ...., z′ iT n iT−¯z ′ i.. , and let u itk = y itk − ˆb ′ i z itk. Then ū i.. = n −1 ∑ T∑ niti t=1 k=1 u itk is unbiased forµ i and asymptotically normal, and its variance can be estimated consistently byˆV i =1n i (n i − 1)T∑ ∑n it(u itk − ū i.. ) 2 .t=1 k=1Theorem 2. Suppose that, for patients dropping out after visit t, the y itk ’s areindependent with means µ it +b ′ z itk and variances σ 2 it > 0. Under (1), as n i → ∞for all i, W → d χ 2 I−1 , whereW =I∑i=1( ∑1Ii=1ū i.. −ūi../ ˆV) 2i∑ˆV Ii i=1 1/ ˆV.iConsequently, an asymptotic size α test rejects (1) if and only if W > χ 2 I−1,α .Note that we do not assume the variances of y itk ’s are equal in Theorem 2.When (1) is rejected, we can make inference (such as pairwise or multiplecomparison on µ i ’s) using the asymptotic results based on ū i.. and ˆV i .3. Tests for Interaction in Two-Way ModelsTwo-way ANOVA or ANCOVA is often used in clinical trials. In additionto the treatment effect, a common factor in a two way ANOVA or ANCOVA isthe center effect in a multicenter trial. Consider a clinical trial carried out inJ centers with I treatments, n ij patients randomized to treatment group i atcenter j, and T scheduled visits for each patient. The total number of patientsis n = ∑ I ∑ Ji=1 j=1 n ij. Suppose that under treatment i at center j, n ijt patientsdrop out after visit t. Then (n ij1 , . . . , n ijT ) has the multinomial(n ij , p ij1 , ..., p ijT )distribution, where p ijt is the population proportion of patients dropping outafter visit t under treatment i at center j. Let y ijtk be the last observed response

LAST OBSERVATION ANALYSIS IN ANOVA AND ANCOVA 863variable of interest from patient k under treatment i at center j who droppedout after the tth visit.Under the two-way ANOVA model, for patients dropping out after visit t,we assume that y ijtk ’s are independent with means µ ijt . Similar to the one-waycase, we use theT∑µ ij = p ijt µ ijt (8)t=1as measures for treatment and center effects. Under two-way models, the µ i andµ it of previous sections should be replaced by µ ij and µ ijt , respectively. Considerthe decompositionµ ij = µ + α i + β j + γ ij ,where µ is an overall mean, α i ’s are fixed treatment effects (α 1 +· · ·+α I = 0), β j ’sare fixed center effects (β 1 + · · · + β J = 0), and γ ij ’s are fixed interaction effects(γ i1 + · · · + γ iJ = γ 1j + · · · + γ Ij = 0 for any i and j). Although the treatmenteffects α i ’s are of primary interest, the analysis in two-way ANOVA often startswith a test for the treatment-by-center interaction with the null hypothesisH 0 : γ ij = 0, for all i and j. (9)To test (9), the LOCF test treats y ijtk as the observation in the end of the trialand rejects H 0 when MSAB/MSE > F (I−1)(J−1),n−IJ,α , whereMSAB =MSE =1(I − 1)(J − 1) ȳ′ L(L ′ ΛL) −1 L ′ ȳ,1n − IJI∑J∑i=1 j=1 t=1 k=1n T∑ ∑ ijt(y ijtk − ȳ ij.. ) 2 ,ȳ ij.. is the average of y ijtk ’s over t and k, ȳ = (ȳ 11.. , . . . , ȳ I1.. , . . . , ȳ 1J.. , . . . , ȳ IJ.. ) ′ ,Λ = diag(n −111 , . . . , n−1 I1, ..., n−11J , . . . , n−1 IJ ),( ) ( )1 ′L =(J−1)1 ′⊗(I−1),−I (J−1) −I (I−1)1 m is the m-vector of ones, I m is the identity matrix of order m, and ⊗ is theKronecker product. The following result shows what this tests for, and when itis asymptotically valid. The proof can be found in Cheng (2004).Theorem 3. Assume that, for patients dropping out after visit t, y ijtk ’s areindependent with means µ ijt and variance σ 2 > 0.

864 BIN CHENG, JUN SHAO AND BOB ZHONG(i) As n ij → ∞ for all i, j, MSE → p σ 2 + η, whereη = lim 1 nI∑i=1 j=1J∑n ij τij 2 and τij 2 =T∑p ijt (µ ijt − µ ij ) 2 .(ii) Under (9), MSAB converges in distribution to a linear combination of (I −1)(J − 1) independent chi-square random variables with 1 degree of freedom.The LOCF test for interaction is asymptotically valid (i.e., MSAB/MSE isasymptotically distributed as χ 2 (I−1)(J−1)) if and only ift=1L ′ VL = ηL ′ ΛL, (10)where V = diag(n −111 τ 11 2 , . . . , n−1 I1 τ I1 2 , . . . , n−1 1J τ 1J 2 , . . . , n−1 IJ τ IJ 2 ).(iii) When I = J = 2, (10) becomes∑ 2 ∑ 2i=1 j=1limn ijτij2 ∑ 2∑ 2i=1 j=1∑ 2 ∑ 2i=1 j=1 n = limn−1 ij τ ij2∑ 2 ∑ 2. (11)iji=1 j=1 n−1 ijWhen I = 2 and J ≥ 3, (10) becomesn −11j τ 2 1j + n−1 2j τ 2 2jn −11j+ n−1 2j=∑ Jj=1 n ijτij2∑ 2∑ Ji=1 j=1 n , for all j. (12)ij∑ 2i=1When I ≥ 3 and J = 2, (10) becomesn −1i1 τ 2 i1 + n−1 i2 τ 2 i2n −1i1+ n−1 i2=∑ Ii=1∑ 2j=1 n ijτij2∑ I ∑ 2i=1 j=1 n , for all i. (13)ijWhen I ≥ 3 and J ≥ 3, (10) becomesτ 2 ij = constant. (14)When I = J = 2 and µ ijt ’s are different for given i and j (so that the τ ij ’s aredifferent), (11) implies that the LOCF test for treatment-by-center interactionis asymptotically valid for testing (9) if the design is balanced in the sense thatlim n ij /n = 1/4 for any i and j. Although in many applications the numberof treatments I = 2, the number of centers J is often more than 2. From(12) through (14), we know that when either I or J is more than 2, the onlypractical situation that a LOCF procedure is valid is when the τij 2 are all thesame or, equivalently, the µ ijt ’s are the same for fixed i and j. A result similarto Theorem 3 for a two-way ANCOVA model can be derived, but it is omittedsince a necessary condition for the validity of the LOCF test is I = J = 2, whichis a limited special case.

LAST OBSERVATION ANALYSIS IN ANOVA AND ANCOVA 865An asymptotically valid test for (9) can be derived based on cell mean estimatorsȳ ij.. and their consistent variance estimators. We now consider the generaltwo-way ANCOVA model (which includes the ANOVA model as a special case)in which the mean of y ijtk (for patients dropping out after visit t) isµ ijt + b ′ z ijtk , (15)where z ijtk is a q-vector of covariates observed for each patient, and b is a q-vectorof unknown parameters. For any i and j, let ˆb ij be the least squares estimatorof b based on the data from the patients who received the ith treatment in the∑ Tt=1∑ nijtk=1 u ijtk is anjth center and let u ijtk = y ijtk − ˆb ′ ij z ijtk. Then ū ij.. = n −1ijunbiased and asymptotically normal estimator of µ ij with a consistent varianceestimatorˆV ij =1n ij (n ij − 1)n T∑ ∑ ijt(u ijtk − ū ij.. ) 2 .t=1 k=1Theorem 4. Suppose that, for patients dropping out after visit t, the y ijtk ’sare independent with means given by (15) and variances σijt 2 . Under (9), asn ij → ∞ for all i, j, W → d χ 2 (I−1)(J−1) , where W = ū′ L(L ′ ˆVL) −1 L ′ ū, ˆV =diag( ˆV 11 , . . . , ˆV I1 , . . . , ˆV 1J , . . . , ˆV IJ ), and ū=(ū 11.. , . . . , ū I1.. , . . . , ū 1J.. , . . . , ū IJ.. ) ′ .Consequently, a test of (9) with asymptotic size α rejects H 0 if W >χ 2 (I−1)(J−1),α. It is clear that the result in Theorem 4 can be extended to K-wayANCOVA models.4. Additive Two-Way ANCOVAIn a multicenter clinical trial, treatment-by-center interaction can sometimesbe ignored, especially when covariates related to centers are introduced into themodel. Hence, in this section we consider an additive two-way ANCOVA model,which includes the additive two-way ANOVA model as a special case.Consider the multicenter clinical trial described in Section 3, where the meanof y ijtk for patients dropping out after visit t is µ ijt + b ′ z ijtk , z ijtk is a q-vector ofcovariates observed from every patient, and b is a q-vector of unknown parameters.Let µ ij be given by (8) and assume the additive modelµ ij = µ + α i + β j , (16)where µ is an overall mean, α i ’s are fixed treatment effects (α 1 + · · · + α I = 0),and β j ’s are fixed center effects (β 1 + · · · + β J = 0). The null hypothesis of notreatment effect isH 0 : α 1 = · · · = α I = 0. (17)

866 BIN CHENG, JUN SHAO AND BOB ZHONGThe LOCF procedure for testing (17) treats µ ijt as µ ijT for all t and uses thefollowing model E(Y) = Xθ + Zb, where θ = (µ, α 1 , . . . , α I , β 1 , . . . , β J ) ′ , Y isthe column vector formed by listing the elements y ijtk in the order of i, j, t andk, Z is the matrix formed by listing the row vectors z ijtk in the order of i, j, tand k, and X is the usual design matrix in a two-way additive ANOVA model.Define P = I − X(X ′ X) − X ′ andˆb = (Z ′ PZ) −1 Z ′ PY. (18)The following theorem shows when the LOCF test is asymptotically validfor testing (17). Its proof is similar to that of Theorem 1 and is omitted. First,let( n 2∑ J∑ T∑ ijtn∑J∑ T∑ ∑ ijtMSTR =a 2 ijtk ,i=1 j=1 t=1k=1a ijtk y ijtk) 2/ 2∑i=1j=1 t=1 k=1where the a ijtk ’s are the components of the vector that is the difference betweenthe second and the third rows of (X ′ X) − X ′ (I − Z(Z ′ PZ) −1 Z ′ P), andMSE =1n − (2J + q)2∑J∑i=1 j=1 t=1 k=1n T∑ ∑ ijt [(y ijtk − ȳ ij.. ) − ˆb ′ 2(z ijtk − ¯z ij.. )],where ȳ ij.. and ¯z ij.. are averages of y ijtk and z ijtk over indices t and k, respectively.Theorem 5. Assume that I = 2 and, for patients dropping out after visit t, they ijtk ’s are independent with means µ ijt + b ′ z ijtk and variance σ 2 > 0, and theµ ij ’s have form given in (16).(i) The ANCOVA test based on LOCF rejects hypothesis (17) when the ratioF =MSTR/MSE is larger than F 1,n−2J−q,α .(ii) As n ij → ∞, i = 1, 2 and j = 1, . . . , J, MSE → p σ 2 + η, whereη = lim∑ 2∑ Ji=1 j=1 n ijτij2∑ 2 ∑ Ji=1 j=1 n ij, τ 2 ij =T∑p ijt (µ ijt − µ ij ) 2 . (19)(iii) Under (17), as n ij → ∞, i = 1, 2 and j = 1, . . . , J, MSTR → d (σ 2 + ζ)χ 2 1 ,where∑ 2 ∑ Ji=1 j=1ζ = limw ijτij2 n T∑ ∑ ijt∑ 2∑ Ji=1 j=1 w , w ij = a 2 ijtk . (20)ijt=1t=1 k=1(iv) For testing hypothesis (17), the LOCF procedure described in (i) is asymptoticallyvalid if and only iflim∑ 2i=1∑ Jj=1 n ijτ 2 ij∑ 2 ∑ Ji=1 j=1 n ij= lim∑ 2 ∑ Ji=1 j=1 w ijτij2∑ 2 ∑ Ji=1 j=1 w . (21)ij

LAST OBSERVATION ANALYSIS IN ANOVA AND ANCOVA 867When the design is balanced, i.e., n ij = n 0 for all i, j, (21) can be simplified.Corollary 1. Assume the conditions in Theorem 5 and n ij = n 0 for all i, j. Let˜Z be the (IJn 0 ) × q matrix whose rows are z ′ ijtk − ¯z′ i... − ¯z′ .j.. + ¯z′ ..... Thenw ij = (Jn 0 ) −1 + (¯z 1... − ¯z 2... ) ′ (˜Z ′ ˜Z) −1 S ij (˜Z ′ ˜Z) −1 (¯z 1... − ¯z 2... ),∑ nijtwhere S ij = ∑ Tt=1 k=1 (z ijtk − ¯z i... − ¯z .j.. + ¯z .... ) (z ijtk − ¯z i... − ¯z .j.. + ¯z .... ) ′ , and(21) holds if and only if lim ¯z 1... = lim ¯z 2... or lim n −10 S ij are the same for all iand j.When (21) does not hold, the LOCF procedure described in Theorem 5(i)is no longer valid for testing (17). To derive a general valid test for (17), forany fixed i , let ˆb i be the estimator of b based only on the data from thepatients who received the ith treatment by a formula similar to (18). Defineu ijtk = y ijtk − ˆb ′ i z ijtk. Then ū i... = 1 ∑ J ∑ T ∑ nijtn i j=1 t=1 k=1 u ijtk is an unbiased andasymptotically normal estimator for µ + α i and its variance can be estimatedconsistently byn1J∑ T∑ ∑ ijtˆV i =(u ijtk − ū i... ) 2 .n i (n i − 1)j=1 t=1 k=1Theorem 6. Suppose that, conditional on (n ij1 , . . . , n ijT ), the y ijtk ’s are independentwith means µ ijt + b ′ z ijtk and variances σijt 2 , and the µ ij’s have the decompositiongiven in (16). Under (17), as n ij → ∞ for all i and j, W → d χ 2 I−1 ,where(I∑∑1Ii=1W = ū i... −ūi.../ ˆV) 2i∑ˆV Ii i=1 1/ ˆV.ii=1Consequently, a test of (17) with asymptotic size α rejects H 0 if W > χ 2 I−1,α .Inference about α i ’s after (17) is rejected can be made using the asymptoticresults based on ū i... and ˆV i .The results in Theorems 5 and 6 can be extended to K-way additive AN-COVA models with K ≥ 3. We provide a brief discussion here and omit thedetails. Under any K-way additive ANCOVA model, the LOCF test for the effectof a factor with two levels is asymptotically valid if the design is balanced,and either the covariate averages under the two levels are the same, or the covariatevariability across different cells (combinations of factors) is constant. Inany case, asymptotically valid tests can be derived along the lines of Theorem 6.5. Simulation ResultsA simulation study was performed to study the finite-sample type I error ofthe LOCF test and the proposed test in Theorem 6 in an ANCOVA model. The

868 BIN CHENG, JUN SHAO AND BOB ZHONGsample sizes and population parameters were chosen to be similar to a phase IIclinical trial in a real application with I = 2 treatments, J = 3 centers, and T = 3visits. Responses y ijtk ’s were generated by the following steps. All parametervalues are given in Table 1.Table 1. Population parameters in the simulation.i, j p ij1 , p ij2 , p ij3 µ ij1 , µ ij2 , µ ij3 σ ij1 , σ ij2 , σ ij31, 1 0.30, 0.20, 0.50 −11656.40, −11668.30, −11804.53 251.2, 249.7, 348.71, 2 0.36, 0.36, 0.28 −11682.40, −11660.80, −11758.14 396.1, 131.3, 258.01, 3 0.08, 0.33, 0.59 −12012.36, −11650.40, −11434.70 143.8, 143.8, 591.92, 1 0.33, 0.07, 0.60 −11705.31, −10473.82, −11905.71 510.4, 486.2, 486.22, 2 0.18, 0.27, 0.55 −12160.21, −11500.81, −11637.39 371.7, 807.8, 831.52, 3 0.18, 0.18, 0.64 −11767.81, −10762.31, −11720.27 501.4, 311.6, 716.71. Generate covariates z 1jtk ’s as a random sample of size n 1 = n 11 +n 12 +n 13 fromN(846.6, 514.1 2 ), and z 2jtk ’s as a random sample of size n 2 = n 21 + n 22 + n 23from N(845.2, 367.7 2 ).2. For each fixed i and j, generate (n ij1 , n ij2 , n ij3 ) from Multinomial (n ij , p ij1 ,p ij2 , p ij3 ).3. For each fixed i, j, and t, generate y ijtk ’s as a random sample of size n ijt fromN(µ ijt + bz ijtk , σijt 2 ), where b = 14.7.Note that the means of y ijtk ’s were chosen so that µ 1j = µ 2j for all j,which implies that the model is additive with no treatment effect, that is, (17)is true. The actual type I errors of the LOCF test and the proposed test wereevaluated by simulation with 5,000 runs. Results with different choices of n ij ’sare summarized in Table 2. The results indicate that the proposed test has sizeclose to the nominal level 5% regardless of whether the sample sizes are balancedor not. The results also show that the LOCF test has the right size when thesample sizes are almost balanced, but a wrong size when the sample sizes areunbalanced. These simulation results generally support our asymptotic results.Table 2. Type I errors of the LOCF and proposed tests.Sample SizeType I Errorn 11 , n 12 , n 13 n 21 , n 22 , n 23 LOCF Test Proposed Test30, 33, 36 27, 33, 33 0.0548 0.054460, 66, 72 27, 33, 33 0.1118 0.052015, 17, 18 27, 33, 33 0.0170 0.051830, 33, 36 54, 66, 66 0.0170 0.049630, 33, 36 14, 17, 17 0.1078 0.0576

LAST OBSERVATION ANALYSIS IN ANOVA AND ANCOVA 869AppendixProof of Theorem 1.(ii) Let u itk = y itk − b ′ z itk . The result follows from Lemma 1 of Shao and Zhong(2003) and the fact thatMSE ==(iii) Note that1n 1 + n 2 − q − 21n 1 + n 2 − q − 22∑T∑ ∑n it [] 2(u itk − ū i.. ) − (ˆb − b) ′ (z itk − ¯z i.. )i=1 t=1 k=1n it2∑T∑ ∑(u itk − ū i.. ) 2 + o p (1).i=1 t=1 k=1(ȳ 1.. − ˆb ′¯z 1.. ) − (ȳ 2.. − ˆb ′¯z 2.. ) =T∑ ∑n 1ta 1tk y 1tk −t=1 k=1T∑ ∑n 2ta 2tk y 2tk .t=1 k=1By an argument similar to that of Theorem 1 of Shao and Zhong (2003) andTheorem 3.12 of Shao (2003), we have[(ȳ 1.. − ˆb ′¯z 1.. ) − (ȳ 2.. − b ′¯z ] /√ √ √√ ( 2∑2.. )∑ nitT∑ ∑n iti=1 t=1 k=1a 2 itk)σ 2 + φ 2 1 + φ2 1 → d N(0, 1),where φ 2 i = Var( ∑ Tt=1 k=1 a itk(µ it + b ′ z itk )). Then the result follows if φ 2 i =τi 2w2 i holds for i = 1, 2. We only prove the case when i = 1 as an illustration.Since the covariate value of a patient does not vary with visit, we can rewrite z itkas z ij and a itk as a ij , where j = 1, . . . , n 1 . For t = 1, . . . , T , let X 1jt = 1 if andonly if a 1j is from a patient who dropped out after the tth visit, and X 1jt = 0otherwise. Clearly, X 1j = (X 1j1 , . . . , X 1jT ) i.i.d. ∼ Multinomial(1; p 11 , . . . , p 1T ), j =1, . . . , n 1 . Hence( T∑n 1φ 2 1 = Var ∑)a 1j (µ 1t + b ′ z 1j )X 1jtt=1 j=1( ∑n 1 ( ∑T= Var a 1j µ 1t X 1jt +j=1( ∑n 1= Var( ∑n 1= Vart=1∑Ta 1jj=1 t=1∑Ta 1jj=1 t=1T∑ ))b ′ z 1j X 1jtt=1∑n 1 )µ 1t X 1jt + a 1j b ′ z 1jµ 1t X 1jt)j=1

870 BIN CHENG, JUN SHAO AND BOB ZHONG∑n 1 ( ∑T )= a 2 1jVar µ 1t X 1jtj=1= τ 2 1Acknowledgements∑n 1j=1t=1a 2 1j = τ 2 1T∑ ∑n 1sa 2 1tk = τ 1 2 w2 1 .t=1 k=1This work was partially supported by grant CA53786 from the NationalCancer Institute. The authors would like to thank two referees and an associateeditor for their helpful comments.ReferencesCheng, B. (2004). Some hypothesis testing results in two-way linear models for clinical trials.Ph.D. thesis, University of Wisconsin-Madison.Dawson, J. D. (1994a). Comparing treatment groups on the basis of slopes, areas-under-thecurve,and other summary measures. Drug Infor. J. 28, 723-732.Dawson, J. D. (1994b). Stratification of summary statistic tests according to missing patterns.Statist. Medicine 13, 1853-1863.Dawson, J. D. and Lagakos, S. W. (1993). Size and power of two-sample tests of repeatedmeasures data. Biometrics 49, 1022-1032.FDA (1988). Guideline for Format and Content of the Clinical and Statistical Sections of NewDrug Applications. Center for Drug Evaluation and Research, Food and Drug Administration,Rockville, Maryland.Heyting, A. and Tolboom, J. T. B. M. and Essers, J. G. A. (1992). Statistical handling ofdrop-outs in longitudinal clinical trials. Statist. Medicine 11, 2043-2061.Lavori, P. W. (1992). Clinical trials in psychiatry: Should protocol deviation censor patientdata? Neuropsychopharmacology 6, 39-48.Little, R. (1993). Pattern-Mixure Models for Multivariate Incomplete Data. J. Amer. Statist.Assoc. 88, 125-134.Searle, S. R. (1987). Linear Models for Unbalanced Data. Wiley, New York.Shao, J. (2003). Mathematical Statistics. Second edition. Springer-Verlag, New York.Shao, J. and Zhong, B. (2003). Last Observation Carry-Forward and Last Observation Analysis.Statist. Medicine 22, 2429-2441.Shih, W. and Quan, H. (1998). Stratified testing for treatment effects with missing data.Biometrics 54, 782-787.Ting, N. (2000). Carry-forward analysis. In Encyclopedia of Biopharmaceutical Statistics(Edited by S. Chow), 103-109. Marcel Dekker, New York.Department of Biostatistics, Mailman School of Public Health, Columbia University, New York,NY 10032, U.S.A.E-mail: bc2159@columbia.eduDepartment of Statistics, University of Wisconsin, Madison, WI 53706, U.S.A.E-mail: shao@stat.wisc.eduCentocor, Inc., Malvern, PA 19355, U.S.A.E-mail: BZhong@CNTUS.JNJ.COM(Received December 2003; accepted November 2004)