last observation analysis in anova and ancova - Institute of Statistical

14 Pages·2005·0.15 MB·English

Statistica Sinica15(2005),857-870
LAST OBSERVATION ANALYSIS IN ANOVA
AND ANCOVA
Bin Cheng, Jun Shao and Bob Zhong
Columbia University, University of Wisconsin and
Centocor, Inc.
Abstract: Inclinicaltrialswithmultiplevisits,dropoutsoftenoccurandthepopula-
tion of patientswho droppedoutmaybe di(cid:11)erentfrom the population of patients
who completed the study. To assess treatment e(cid:11)ects over the population of all
randomizedpatients, whichis calledthe intention-to-treatanalysis andis required
byregulatoryagencies,lastobservationanalysis(LOAN)focusesonthelastobser-
vation of each patient prior to dropout. As a type of LOAN, the last observation
carry-forward (LOCF) method treats the last observation prior to dropout as the
missing observation at the end of the trial and applies standard tests designed
for the case of no dropout. Regulatory agencies such as the U.S. Food and Drug
Administration (FDA) have expressed concerns about the validity of the LOCF
methods. In this paper we study the validity of LOAN and LOCF tests under an
analysis of covariance model, which includes the analysis of variance model as a
special case. In situations where LOANis relevant, we provide explicit conditions
under which LOCF tests are asymptotically valid and we derive asymptotically
valid tests when LOCF tests are invalid.
Keywordsandphrases: Dropout,intention-to-treatanalysis,lastobservationcarry-
forward, balanced design, balanced covariates.
1. Introduction
In clinical trials, data are often collected over multiple visits of participating
patients, and statistical analyses focus on observations at the end of the study or
change-of-e(cid:14)cacy measurements from baseline to the end of the trial. Despite a
thoughtful and well-designed studyprotocol, it is frequently the case in a clinical
trial that patients drop out prior to the end of the study. In the presence of
dropout, many regulatory agencies require the intention-to-treat analysis that
focuses on all randomized patients with at least one post-treatment evaluation.
An approach focusing on the last observed visit has received some attention
recently, the so-called last observation analysis (LOAN). Let (cid:22) be the ith treat-
it
ment population mean of the last response y (the primary variable of interest)
of a patient who dropped out after visit t, where i = 1;:::;I, t = 1;:::;T, and
visit T is the end of the study so that (cid:22) is the mean of completers. (Note that
iT
858 BINCHENG,JUNSHAOANDBOBZHONG
(cid:22) is typically di(cid:11)erent from the ith treatment population mean of y at visit t
it
in the case of no dropout.) The LOAN evaluates treatment e(cid:11)ects by comparing
the (cid:22) , t = 1;:::;T, i = 1;:::;I. The oldest LOAN is called last observation
it
carryforward(LOCF),animputationmethodthatimputesthemissingresponse
at the end of the study by using the response at the visit prior to dropout. Once
this carry forward imputation is done, one applies standard statistical tests that
treatall observations(imputedornot) asresponsesat theendofthestudy(Ting
(2000)). Although the LOCF has a long history of application, there has been
concernthat treating carried-forward data asobserved data creates biases insta-
tistical tests for treatment e(cid:11)ects (Heyting, Tolboom and Essers (1992), Lavori
(1992), Dawson (1994a) and Ting (2000)). A second type of LOAN de(cid:12)nes the
overall treatment e(cid:11)ect as a weighted average of D ;:::;D , where, for each
1 T
t, D is a measure assessing the di(cid:11)erence among (cid:22) ;:::;(cid:22) (see, for example,
t it It
Dawson and Lagakos (1993), Dawson (1994) and Shih and Quan (1998)). A
key assumption here is that the missing patterns among di(cid:11)erent treatments are
almost the same (i.e., p = (cid:1)(cid:1)(cid:1) = p for each t, where p is the population
1t It it
proportion of patients dropping out after visit t under treatment i), but this is
not realistic in many applications. The most recent LOAN proposed in Shao
and Zhong (2003) assess treatment e(cid:11)ects by comparing the weighted averages
(cid:22) = p (cid:22) , i= 1;:::;I, whichare unbiasedlyestimated bythesample means
i t it it
based on LOCF data.
P
Notethattheinterpretationoftreatmente(cid:11)ectsintheLOANisverydi(cid:11)erent
from that in the approach of comparing treatment e(cid:11)ects at the end of the
study (in the presence of dropout). In some practical applications, using (cid:22) ’s
i
in the comparison of treatment e(cid:11)ects makes sense (e.g., dropout is caused by
death), whereas in some situations comparing e(cid:11)ects at the end of the study
is more reasonable. When dropout is present and depends on y (observed and
unobserved), however, treatment e(cid:11)ects at the end of the study (even if they are
not hypothetical) may not be estimable unless a strong (typically nonveri(cid:12)able)
assumption is imposed on the dropout mechanism and/or the y-population. For
example, the treatment e(cid:11)ects at the end of the study may be confounded with
other e(cid:11)ects when patients switch to other medications after dropout. Hence,
analysis of the (cid:22) ’s may be used if there is no other more reasonable approach.
i
The purpose of this paper is to study methods for inference on (cid:22) ;:::;(cid:22) ,
1 I
assuming that the (cid:22) ’s can be used to interpret treatment e(cid:11)ects. We adopt the
i
pattern-mixture approach (Little (1993)), which requires very few assumptions
about the dropout mechanism.
Although the sample means based on LOCF data estimate the (cid:22) ’s, the
i
LOCF tests may not be correct. There is a belief that the size of a LOCF test
may be substantially higher or lower than the nominal size (cid:11) unless (cid:22) ;:::;(cid:22)
i1 iT
LASTOBSERVATIONANALYSISINANOVAANDANCOVA 859
arethesame. However, ShaoandZhong(2003) showedthat, inone-way ANOVA
with two treatments and a balanced design (i.e., the designed sample sizes of
treatment groups are the same), the asymptotic size of the LOCF test is still
the nominal size (cid:11) when the null hypothesis is the equality of (cid:22) ’s, regardless of
i
whether (cid:22) ;:::;(cid:22) are the same or not. It is also shown in Shao and Zhong
i1 iT
(2003) that the asymptotic size of the LOCF test is not (cid:11) when the design
is not balanced or more than two treatments are compared. An explanation
for these results is that, based on LOCF data, the mean sum of squares for
treatment (MSTR) in the one-way ANOVA table is asymptotically distributed
as a weighted average of I(cid:0)1 independent chi-square random variables, where I
isthenumberoftreatments; whenI = 2,theMSTRisasymptoticallydistributed
as a scaled chi-square random variable; when the design is balanced, this scale is
exactly the same as the limit of the mean sum of squares for error (MSE), which
ensures the asymptotic validity of the LOCF test; if either I (cid:21) 3 or the design is
not balanced, the ratio MSTR/MSE is asymptotically distributed as a weighted
average of chi-square random variables (not a chi-square random variable) and,
thus, the size of the LOCF test is wrong.
SinceaLOCFtestisoftenusedinclinicaltrials,itisimportanttoknowwhen
itis(asymptotically)validand,inthecasewheretheLOCFtestisnotvalid,what
is a valid testing procedure. The result in Shao and Zhong(2003) only applies to
one-way ANOVA. In clinical trials, an analysis of covariance (ANCOVA) is often
used to incorporate covariates such as the baseline observations. Also, many
clinical trials are multicenter trials, which leads to a two-way (or K-way; K (cid:21) 3)
ANOVA or ANCOVA.
For a better understanding of the problem, we start with the one-way AN-
COVA in Section 2. Our result shows that in order to have an asymptotically
valid LOCF test, not only the designed sample sizes of the two treatment groups
need to be the same, but also the covariates in the model need to satisfy a bal-
ance condition. More precisely, the covariates in two di(cid:11)erent treatment groups
need to have either the same average or the same variability. Furthermore, we
derive a test that is always asymptotically valid and can be used to replace the
LOCFtestwhenitisasymptotically invalid. InSection3,wefocusontestingthe
interaction e(cid:11)ect in a two-way ANOVA model (without covariates). Our results
show that the LOCF test is valid only in some very special situations. A similar
conclusioncanbedrawnforatwo-way ANCOVAmodel. Anasymptoticallyvalid
test for interaction under a two-way ANCOVA model is derived. Based on the
results in Section 3, we consider tests for the main e(cid:11)ect (treatment e(cid:11)ect) in a
two-way additive ANCOVA modelinSection 4. Theresultinone-way ANCOVA
is extended to this model. Extensions of our results to general K-way ANCOVA
are straightforward. Finally, some simulation results are presented in Section 5.
860 BINCHENG,JUNSHAOANDBOBZHONG
2. One-Way ANCOVA
Consideraclinical trialconsistingofI treatments, n patientsrandomizedto
i
treatmentgroupi,andT scheduledpost-baselinevisitsforeachpatient. Suppose
thatundertreatmenti,n patientsdropoutaftervisitt. Hence, n +(cid:1)(cid:1)(cid:1)+n =
it i1 iT
n and (n ;:::;n ) has the multinomial(n ;p ;:::;p ) distribution, where p
i i1 iT i i1 iT it
isthepopulationproportionofpatientsdroppingoutaftervisittundertreatment
i. We assume a pattern-mixture one-way ANCOVA model, i.e., for patients who
drop out after visit t, their last observed responses y ’s are independent with
itk
0
means (cid:22) + bz , where k = 1;:::;n , t = 1;:::;T, i = 1;:::;I, the (cid:22) ’s
it itk it it
are unknown (cid:12)xed treatment e(cid:11)ects, z is a q-vector of covariates observed for
itk
each patient, and b is a q-vector of unknown parameters. Unlike the y-response
variable, the z-covariate for a patient does not vary with t, although we use
the notation z to specify it. No other condition is imposed on the dropout
itk
mechanism (i.e., dropout may be nonignorable).
When there is no dropout, testing for treatment e(cid:11)ect may be carried out
by 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 the
hypothesis
H : (cid:22) = (cid:1)(cid:1)(cid:1) = (cid:22) ; (1)
0 1 I
where (cid:22) = p (cid:22) +(cid:1)(cid:1)(cid:1)+p (cid:22) .
i i1 i1 iT iT
The LOCF test is the ANCOVA test that treats y as the observation at
itk
the end of the trial. Since (cid:22) usually changes with t, one wonders what the
it
LOCF tests for. If (1) is the hypothesis of interest, there is the question of
validity of the LOCF test. The following result shows when the LOCF test is
asymptotically valid for (1). Since Shao and Zhong (2003) showed that when
I (cid:21) 3, the LOCF test is asymptotically wrong for testing (1) in a one-way
ANOVA without covariates, we only consider the case of I = 2 treatments.
Inthispaper,(cid:31)2 denotesthechi-squaredistributionwithddegreeoffreedom,
d
while (cid:31)2 and F denote, respectively, the 1(cid:0)(cid:11) quantiles of (cid:31)2 and the F-
d;(cid:11) l;m;(cid:11) d
distribution with degrees of freedom l and m, where (cid:11) is a given nominal level.
Let
0 0 2 2 T nit
MSTR= (y(cid:22) (cid:0)b^ (cid:22)z )(cid:0)(y(cid:22) (cid:0)b^ z(cid:22) ) a2 ;
1:: 1:: 2:: 2:: itk
h i (cid:30)Xi=1Xt=1Xk=1
1 2 T nit 0 2
MSE= (y (cid:0)y(cid:22) )(cid:0)b^ (z (cid:0)(cid:22)z ) ;
n +n (cid:0)q(cid:0)2 itk i:: itk i::
1 2
Xi=1Xt=1Xk=1h i
wherez isthecovariate valueassociated withy ,y(cid:22) andz(cid:22) are, respectively,
itk itk i:: i::
the averages of y and z over the indexes t and k,
itk itk
b^ = (Z~0Z~)(cid:0)1 2 T nit(y (cid:0)y(cid:22) )(z (cid:0)(cid:22)z );
itk i:: itk i::
i=1 t=1 k=1
XXX
LASTOBSERVATIONANALYSISINANOVAANDANCOVA 861
Z~ is the (n + n ) (cid:2) q matrix whose (cid:12)rst n rows are z0 (cid:0) (cid:22)z0 ;:::;z0 (cid:0)
1 2 1 111 1:: 11n11
(cid:22)z0 ;:::;z0 (cid:0)(cid:22)z0 and whose last n rows are z0 (cid:0)(cid:22)z0 ;:::;z0 (cid:0)(cid:22)z0 ;:::,
z10:: (cid:0)z(cid:22)1T0n,1Tand a1:: = n(cid:0)1+((cid:0)1)i(z(cid:22)2 (cid:0)(cid:22)z )0(Z~201Z~1)(cid:0)1(2z:: (cid:0)(cid:22)z21)n.21 2::
2Tn2T 2:: itk i 1:: 2:: itk i::
Theorem 1. Assume I =2 and, for patients dropping out after visit t, the y ’s
itk
are independent with means (cid:22) +b0z and variance (cid:27)2 > 0.
it itk
(i) TheANCOVAtestbasedonLOCFrejects(1)whentheratioF=MSTR/MSE
is larger than F1;n1+n2(cid:0)q(cid:0)2;(cid:11).
(ii) As n ! 1, i = 1;2, MSE ! (cid:27)2 + (cid:17), where ! denotes convergence in
i p p
probability, with
n (cid:28)2 +n (cid:28)2
(cid:17) =lim 1 1 2 2 (2)
n +n
1 2
and (cid:28)2 = T p ((cid:22) (cid:0)(cid:22) )2.
i t=1 it it i
(iii)Under (1), as n ! 1, i = 1;2, MSTR ! ((cid:27)2 +(cid:16))(cid:31)2, where ! denotes
P i d 1 d
convergence in distribution, and
w (cid:28)2 +w (cid:28)2
(cid:16) =lim 1 1 2 2; (3)
w +w
1 2
w = T nit a2 = 1 +(z(cid:22) (cid:0)z(cid:22) )0(Z~0Z~)(cid:0)1S (Z~0Z~)(cid:0)1(z(cid:22) (cid:0)z(cid:22) ); (4)
i itk n 1:: 2:: i 1:: 2::
i
t=1 k=1
XX
with S = T nit (z (cid:0)(cid:22)z )(z (cid:0)z(cid:22) )0.
i t=1 k=1 itk i:: itk i::
TheproofPofThePorem1(i)isbasedontheformula(B)inSearle(1987, p.425).
It is straightforward and therefore omitted. The proofs for Theorem 1(ii) and
(iii) are given in the Appendix.
It follows from Theorem 1 that the LOCF test is asymptotically valid for
testing (1) if and only if (cid:17) = (cid:16). Examining (2) and (3), we (cid:12)nd that the LOCF
test is asymptotically valid if either (cid:28)2 and (cid:28)2 are asymptotically the same, or
1 2
n =(n +n ) and w =(w +w ) are asymptotically the same. The only practical
1 1 2 1 1 2
situation in which (cid:28)2 and (cid:28)2 are asymptotically the same is when (cid:28)2 = (cid:28)2 = 0,
1 2 1 2
which corresponds to the case of (cid:22) = (cid:22) for all t. Hence, if (cid:22) ’s for a given i
it iT it
are di(cid:11)erent, the asymptotic validity of the LOCF test dependson the condition
n w
1 1
lim = lim : (5)
n +n w +w
1 2 1 2
We (cid:12)nd that two practical situations in which (5) holds are
n
lim 1 = 1 and lim(z(cid:22) (cid:0)z(cid:22) ) = 0; (6)
1:: 2::
n
2
n S S
lim 1 = 1 and lim 1 (cid:0) 2 = 0: (7)
n n n
2 (cid:18) 1 2(cid:19)
862 BINCHENG,JUNSHAOANDBOBZHONG
That is, the LOCF test is asymptotically valid for testing (1) when (6) or (7)
holds. NotethatShaoandZhong(2003) showedthattheconditionlim(n =n ) =
1 2
1 ensures the asymptotic validity of the LOCF test in one-way ANOVA. When
there are covariates, our result shows that in addition to the balance condition
limn =n = 1, the validity of the LOCF test requires the covariates to be bal-
1 2
anced in the sense that either the means of the covariates under two treatments
are asymptotically the same (condition (6)) or the covariance matrices of the
covariates under two treatments are asymptotically the same (condition (7)).
When there are I (cid:21) 3 treatments or the design is not balanced, the LOCF
testhasthewrongasymptoticsizefortesting(1). Anasymptotically validtestof
(1)isderivedasfollows. Fortheithtreatment,let b^ =(Z~0Z~ )(cid:0)1 T nit z y ,
i i i t=1 k=1 itk itk
whereZ~ isthen (cid:2)qmatrixwhosen rowsarez0 (cid:0)(cid:22)z0 ;:::;z0 (cid:0)(cid:22)z0 ;::::;z0
(cid:0)z(cid:22)0 , anid let u i = y (cid:0)b^0z . Thien u(cid:22) = n(cid:0)i111 Ti:: niti1unPi1 isPiu::nbiaseidTnfoiTr
i:: itk itk i itk i:: i t=1 k=1 itk
(cid:22) and asymptotically normal, and its variance can be estimated consistently by
i
P P
1 T nit
V^ = (u (cid:0)u(cid:22) )2:
i n (n (cid:0)1) itk i::
i i
t=1 k=1
XX
Theorem 2. Suppose that, for patients dropping out after visit t, the y ’s are
itk
independent with means (cid:22) +b0z and variances (cid:27)2 > 0. Under (1), as n !1
it itk it i
for all i, W ! (cid:31)2 , where
d I(cid:0)1
I 1 I u(cid:22) =V^ 2
W = u(cid:22) (cid:0) i=1 i:: i :
i=1 V^i i:: P Ii=11=V^i !
X
P
Consequently, anasymptoticsize(cid:11)testrejects(1)ifandonlyifW >(cid:31)2 .
I(cid:0)1;(cid:11)
Note that we do not assume the variances of y ’s are equal in Theorem 2.
itk
When (1) is rejected, we can make inference (such as pairwise or multiple
comparison on (cid:22) ’s) using the asymptotic results based on u(cid:22) and V^.
i i:: i
3. Tests for Interaction in Two-Way Models
Two-way ANOVA or ANCOVA is often used in clinical trials. In addition
to the treatment e(cid:11)ect, a common factor in a two way ANOVA or ANCOVA is
the center e(cid:11)ect in a multicenter trial. Consider a clinical trial carried out in
J centers with I treatments, n patients randomized to treatment group i at
ij
center j, and T scheduled visits for each patient. The total number of patients
I J
is n= n . Suppose that under treatment i at center j, n patients
i=1 j=1 ij ijt
dropoutafter visit t. Then(n ;:::;n ) hasthemultinomial(n ;p ;:::;p )
ij1 ijT ij ij1 ijT
P P
distribution, where p is the population proportion of patients dropping out
ijt
after visit t undertreatment i at center j. Let y bethe last observed response
ijtk
LASTOBSERVATIONANALYSISINANOVAANDANCOVA 863
variable of interest from patient k under treatment i at center j who dropped
out after the tth visit.
Under the two-way ANOVA model, for patients dropping out after visit t,
we assume that y ’s are independent with means (cid:22) . Similar to the one-way
ijtk ijt
case, we use the
T
(cid:22) = p (cid:22) (8)
ij ijt ijt
t=1
X
as measures for treatment and center e(cid:11)ects. Under two-way models, the (cid:22) and
i
(cid:22) ofprevioussections shouldbereplaced by(cid:22) and(cid:22) , respectively. Consider
it ij ijt
the decomposition
(cid:22) = (cid:22)+(cid:11) +(cid:12) +(cid:13) ;
ij i j ij
where(cid:22)isanoverallmean,(cid:11) ’sare(cid:12)xedtreatmente(cid:11)ects((cid:11) +(cid:1)(cid:1)(cid:1)+(cid:11) = 0),(cid:12) ’s
i 1 I j
are (cid:12)xed center e(cid:11)ects ((cid:12) +(cid:1)(cid:1)(cid:1)+(cid:12) = 0), and (cid:13) ’s are (cid:12)xed interaction e(cid:11)ects
1 J ij
((cid:13) +(cid:1)(cid:1)(cid:1)+(cid:13) = (cid:13) +(cid:1)(cid:1)(cid:1)+(cid:13) = 0 for any i and j). Although the treatment
i1 iJ 1j Ij
e(cid:11)ects (cid:11) ’s are of primary interest, the analysis in two-way ANOVA often starts
i
with a test for the treatment-by-center interaction with the null hypothesis
H :(cid:13) = 0; for all i and j: (9)
0 ij
To test (9), the LOCF test treats y as the observation in the end of the trial
ijtk
and rejects H0 when MSAB=MSE> F(I(cid:0)1)(J(cid:0)1);n(cid:0)IJ;(cid:11), where
1
0 0 (cid:0)1 0
MSAB= y(cid:22) L(L(cid:3)L) Ly(cid:22);
(I (cid:0)1)(J (cid:0)1)
I J T nijt
1
MSE= (y (cid:0)y(cid:22) )2;
n(cid:0)IJ ijtk ij::
i=1 j=1 t=1 k=1
XXXX
0
y(cid:22) is the average of y ’s over t and k, y(cid:22) = (y(cid:22) ;:::;y(cid:22) ;:::;y(cid:22) ;:::;y(cid:22) ),
ij:: ijtk 11:: I1:: 1J:: IJ::
(cid:3)= diag(n(cid:0)1;:::;n(cid:0)1;:::;n(cid:0)1;:::;n(cid:0)1),
11 I1 1J IJ
0 0
1 1
L = (J(cid:0)1) (cid:10) (I(cid:0)1) ;
(cid:0)I(J(cid:0)1)! (cid:0)I(I(cid:0)1)!
1 is the m-vector of ones, I is the identity matrix of order m, and (cid:10) is the
m m
Kronecker product. The following result shows what this tests for, and when it
is asymptotically valid. The proof can be found in Cheng (2004).
Theorem 3. Assume that, for patients dropping out after visit t, y ’s are
ijtk
independent with means (cid:22) and variance (cid:27)2 > 0.
ijt
864 BINCHENG,JUNSHAOANDBOBZHONG
(i) As n !1 for all i;j, MSE! (cid:27)2 +(cid:17), where
ij p
I J T
1
(cid:17) = lim n (cid:28)2 and (cid:28)2 = p ((cid:22) (cid:0)(cid:22) )2:
n ij ij ij ijt ijt ij
i=1 j=1 t=1
XX X
(ii) Under (9), MSAB converges in distribution to a linear combination of (I (cid:0)
1)(J (cid:0)1) independent chi-square random variables with 1 degree of freedom.
The LOCF test for interaction is asymptotically valid (i.e., MSAB/MSE is
asymptotically distributed as (cid:31)2 ) if and only if
(I(cid:0)1)(J(cid:0)1)
0 0
LVL= (cid:17)L(cid:3)L; (10)
where V = diag(n(cid:0)1(cid:28)2 ;:::;n(cid:0)1(cid:28)2 ;:::;n(cid:0)1(cid:28)2 ;:::;n(cid:0)1(cid:28)2 ).
11 11 I1 I1 1J 1J IJ IJ
(iii)When I = J = 2, (10) becomes
2 2 n (cid:28)2 2 2 n(cid:0)1(cid:28)2
i=1 j=1 ij ij i=1 j=1 ij ij
lim = lim : (11)
2 2 n 2 2 n(cid:0)1
P i=1P j=1 ij P i=1P j=1 ij
When I = 2 and J (cid:21)P3, (1P0) becomes P P
n(cid:0)1(cid:28)2 +n(cid:0)1(cid:28)2 2 J n (cid:28)2
1j 1j 2j 2j i=1 j=1 ij ij
= ; for all j: (12)
n(cid:0)1 +n(cid:0)1 2 J n
1j 2j P i=1P j=1 ij
When I (cid:21) 3 and J = 2, (10) becomPes P
n(cid:0)i11(cid:28)i21+n(cid:0)i21(cid:28)i22 = Ii=1 2j=1nij(cid:28)i2j; for all i: (13)
n(cid:0)1+n(cid:0)1 I 2 n
i1 i2 P i=1P j=1 ij
When I (cid:21) 3 and J (cid:21) 3, (10) becomPes P
(cid:28)2 = constant: (14)
ij
WhenI = J = 2and(cid:22) ’saredi(cid:11)erentforgiveniandj (sothatthe(cid:28) ’sare
ijt ij
di(cid:11)erent), (11) implies that the LOCF test for treatment-by-center interaction
is asymptotically valid for testing (9) if the design is balanced in the sense that
limn =n = 1=4 for any i and j. Although in many applications the number
ij
of 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 only
practical situation that a LOCF procedure is valid is when the (cid:28)2 are all the
ij
same or, equivalently, the (cid:22) ’s are the same for (cid:12)xed i and j. A result similar
ijt
to Theorem 3 for a two-way ANCOVA model can be derived, but it is omitted
since a necessary condition for the validity of the LOCF test is I = J =2, which
is a limited special case.
LASTOBSERVATIONANALYSISINANOVAANDANCOVA 865
An asymptotically valid test for (9) can be derived based on cell mean esti-
matorsy(cid:22) andtheirconsistentvarianceestimators. Wenowconsiderthegeneral
ij::
two-way ANCOVA model (which includes the ANOVA model as a special case)
in which the mean of y (for patients dropping out after visit t) is
ijtk
0
(cid:22) +bz ; (15)
ijt ijtk
wherez isaq-vectorofcovariatesobservedforeachpatient,andbisaq-vector
ijtk
of unknown parameters. For any i and j, let b^ be the least squares estimator
ij
of b based on the data from the patients who received the ith treatment in the
jth center and let u = y (cid:0)b^0 z . Then u(cid:22) = n(cid:0)1 T nijt u is an
ijtk ijtk ij ijtk ij:: ij t=1 k=1 ijtk
unbiased and asymptotically normal estimator of (cid:22) with a consistent variance
ij
P P
estimator
T nijt
1
V^ = (u (cid:0)u(cid:22) )2:
ij n (n (cid:0)1) ijtk ij::
ij ij
t=1 k=1
XX
Theorem 4. Suppose that, for patients dropping out after visit t, the y ’s
ijtk
are independent with means given by (15) and variances (cid:27)2 . Under (9), as
ijt
n ! 1 for all i;j, W ! (cid:31)2 , where W = u(cid:22)0L(L0V^L)(cid:0)1L0u(cid:22), V^ =
ij d (I(cid:0)1)(J(cid:0)1)
diag(V^ ;:::;V^ ;:::;V^ ;:::;V^ ), and u(cid:22)=(u(cid:22) ;:::;u(cid:22) ;:::;u(cid:22) ;:::;u(cid:22) )0.
11 I1 1J IJ 11:: I1:: 1J:: IJ::
Consequently, a test of (9) with asymptotic size (cid:11) rejects H if W >
0
(cid:31)2 . It is clear that the result in Theorem 4 can be extended to K-way
(I(cid:0)1)(J(cid:0)1);(cid:11)
ANCOVA models.
4. Additive Two-Way ANCOVA
Inamulticenter clinical trial, treatment-by-center interaction cansometimes
be ignored, especially when covariates related to centers are introduced into the
model. Hence, in this section we consider an additive two-way ANCOVA model,
which includes the additive two-way ANOVA model as a special case.
Considerthemulticenterclinical trialdescribedinSection3,wherethemean
0
of y for patients dropping out after visit t is (cid:22) +bz , z is a q-vector of
ijtk ijt ijtk ijtk
covariates observed from every patient, and b is a q-vector of unknown parame-
ters. Let (cid:22) be given by (8) and assume the additive model
ij
(cid:22) = (cid:22)+(cid:11) +(cid:12) ; (16)
ij i j
where (cid:22) is an overall mean, (cid:11) ’s are (cid:12)xed treatment e(cid:11)ects ((cid:11) +(cid:1)(cid:1)(cid:1)+(cid:11) = 0),
i 1 I
and (cid:12) ’s are (cid:12)xed center e(cid:11)ects ((cid:12) +(cid:1)(cid:1)(cid:1)+(cid:12) = 0). The null hypothesis of no
j 1 J
treatment e(cid:11)ect is
H :(cid:11) = (cid:1)(cid:1)(cid:1) = (cid:11) = 0: (17)
0 1 I
866 BINCHENG,JUNSHAOANDBOBZHONG
The LOCF procedure for testing (17) treats (cid:22) as (cid:22) for all t and uses the
ijt ijT
0
following model E(Y) = X(cid:18) +Zb, where (cid:18) = ((cid:22);(cid:11) ;:::;(cid:11) ;(cid:12) ;:::;(cid:12) ), Y is
1 I 1 J
the column vector formed by listing the elements y in the order of i;j;t and
ijtk
k, Z is the matrix formed by listing the row vectors z in the order of i;j;t
ijtk
and k, and X is the usual design matrix in a two-way additive ANOVA model.
De(cid:12)ne P = I(cid:0)X(X0X)(cid:0)X0 and
b^ = (Z0PZ)(cid:0)1Z0PY: (18)
The following theorem shows when the LOCF test is asymptotically valid
for testing (17). Its proof is similar to that of Theorem 1 and is omitted. First,
let
2 J T nijt 2 J T nijt
2
MSTR= a y a2 ;
ijtk ijtk ijtk
(cid:16)Xi=1Xj=1Xt=1Xk=1 (cid:17) (cid:30)Xi=1Xj=1Xt=1Xk=1
where the a ’s are the components of the vector that is the di(cid:11)erence between
ijtk
the second and the third rows of (X0X)(cid:0)X0(I(cid:0)Z(Z0PZ)(cid:0)1Z0P), and
1 2 J T nijt 0 2
MSE= (y (cid:0)y(cid:22) )(cid:0)b^ (z (cid:0)z(cid:22) ) ;
n(cid:0)(2J +q) ijtk ij:: ijtk ij::
Xi=1Xj=1Xt=1Xk=1h i
wherey(cid:22) andz(cid:22) areaveragesofy andz overindicestandk,respectively.
ij:: ij:: ijtk ijtk
Theorem 5. Assume that I = 2 and, for patients dropping out after visit t, the
y ’s are independent with means (cid:22) +b0z and variance (cid:27)2 > 0, and the
ijtk ijt ijtk
(cid:22) ’s have form given in (16).
ij
(i) The ANCOVA test based on LOCF rejects hypothesis (17) when the ratio
F =MSTR/MSE is larger than F1;n(cid:0)2J(cid:0)q;(cid:11).
(ii) As n !1, i = 1;2 and j = 1;:::;J, MSE! (cid:27)2 +(cid:17), where
ij p
2 J n (cid:28)2 T
(cid:17) = lim i=1 j=1 ij ij; (cid:28)2 = p ((cid:22) (cid:0)(cid:22) )2: (19)
2 J n ij ijt ijt ij
P i=1P j=1 ij t=1
X
(iii)Under (17), as n !P1, Pi = 1;2 and j = 1;:::;J, MSTR ! ((cid:27)2 +(cid:16))(cid:31)2,
ij d 1
where
2 J w (cid:28)2 T nijt
(cid:16) =lim i=1 j=1 ij ij; w = a2 : (20)
2 J w ij ijtk
P i=1P j=1 ij t=1 k=1
XX
(iv)For testing hypothesis (1P7), thPe LOCF procedure described in (i) is asymptot-
ically valid if and only if
2 J n (cid:28)2 2 J w (cid:28)2
i=1 j=1 ij ij i=1 j=1 ij ij
lim = lim : (21)
2 J n 2 J w
P i=1P j=1 ij P i=1P j=1 ij
P P P P