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\footnotesize
\noindent Statistica Sinica {\bf 15}(2005), 857-870

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\centerline{\large\bf LAST OBSERVATION ANALYSIS IN ANOVA}
\vspace{2pt}
\centerline{\large\bf  AND ANCOVA}
\vspace{.25cm}
\author{Bin Cheng, Jun Shao and Bob Zhong}
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\centerline{\it Columbia University, University of Wisconsin and}
\centerline{\it Centocor, Inc.}
%Eli Lilly {\rm \&} Company}
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\begin{quotation}
\noindent {\it Abstract:}
In clinical trials with multiple visits, dropouts often occur
and the population of patients who dropped out may be different from
the population of patients who completed the study.
To assess treatment effects
over the population of all randomized patients,
which is called the intention-to-treat analysis
and is required by regulatory agencies,
 last observation analysis (LOAN) focuses
on the last observation 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  LOAN is relevant, we provide
explicit conditions under which  LOCF tests are asymptotically valid
and we derive asymptotically valid tests
when LOCF tests are invalid.\par
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\noindent {\it Key words and phrases:}
 Dropout, intention-to-treat analysis, last observation carry-forward,
balanced design, balanced covariates.\par
\end{quotation}

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