STATISTICS IN MEDICINEStatist. Med. 2004; 23:3241–3244

LETTER TO THE EDITOR

Last observation carry-forward and last observation analysis

by J. Shao and B. Zhong, Statistics in Medicine 2004; 22:2429–2441

From: James CarpenterSenior Lecturer in Medical Statistics, LSHTM, U.K.Mike KenwardGSK Professor of Biostatistics, LSHTM, U.K.Stephen EvansProfessor of Pharmaco-epidemiology, LSHTM, U.K.Ian WhiteSenior Scientist, MRC Biostatistics Unit, Cambridge, U.K.

The recent paper ‘Last observation carry-forward and last observation analysis’ by J. Shaoand B. Zhong, Statistics in Medicine 2004; 22:2429–2441 is highly misleading.The authors claim in the abstract that the ‘LOCF one-way ANOVA test is asymptoti-

cally valid... in the special but important case where only two treatments are compared’ Validfor testing which null hypothesis? Not the one which readers would naturally assume, thatthe e�ect of two drugs at the end of a trial is indistinguishable, for which LOCF has beenadvocated.Speci�cally, suppose a trial with I arms, i=1; 2; : : : ; I follows up patients at times

j=1; : : : ; J . Let �ij be the mean response of patients under treatment i who drop outafter follow-up j. Let

�i=J∑

j=1pij�ij

where pij=E[nij=ni], the average proportion of patients on treatment i dropping out followingvisit j. The null hypothesis considered by the authors is

H0 : �1 =�2 = · · · =�I

which is an end-point analysis. Thus, not only is this not a missing data problem (becauseour best estimate of �ij is the mean response at time j of the set of patients under treatmenti who are last seen at visit j), but H0 can be satis�ed when the e�ect of treatment is quitedi�erent in the two arms.For instance, suppose there are three follow-up visits, that response to the active drug

peaks around visit 2 and that placebo response is constant. Further, suppose that this causes asubstantial, but random, subset of patients in the active treatment arm to drop out at visit 2.Suppose that the response of those who continue to the end declines towards their baseline.

Copyright ? 2004 John Wiley & Sons, Ltd.

3242 LETTER TO THE EDITOR

Then the authors will tend to reject their H0, and report a treatment e�ect, while there is nolong term di�erence between the treatments.This illustrates that the authors’ null hypothesis is likely to be rejected when there is a

transitory e�ect of treatment. Although transitory e�ects might be of interest to the trialists,if they were of primary interest a di�erent design, with shorter follow-up, would have beenemployed. Further, transitory e�ects are of little interest to the regulators or patients.However, as the dropout mechanism is missing at random, the correct (unbiased) estimate

of treatment e�ect at the end of the trial can be readily estimated by �tting a linear mixedmodel. Simply �t (i) visit as a factor, (ii) a full baseline ∗ visit interaction, (iii) a fulltreatment ∗ visit interaction and (iv) an unstructured covariance matrix.In addition, the authors use the term ‘informative dropout’ in an idiosyncratic and misleading

manner. Usually, it means that the probability of a patient dropping out, conditional on all theobserved information on that patient, nevertheless depends on the unseen response. Althoughthe authors’ de�nition is not formally given, it appears to be that the expected value of theunseen response is di�erent from the expected value of the last seen response: a situationwhich can occur under missing completely at random, missing at random and informativedropout.Note further that the authors admit (p. 2430, L11) that LOCF should not be used when

dropout is a potential e�ect of treatment. To assume otherwise, however, is unrealistic.To conclude, this article attempts to promote LOCF as a valid statistical method for

analysing trials with dropout. This is misleading since the hypothesis suggested by the authorshas no meaning in clinical terms. Statisticians and regulators, please don’t be misled.

(DOI: 10.1002/sim.1891)

AUTHORS’ REPLY

In responding to the letter from Carpenter, Kenward, Evans, and While (CKEW), we wouldlike to point out that, unfortunately, some readers think that a research article is misleadingbecause they draw a misleading conclusion by themselves. The letter from CKEW is anexample.CKEW concluded that ‘this article attempts to promote LOCF as a valid statistical method

for analyzing trials with dropout’. This is purely their own conclusion. It is clearly stated inthe Introduction of our article that the ‘purpose of this article is to provide some theoreticalresults showing when the one-way ANOVA test based on the LOCF is correct or incorrect’and ‘to propose a di�erent test with a correct asymptotic size when the LOCF test is incorrect’.The readers should read details to �nd out when the LOCF is correct and the correctness is forwhich hypothesis. For almost every statistical method, its correctness depends on the problem,the parameter or hypothesis of interest, and some regularity conditions. It is stated in Section 2of our article that we are interested in hypothesis (2) (i.e. �i’s are the same for all treatments)when the global mean �i is used as the measure of drug e�ect i. Whether or not hypothesis

Copyright ? 2004 John Wiley & Sons, Ltd. Statist. Med. 2004; 23:3241–3244