testing for random dropouts in repeated measurement data

Testing for Random Dropouts in Repeated Measurement DataAuthor(s): Peter J. DiggleSource: Biometrics, Vol. 45, No. 4 (Dec., 1989), pp. 1255-1258Published by: International Biometric SocietyStable URL: http://www.jstor.org/stable/2531777 .

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BIOMETRICS 45, 1255-1258 December 1989

Testing for Random Dropouts in Repeated Measurement Data

Peter J. Diggle

Department of Mathematics, University of Lancaster, Bailrigg, Lancaster LA 1 4YL, England

SUMMARY

Dropouts in repeated measurement data occur when the sequences of measurements on some experimental units are terminated prematurely. A number of methods which have been proposed for the analysis of repeated measurements with dropouts assume that the dropouts occur at random within experimental treatment groups. This paper develops a method of testing the hypothesis of random dropouts within groups, and applies the method to published data concerning the effect of halothane on the blood pressure of rats following experimentally induced heart attack.

1. Introduction

Repeated measurement data arise when a time-ordered sequence of measurements is made on each of a number of experimental units. Typically, the units fall into two or more different experimental groups, the groups representing different treatment regimes. We shall call the data complete if measurements are made on each unit at a common set of times tj (j = 1, ..., n). This leads to a complete two-way array of measurements

yij (i = 1, .. ., m; j = 1, .. ., n) in which yij represents the jth measurement on the ith of m units. Several authors have proposed methods for analysing incomplete repeated measurement data. See, for example, Crepeau et al. (1985), Jones (1987), Kenward (1987), and Diggle (1988). These authors argue, explicitly or implicitly, that incomplete sequences of measurements on particular units should be retained in the analysis, provided that the data within these incomplete sequences are not atypical.

A common form of incomplete data is that which arises through dropouts, or premature termination of some of the sequences of measurements. The resulting incomplete array of measurements is of the form yij (j = 1, . .. , ni; i = 1, . .. , m), where each yij corresponds to time tj and each ni - n. Our objective in this paper is to develop a method of testing the hypothesis that dropouts occur at random within groups, in the sense of being unrelated to measurement history. Our method consists of applying separate tests at each tj within each group and analysing the resulting sample of P-values for departure from the uniform distribution on (0, 1). The method is intended to be useful for preliminary screening of the data. Nonrandom dropouts imply that the censoring of the data may be informative, which raises interesting and complex modelling issues. See, for example, Wu and Bailey (1988) or Wu and Carroll (1988). Note also that Rubin (1976) uses the term "missing at random" in a wider sense than is intended here.

2. A Class of Tests for Random Dropouts

We wish to test the hypothesis that within each group and for each j - n - 1, the units with ni = j are a random sample from the units with ni > j. A general class of tests can be constructed by the following method.

Key words: Missing values; Monte Carlo tests; Probability plots; Repeated measurements.

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1256 Biometrics, December 1989

First, for each j define a function hj ( Yi, . . ., yj ). Now, within each group and for each time point tj, identify those Rj units which have ni > j and compute the corresponding scores hij = hj(yil, . . ., yij) (i = 1, . . ., Rj). Within this set of Rj scores, identify those rj scores which correspond to units with ni = j. Assuming that 1 < rj < Rj, test the hypothesis that the rj scores so identified are a random sample from the Rj scores hij. Finally, test the hypothesis that the set of P-values obtained by applying the above procedures to each time point within each group is a random sample from a uniform distribution on (0, 1).

Note that for the final stage of the test, the independence of the P-values under the hypothesis of random dropouts within groups relies on the fact that once a unit drops out it never returns. The method as presented would not be valid for testing whether general patterns of missing values are random, although any pattern of missing values can be converted to the present form by further deletions.

Two technical questions concern the construction of the individual P-values and the mode of their subsequent analysis. On the first question, the aim should be to choose a set of functions hj(-) so that scores which are extreme in a particular direction cast doubt on the hypothesis of random dropouts. A natural test statistic is then hj, the mean of the rj scores corresponding to the units with ni = j which are about to drop out. The approxi- mate null sampling distribution of hj is normal with mean Hj = R7- z&' hij and variance SJ2 (Rj - rj )/(rjRj), where Sj2 = (Rj - 1)-i I l (h,j - Hj )2. See, for example, Cochran (1977).

However, in many applications some of the rj and Rj will be small, and the normal approximation may then be poor. For an exact test, we can either evaluate the complete randomisation distribution of hj under the null hypothesis or, when (Ri ) is too large for this to be practical, recompute hj after s - 1 independent selections of rj scores chosen without replacement from the set hj (i = 1, . .. , Rj), and quote the P-value as kls, where k denotes the rank of the original hj amongst the recomputed values. The latter strategy gives an example of a Monte Carlo test (Barnard, 1963). The choice of the functions hj(.) should reflect, at least in a qualitative sense, the alternative hypothesis in mind. For example, if dropout is suspected to be the result of an abnormally low value of the quantity being measured, a natural choice is hj (y, . .., yj ) = yj. If it is suspected to be the culmination of a sustained low measurement, a better choice is

hj ( y, ..., yj) = yk k=1

The second question concerns the analysis of the resulting set of P-values, say Pu (u = 1, ..., t). As a general aid to interpretation, we recommend inspection of a prob- ability plot of the Pu. If the dropouts are random, the pu should behave like a random sample from a uniform distribution on (0, 1). For a formal test, we use the one-sided or two-sided Kolmogorov-Smirnov statistic as appropriate, i.e., D+ = sup{F(p) - p }, D_ = -inf{F(p) - p}, or D = max{D+, D_}. An important proviso is that the null distribution of pu is a discrete uniform on the points i/ku (i = 1, 2, ..., ku), where each ku is a binomial coefficient, (Rrf ) for some j. Furthermore, this assumes that there are no tied values. Note that ties can occur in the hi even when there are no ties in the hii. For example, if rj = 2, Rj = 5, and h,j = 7, 8, 9, 10, 12, the ten possible values of hj are 15, 16, 17, 19, 17, 18, 20, 19, 21, 22. If we consider small values of hj to be extreme, the null distribution of pu has support S = {.1 .2, .4, .5, .7, .8, .9, 1.0} with Pr(pu = .4) = Pr(pu = .7) = .2 and Pr( p, = x) = .1 for all other x E S. In such cases, it remains true that Pr(pu < x) =

x, Vx E S. These considerations complicate the null sampling distribution of the Kolmogorov-

Smirnov statistic and invalidate the use of critical values appropriate to a continuous

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Dropouts in Repeated Measurements 1257

uniform. However, a Monte Carlo implementation is straightforward. We simply rank the observed value of the test statistic amongst s - 1 simulated values based on random samples drawn from the appropriate set of discrete uniforms.

3. An Application

Crepeau et al. (1985) present data from an experiment to determine the effect of halothane on the blood pressure of rats following experimentally induced heart attack. There are five experimental groups, corresponding to a control and four increasing doses of halothane, with 10 or 11 rats in each group. Up to nine measurements of blood pressure were made on each rat. However, 29 of the 54 rats, including all 11 in the highest-dose group, died before the end of the experiment. Crepeau et al. (1985), and subsequently Diggle (1988), discarded the highest-dose group and analysed the remainder of the data assuming random dropouts.

A glance at the data suggests that a sufficiently high dose of halothane both decreases the mean blood pressure and increases the proportion of deaths. It follows that dropouts are not random over the whole experiment, but this does not rule out the possibility that they may be random within groups. We use the functions hj(yi, . .., yj) = yj to define the scores hij and evaluate each Pu by complete enumeration of its randomisation distribution. Figure 1 shows a probability plot of the 16 pu-values, with the four values from the highest- dose group highlighted. We use the one-sided Kolmogorov-Smirnov statistic D+ to test for departure from uniformity, implemented via Monte Carlo with s = 1,000. For the

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Figure 1. A probability plot of the 16 p4-values obtained from the halothane data of Crepeau et al. (1985). The circled points correspond to the highest-dose group.

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1258 Biometrics, December 1989

16 p1-values, D+ - .215 with a P-value of .034. Omitting the four pa-values from the highest-dose group, we obtain D+ = .194 with a P-value of .195. The two-sided statistic D gave a nonsignificant result in both cases. Using hj ( y, ..., )= zk=, Yk gave corresponding P-values of .174 and .241.

This analysis suggests that the dropouts can be treated as random within groups if we exclude the highest-dose group, a conclusion which supports the analyses of the data reported in Crepeau et al. (1985) and Diggle (1988).

ACKNOWLEDGEMENT

I thank an anonymous associate editor, whose perceptive comments led to a substantial improvement in the paper.

RE~SUME~

Des sorties d'etude dans des ilans a mesures repetees ont lieu des que des sequences de mesures sont interrompues prematur6ment sur certaines unites experimentales. Un certain nombre de m6thodes qui ont et proposees poiv l'ianalyse des mesures r6petees avec sorties d'etude supposent que celles-ci aient lieu de favon aleatoire 'a l'int6rieur des groupes traites. Cet article propose une methode pour tester une telle hypothese, a savoir le caractere aleatoire des sorties d'etude dans les groupes; il applique ensuite cette methode a des donnees publiees concernant l'effet du halothane sur la pression sanguine de rats qui ont subi une crise cardiaque provoqu6e de fa~on experimentale.

REFERENCES

Barnard, G. A. (1963). Contribution to the Discussion of Professor Bartlett's paper. Journal of the Royal Statistical Society, Series B 25, 294.

Cochran, W. G. (1977). Sampling Techniques, 3rd edition. New York: Wiley. Crepeau, H., Koziol, J., Reid, N., and Yuh, Y. S. (1985). Analysis of incomplete multivariate data

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Received December 1988; revised April and June 1989.

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