A new parameterization for pattern mixture models oflongitudinal data with informative dropoutMichael J. Daniels 1Joseph W. Hogan 21 Department of Statistics, Iowa State University2 Center for Statistical Sciences, Brown UniversityMichael J. DanielsDepartment of StatisticsIowa State University102G Snedecor HallAmes, IA 50011e-mail: mdaniels@iastate.eduphone: (515) 294 7775fax: (515) 294 4040

SummaryPattern mixture models are frequently used to analyze longitudinal data where missingnessis induced by dropout. For measured responses, it is typical to model the complete dataas a mixture of multivariate normal distributions, where mixing is done over the dropoutdistribution. Fully parameterized pattern mixture models are not identi�ed by incompletedata; Little (1993) has characterized several identifying restrictions which can be used formodel �tting.We propose a re-parameterization of the pattern mixture model which allows investi-gation of sensitivity to assumptions about non-identi�ed parameters in both the mean andvariance, and allows consideration of a wide range of non-ignorable missing data mechanisms.The parameterization makes clear an advantage of pattern mixture models over parametricselection models, namely that the missing data mechanism can be varied without a�ectingthe marginal distribution of the observed data.To illustrate the utility of the new parameterization, we analyze data from a recentclinical trial of growth hormone for maintaining muscle strength in the elderly. Dropoutoccurs at a high rate and is potentially informative. We undertake a detailed sensitivityanalysis to understand the impact of missing data assumptions on the inference about thee�ects of growth hormone on muscle strength.Key Words: sensitivity analysis; selection model; missing data; nonignorable nonre-sponse; clinical trial, repeated measures.1

1 Introduction and BackgroundIn analyzing repeated measures data, scientists are often confronted with the problem ofdropout, which may be informative in the sense that it is related to unobserved outcomes,even after conditioning on observed information. Recently, many di�erent approaches toanalyzing repeated measures with informative dropout have been proposed; they includeparametric (Wu and Carroll, 1988; Little, 1993; Diggle and Kenward, 1994; Follmann andWu, 1995), semiparametric (Robins and Rotnitzky, 1995; Scharfstein and Robins, 1999),and nonparametric (Yao, Wei, and Hogan, 1998) methods. See Wu, Hunsberger, and Zucker(1994), Little (1995), and Hogan and Laird (1997b) for recent reviews.Two widely studied approaches to modeling repeated measures with dropout are mixturemodels and selection models. In both, the joint distribution of a repeated measures vectoryi and a dropout time (or more generally, missing data indicator) ui are modeled. Selectionmodeling, which appeals directly to Rubin's missing data hierarchy, calls for speci�cationof the complete data distribution f(y) and the missing data mechanism f(ujy). Mixturemodeling assumes that f(y) is a mixture over dropout times, and calls for speci�cation ofpattern-speci�c distributions f(yju). Pattern mixture models (Little, 1993, 1994) specifyf(y) as a mixture over missing data patterns, which in longitudinal data usually are takento be dropout times.Diggle and Kenward (1994) introduced de�nitions of informative dropout for repeatedmeasures data, which correspond to Rubin's (1976) general taxonomy for classifying missingdata mechanisms. For inferences based on a likelihood, an important distinction is whetherthe missing data mechanism is ignorable or non-ignorable (NI). Ignorability consists of twoconditions: (1) the probability of missingness, conditional on observed outcomes, is indepen-dent of missing outcomes; and (2) the parameter spaces characterizing the complete data2

and the missing data process are non-intersecting (Rubin, 1976). Under ignorability, validinference about the complete data can be obtained from the observed-data likelihood. Undernon-ignorable missingness, it is necessary to correctly specify the missing data mechanismto obtain valid inference. An inherent di�culty facing the data analyst is that, ignorabil-ity in general is not a testable hypothesis. Many models which incorporate NI missingnessare identi�ed only through parameter restrictions or distributional assumptions (see Little1994; Scharfstein and Robins, 1999). In practice, lack of knowledge about the true missingdata mechanism is frequently handled by assessing the sensitivity of inferences under variousstructural and parametric assumptions about the missing data mechanism.An appealing feature of selection models is that commonmissing data mechanisms such asMAR and MCAR have an explicit representation through f(ujy); sensitivity analyses can behandled by varying parameters in f(ujy). However, selection model estimates and inferencescan be highly sensitive to the assumed structure of f(ujy) (Laird, 1988). By contrast, themissing data mechanism implied by a mixture model is not always obvious and rarely takesa closed form. Sensitivity analyses can be constructed by considering various identifyingrestrictions, some of which correspond to familiar missing data mechanisms (Little, 1995).In this paper, we introduce a new parameterization of the pattern mixture model whichcan be used to represent various missing data mechanisms and construct sensitivity analyses.The re-formulated model is cast in terms of changes in location and scale between patterns.We consider PMM's which are mixtures of multivariate normal distributions, where dropoutde�nes pattern. For these models, it is relatively straightforward to separate the complete-data parameter space into its identi�ed and non-identi�ed subsets, and to derive identifyingrestrictions corresponding to MCAR, MAR, and to any NI mechanism. Sensitivity analysesfollow by varying the non-identi�ed parameters, thus allowing the analyst to examine allnon-ignorable speci�cations along a (plausible) continuum. This parameterization allows3

separate investigation of assumptions about non-identi�able parameters in both the meanand variance.In the next section, we describe the pattern mixture model for mixtures of normal dataand discuss several aspects of identi�ability and impliedmissing data mechanisms. In Section3 we describe the new parameterization and derive representations of MCAR, MAR, and NImechanisms for the pattern mixture model. Section 4 contains a detailed analysis of a recentclinical trial of recombinant human growth hormone (rhGH) for increasing muscle strengthin the elderly, including several sensitivity analyses. In the �nal section we discuss severalimportant points related to modeling incomplete data, and give recommendations for furthertopics of investigation.2 The pattern mixture model for incomplete repeatedmeasures dataThe pattern mixture model assumes the distribution of the complete data is a mixtureover missing data patterns. Let yi = (yi1; : : : ; yin)0 represent the complete-data vector ofresponses for subject i, where i = 1; : : : ; N . We assume missingness is induced by dropout,which occurs at occasion ui 2 f1; : : : ;Kg. If dropout can happen at each measurement time,then K = n; otherwise, the set f1; : : : ;Kg are simply labels. The pattern mixture modelassumes separate multivariate normal (MVN) distributions for each pattern, i.e.,(yijui = k) � N ��(k);�(k)�Little and Wang (1996) describe a version where component means can depend linearlyon covariates through a K � p covariate matrix Xi. The variances also can depend oncovariates, usually by assuming separate variances across levels of a discrete covariate likegroup assignment. Allowing the dropout distribution to be a function of covariates such astreatment group also is straightforward (and usually necessary).4

The PMM is attractive because it is possible to leave the marginal dropout distributionunspeci�ed. However, the conditional distribution of dropout as a function of responsedata is not usually transparent. Sensitivity analyses suggested in prior work are framed interms of a selection mechanism where the relationship between dropout and nonresponseis captured in one (possibly vector-valued) parameter which a�ects the mean and variancesimultaneously (see Little and Wang, 1996, for an example).The fully general PMM is always under-identi�ed; identifying restrictions can be char-acterized and understood in terms of implied selection mechanisms (Little, 1994; Little andWang, 1996). Consider the simple case of bivariate outcome yi = (yi1; yi2)0, where data onyi1 is complete but some subjects are missing yi2. For dropout patterns k = 1; 2 where k = 2denotes those with complete data,�(k) = ��(k)1 ; �(k)2 �0 and �(k) = " �(k)11 �(k)12�(k)12 �(k)22 #represent the pattern speci�c mean vectors and variance matrices. Both �(2) and �(2) arefully identi�ed, but certain components of �(1) and �(1) are not. In particular, withoutmodeling restrictions, there are no observations to identify �(1)2 , �(1)12 , and �(1)22 . For higher-dimensional response under monotone missingness, it is straightforward to see which modelparameters are not identi�ed.Various modeling restrictions can be used to identify these parameters under both ignor-ability and non-ignorability of the missing data mechanism. Little (1993) and Mohlenberghset al (1998) have characterized several identi�cation restrictions for PMM's under ignor-ability. Complete-case restrictions equate non-identi�able parameters in various dropoutpatterns to their counterparts in the completers group. Available case (AC) restrictions areconstructed similarly, but make use of all available data for identi�cation rather than justthat from completers. Molenberghs et al (1998) show that AC restrictions have a one-to-one5

correspondence with MAR under the pattern mixture speci�cation.Little (1994) and Little and Wang (1996) use selection model restrictions to characterizeidentifying constraints for a variety of PMM's under non-ignorable dropout. For the bivariatecase, the selection-model identifying restriction ispr(u = 1jy1; y2) = f(y1; y2)for some unspeci�ed f(�). A multivariate generalization follows directly. Little and Wang(1996) considered the special case where yi = (y1i; : : : ; yin)0 and only yin is subject to non-response. The identifying restriction ispr(yin missingjLy?i ; yin) = f(Ly?i + �yin)where L is a 1 � n � 1 known matrix and y?i = (yi1; : : : ; yi;n�1). Di�erent forms for f(usually linear combinations of yi) and di�erent values of L and � characterize assumptionsabout the pattern-speci�c regressions of yin on yi1; : : : ; yi;n�1. For example, � = 1 is anonignorable missingness speci�cation which identi�es the model through the assumptionthat the pattern-speci�c regressions are equivalent between those with missing and observedyin. This approach makes clear the nature of the missing data mechanisms, but constrainsthe mean and variance simultaneously and in ways which are not transparent.3 Parameterizing in terms of location and scale changesThe PMM can be identi�ed through between-pattern location shifts and scale changes, whichappeals directly to its underlying structure. This parameterization makes it straightforwardto derive the restrictions corresponding to MCAR and MAR, and to specify the full range ofnon-ignorable missing data mechanisms implied by the PMM (under multivariate normalitywithin pattern, the missing data mechanism has a convenient and intuitive representationas a polytomous regression). 6

The implication for sensitivity analyses is that one can separately examine underlyingassumptions about mean and variance parameters which are not identi�ed by the observeddata. Furthermore, the location-scale parameterization reduces sensitivity analyses to aseries of complete-data problems, requiring very little e�ort beyond the usual �tting of mul-tivariate normal regressions; thus, in principle, no special software is required for sensitivityanalyses.Consider �rst the simple pattern mixture model in which all dropouts occur at the sametime point. The complete data for subject i is an n-dimensional vector yi = (yi1; : : : ; yin), anddropout status is characterized by ui 2 f1; 2g (1=dropout; 2=completer). The correspondingpattern mixture model is (yijui = k) � N(�(k);�(k)) (1)pr(ui = k) = � � if k = 11� � if k = 2Elements of �(1) and �(1) beyond the dropout time are not identi�able.When ui is Bernoulli and all dropouts happen at a common time point, the PMM can bespeci�ed by assuming dropouts di�er in distribution from completers by a shift in locationand scale, yielding �(1) = �(2) + � and �(1) = C�(2)C 0 where � is an n� 1 vector and C is an�n scaling matrix. This is an over-parameterization because knowing � and C identi�es �(1)and �(1). This parameterization is motivated by the structure of the PMM, which explicitlyseparates completers and dropouts (or more generally, patterns of dropouts). In this simplecase, � and C measure quantitative di�erences between completers and dropouts.Consider the case where dropout is monotone and has more than two patterns. Thenui 2 f1; : : : ;Kg, where ui = K denotes a completer and �(k) = pr(ui = k). In manysettings, dropout may occur at each measurement time, so that ui represents the numberof observed measurements and K = n. The location-scale parameterization gives rise to a7

set of di�erence vectors �(1); : : : ; �(K�1) and scaling matrices C(1); : : : ; C(K�1) which relatemeans and variances from adjacent dropout patterns. In general,�(k) = �(k+1) + �(k) and �(k) = C(k)�(k+1)C 0(k);which implies �(k) = �(k+1) � �(k) and C(k) = ��(k)�1=2 ��(k+1)��1=2 (2)The target for inference is typically the marginal mean of y, which follows directly asE(y) = �(K) + �1 � �(K)� �(K�1) + �1� �(K) � �(K�1)� �(K�2) + � � �+ �(1)�(1): (3)Sensitivity analyses based on the C and � parameters are based on assumed or suspecteddi�erences between unobserved data within a pattern relative to patterns with more completedata.3.1 Representation of missing data mechanismsUnder MCAR, all pattern-speci�c distributions are equal to that of the completers, giving�(1) = �(2) = � � � = �(K�1) = 0 and C(1) = C(2) = � � � = C(K�1) = I.To derive identifying restrictions under MAR, we use the following representation ofMAR given by Molenberghs et al. (1998). Subscripts i are suppressed for clarity. For anyt � 2 and for any k < t, dropout is an MAR process if and only iff(ytjy1; : : : ; yt�1; u = k) = f(ytjy1; : : : ; yt�1; u � t); (4)where k = 1; : : : ;K. Restriction (3.3) equates, between those who have dropped out priorto t and those who have not, the conditional distribution of yt given the past. When K = n,the yt in (3.3) is univariate; when K < n, yt is de�ned by grouping on dropout time and canbe multivariate. 8

Example 1: Two-component mixture of bivariate normals. Under MAR, (3.3) impliesthat the conditional distribution of the missing data given the observed data is the same inboth patterns. Multivariate normality implies that �(1)2 , the mean of yi2 among dropouts, isidenti�ed by �(1)2 = �(2)21 ��(2)11 ��1 ��(2)1 � �(1)1 � ; (5)a function of identi�able parameters. The corresponding variance parameters in �(1) areidenti�ed by �(1)21 = �(2)21 ��(2)11 ��1�(1)11 (6)�(1)22 = �(2)22 � �(2)21 ��(2)11 ��1 ��(2)12 � �(1)12 � ; (7)and (2) is used to �nd C.Example 2: K-component mixture of multivariate normals. For illustration, we considercase where K = n = 3 (the scenario in our example). Restriction (3.3) can be expanded asfollows: Pattern 1 (u = 1)f(y2jy1; u = 1) = f(y2jy1; u � 2) (8)f(y3jy1; y2; u = 1) = f(y3jy1; y2; u = 3) (9)Pattern 2 (u = 2)f(y3jy1; y2; u = 2) = f(y3jy1; y2; u = 3) (10)Each of (8) { (10) yields a set of identifying restrictions. For example, (8) identi�es mean9

and variance parameters for y2 among those with u = 1:�(1)21 = �?2�(1)11 �(2)21 ��(2)11 ��1 + �?3�(1)11 �(3)21 ��(3)11 ��1�(1)22 = �?2 ��(2)22 � �(2)21 ��(2)11 ��1 �(2)12 � + �?3 ��(3)22 � �(3)21 ��(3)11 ��1 �(3)12 �+ �(1)21 ��(1)11 ��1 �(1)12�(1)2 = �?2 ��(2)2 + �(2)21 ��(2)11 ��1 �(2)1 �+ �?3 ��(3)2 + �(3)21 ��(3)11 ��1 �(3)1 �� �(1)21 ��(1)11 ��1 �(1)1where �?k = �k=(�2+ �3) for k = 2; 3. Restrictions (9) and (10) work similarly to Example 1and utilize the appropriate partitions of the mean vectors and variance matrices. The locationand scale change parameters corresponding to MAR are found using (2). It is important tonote that MAR imposes no constraints on parameters identi�ed by the observed data. Inaddition, appropriate modi�cations to Equations (5)-(7) can be used to calibrate a selectionparameter (such as that proposed by Little and Wang (1996)) in terms of location and scalechanges.3.2 Selection mechanism implied by the PMMUnder normality within pattern, the selection mechanism can be represented as the polyto-mous regressionlog � pr(u = kjy)pr(u = Kjy)� = log� �(k)�(K)�+ 12 log j�(k)jj�(K)j � 12e(k)0 ��(k)��1 e(k) + 12e(K)0 ��(K)��1 e(K);where e(k) = �y � �(k)� and k = 1; : : : ;K � 1. This formulation gives the probability ofdropping out in pattern k | as opposed to completing the study | as a function of bothobserved and missing y data. The logit of the fully general selection mechanism is quadraticin y; the assumption of common variance across patterns implies that the logit is linear in10

y. As part of a sensitivity analysis, the dropout probability implied by the location-scalechanges can be plotted as a function of missing y to assess its plausibility.3.3 Sensitivity analyses based on the location-scale parameteriza-tionA major motivation for parameterizing the normal-mixture PMM using location and scalechanges is to exploit one of its important features, namely that one can keep the marginaldistribution of the observed data �xed (or nearly so) while examining a wide range of non-ignorable missing data mechanisms. Speci�cally, sensitivity analyses based on varying C and� can be constructed so as to place little or no restrictions on parameters identi�ed solely bythe observed data.This property of a PMM distinguishes it from selection models which assume a singlemultivariate normal distribution for the complete data (as in Diggle and Kenward, 1994),where the marginal distribution of the observed data depends explicitly on the assumedmissing data mechanism. Even under the correct complete data model, lack of �t to observeddata can be in uenced by the choice of missing data mechanism. This di�culty can beavoided by dropping parametric restrictions on the observed data (Scharfstein and Robins,1999).Sensitivity analyses for the PMM can be carried out either in a frequentist (maximumlikelihood) or Bayesian framework. In the example below, we take a Bayesian approach,comparing posterior inferences about a quantity of interest at a variety of pre-speci�edcombinations of ��(k); C(k). Because MCAR is rarely tenable in practical applications,values of C and � corresponding to MAR provide a reasonable starting point. We computethem using sample means and covariances together with the identifying restrictions impliedby (). The scope of a sensitivity analyses usually depends on the application, and shouldmake use of available subject-matter knowledge.11

3.4 Parameter estimation using Gibbs samplingWe consider a fully Bayesian implementation of our approach for reasons discussed below.Flat priors are used for the marginal mean and variance for each pattern. Given C and�, all unidenti�ed parameters are deterministic functions of identi�able model parameters,and are determined by the assumed missing data mechanism. To simplify our descriptionof the estimation routine, we introduce some additional notation. Let yobs correspond tothe observed data and ymis correspond to the missing data. Similarly, let �I represent allmean and variance parameters identi�ed by observed data in the general PMM speci�cationwith no restrictions (equation 2), and let �U represent the non-identi�ed parameters. Forexample, in the bivariate normal case, �I = n�(2)1 ; �(2)2 ;�(2)11 ;�(2)12 ;�(2)22 ; �(1)1 ;�(1)11 o ; and �U =n�(1)2 ;�(1)12 ;�(1)22 o :The Bayesian analysis is implemented using the following algorithm.1. Sample � from a Dirichlet distribution using a at prior on �.2. Sample �I for each pattern of missingness using a Gibbs sampler (sample �I jyobs). Thefull conditionals of the identi�ed mean parameters are (multivariate) normal and theidenti�ed submatrices of the pattern-speci�c variances are inverse Wishart.3. If MAR is assumed, compute C(1); : : : ; C(K�1) and �(1); : : : ; �(K�1) using the sampledvalue of �I and equations (5-??). If the missing data mechanism has been speci�ed bysetting the C's and �'s in advance, proceed to the next step.4. Compute �U for each pattern of missingness using the values from Steps 2 and 3. Thiscan be viewed as sampling from [�U jC; �; �I; yobs], where the conditional has a pointmass determined by the missing data mechanism.5. Given the complete set of parameters and the observed data, sample from the full12

conditional distribution of the missing data, i.e., ymisj�U ; �I ; yobs. Steps 1 { 5 yields asample from ymisjyobs.6. Using the Gibbs sampler, sample the mean and covariance parameters for the fullmodel, i.e., �U ; �Ijyobs; ymis. Combining Steps 5 and 6 gives the desired posterior dis-tribution, �;�jyobs conditional on the hypothesized missing data mechanism.7. To draw from the posterior marginal mean of y, use the sampled values of � from Step1 and Equation (3).Inferences from a model which �xes C and � in advance are necessarily conditioned onthe assumed missing data mechanism. An advantage to a Bayesian approach, as opposedto straight maximum likelihood, is the facility to incorporate uncertainty associated withthe \�lling in" process and with the unknown parameters. (In the frequentist setting, thisuncertainty can be handled by using multiple imputation). Although the Bayesian analysiscan be implemented using a combination of BUGS and S-Plus, we use code written inFortran.4 Analysis of growth hormone dataIn this section we analyze data from a randomized clinical trial of the e�ects of recombinanthuman growth hormone (rhGH) therapy for building and maintaining muscle strength inthe elderly (Kiel et al, 1998). In addition to the usual MAR analysis, we consider inferencesunder a wide range of NI mechanisms de�ned by between-pattern location and scale changes,and judge the impact of the missing data assumptions using a sensitivity analysis.The trial enrolled 161 subjects and randomized them to one of four treatments: placebo,growth hormone only, exercise plus placebo, and exercise plus growth hormone. Bothplacebo and growth hormone were administered via daily injections; those on the GH arm13

received 0.015 mg/kg rhGH. Participants had a variety of isokinetic muscle strength mea-sures recorded at baseline, six months and twelve months. For our illustration, we comparemean quadriceps strength at one year between subjects assigned to placebo plus exercise (E)growth hormone plus exercise (GH). Strength is measured as the maximum foot-pounds oftorque which can be exerted against resistance provided by a mechanical device (Kiel et al,1998).Table 1 gives sample means and standard deviations of quad strength using availabledata at each time. A fairly high dropout rate is seen on each arm (9/40 on E, 16/39 on GH),with most subjects dropping out between 0 and 6 months. Among completers, those on GHhave greater but more variable quadriceps strength.Two analyses are given below. In the �rst, we consider the simple case where only thethe baseline and 12-month measurements are used, which is intended to illustrate the basicconcepts of the sensitivity analyses in a simpli�ed setting. The second analysis uses all threemeasures and provides a wider range of sensitivity analyses. The number of non-identi�edparameters increases with the number of dropout patterns, making fully general sensitivityanalyses cumbersome. We give some recommendations for keeping things manageable.Our sensitivity analyses are informed by the summary in Table 1. In the exercise groupat baseline, for example, dropouts have similar mean and slightly higher variance than com-pleters (Table 1). However, on the GH arm, the baseline mean among dropouts is nearlyone standard deviation lower than among completers. In addition, the dropout rate is nearlytwice as high on the GH arm (42% vs. 23%).4.1 Analysis 1: bivariate pattern mixture modelLet yi = (yi1; yi2)0 represent the baseline and 12-month quad strength measures for subjecti, let xi 2 fG;Eg denote treatment group, and let ui 2 f1; 2g represent the binary indicator14

of dropout status (1 = dropout, 2 = completer). The bivariate normal PMM is(yi j xi; ui = k) � N(�(k)xi ;�(k)xi )pr(ui = 2 j xi) = �xiThe 12-month mean for dropouts, �(1)2xi, is identi�ed via �(2)2xi+�2xi; where �2xi is the di�erencebetween completers and dropouts at month 12. Similarly, the 12-month variance is identi�edthrough the scaling factor C22xi Our target of inference is pG, the posterior probability that12-month mean quad strength in the GH arm is greater than that in the E arm.The MAR assumption provides a reasonable starting point for examining the e�ects ofvarious missing data mechanisms. Under MAR, we have�E = ��4:6�3:4� CE = � 1:306 0:1160:210 1:085 ��G = ��20:7�21:5� CG = � 0:985 �0:004�0:012 0:995 � :The �rst element in each � vector is the di�erence in observed-data means between dropoutsand completers at baseline. If the correlation structure is identical across patterns (i.e.,the o�-diagonal elements of the C matrix are zero), then the diagonal elements of the Cmatrix are equal to the ratio of standard deviations. Here, the o�-diagonal elements areclose to zero, so the upper left entry is close to the ratio of observed standard deviations.The remaining parameters are identi�ed by the MAR restrictions given in Section 3.1. Thus,MAR implies a 3.4-unit di�erence in means at month 12 between dropouts and completerson the exercise arm, and a 21.5-unit di�erence on the GH arm. Under MAR, the di�erence in12-month mean quad strength is 5.8, with pG = :72. Under the unlikely MCAR assumption,the di�erence is 15:8 with pG = :95.To examine sensitivity to other missing data assumptions, we consider two scenarios. Inthe �rst, we assume MAR for the E arm and allow NI missingness on the GH arm. The NI15

mechanisms are constructed by varying �2G from {30 (dropouts much worse than completers)to 10 (dropouts slightly better than completers) and C22G from 0.5 to 2.0. In order to considervarious choices of the non-identi�ed variance parameter �(1)22 without simultaneously a�ectingcorrelations, we hold C12G = C21G = 0, which equates the correlations among dropoutsto those among completers. Under the second scenario, we allow NI missingness in bothtreatment arms. The ranges for � and C are chosen using Table 1 and the MAR analysis asa guide.A sensitivity analysis for the �rst scenario is illustrated in Figure 1, is a contour plotof pG as a function of �2G and C22G. The posterior probability is seen to depend moreon assumptions about the mean than about the variance. When �2G = 0 (i.e. completersand dropouts have identical 12-month means), pG is high, favoring GH regardless of theassumption about variance. If instead (and perhaps more realistically), dropouts are assumedto have lower 12-month quad strength than completers, evidence for GH becomes weaker.A di�erence of 20 ft-lbs gives pG between 0.7 and 0.8; a 30 ft-lb di�erence gives pG around0.6. Naturally the degree to which �2G a�ects the �nal inference depends on the proportionof dropouts, which is 0.42 for the GH arm.Figure 2 is a series of contour plots of pG as a function of �2G; �2E, C22G; and C22E,showing inference under treatment-speci�c NI mechanisms. These contour plots suggestthat the �nal inference is still largely dependent on the assumed di�erence between dropoutsand completers on the GH arm. Consider the case where C22G = C22E = 1. If �2G = �10,then pG remains between 0.90 and 0.95 as �2E ranges from 0 to {30. On the other hand, if�2E = �10 then pG ranges from .75 for �2G = �30 to nearly 1.0 for �2G = 0. The greatersensitivity of pG to di�erences between completers and dropouts in the GH is a function ofits higher proportion of dropouts; graphically, this is seen in steeper slopes of the contoursemanating from the horizontal (GH) axes. If both arms had equal proportions of dropouts,16

the contours would be more parallel to a 45-degree line. Figure 2 also demonstrates thatposterior inferences are sensitive to changes in the month-12 SD ratio between 0.5 and 2.0.There is greater sensitivity to C(1)22G due to a higher proportion of dropouts on the GH arm.How can these analyses be used to draw conclusions? Those who completed the trial onGH started with greater quad strength than dropouts (78 ft-lbs vs. 57 ft-lbs). One approachis to assume the 20-ft-lb di�erence observed at baseline between dropouts and completersin the GH arm as a persists throughout the study, e�ectively giving parallel pro�les acrosstime. This can be viewed as a `best case scenario' for dropouts, and actually allows fora 10-ft-lb increase. Across the NI mechanisms represented in Figure 2, we see that pG ishighest (roughly 0.75 to 1.0) when �2E = �30. Thus, unless the di�erence between dropoutsand completers in the E group is very large, the evidence in favor of GH is not strong.4.2 Analysis 2: multivariate PMMwith multiple dropout patternsA multivariate PMM with multiple dropout times allows a more wide-ranging examination ofmodeling assumptions. Let yi = (yi1; yi2; yi3)0 represent the vector of quad strength measuresat months 0, 6, and 12, and let ui 2 f1; 2; 3g represent dropout pattern. In our example,which has monotone dropout with no empty patterns, ui corresponds to the number ofobserved quad strength measures. The PMM is similar to the bivariate PMM, except thatui is multinomial with �(k)xi = pr(ui = k j xi) and Pk �(k)xi = 1 for k = 1; 2; 3, xi 2 fG, Eg.For the three-dimensional PMM, for each treatment group, there are three nonidenti�edparameters in the mean and eight in the covariance matrix. Re-parameterizing patternspeci�c means in terms of their di�erences, we have the following. The vector�(2)x = ��(2)1x ; �(2)2x ; �(2)3x �0 = �(3)x � �(2)xrepresents di�erences in mean between completers (ui = 3) and month-6 dropouts (ui =2), where only �(2)1x and �(2)2x are identi�ed by observed data; specifying �(2)3x identi�es �(2)3x .17

Similarly, �(1)x = ��(1)1x ; �(1)2x ; �(1)3x �0 = �(2)x � �(1)x ;is the di�erence in means between those in dropout patterns 2 and 1. Only �(1)1x is identi�edby observed data; �(1)2x and �(1)3x must be speci�ed. The scaling matrices C(1)x and C(2)x aresimilarly de�ned.Our analysis under the multivariate model follows along the same lines as the bivariatemodel. First, we assume MAR in the exercise group and examine the e�ects of variousNI mechanisms in the GH group. This aim of this analysis is to identify pattern-speci�ce�ects on posterior inferences. We then examine inferences under group-speci�c NI missing-ness, where the NI missingness is induced by assuming pattern-speci�c di�erences betweendropouts and completers at month 12.Analysis 2a: MAR for E group, NI missingness for GH group. NI mechanisms in the GHgroup are generated via pattern-speci�c di�erences in month 12 mean and variance. Speci�-cally, we characterize month-12 di�erences between completers and dropouts by varying �(2)3Gbetween {30 and 10, and characterize month-12 di�erences in the two dropout patterns byvarying �(1)3G between {10 and 10. The appropriate elements of the C matrices are used tovary between-pattern ratio of month-12 SD from 0.5 to 2.0. Non-identi�ed parameters cor-responding to pattern di�erences at month 6 are held at their MAR values (using Equation(8)), which preserves the values of the identi�ed parameters, and C(1)3jE, C(2)3jE, C(1)j3E, and C(2)j3E,j = 1; 2 are set to zero (i.e., non-completers have same correlation structure as completersfor the 12 month values).Figure 3 shows contour plots of pG at various combinations of the � and C parameters.If MAR is assumed for the E group, then most situations leading to superiority of GH(i.e. pG > :90) correspond to early dropouts performing better than later dropouts andcompleters. For example, if 12-month mean quad strength is equal among the two dropout18

patterns (�(1)3G = 0), then pG > :90 only under the unlikely situation where quad strengthamong dropouts is greater than or equal to that among completers. In the GH arm, thosewho remain in the trial through month 6 seem to bene�t from the partial therapy (Table1) via a 10-point increase. Under the assumption that the 10-point di�erential continuesthrough month 12, none of the scenarios on the contour plots indicate superiority for GH.The nearly horizontal slope of the contour plot indicates that pG is less sensitive to di�erencesbetween patterns among dropouts than between dropouts and completers. The sensitivityof inferences to the variances can also be seen clearly from Figure 3 with pG decreasing asthe variances increase.Analysis 2b: NI mechanisms in both groups. We induce NI missingness through di�er-ences between dropouts and completers at month 12. Non-identi�ed parameters in pattern 1are equated to their pattern-2 counterparts. Operationally, this means we need only to vary�(2)3x and C(2)33x because the other non-identi�ed parameters are handled as in Analysis 2a.Contour plots of pG are given in Figure 4, and again indicate weak evidence in favorof GH. The conclusions are synonymous with Analysis 1; that is, to achieve a signi�cantdi�erence, the non-completers in the E group much do much worse than the completers (byabout 1 standard deviation) while the non-completers in the GH group must do at leastas well as the completers. To illustrate using a speci�c scenario, assume a 15-ft-lb baselinemean di�erence between completers and dropouts in the E arm giving �(2)3E = �15 (this is themaximumdi�erence observed in Table 2 and a more severe di�erence than implied by MAR).The contour plots indicate that for pG > 0:9, completers in the GH arm can be superior todropouts by at most 10-15 ft-lbs. This would require roughly a 15 ft-lb improvement amongmonth 0 dropouts and a 5 ft-lb improvement among month 6 dropouts, both presumablywithout the bene�t of rhGH. As the relative variability of dropouts to completers increases,the �'s must increase in magnitude to maintain high values of pG.19

4.3 Estimates of mean quad strengthTable 2 lists, for various location-scale speci�cations, posterior means and standard devi-ations for group-speci�c 12-month quad strengths and the group di�erence. Under mostcon�gurations, mean quad strength on GH is higher; however, for the di�erence to be sta-tistically signi�cant, the dropouts in the E group must do much worse than the completersand dropouts on GH must rise to the level of completers. The observed data in Table 1 sug-gests this is unlikely. Table 2 also shows evidence of increased uncertainty associated withthe treatment di�erence as the variances among dropouts increases relative to that amongcompleters.The treatment di�erences listed in Table 2 can be viewed as intention to treat comparisonsbecause they quantify di�erences which would have been observed had everyone completedthe trial. A di�culty with intention to treat is that it represents method e�ectiveness onlyunder a particular compliance pattern (namely the one for this trial), and compliance isusually unobserved for dropouts. Assumptions about compliance can be used to inform thechoices of the location scale parameters.4.4 Missing data mechanismFigure 5 shows the graphical representation of the missing data mechanism on the G armunder four di�erent missing data assumptions in the bivariate PMM (Analysis 1). Each plotshows the probability of dropout as a function of 12-month quad strength (missing for somesubjects), conditional on various values of baseline quad strength (observed for all subjects).We condition on baseline quad strength values 30, 60, 90, and 120 ft-lbs. The �rst plotgives dropout probability under MCAR, which is a constant function of both baseline and12-month outcome. The second corresponds to MAR, illustrating that dropout probabilityis constant conditional on baseline (observed) outcome. The third and fourth plots show two20

di�erent NI mechanisms based on the di�erence in month-12 means between dropouts andcompleters; the NI mechanisms correspond to �G = �30 and �G = �10, respectively (recallunder MAR, �G = �21:5). Although both mechanisms imply that dropouts are worse o�than completers at month 12, the dropout mechanism when �G = �10 implies that dropoutis an increasing function of month-12 outcome conditional on baseline. This plot supportsthe notion that �G > �10 is unlikely.5 DiscussionWe have described a parameterization of the pattern mixture model for continuous datawhich makes explicit all non-identi�ed parameters in both the mean and variance. The non-identi�ed parameters for patterns with incomplete data are functions of the fully identi�edparameters in the completers pattern through additive terms (�) for the mean and multi-plicative terms (C) for the variances and correlations. The model is fully identi�ed by �xing� and C, which characterize di�erences between completers and dropouts. We have shownthat MCAR and MAR mechanisms can be represented in terms of these parameters, provid-ing a useful alternative to selection-model representations. The proposed parameterizationfacilitates sensitivity analyses based on distributional di�erences between patterns, whichappeals to the structure of pattern mixture models.5.1 A natural parameterization for sensitivity analysesAn important advantage of sensitivity analyses based on the new parameterization | asopposed to those based on selection model restrictions | is that one can consider a widerange of missing data mechanisms without a�ecting the marginal distribution of the observeddata.Sensitivity analyses based on selection model restrictions appeal to Rubin's (1976) char-21

acterization of missing data mechanisms, and in simple cases (e.g. one dropout time) it ispossible to characterize dropout using a single selection parameter. A limitation of this ap-proach is that mean and variance parameters are simultaneously constrained in ways whichare not always obvious.Our parameterization allows a fully general sensitivity analysis because it explicitly char-acterizes all possible models, nonignorable and otherwise. Further, it is consistent with thebasic structure of the pattern mixture model. That is, the PMM speci�es di�erent distri-butions for each pattern; the � and C terms characterize di�erences between the patterns;and the analyst uses � and C to construct sensitivity analyses based on assumed di�erencesbetween dropouts and completers. The di�erences can be speci�ed separately for the meanand variance while holding the correlations at MAR values or constant across pattern. Fur-ther work will explore the role of the correlations in NI speci�cations; this is related to theapproach taken by Little and Wang (1996), who specify regressions of measures taken laterin the study on those taken earlier. Clearly the analyst needs to consider a restricted andsensible range of sensitivity analyses which are consistent with the application.5.2 Di�culties with sensitivity analyses and possibilities for a sin-gle analysisThere is a conundrum inherent in drawing inference under non-ignorable missingness becausethe general PMM is underidenti�ed. A sensitivity analysis looks at inferences for variousvalues of the nonidenti�ed parameters; in the absence of prior information about theseparameters, the sensitivity analysis can be viewed as an in�nite number of (correlated)multiple comparisons.In many settings, the statistician turns to the subject matter expert for advice on aplausible range of values for the non-identi�ed parameters. In a Bayesian framework, onecan plausibly ask the expert to assign weights as well; this information can be eventually22

transformed to a prior which identi�es the relevant parameters. Prior elicitation is nontrivial(e.g., see Chaloner et al., 1993), but represents a plausible step toward drawing scienti�callyreasonable conclusions in missing data settings. This is especially true in clinical trials insettings where the characteristics of study dropouts can either be directly observed (suchas those involving psychiatric inpatients) or are somewhat well understood as a result ofconsiderable collective experience (such as in AIDS or cancer clinical trials).Our parameterization provides a natural framework for using informative prior distri-butions, which would be placed on parameters which characterize di�erences in responsebetween patterns. Incorporation of informative priors would require only minor modi�ca-tions to our Gibbs sampler described in Section 3.4.5.3 Extensions and further researchPattern mixture models can, in principle, be applied to any regression setting (Eckholmand Skinner, 1998). Work in progress addresses formulations of pattern mixture models forbinary data and more generally, for distributions which are exponential families.We have assumed monotone missingness, but in general this is not necessary. For non-monotone missingness, one can group together subjects on their dropout time only. Then,conditional on dropout time, the intermittent missingness is assumed to be MAR (a typicallyreasonable assumption). This modi�cation (termed pattern-set mixture modeling by Little,1993) would require only one extra step in the Gibbs sampler.Another common issue in analyzing incomplete longitudinal data is handling di�erenttypes of dropouts or non-completers. One example is an interim analysis in which subjectswith incomplete data can be classi�ed as either dropout or late entrant. Another exampleis a study where some subjects drop out due to side e�ects (perhaps not outcome related)and others due to lack of e�cacy (probably outcome related). Although this seems more23

naturally handled in a selection model, one can treat non-informative dropout time as acensoring time, which introduces very little complication to parameter estimation for thepattern mixture model (Hogan and Laird, 1997a).AcknowledgmentThe authors thank Dr. David MacLean of Memorial Hospital of Rhode Island and P�zer forproviding the data from the growth hormone study. The data were collected under a grantfrom the NIH.ReferencesChaloner, K., Church, T., Louis, T.A., Matts, J.P. (1993) Graphical elicitation of a priordistribution for a clinical trial, The Statistician, 42:341-353.Diggle, P. and Kenward, M.G. (1994) Informative drop-out in longitudinal data analysis,Applied Statistics, 43: 49-93.Eckholm, A. and Skinner C. (1998) The Muscatine children's obesity data reanalysed usingpattern mixture models, Applied Statistics, 47:251-263.Follmann, D. and Wu, M. (1995) An approximate generalized linear model with randome�ects for informative missing data, Biometrics, 51:151-168.Hogan JW, Laird NM (1997a). Mixture models for the joint distribution of repeatedmeasures and event times. Statistics in Medicine, 16:239-257Hogan JW, Laird NM (1997b). Model based approaches to analyzing incomplete longitu-dinal and failure time data. Statistics in Medicine, 16:259-272.24

Kiel DP, Puhl J, Rosen CJ, Berg K, Murphy JB, MacLean DB (1998). Lack of associationbetween insulin-like growth factor I and body composition, muscle strength, physicalperformance or self reported mobility among older persons with functional limitations.Journal of the American Geriatrics Society, 46:822-828.Laird NM (1988). Missing data in longitudinal studies. Statistics in Medicine 7:305-315.Little, R.J.A. (1993) Pattern-mixture models for multivariate incomplete data, Journal ofthe American Statistical Association, 88: 125-134.Little, R.J.A. (1994) A class of pattern mixturemodels for normal incomplete data, Biometrika,81:471-483.Little, R.J.A. (1995) Modelling the dropout mechanism in repeated-measures studies, Jour-nal of the American Statistical Association, 90:1112-1121.Little, R.J.A., and Rubin, D.B. (1987) 'Statistical Analysis with missing data', New York:John Wiley & Sons.Little, R.J.A. and Wang, Y. (1996) Pattern-mixture models for multivariate incompletedata with covariates, Biometrics, 52: 98-111.Molenberghs, G., Michiels, B., Kenward, M.G., Diggle, P.J. (1998) Monotone missing dataand pattern mixture models. Statistical Neerlandica, To appear.Robins, J. and Rotnitzky, A. (1995). Analysis of semiparametric regression models forrepeated outcomes in the presence of missing data. Journal of the American StatisticalAssociation 90, 106-121.Rubin, D.B. (1976) Inference and missing data, Biometrika, 63: 581-590.25

Scharfstein DO, Robins JM (1999). Adjusting for non-ignorable non-response using semi-parametric non-response models with time-dependent covariates. Journal of the Amer-ican Statistical Association, to appear.Wu, M.C., Hunsberger, S. and Zucker, D. (1994) Testing for di�erences in changes in thepresence of censoring: parametric and non-parametric methods, Statistics in Medicine,13:635-646.Wu, M.C. and Carroll, R.J. (1988) Estimation and comparison of changes in the presence ofinformative right censoring by modeling the censoring process, Biometrics, 44: 175-188.Yao Q, Wei LJ, Hogan JW (1998). Analysis of incomplete repeated measurements withdependent censoring times. Biometrika 85:139-150.

26

Table 1: Sample means (standard deviations), by treatment group (E = exercise; G = rhGHplus exercise), and dropout pattern. MonthTreatment k nk �̂k 0 6 12E 1 7 0.18 64.5 (32.1)2 2 0.05 86.6 (51.4) 86.3 (51.1)3 31 0.78 64.8 (23.8) 81.3 (25.1) 72.5 (21.0)All 40 65.9 (26.2) 81.6 (26.0) 72.5 (21.0)GH 1 12 0.32 57.8 (25.6)2 4 0.11 56.7 (15.2) 67.9 (25.8)3 22 0.58 78.1 (23.7) 90.2 (31.9) 88.4 (32.1)All 38 69.7 (25.0) 86.8 (31.6) 88.4 (32.1)27

Table 2: Estimates of the posterior mean (posterior standard deviation) for the response at12 months on rhGH plus exercise (GH) and exercise only (E) arms, for various non-ignorablemissing data mechanisms. In each model, �(2)3xi and C(2)33xi are varied while holding �(1)3xi = 0and C(1)33xi = 1 (xi 2 fG;Eg). For additional details, see analysis 2b. in the text.Missing data mechanism Posterior Mean (SD) of Week-12 Quad Strength (ft-lbs)�(2)3G C(2)33G �(2)3E C(2)33E rhGH + Exercise Exercise Di�erenceMAR MAR 82.5 (8.7) 72.2 (6.7) 10.4 (10.6)0 1 0 1 88.6 (8.6) 72.6 (5.6) 15.9 (10.0){15 1 0 1 82.1 (9.1) 72.7 (6.0) 9.5 (11.0){30 1 0 1 76.1 (9.2) 72.5 (6.0) 3.6 (11.1)0 1 {15 1 88.3 (8.7) 69.3 (6.0) 18.9 (10.5){15 1 {15 1 81.7 (9.0) 68.9 (6.0) 12.8 (11.0){30 1 {15 1 75.7 (9.0) 69.0 (6.2) 6.6 (10.8)0 1 {15 2 88.4 (9.1) 69.1 (7.0) 19.3 (11.3){15 1 {15 2 82.3 (9.5) 69.1 (7.1) 13.3 (11.8){30 1 {15 2 75.6 (9.1) 69.2 (6.9) 6.4 (11.3)0 2 {15 2 88.2 (12.1) 68.9 (6.8) 19.4 (13.9){15 2 {15 2 82.0 (11.8) 69.2 (7.1) 12.8 (13.9){30 2 {15 2 76.0 (12.6) 69.3 (7.0) 6.7 (14.2)28

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Figure 5: Selection mechanisms: The probability of dropout as a function of Month 12 quadstrength for MCAR, MAR, and two NI models (see Section 4.4 for further details) for fourbaseline quad strengths (ft-lbs): � � � 30, - - - - 60, { { { 90, | | | 120.33