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The Prevention and Treatment of Missing Data in Clinical Trials 4 Drawing Inferences from Incomplete Data In this chapter, we review and comment on several approaches for drawing inferences from incomplete data. A substantial literature on this topic has developed over the last 30 years, and the range of approaches to modeling and inference is extremely broad. We make no attempt here to summarize that entire literature; rather, we focus on those methods that are most directly relevant to the design and analysis of regulatory clinical trials. We begin by presenting a set of principles for drawing inference from incomplete data. A major theme that we reiterate throughout the chapter is that inference from incomplete data relies on subjective, untestable assumptions about the distribution of missing values. On its face, this statement seems obvious. However, for a number of commonly used methods, users are not always aware of the assumptions that underlie the methods and the results drawn from applying them. This lack of awareness is particularly true of single imputation methods—such as last or baseline observation carried forward (LOCF or BOCF) and random effects (mixed effects) regression models—that rely on strong parametric assumptions. In the second section of the chapter, we introduce a set of notation that is used throughout (and in Chapter 5). The third section summarizes the assumptions that underlie inference from incomplete data (missing completely at random, missing at random, etc.). The remaining sections describe commonly-used methods of analysis and offer comments and recommendations about their use in practice. In some cases, we offer recommendations for further research and investigation. For both this chapter and the next, it is important to note the role of software. None of the techniques for either the primary analysis of clini-
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The Prevention and Treatment of Missing Data in Clinical Trials cal trial data or for the subsequent sensitivity analysis that are described in the next chapter can be widely used, either at the U.S. Food and Drug Administration (FDA) or by trial sponsors, unless they are made available in one or more of the standard statistical software packages. It is beyond the scope of this report to describe and review specific software packages or routines. Many of the commonly used commercial and open-source packages used in the analysis of trials for the regulatory setting (SAS, SPSS, Stata, and R) allow for the analysis of incomplete data, using methods such as direct likelihood, Bayesian analysis, generalized estimating equations, inverse probability weighting, and multiple imputation. Statistical software is evolving at a rapid pace to keep up with new developments in methodology and to implement proven methods. However, although progress is being made, the current suite of available tools remain lacking regarding augmented inverse probability weighting (IPW), missing not at random (MNAR) models, and analysis of the sensitivity to assumptions concerning the mechanism for missing outcome data. Given the urgency of the greater application of MNAR models and sensitivity analysis, we encourage the development and release of software tools to address these deficiencies. We again emphasize the importance of understanding and communicating the assumptions underlying analyses that are implemented in whatever software package is being used to draw inference about treatment effects. In most cases, communication of this information will necessitate referring to technical documentation for a specific analysis routine or procedure. PRINCIPLES There is no universal method for handling incomplete data in a clinical trial. Each trial has its own set of design and measurement characteristics. There is, however, a set of six principles that can be applied in a wide variety of settings. First, it needs to be determined whether missingness of a particular value hides a true underlying value that is meaningful for analysis. This may seem obvious but is not always the case. For example, consider a longitudinal analysis of CD4 counts in a clinical trial for AIDS. For subjects who leave the study because they move to a different location, it makes sense to consider the CD4 counts that would have been recorded if they had remained in the study. For subjects who die during the course of the study, it is less clear whether it is reasonable to consider CD4 counts after time of death as missing values. Second, the analysis must be formulated to draw inference about an appropriate and well-defined causal estimand (see Chapter 2). The causal estimand should be defined in terms of the full data (i.e., the data that were
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The Prevention and Treatment of Missing Data in Clinical Trials intended to be collected). It is important to distinguish between the estimand and the method of estimation, the latter of which may vary depending on assumptions. Third, reasons for missing data must be documented as much as possible. This includes full and detailed documentation for each individual of the reasons for missing records or missing observations. Knowing the reason for missingness permits formulation of sensible assumptions about observations that are missing, including whether those observations are well defined. Fourth, the trial designers should decide on a primary set of assumptions about the missing data mechanism. Those primary assumptions then serve as an anchor point for the sensitivity analyses. In many cases, the primary assumptions can be missing at random (MAR) (see Chapter 1). Assumptions about the missing data mechanism must be transparent and accessible to clinicians. Fifth, the trial sponsors should conduct a statistically valid analysis under the primary missing data assumptions. If the assumptions hold, a statistically valid analysis yields consistent estimates, and standard errors and confidence intervals account for both sampling variability and for the added uncertainty associated with missing observations. Sixth, the analysts should assess the robustness of the treatment effect inferences by conducting a sensitivity analysis. The sensitivity analysis should relate treatment effect inferences to one or more parameters that capture departures from the primary missing data assumption (e.g., MAR). Other departures from standard assumptions should also be examined, such as sensitivity to outliers. NOTATION Throughout this and the next chapter, we use the following conventions. Let X represent treatment indicators and baseline (i.e., pretreatment) covariates that are fully observed and conditioned on in the primary statistical analysis (such as study center and stratification variables). Another way to characterize X is as the design variables that would be adjusted for or conditioned on in the final analysis. Let Y denote the primary outcome variable, which may be a single outcome, a vector of repeated measurements, or a time to event. Auxiliary variables are denoted by V; these variables are distinct from design variables X and may represent individual-level characteristics (either pre- or posttreatment) that aid in drawing inference from incomplete response data. Information on compliance or side effects of treatments that may be useful for modeling the missing data but are not included in the primary analytic model may be included in V. (We note that the collection and use of all available covariate information that is predic-
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The Prevention and Treatment of Missing Data in Clinical Trials tive of the outcome in the full data model, and the occurrence of missing outcome data in the missing data model, is important and can dramatically improve the associated inference.) In the absence of missing data, let Z denote the values of (V,Y) for an individual participant. For simplicity, we assume throughout that observations on (V,Y) are independent within levels of X. To distinguish between missing and observed data, let M denote the indicator of whether Y is missing. In repeated measures studies, we include a subscript for repeated measures. That is, if the intended outcome measures are Y = (Y1, Y2,…,YK), the corresponding missingness indicators are M = (M1,M2,…,MK), where Mj = 1 if Yj is missing, and Mj = 0 if it is observed. We generally will assume that Y and V have the same missing data pattern, though in practice this restriction can be relaxed. In many situations, missing values can be denoted by a single value, such as M = 1; in other settings, it may be useful to allow more than one missing-value code to indicate different types of missing data, such as M = 1 for lack of efficacy, M = 2 for inability to tolerate a drug because of side effects, M = 3 for a missed clinic visit, and so on. This notation allows for different modeling assumptions for the different causes of missing data. ASSUMPTIONS ABOUT MISSING DATA AND MISSING DATA MECHANISMS The general missing data taxonomy described in this section is fully presented in Rubin (1976) and Little and Rubin (2002). Elaboration on the sequential versions of these for longitudinal data can be found in Robins et al. (1995) and Scharfstein et al. (1999). Discussion of the more general notion of coarsening can be found in Heitjan (1993) and Tsiatis (2006). Missing Data Patterns and Missing Data Mechanisms It is useful to distinguish the pattern of the missing data from the missing data mechanism. The pattern simply defines which values in the data set are observed and which are missing, as described for an individual by the vector of indicators M = (M1,…,MK). Some methods for handling missing data apply to any pattern of missing data; other methods assume a special pattern. A simple example of a special pattern is univariate missing data, where missingness is confined to a single variable. Another special pattern is monotone missing data, where the variables can be arranged so that Yj+1 is missing for all cases where Yj is missing. This pattern commonly arises in longitudinal data, when the sole cause of missingness is attrition or dropouts, and there are no intermittently missing values.
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The Prevention and Treatment of Missing Data in Clinical Trials The missing data mechanism relates to why values are missing and the connection of those reasons with treatment outcomes. The missing data mechanism can be represented in terms of the conditional distribution [M | X,V,Y]1 for the missing data indicators given the values of the study variables that were intended to be collected. To emphasize that this distribution may depend both on observed and missing values of V and Y, this is sometimes written as [M | X,Vobs,Vmis,Yobs,Ymis]. Missing Completely at Random Missing data are missing completely at random (MCAR) if missingness does not depend on values of the covariates, auxiliary and outcome variables (X,V,Y). That is, (1) MCAR is generally a very strong assumption, unlikely to hold in many clinical trials. Situations in which MCAR might be plausible include administrative censoring, when outcomes are censored because a study is terminated at a planned date, and the outcome has not yet occurred for late accruals; and designed missing data, when expensive or onerous measurements are recorded only for a random subsample of participants. A closely related concept is conditional MCAR, which allows for the independence of the missing values, but is conditional on covariates X. Finally, it is useful to mention that MCAR is unique in that one can test whether the missing outcomes are MCAR if they are at least missing at random, which is discussed below. Missing at Random A more realistic condition than MCAR for many studies is MAR, which requires that missingness is independent of missing responses Ymis and Vmis, conditionally on observed responses (Yobs, Vobs) and covariates X. That is, (2) If Y and V are considered to be random variables with distributions based on a model, then one can show that condition (2) is equivalent to (3) 1 The notation [a | b, c] (e.g., [M | X,V,Y]) is used to denote the conditional distribution of a given the joint distribution of b and c.
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The Prevention and Treatment of Missing Data in Clinical Trials which implies that the predictive distribution of the missing variables given the observed variables does not depend on the pattern of missing values. This version of MAR is relevant from an analysis perspective because it characterizes the predictive distribution of the missing values, which is the basis for principled methods of imputation. As we describe below, many standard analysis methods for incomplete data operate under the MAR assumption. It is therefore imperative that both the MAR assumption and the assumptions underlying the full data model (e.g., multivariate normality) be thoroughly justified before results from these models can be considered valid for treatment comparisons. In general: (a) even under MAR, different assumptions about the full data model will lead to different predictive distributions; (b) with incomplete data, assumptions about both the missing data mechanism and the full data model are unverifiable from the data; and (c) nevertheless, inference and therefore decisions about treatment effect often crucially depend upon them. MAR for Monotone Missing Data Patterns With longitudinal repeated measures, and even for event time outcomes, the MAR assumption is not always intuitive for a general pattern of missing values. However, it has a simple interpretation in the case of monotone missing data, such as that caused by dropouts. Suppose the data intended to be collected comprise repeated measures on an outcome Y, denoted by Y1,…,YK. Let Mj = 1 if Yj is missing, and let Mj = 0 if Yj is observed. Under monotone missingness, if observation j is missing (Mj = 1), then all subsequent observations also are missing (Mj+1 = … = MK = 1). At any given time j, let denote the history of measurements up to but not including time j, and let denote the future measurements scheduled, including and after time j. At time j, the predictive distribution of future values given the observed history is denoted by . The MAR condition holds if predictions of future measurements for those who drop out at time j are equivalent in distribution to predictions for those who have observed data at and after time j. Formally, MAR is equivalent to (4) Hence, under MAR, missing values at time j and beyond can be predicted sequentially from the histories of participants still in the study at time j. MAR for monotone missing data patterns also can be written in terms of the probability of dropouts at each measurement occasion. At time j,
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The Prevention and Treatment of Missing Data in Clinical Trials the dropout probability is P(Mj = 1 | Mj–1 = 0). In general, this probability can depend on any aspect of the observations intended to be collected. MAR states that the dropout probability can only depend on observed data history, (5) This representation shows that one can think of the MAR assumption as a sequentially random dropout process, where the decision to drop out at time j is like the flip of a coin, with probability of ‘heads’ (dropout) depending on the measurements recorded through time j – 1. Both (4) and (5) can be generalized by allowing the past measurements to include auxiliary covariates. Specifically, let denote the observed history of both outcomes and auxiliaries. Then MAR can be restated by replacing with in (4) and (5). In fact, the MAR assumptions (4, 5) change depending on the set of auxiliary variables V included in the analysis. The validity of the MAR assumption can be improved by measuring and including auxiliary variables that are predictive of whether the outcome variables are missing and predictive of the values of the missing variables. Missing Not at Random MAR will fail to hold if missingness or dropout depends on the values of missing variables after conditioning on the observed variables. When MAR fails to hold, missing data are said to be MNAR. For a monotone missing data pattern, missingness will be MNAR if there exists, for any j, at least one value of for which (6) or equivalently, there exists, for any j, at least one value of , such that (7) For (6), the consequence of MNAR is that the prediction of future observations for those who drop out cannot be reliably predicted using data observed prior to dropping out; or, that the distribution  differs between those who do and do not drop out at time j. Because these differences cannot be estimated from the observed data, they are entirely assumption driven. This is the central problem of missing data analysis in clinical trials.
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The Prevention and Treatment of Missing Data in Clinical Trials Example: Hypertension Trial with Planned and Unplanned Missing Data Murray and Findlay (1988) describe data from a large multicenter trial of metopropol and ketanserin, two antihypertensive agents for patients with mild to moderate hypertension, with diastolic blood pressure as the outcome measure of interest. The double-blind treatment phase lasted 12 weeks, with clinic visits scheduled for weeks 0, 2, 4, 8, and 12. The protocol stated that patients with diastolic blood pressure exceeding 110 mmHg at either the 4- or 8-week visit should “jump” to an open follow-up phase—a form of planned dropout. In total, 39 of the 218 metopropol patients and 55 of the 211 ketanserin patients jumped to open follow-up. In addition, 17 metopropol patients and 20 ketanserin patients had missing data for other reasons, including side effects. Analyses of the observed data clearly showed that those with missing blood pressure readings differed systematically from the patients who remained in the study, as would be predicted by the protocol for jumping to the open phase. This example provides an illustration of the importance of defining what is represented by a missing outcome. For the participants who were removed from protocol, it is possible to treat the missing values as values that would be observed had the individuals remained on treatment. The mechanism for those with missing values is MAR because missing outcomes resulted from the value of a recorded intermediate outcome variable for blood pressure, and are therefore a function of an observed value. Summary Inferences from incomplete data, whether model-based or not, rely on assumptions—known as missing data mechanisms—that cannot be tested from the observed data. A formal taxonomy exists for classifying missing data mechanisms, including for longitudinal and event history data. The mechanisms can be classified as MCAR, MAR, and MNAR. Missing data mechanisms describe the relationship between the missing data indicator(s) M, the full outcome data Y = (Yobs,Ymis), design variables X, and auxiliary covariates V. Traditionally, these assumptions characterize restrictions on the distribution of M given (Yobs,Ymis,X,V). Each has an equivalent representation in terms of the predictive distribution of missing responses, namely Ymis given (M,Yobs,X,V). COMMONLY USED ANALYTIC METHODS UNDER MAR Three common approaches to the analysis of missing data can be distinguished: (1) discarding incomplete cases and analyzing the remainder
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The Prevention and Treatment of Missing Data in Clinical Trials (complete-case analysis); (2) imputing or filling in the missing values and then analyzing the filled-in data; and (3) analyzing the incomplete data by a method that does not require a complete (i.e., a rectangular) data set. Examples of (3) include likelihood-based methods, such as maximum likelihood (ML), restricted ML, and Bayesian methods; moment-based methods, such as generalized estimating equations and their variants; and semiparametric models for survival data, such as the Cox proportional hazards model. Multiple imputation (Rubin, 1987; Little and Rubin, 2002), an extension of single imputation that allows uncertainty in the imputations to be reflected appropriately in the analysis, is closely related to Bayesian methods (discussed later in this chapter). Deletion of Cases with Missing Data A simple approach to missing data is complete-case analysis, also known as listwise deletion, in which incomplete cases are discarded and standard analysis methods are applied to the complete cases. In many statistical packages, it is the default analysis. Although it is possible to list conditions under which an analysis of complete cases provides a valid inference (essentially, conditional MCAR), this method is generally inappropriate for a regulatory setting. When missingness is in the outcome, the MAR assumption is generally weaker and can reduce bias from deviations from MCAR by making use of the information from incomplete data. Furthermore, when missingness is appreciable, rejection of incomplete cases will involve a substantial waste of information and increase the potential for significant bias.2 In addition, if data are not collected after withdrawal from treatment, then the MAR assumption relies only on information accumulated while subjects are on treatment. Hence, any method that relies on MAR is estimating the mean under the condition that everyone had remained on treatment. This generally will not provide a valid estimator of the intention-to-treat effect. On the other hand, if data are collected after withdrawal from treatment, this information can be used either within inverse probability weighting (IPW) or in an imputation context to estimate an intention-to-treat effect under MAR (Hogan and Liard, 1996). It is for this 2 When data are not MCAR, the bias of complete-case analysis depends on the degree of deviation from MCAR, the amount of missing data, and the specifics of the analysis. In particular, the bias in estimating the mean of a variable is the difference in the means for complete and incomplete cases multiplied by the fraction of incomplete cases. Thus, the potential for bias increases with the fraction of missing data. With respect to regression models, complete-case analysis yields valid inferences in regression if the model is correctly specified and missingness depends on the predictor variables, observed or missing, but not on the outcome. (For details, see Little and Rubin .)
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The Prevention and Treatment of Missing Data in Clinical Trials reason that we emphatically recommend aggressive collection of outcome data after individuals withdraw from treatment. Inverse Probability Weighting Univariate Outcome When data are MAR but not MCAR, a modification of complete-case analysis is to assign a sampling weight to the complete cases. This tends to reduce bias, to the extent that the probability of being observed is a function of the other measured variables. Consider the simple case in which the intended outcome is Y, the design variables are X, and some auxiliary variables V are available. As usual, M = 1 indicates that Y is missing. To describe IPW, it is useful to introduce a response indicator, R = 1 – M, such that R = 1 when Y is observed and R = 0 when it is missing. An IPW estimator for the mean of Y can be computed as follows: Specify and fit a model for π(X,V,θ) = Pθ(R = 1 | X,V), for example using logistic regression. Estimate the mean of Y using the weighted average (8) that is, the average of the observed Y weighted inversely by the probability of being observed. Standard error estimators can be computed analytically or by bootstrap methods. (For details on the bootstrap estimator of variance, see Efron and Tibshirani, 1993.) For large samples, this method properly adjusts for bias when the data are MAR, provided the model for π(X,V,θ) is correctly specified. In finite samples, the method can yield mean estimates that have high variance when some individual-specific weights are high (i.e., when π is close to zero). An alternative is to create strata based on the predicted probability of being complete and then weight respondents by the inverse of the response rate within these strata. Strata can be chosen to limit the size of the weights and hence control variance. In addition to the MAR assumption, the IPW method requires two other key assumptions: (1) there are no covariate profiles (X,V) within which Y cannot be observed and (2) the support of the missing data distri-
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The Prevention and Treatment of Missing Data in Clinical Trials bution is the same as that for the observed data distribution. Technically, (1) stipulates that P(R = 1 | X,V) > 0 for all possible realizations of (X,V). A potential restriction imposed by (2) is that individual missing values cannot be imputed outside the range of observed values. IPW Regression for Repeated Measures With repeated measures, a convenient way to estimate the treatment effect is through a regression model for the mean of the outcome vector conditional on the design variables X. With fully observed data, repeated measures regression models can be fit using generalized estimating equations (GEE) (Zeger and Liang, 1986). With fully observed data, a desirable property of regression parameter estimates from GEE is that they retain such properties as consistency and asymptotic normality regardless of the assumed within-subject (longitudinal) correlation structure. When data are missing, this property no longer holds, and regression estimates may depend strongly on the assumed correlation structure (see Hogan et al., 2004, for an empirical example). When missingness is MAR and follows a monotone pattern, the IPW method can be used to obtain consistent estimates of regression parameters using a specified procedure. Here, we emphasize that auxiliary information should be included in the observed-data history, Zj− = (Y1,…,Yj−1, V1,…,Vj−1) and the model for π(X,V,θ). The procedure is as follows: Specify the regression model that would be used had all the intended data been collected. Let denote the probability that Yj is observed. Specify and fit a model for ; denote the estimated parameters by . Let denote the probability that an individual has remained in the study to time j. Fit the regression specified in Step 1, and weight individual contributions to the model by . Use the assumed independence correlation structure. Use the bootstrap technique for standard error estimation. In large samples, the IPW GEE yields consistent estimators when the response probability model is correctly specified, but again may have high variance when individual weights are large. The augmented IPW GEE procedure (discussed below) can be used to partially remedy this weakness.
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The Prevention and Treatment of Missing Data in Clinical Trials Target Distribution (or Parameter) Most often in clinical trials, primary interest centers on the distribution [Y | X] = [Yobs,Ymis | X], where X includes the treatment group and possibly other design variables. The target distribution is related to the full-data distribution through the identity: (19) Hence, inference about the target distribution relies critically on the untestable assumptions being made about p(ymis | yobs,m,x). Selection and Pattern Mixture Models Two broad classes of models for the joint distribution of Y and M are selection models, which factor the full data distribution as (20) and pattern mixture models, which factor the full-data distribution as (21) Pattern mixture models can be factored to make the missing data extrapolation explicit within missing data pattern M, that is (22) Selection Models Selection models can be divided into two types, (1) parametric and (2) semiparametric. Parametric selection models were first proposed by Rubin (1974) and Heckman (1976), based on parametric assumptions for the joint distribution of the full data (usually, a normal distribution for responses and a probit regression for the missing data indicators). For repeated measures, parametric selection models were described by Diggle and Kenward (1994), and semiparametric models were proposed by Robins et al. (1995) and Rotnitzky et al. (1998). To illustrate a standard formulation, assume the full-response data comprise (Y1,Y2), and the objective is to capture the mean of Y2 in each treatment group. Further, assume Y2 is missing on some individuals. A parametric selection model might assume that the full-response data follows a bivariate normal distribution: (23)
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The Prevention and Treatment of Missing Data in Clinical Trials and the “selection mechanism” part of the model follows a logistic regression (24) Parametric selection models can be fit to observed data, even though there appears to be no empirical information about several of the model parameters. Specifically, there is no information about the association between M and Y2 because Y2 is missing. Likewise, there is no information about the mean, variance, and covariance parameters involving Y2. The model can be fit because of the parametric and structural assumptions being imposed on the full-data distribution. This can be seen as both beneficial or as a reason to exercise extreme caution. Convenience is the primary benefit, especially if the model can be justified on scientific grounds. The reason for caution is that, again, none of the assumptions underlying this parametric model can be checked from the observed data. In parametric selection models fit under the MNAR assumption, identification of parameters and sensitivity to assumptions raises serious problems: see, for example, Kenward (1998), Little and Rubin (2002, Chapter 15), the discussion of Diggle and Kenward (1994), and Daniels and Hogan (2008, Chapter 9). Semiparametric selection models do not assume a parametric model for the full-data response distribution, so they are therefore somewhat less sensitive to these assumptions. These models are discussed in greater detail in Chapter 5. Pattern Mixture Models Pattern mixture models were proposed for repeated measures data by Little (1993, 1994); a number of extensions and generalizations have followed. The connection between pattern mixture and selection models is described in Little and Wang (1996), in Molenberghs et al. (1998), and in Birmingham et al. (2003). The models can be viewed from an imputation perspective, in which missing values Ymis are imputed from their predictive distribution given the observed data including M; that is, (25) Under MAR, this equals p(ymis | yobs ,x). However, if data are not MAR, the predictive distribution (25) is a direct by-product of the pattern mixture formulation because it conditions on the missing data indicators. This more direct relationship between the pattern mixture formulation and the
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The Prevention and Treatment of Missing Data in Clinical Trials predictive distribution for imputations yields gains in transparency and computational simplicity in some situations, as illustrated in Kenward and Carpenter (2008, Section 4.6). Under MNAR, the selection model factorization requires full specification of the model for the missing data mechanism. Some pattern mixture models avoid specification of the model for the missing data mechanism in MNAR situations by using assumptions about the mechanism to yield restrictions on the model parameters (Little, 1994; Little and Wang, 1996; Hogan and Laird, 1997). Many pattern mixture formulations are well suited to sensitivity analysis because they explicitly separate the observed data distribution from the predictive distribution of missing data given observed data. Sensitivity analyses can be formulated in terms of differences in mean (or other parameter) between those with observed and those with missing responses. Advantages and Disadvantages of Selection and Pattern Mixture Models Substantively, it seems more natural to assume a model for the fulldata response, as is done in selection models. For example, if the outcome is blood pressure, it may seem natural to assume the combined distribution of blood pressures over observed and missing cases follows a single distribution, such as the normal distribution. Moreover, if MAR is plausible, a likelihood-based selection formulation leads directly to inference based solely on the model for the full-data response, and inference can proceed by ML. However, it may not be intuitive to specify the relationship between nonresponse probability and the outcome of interest, which typically has to be done in the logit or probit scale. Moreover, the predictive distribution of missing responses typically is intractable, so it can be difficult to understand in simple terms how the missing observations are being imputed under a given model. And, as indicated above, selection models are highly sensitive to parametric assumptions about the full data distribution. This concern can be alleviated to some degree by the use of semiparametric selection models. Specification of pattern mixture models also appeals to intuition in the sense that it is natural to think of respondents and nonrespondents having different outcome distributions. The models are transparent with respect to how missing observations are being imputed because the within-pattern models specify the predictive distribution directly. Pattern mixture models can present computational difficulties for estimating treatment effects because of the need to average over missing data patterns; this is particularly true of pattern mixture specifications involving regression models within each pattern.
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The Prevention and Treatment of Missing Data in Clinical Trials Examples: Pattern Mixture Model for Continuous Outcomes Daniels and Hogan (2008, Chapter 10) use pattern mixture models to analyze data from a randomized trial of recombinant human growth hormone (rHGH) on muscle strength in elderly people. More than 120 people were randomized to four different treatment arms. The primary outcome in this trial was quadriceps strength, assessed at baseline, 6 months, and 12 months. A pattern mixture model was fit under MAR and parameterized to represent departures from MAR. The example shows how to construct sensitivity plots to assess the effect of departures from MAR on the inferences about treatment effect. An important feature of the model is that the fit to the observed data is unchanged at different values of the sensitivity parameters. However, the model does rely on parametric assumptions, such as normality. These assumptions can be checked for the observed data, but have to be subjectively justified for the missing data. Example: Pattern Mixture Model for Binary Outcomes Daniels and Hogan (2008, Chapter 10) use pattern mixture models to analyze data from an intervention study for smoking cessation among substance abusers. The primary outcome was smoking status, assessed at baseline, 1 month, 6 months, and 1 year. A pattern mixture model was fit under MAR and expanded to allow for MNAR missingness. In addition to presenting sensitivity analysis, the example shows how to incorporate prior information about the smoking rate of dropouts to obtain a summary inference about treatment effect. Sensitivity of Parametric Selection Models The sensitivity of MNAR selection models to distributional assumptions is illustrated by Verbeke and Molenberghs (2000, Chapter 17), who show that, in the context of an onychomycosis study, excluding a small amount of measurement error drastically changes the likelihood ratio test statistics for the MAR null hypothesis. In a separate example, Kenward (1998) revisited the analysis of data from a study on milk yield performed by Diggle and Kenward (1994). In this study, the milk yields of 107 cows were to be recorded during 2 consecutive years. Data were complete in the first year, but 27 measurements were missing in year 2 because these cows developed mastitis, which seriously affected their milk yield and therefore deemed missing for the purposes of the study. Although in the initial paper there was some evidence for MNAR, Kenward (1998) showed that removing two anomalous profiles from the 107 completely eliminated this evidence. Kenward also showed that changing the conditional distribution
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The Prevention and Treatment of Missing Data in Clinical Trials of the year 2 yield, given the year 1 yield, from a normal to a heavy-tailed t distribution led to a similar conclusion. Several authors have advocated using local influence tools for purposes of sensitivity analysis (Thijs et al., 2000; Molenberghs et al., 2001; Van Steen et al., 2001; Verbeke et al., 2001; Jansen et al., 2006). In particular, Molenberghs et al. (2001) revisited the mastitis example. They were able to identify the same two cows also found by Kenward (1998), in addition to another one. However, it is noteworthy that all three are cows with complete information, even though local influence methods were originally intended to identify subjects with other than MAR mechanisms of missingness. Thus, an important question concerns the combined nature of the data and model that leads to apparent evidence for an MNAR process. Jansen et al. (2006) showed that a number of features or aspects, but not necessarily the (outlying) nature of the missingness mechanism in one or a few subjects, may be responsible for an apparent MNAR mechanism. Selection and Pattern Mixture Models: Literature The literature covering selection and pattern mixture models is extensive. Review papers that describe, compare, and critique these models include Little (1995), Hogan and Laird (1997, 2004), Kenward and Molenberghs (1999), Fitzmaurice (2003), and Ibrahim and Molenberghs (2009). The models are also discussed in some detail in Little and Rubin (2002), Diggle et al. (2002), Fitzmaurice et al. (2004), Molenberghs and Kenward (2007), and Daniels and Hogan (2008). An extensive literature also exists on extensions of these models involving random effects, sometimes called shared-parameter or random-coefficient-dependent models. Reviews are given by Little (1995) and Molenberghs and Kenward (2007). Although these models can be enormously useful for complex data structures, they need to be used with extreme caution in a regulatory setting because of the many layers of assumptions needed to fit the models to data. Recommendations Recommendation 9: Statistical methods for handling missing data should be specified by clinical trial sponsors in study protocols, and their associated assumptions stated in a way that can be understood by clinicians. Since one cannot assess whether the assumptions concerning missing data are or are not valid after the data are collected, one cannot assert that the choice of missing data model made prior to data collection needs to be
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The Prevention and Treatment of Missing Data in Clinical Trials modified as a result of a lack of fit. Thus, one needs to carry out a sensitivity analysis. Of course, model fitting diagnostics can be used to demonstrate that the complete data model may need to be adjusted, but the missing data model raises no additional complexities. Recommendation 10: Single imputation methods like last observation carried forward and baseline observation carried forward should not be used as the primary approach to the treatment of missing data unless the assumptions that underlie them are scientifically justified. Single imputation methods do not account for uncertainty associated with filling in the missing responses. Further, LOCF and BOCF do not reflect MAR data mechanisms. Single imputation methods are sometimes used not as a method for imputation but rather as a convenient method of sensitivity analysis when they provide a clearly conservative treatment of the missing data. This can obviously be accomplished by using a best possible outcome for the missing values in the control group and a worst possible outcome for the missing values in the treatment group. If the result of such a technique is to demonstrate that the results of the primary analysis do not depend on the treatment of the missing data, such an approach can be useful. However, techniques that are often viewed as being conservative and therefore useful in such an approach, are sometimes not conservative and so care is required. Recommendation 11: Parametric models in general, and random effects models in particular, should be used with caution, with all their assumptions clearly spelled out and justified. Models relying on parametric assumptions should be accompanied by goodness-of-fit procedures. We acknowledge that this is an area where the current toolkit is somewhat lacking, and therefore more research is needed. Some contributions to this area include Verbeke et al. (2001, 2008), Gelman et al. (2005), and He and Raghunathan (2009). Recommendation 12: It is important that the primary analysis of the data from a clinical trial should account for the uncertainty attributable to missing data, so that under the stated missing data assumptions the associated significance tests have valid type I error rates and the confidence intervals have the nominal coverage properties. For inverse probability weighting and maximum likelihood methods, this can be accomplished by appropriate computation of standard errors, using either asymptotic results or the bootstrap. For imputation, it
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The Prevention and Treatment of Missing Data in Clinical Trials is necessary to use appropriate rules for multiply imputing missing responses and combining results across imputed datasets because single imputation does not account for all sources of variability. Recommendation 13: Weighted generalized estimating equations methods should be more widely used in settings when missing at random can be well justified and a stable weight model can be determined, as a possibly useful alternative to parametric modeling. Recommendation 14: When substantial missing data are anticipated, auxiliary information should be collected that is believed to be associated with reasons for missing values and with the outcomes of interest. This could improve the primary analysis through use of a more appropriate missing at random model or help to carry out sensitivity analyses to assess the impact of missing data on estimates of treatment differences. In addition, investigators should seriously consider following up all or a random sample of trial dropouts, who have not withdrawn consent, to ask them to indicate why they dropped out of the study, and, if they are willing, to collect outcome measurements from them. INSTRUMENTAL VARIABLE METHODS FOR ESTIMATING TREATMENT EFFECTS AMONG COMPLIERS Estimates of treatment effects for all individuals randomized as in intention-to-treat analysis are protected against bias by the randomization. In this estimand, individuals who are assigned a treatment but never comply with it, perhaps because they cannot tolerate treatment side effects, are treated in the same way as individuals who comply with the treatment. Sometimes, particularly in secondary analyses, interest lies in the treatment effect in the subpopulation of individuals who would comply with a treatment if assigned to it. The average treatment effect in this population is called the complier-average causal effect (CACE) (Baer and Lindeman, 1994; Angrist et al., 1996; Imbens and Rubin, 1997a, 1997b; Little and Yau, 1998; White, 2005). An alternative estimand to the CACE is the average treatment effect (ATE) (Robins, 1989; Robins and Greenland, 1996). It is defined as the difference in mean outcome if all individuals had been assigned and complied with the treatment (T = 1) and the mean if all individuals had been assigned and complied with the control treatment (T = 0). The ATE is defined for the whole target population, and it requires assumptions about the treatment outcome for noncompliers had they complied with the treatment. Whether this counterfactual event is meaningful typically depends on context. For example, noncompliance to a behavioral treatment, such as an exercise
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The Prevention and Treatment of Missing Data in Clinical Trials regime, might plausibly be changed by increased motivation, as might occur if evidence of success of the treatment becomes widely known. In contrast, if noncompliance to a drug is the result of intolerable side effects, then compliance may require a reformulation of the drug to remove the side effects. Such reformulation may change the properties of the drug, and estimation of the ATE is consequently more speculative. Simple approaches to estimating the CACE or the ATE include astreated analysis, in which participants are classified according to the treatment actually received, and per-protocol analysis, which restricts analysis to participants who comply with the assigned treatment. These analyses are subject to selection bias in that participants who comply with a treatment may be a biased sample of participants randomized to that treatment. The bias may be reduced by adjustment for covariates, but it remains a major concern. Although this is often characterized as a problem of selection bias, recent approaches have suggested alternatives to as-treated and per-protocol analyses by applying a missing-data perspective. Consider a binary variable C(T) taking the value 1 if an individual would comply with a particular treatment T if assigned to it, and 0 otherwise. We call this variable principal compliance, to distinguish it from observed compliance, which depends on the treatment actually assigned. It is a special case of principal stratification (Frangakis and Rubin, 2002). Principal compliance C(T) is observed for participants who are assigned to treatment T, but it is not observed for participants assigned other treatments, T′, so for these individuals the values of C(T) can be regarded as missing. In simple trials involving an active treatment and a control treatment, an alternative to as-treated and per-protocol estimates is based on the idea of treating the randomization as an instrumental variable (IV), in economic parlance. The IV estimator yields a direct estimate of the CACE, and it is protected from selection bias by the randomization. However, it requires certain assumptions to be valid, and it also yields estimators with potentially high variance, particularly if the treatment compliance rate is low. Model-based versions of the IV estimator based on treating C as missing for some participants have been proposed that are potentially more efficient, although they make stronger distributional assumptions. For a nontechnical article comparing this approach with as-treated and per-protocol estimates, see Little et al. (2009) for a discussion of extensions to two or more active treatments, see Long, Little, and Lin (in press). An example illustrating the above discussion and a number of associated issues is provided by the evaluation of a trial to assess the effect of an influenza vaccine (Hirano et al., 2000). The trial randomly assigned physicians to encouragement (T1) or no encouragement (T2) to vaccinate their patients against influenza. The primary endpoint was hospitalization, and
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The Prevention and Treatment of Missing Data in Clinical Trials the intention-to-treat estimates showed of those encouraged, 7.8 percent were hospitalized and of those not encouraged 9.2 percent were hospitalized. However, the trial had only a weak effect on the actual taking of the vaccine: of those encouraged, 31 percent of patients received the vaccine; of those not encouraged, 19 percent of patients received the vaccine. Therefore, to better understand the trial results, at least a secondary estimand of interest was CACE, that is, in this case, the effect of encouragement on hospitalization for the patients who would have been vaccinated if their physician had been encouraged but not vaccinated if their physician had not been encouraged. Assuming the standard exclusion restrictions of IV, CACE was estimated as an 8.2 percent reduction in hospitalization. Yet, even this turned out to represent only part of a better understanding of the trial results. In this study, there were a number of good baseline predictors of compliance under both arms, C(T1) and C(T2), and thus, the effect of compliers could be in part identified without the need of exclusion restrictions. When these restrictions were relaxed, the effect of encouragement on compliers was estimated at 3.7 percent, but there was at least as large of an estimated effect (5.3 percent) of encouragement on hospitalization for always-takers. Later commentaries on these results suggested that the latter effect is explainable by the earlier time in the season at which the always-takers likely receive the vaccine when encouraged, compared to when not encouraged. Since this effect is comparable to CACE, it suggested that the effect of vaccination lies more in its timing and not only on its receipt. To further explicate this method, we offer an example of coprimary outcomes that induce missing data. For randomized controlled trials with two (or more) coprimary outcomes, say E and Y, values of E can determine whether Y has a meaning as a measurement. This effect presents a challenge in the very definition of the effect between the two interventions, say T1 and T2, on Y, because the existence of Y is determined after the intervention. This problem can be treated in principle in the context of missing information, not of Y (which is sometimes undefined) but of certain strata, called principal strata. Our example involves clinical trials for HIV. The idea of cell-mediated immunity is to train the killer cells to recognize and attack a protein that human CD4 cells create when the CD4 cells are infected (as opposed to targeting the virus directly, whose identification is difficult due to mutations over time). For this reason, randomized trials for cell-mediated immunity vaccines should be designed to assess two coprimary outcomes: reducing primary infection (say, E), and, if a person is infected (E = 1), keeping low viral load (say Y). Work by Gilbert et al. (2003) and then by Mehrotra et al. (2006) showed how principal stratification (Frangakis and Rubin, 2002) can be used to formulate the target hypotheses with such coprimary outcomes. Specifically, the first coprimary
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The Prevention and Treatment of Missing Data in Clinical Trials research hypothesis is that changing treatment T1 (placebo) to T2 (vaccine) changes the primary infection rate E. The second coprimary research hypothesis should capture that the vaccine can also affect viral load when infected. However, the viral load distributions between infectees under the placebo condition and those infected under the vaccine condition could be different simply because the immune system is inherently different between the two groups. (In fact, if the vaccine prevents some primary infections, infectees under vaccine are expected to have weaker baseline immune system than infectees under placebo.) One can disentangle baseline differences from vaccine effects if one focuses on the people who would have been infected regardless of receiving the vaccine or the placebo. This stratum is known as a principal stratum because membership to it does not change depending on assignment to different interventions. Thus, the second coprimary research hypothesis can be that changing treatment T1 (placebo) to T2 (vaccine) will change the viral load for those for whom changing T1 to T2 does not prevent primary infection. For a person under placebo who gets infected (E(T1) = infected), one does not know if the person would have been also infected under vaccine (E(T2) = infected), so membership to the principal stratum—E(T1) = E(T2) = infected—is partly missing. (Estimation of the effect of vaccine on viral load Y for this stratum is discussed above.) Additional examples of randomized controlled trials with coprimary outcomes using principal stratification include determining if the immune response to a vaccine is causing reduction in infection rates (Follmann, 2006); assessing more general surrogate outcomes in vaccine trials (Qin et al., 2008); and evaluating the effect of an intervention on severity of a disease (e.g., of prostate cancer) when a person does get the disease (Shepherd et al., 2008). MISSING DATA IN AUXILIARY VARIABLES The assumptions and models discussed above have been limited to outcome variables. Usually, there are many auxiliary variables collected at each visit that can be useful to incorporate into the analysis. Specifically, these variables are useful because they both help explain the reasons for future nonresponse as well as help predict the missing outcomes (and so help improve the efficiency with which the treatment effects are estimated). They can also serve to make the MAR assumption more tenable. We have assumed throughout that the collection of auxiliary variable data is complete, which is clearly not always the case. We do note that the above approaches can be modified to incorporate missing auxiliary data by augmenting the missing outcome variable with a missing V. Although including V along with the missing outcome variable will often address the problem,
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The Prevention and Treatment of Missing Data in Clinical Trials the literature on missing data in longitudinal settings is fairly limited, and more research on dealing with missing auxiliary data would be useful. We do believe that many of the above approaches can be easily modified to incorporate auxiliaries by replacing in the conditional means and probabilities with , which includes (Y1,…,Yk–1,V1,…,Vk–1). An excellent example of the use of this method is Liu et al. (2009).