The Prevention and Treatment of Missing Data in Clinical Trials (2010) / Chapter Skim
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4 Drawing Inferences from Incomplete Data
4 Drawing Inferences from Incomplete Data
Pages 47-82
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From page 47... ...
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.
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From page 48... ...
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.
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From page 49... ...
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.
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From page 50... ...
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.
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From page 51... ...
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
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From page 52... ...
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.
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From page 53... ...
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.
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Summary 1. 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.
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From page 55... ...
, 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)
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From page 56... ...
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.
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From page 57... ...
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
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The authors used inverse probability weighting to estimate the effect of a behavioral intervention involving supervised exercise on the rate of smoking cessation among 300 women. The primary outcome was smoking status, assessed weekly over 12 weeks.
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Advantages and Disadvantages of IPW Methods The IPW method is generally simple to implement when the missing values have a monotone pattern, and can be carried out in any software package that allows weighted analyses. A key advantage is that, under a correctly specified model for missingness, information on many auxiliary variables can be accommodated, including information on previously observed outcomes.
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Now let D = (Yobs,X,M) denote the observed data for an individual.
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From page 61... ...
( p y x;θ of the full response data, generating from the posterior can be made easier by embedding a data augmentation step within the sampling algorithm; this approach effectively imputes the missing responses under 3 Note, however, that assessing sensitivity to the prior is an important part of any Bayesian analysis.
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The predicting distribution of the missing responses can differ substantially according to the specification. In fact, with incomplete data, two models having the same mean specification but different specifications of S will yield different inferences about the mean.
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From page 63... ...
When both the error and random effects distributions are normal, the random effects model coincides with a multivariate normal regression with a constrained parameterization of S In this special case, the between- and 4 This may seem counterintuitive because for datasets having no missing values, estimates of the mean typically do not depend on specification of the variance matrix.
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From page 64... ...
Random effects models can be very useful for simplifying a highly multivariate distribution using a small number of parameters. However, with incomplete data, there are several reasons that inference about a treatment effect should not be limited to a single likelihood-based model: • In addition to the ignorability assumption, inference relies on para )
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From page 65... ...
hot deck imputation, which matches the case with missing values to a case with values observed that is similar with respect to observed variables and then imputes the observed values of the respondent; and (c) LOCF or BOCF methods for repeated measures, which impute the last observed value or the baseline value of the outcome variable.
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From page 66... ...
that is, that the predictive distribution of YK, based on design variables X and observed response data Yobs = (Y1,…,YK–1) , is the same for those with missing and observed YK.
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This aspect has been addressed in a variety of widely-available software packages. We emphasize here the need to fully understand and communicate the assumptions underlying any imputation procedures used for drawing inferences about treatment effects.
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However, if the auxiliary variables are incomplete but measured in a substantial number of cases for which CD4 is missing, then multiple imputation can still be applied productively, multiply imputing the missing values of both CD4 and the missing auxiliary variable values in the imputation step. The predictive distribution of the missing values can be based on a parametric model for the joint distribution of V and Y given X, using the Bayesian paradigm described in Section 4.3.2.
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From page 69... ...
. MCAR-MAR Diagnostics As indicated above, while the observed data cannot be used to distinguish between MAR and MNAR missing data mechanisms, they can be used to distinguish between MCAR and MAR models and between competing MAR models.
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From page 70... ...
However, the MAR assumption cannot be verified from observed data, and even with modeling assumptions, the information to simultaneously estimate the parameters of the missing-data mechanism and the parameters of the complete-data model is very limited. Hence, model-based estimates tend to be very sensitive to misspecification of the model.
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From page 71... ...
The second factor on the right-hand side is the model for the distribution of missing observations, and it cannot be inferred from observed data alone for the simple reason that no assumptions about the distribution can be checked from observed data. This factorization makes clear that inference from incomplete data requires the analyst to specify a model (or set of assumptions)
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From page 72... ...
. 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 Yobs , Ymis , M X = M Yobs , Ymis , X × Yobs , Ymis X (20)
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From page 73... ...
. 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, p(ymis | yobs,x,M)
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From page 74... ...
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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From page 75... ...
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)
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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.
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From page 77... ...
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.
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From page 78... ...
Recommendation 14: When substantial missing data are anticipated, auxiliary information should be collected that is believed to be associ ated with reasons for missing values and with the outcomes of interest. This could improve the primary analysis through use of a more appro priate missing at random model or help to carry out sensitivity analyses to assess the impact of missing data on estimates of treatment differ ences.
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From page 79... ...
DRAWING INFERENCES FROM INCOMPLETE DATA 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.
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From page 80... ...
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)
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From page 81... ...
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)
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From page 82... ...
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 Yk in the conditional means and – probabilities with Zk , which includes (Y1,…,Yk–1,V1,…,Vk–1)
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