5.2.1 IMPACT Study

The Improving Mood and Promoting Access to Collaborative Treatment (IMPACT) study has been introduced in Chapters 1 and 4. Briefly, to determine the effectiveness of the IMPACT collaborative care management program for late life depression, a total of 1801 patients aged 60 years or older with major depression (17%), dysthymic disorder (30%), or both (53%) were enrolled and randomized to IMPACT care or usual care. Intervention patients had access for up to 12 months to a depression care manager who was supervised by a psychiatrist and a primary care expert and who offered education, care management, and support of antidepressant management by the patient's primary care physician or a brief psychotherapy for depression. Depression symptoms were assessed by the mean score of the 20 depression items from the Symptom Checklist-90 (SCL-20) at baseline, 3, 6, 12, 18, and 24 months. For details about the study design and primary analysis, see Unützer et al. (2002).

In the following, we conduct longitudinal data analysis using the outcome, the SCL-20 depression score, and contrast various missing data methods. Figure 5.1 shows trajectories of 40 randomly selected individuals and the group mean trajectories over time. It is pretty obvious that individuals from the intervention arm tend to have lower SCL-20 scores.

We consider an analysis that addresses the question whether or not the IMPACT treatment reduces the depression symptoms as measured by the SCL-20 scores. To address this question, we compare the subject-specific changes, from baseline to follow-up, in the SCL-20 scores for the patients in the two study arms. We consider the following linear mixed effects regression model for the subject-specific mean SCL-20 scores:

(5.1)

where Y_ij is the SCL-20 score for the ith patient at the jth time point of assessment (j = 0, 3, 6, 12, 18, 24). The variable group is an indicator variable for the treatment group, with group = 0 if an individual was randomized to the usual care group and group = 1 if randomized to the IMPACT treatment group. The binary variable period denotes the baseline and follow-up periods, with period = 0 for the baseline and period = 1 for the follow-up periods (3–24 months). This allows us to assess the time-averaged intervention effect. We can also replace period with the actual month variable, which then allows us to assess treatment effect at specific months. Although the SCL-20 score is bounded between 0 and 4, because it is an average of 20 items, we assume that it follows a normal distribution in our analysis. Finally, we assume that the random intercepts and slopes, b_i, follow a bivariate normal distribution, with zero mean and an unstructured 2 × 2 covariance matrix G.

We decided to illustrate our analysis using the Stata software version 13.0. The point is that many missing data methods have been implemented in various software packages, and the readers can choose the ones that they are already familiar with or more comfortable using.

The misstable command in Stata allows us to examine the missing data pattern in SCL-20 scores, which is shown in Table 5.1. It can be seen that 162 (9%) subjects had intermittent missingness, 370 (20%) subjects dropped out during the 24-month period, and 1269 (71%) patients had complete data.

5.2.2 NACC UDS Data

The National Alzheimer's Coordinating Center (NACC) Uniform Data Set has already been introduced in Chapter 1. The data set was also analyzed by Monsell et al. 2012). Our aim is to model the decline of the outcome, mini-mental state examination (MMSE), since the first diagnosis of amnestic mild cognitive impairment (aMCI).

Missing data occur in the outcome MMSE as well as in the covariates education, Hachinski ischemic score (HIS), and apolipoprotein E (APOE) e4 alleles. The missing pattern is nonmonotone, as shown in Table 5.2.

The data set is in a “long format,” meaning that each row corresponds to one person-visit, and each subject contributes multiple rows. We observe that HIS was missing for 300 clinical visits and MMSE for 404 visits. Education and APOE e4 are on the subject level; it turns out that 4 subjects had missing education and 949 subjects had missing APOE e4.

5.3.1 Linear Mixed Models for Continuous Longitudinal Data

5.3.1.1 Model Setting

Suppose the outcome vector Y_i takes continuous values, so that we could consider a linear regression model. A linear mixed model (LMM) could be illustrated from a two-stage model formulation. We first fit a regression line for each subject separately:

(5.2)

where β_i is a q-dimensional vector of subject-specific regression coefficients, Z_i is the design matrix of covariates, and ε_i ~ N(0, Σ_i) is the Gaussian residual. Note that ε_i describes the within-subject variability, and usually we assume that the elements of ε_i are independent, that is, . The second-stage model imposes a structure on β_i:

(5.3)

where K_i is the design matrix, β is the vector of regression parameters of interest, and b_i ~ N(0, D) is the residual. Note that b_i represents the variability between subjects, whose elements are usually correlated. We could regard β as the “average” covariates effect, and then b_i is the effect deviation from the population average for subject i.

The two-stage model, defined by (5.2) and (5.3), could be combined into the following one-stage model:

We call β the fixed effects and b_i the random effects due to the fact that β is the unknown parameter vector, whereas b_i is a random vector. More generally, the following four assumptions constitute an LMM:

where and are called the variance components and α is a vector of nuisance parameters specifying these variance components. The LMM can be equivalently viewed as a hierarchical model; that is, a conditional model for Y_i given b_i and a marginal model for b_i:

(5.4)

(5.5)

Marginally, Y_i also follows a multivariate normal (MVN) distribution:

(5.6)

The LMM implies the marginal model, but the converse is not true. So, the parameter vector β may have two interpretations: the population-average covariate effects, or the covariate effects conditional on a given subject. The compatibility of the marginal and conditional model interpretation holds only for the LMM, not the generalized linear mixed models (GLMM), as we will see in the later sections.

5.3.1.2 Model Estimation and Inference

We first assume that the vector of nuisance parameters in the variance components, α, is known, and write . Now β could be solved from weighted least squares:

(5.7)

It is easy to verify that (5.7) is the maximum likelihood estimator (MLE) since the likelihood function is

(5.8)

We should note that the likelihood inference utilizes only the marginal model (5.6), but not the hierarchical models (5.4) and (5.5).

In most cases, however, α is unknown and needs to be estimated. Two common methods for estimating α are maximum likelihood (ML) and restricted maximum likelihood (REML). The ML method estimates α and β simultaneously by maximizing the likelihood function (5.8). The score equations can be derived as follows:

where α_j is the jth element of α and tr(A) is the trace of matrix A.

The ML estimation may result in a substantial amount of bias for α in small samples, due to the estimation of β. A simple analogy is that the ML estimator is a biased estimation of the population variance. Bias correction involves replacing N⁻¹ by (N − 1)⁻¹. The REML approach uses a similar technique, and gives a better estimation of the variance components. The idea is to construct a likelihood function that is no longer a function of β. Note that in (5.7) is the sufficient statistic for β, and REML is indeed a conditional likelihood function . We give the form of the restricted likelihood as follows, and more details and derivations can be found in Patterson and Thompson 1971):

With α estimated using the ML or REML method, β is estimated by plugging into (5.7). Conditional on α, follows a multivariate normal distribution with mean β and variance

In practice, we again replace α by its ML or REML estimate to evaluate this variance.

5.3.1.3 Empirical Bayes Estimates for Random Effects

In some situations, we may wish to know the value of the random effects b_i, so that the individual trajectories could be portrayed. Since b_i is random, the Bayesian idea is a natural choice in order to “guess” the most plausible value of a random variable. If we can find the posterior distribution , the random effects would be easily estimated by the posterior mean , with the parameters replaced by the ML or REML estimates. The estimation of b_i relies on the hierarchical model, and we have

This leads to the conditional distribution,

where

So the empirical Bayes estimator for b_i is

This is also referred to as the best linear unbiased predictor for b_i (Robinson, 1991).

5.3.2 Generalized Estimating Equations

5.3.2.1 Model and Inference

In cross-sectional studies, if we suppose that the scalar response Y_i comes from the exponential family, then the score equation has the following form:

(5.9)

where . With longitudinal data, the idea of generalized estimating equations (GEE) comes from analogy of the score equation (5.9):

(5.10)

where

and V_i is a working covariance matrix for Y_i. We usually decompose the working covariance as , where is a diagonal matrix and R_i(α) is a working correlation matrix with a vector of unknown parameters α. For the exponential family, S_i follows from the mean–variance relationship, and we only need to specify a plausible guess of R_i(α). Common choices of R_i are listed in Table 5.3.

Independence	AR-1
Exchangeable	Unstructured

When the sample size N is large,, defined as the solution to (5.10), approximately follows the normal distribution with mean β and variance , where

and

In practice, we could replace by . Although this is a rather crude estimator for every single , it leads to a good estimate of I₁ with many replications. The variance is also referred to as the sandwich variance estimator. One advantage of GEE is that the inference is always valid even if the working correlation matrix is incorrect. In other words, the only assumption of GEE is the mean structure:

with the link function g, and the working correlation is not so important. With the correct working correlation, GEE is similar to the likelihood inference, and the sandwich variance simplifies to , which can be seen as the inverse of the information matrix. The vector of parameters, β, is interpreted as the population-average effect of the covariates.

5.3.2.2 Estimation of Working Correlation

The working correlation may be specified up to the unknown vector of nuisance parameters, α, which needs to be estimated from the data. Liang and Zeger 1986) proposed to use a moment estimator for α. Their estimator is based on the Pearson residual:

where is the estimated variance for Y_ij. For example, under the exchangeable working correlation, the nuisance parameter vector α becomes a scalar of within-cluster correlation coefficient α. The moment estimator for α is given by

The standard GEE procedure implemented in most software packages is as follows:

1. Compute the initial estimate of using generalized linear models.
2. Compute the Pearson residual r_ij and estimate for (and thus R_i and V_i).
3. Update the estimate for using the estimated correlation from step 2.
4. Iterate steps 2 and 3 until convergence.

More advanced estimators for α are also proposed in the literature. Interested readers should refer to Prentice 1988), Lipsitz et al. 1991), Carey et al. 1993), and Prentice and Zhao 1991). These approaches generally need to specify another set of estimating equations for the second moments, in order to estimate α.

5.3.3 Generalized Linear Mixed Models

5.3.1 Model Setting

In Section 5.3.1, we introduced the mixed model for linear regression. The same idea could be applied to generalized linear models, when multiple observations on each subject are taken over time. Suppose that given the random effects b_i, all the responses Y_ij are independent and follow a distribution in the exponential family. The generalized linear mixed model assumes that

where “⊥” denotes independence. Note that the vector of parameters, β, is interpreted as the subject-specific effect of the covariates (conditional on the random effects). In contrast, the parameters in GEE have a completely different marginal interpretation—the population-average effect of the covariates. Generally, GEE and GLMM are not compatible models: if one integrates out b_i in according to the GLMM specification, then the marginal mean usually does not possess the generalized linear form.

Let be the conditional density of Y_ij given b_i, and be the marginal density of b_i. Then, the likelihood function of β and D is

(5.11)

If f_ij is the normal density, the integral could be calculated analytically, which is the LMM in Section 5.3.1. Otherwise, approximations are needed to evaluate the likelihood function.

5.3.4 Time-Dependent Covariates

In the discussion of the methods for longitudinal data analysis, so far we are interested in the mean of Y_ij given X_ij, which is also called the cross-sectional mean. In practice, however, it is often the case that the mean of Y_ij can be affected by previous observations of covariates, X_i1, ..., X_i,j−1. For example, when studying the association between lung function and air pollution, there is usually a lag between the peak of the pollutant and the change of lung function, so the quantity of interest might be E(Y_ij|X_i,j−1). When studying the effect of a new drug in clinical trials, it is more reasonable to assume a cumulative treatment effect, so the interest is in E(Y_ij|X_i1, ..., X_ij), where we condition on the whole treatment history.

We introduce an important concept for the time-dependent covariates—exogeneity. A covariate process is exogenous to the outcome process if the distribution of current covariate value is completely determined by its previous values, that is,

where and are the past observations of Y and X until time j and W_i is the vector of baseline or time-invariant covariates. If the conditional independence is not satisfied, then the covariate process is said to be endogenous. For example, in a clinical trial with a crossover design, if the treatment is prescribed at the start of the trial, then the treatment variable is exogenous. On the contrary, if the treatment assignment changes according to the patients' response to the drug, then the treatment variable is endogenous. In general, one can check for endogeneity by regressing X_ij on both and .

The advantage of exogeneity is that the joint likelihood of Y and X can be factored out:

where n_i is the number of observations for subject i, , and . Therefore, if one is interested only in f_Y|X, no assumptions on the covariate process are needed. Exogeneity also suggests that

When a covariate is endogenous, we need to be careful in choosing the meaningful model of interest.

Pepe and Anderson 1994) have shown that when including endogenous variables into the regression model, GEE could be biased unless either of the following assumptions holds:

1. Full covariate conditional mean assumption:
   
2. A working-independent correlation assumption.

5.4.1 Setup

In Section 5.3, we summarized commonly used methods for analyzing longitudinal data without missingness. In this section, we will introduce statistical methods to handle longitudinal missing data. Sections 5.5–5.7 will focus on missing outcome only. The scenario of missing both outcome and covariates is discussed in Sections 5.8–5.10.

The pattern of the missing data can be classified as monotone and nonmonotone missingness. As a special case of monotone missingness, dropout means that if Y_ij is missing, all the observations after time j are missing. Otherwise, the missingness is said to be intermittent.

In Chapter 1, we distinguished the missingness mechanisms as missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). For longitudinal missing data, we further define covariate-dependent missingness (CDM): an MAR missingness process is CDM if the probability of nonresponse depends only on covariate X and not on Y. Chapter 5 is entirely based on the MAR assumption, and MNAR models will be introduced in Chapter 7.

5.4.2 Simple Methods

approach for handling missing longitudinal data is simply to remove all the subjects with any missing observations, known as the complete-case analysis. Although easily implemented, this method is usually biased when the missingness mechanism is MAR or MNAR. Even under the MCAR situation, deleting subjects with incomplete observations could lead to substantial loss of information, and hence result in inefficient estimators.

Another way to handle the missing data is single imputation, that is, to replace the missing value with some kind of “guess.” One of the most widely used single imputation methods is last observation carried forward (LOCF). This method replaces every missing value by the last observed value of the same subject. The LOCF method makes a very strong assumption that the value of the outcome remains unchanged after missing. When the missing data are due to cure, this assumption might be appropriate. But in many other settings, LOCF is likely to yield biased results. Saha and Jones (2009) derived an explicit expression for the bias of LOCF under a linear mixed effects model. They found that the bias for the two-group mean comparison is more severe if the time to drop out is shorter, or the dropout probability is more strongly associated with the individual random effects. They also noticed that the variance is always underestimated. Other single imputation methods, such as mean imputation or hot deck imputation, could be used as well. The disadvantage of all these single imputation methods is that the imputed observations are treated as known, so the inference does not reflect the sample variability without special adjustments.

5.5.1 Example: IMPACT Study

Table 5.4 lists percentages of missingness by treatment and month. It appears that the intervention arm had a slightly lower percentage of missing data in SCL-20 scores, but the differences did not appear to be significant between the two arms.

Note that we model the IMPACT data by (5.1). We carry out some inferences in the following sections. There are more efficient imputation techniques available for monotone missingness. To illustrate the method, we first drop the subjects with intermittent missing outcome values and then return to the full sample toward the end.

5.5.1.1 Complete-Case Analysis

We first consider the subjects with complete data (N = 1269). The intervention arm is estimated to have 0.3115 more reduction in change from baseline for the SCL-20 score than the usual care group.

For the complete-case analysis, the estimated fixed effects and variance components from the linear mixed effects model are displayed in Table 5.5. A test of the null hypothesis, H₀ : β₄ = 0, indicates that there is a significant group × period interaction at the 0.05 level. These results suggest that there are differences between the two treatment arms in terms of subject-specific changes in the expected SCL-20 scores. In particular, there is a greater reduction in the expected SCL-20 score from baseline for patients treated with IMPACT care (compared with patients who received the usual care).

Parameter	Estimate	Robust SE	95% CI
β1	1.689	0.025	(1.639, 1.738)
β2	−0.347	0.024	(−0.395, −0.299)
β3	−0.006	0.034	(−0.073, 0.060)
β4	−0.312	0.034	(−0.378, −0.245)
	0.452	0.016	(0.422, 0.484)
	0.412	0.019	(0.377, 0.450)
Corr(b1i, b2i)	−0.235	0.051	(−0.332, −0.134)

The estimated variance components for the random intercept and slope indicate that there is substantial variability in the baseline SCL-20 score in the study population and also substantial variability in the patient-to-patient changes in the SCL-20 score in response to the treatment. For example, the estimated standard deviation of the random intercepts, , implies that there is substantial patient-to-patient variability in terms of their baseline SCL-20 score, since ~95% of the patients have a baseline SCL-20 score that varies from 1.689 − 1.96 × 0.452 to 1.689 + 1.96 × 0.452, or from 0.803 to 2.575. Similarly, there is moderate heterogeneity in the patient-to-patient changes in the SCL-20 score. Finally, the negative correlation among the random intercepts and slopes is significant, indicating that the expected change in the SCL-20 score is directly related to the baseline SCL-20 score, and patients with higher baseline SCL-20 scores tend to have smaller changes in SCL-20 scores, compared with those with lower baseline SCL-20 scores.

5.5.1.2 Available Data Analysis

One advantage with the mixed effects model is that it allows all available data to be included in the analysis. In the following example, we again consider the sample with monotone missingness (N = 1639), and the results based on the available data analysis are reported in Table 5.6. Compared to the complete-case analysis, the estimated intervention effect (group × period) is smaller with smaller standard error too.

Parameter	Estimate	Robust SE	95% CI
β1	1.686	0.022	(1.644, 1.729)
β2	−0.335	0.022	(−0.378, −0.291)
β3	−0.003	0.030	(−0.056, 0.062)
β4	−0.295	0.031	(−0.356, −0.235)
	0.455	0.014	(0.428, 0.484)
	0.414	0.018	(0.380, 0.451)
Corr(b1i, b2i)	−0.200	0.048	(−0.292, −0.105)

5.5.1.3 Last Observation Carried Forward Analysis with Monotone Missingness

A popular and less sophisticated approach is the last observation carried forward approach. By its name, the last observed value is used to replace all missing values if a subject drops out. The results of the LOCF analysis are listed in Table 5.7. Note that the estimated intervention effect is much smaller than both complete-case analysis and available data analysis, suggesting that such an approach is very prone to bias.

Parameter	Estimate	Robust SE	95% CI
β1	1.686	0.022	(1.644, 1.729)
β2	−0.313	0.021	(−0.354, −0.271)
β3	−0.003	0.030	(−0.056, 0.062)
β4	−0.264	0.030	(−0.323, −0.206)
	0.481	0.013	(0.458, 0.508)
	0.449	0.016	(0.419, 0.482)
Corr(b1i, b2i)	−0.193	0.040	(−0.270, −0.113)

5.6.1 Modeling the Selection Probability

The dropout pattern implies that if R_ij = 0, then R_i,j+1 = 0. Assume that R_i1 = 1 for the first time point. Let , which indicates the time of dropout and takes the value between 2 and n_i + 1. The maximum value M_i = n_i + 1 corresponds to a subject with complete longitudinal observations.

The continuation ratio model (CRM) can be used to estimate the selection model. Define

Assume that

where W_ij is the design matrix expanded by X_i, Y_i1, ... , Y_i,j−1. The CRM can be regarded as a discrete-time survival model, which estimates the dropout probability at the current time point, given the dropout not occurring at previous time points. Note that

which is the likelihood contribution for the ith subject in estimating the selection model. The MLE for the continuation ratio model are readily available from many standard statistical packages.

5.6.2 IPWGEE1 and IPWGEE2

Robins et al. (1995) proposed to weight the observation Y_ij using

in which the denominator is denoted as π_ij. Note that

The second equation is due to factoring out the joint distribution of R_i1, ..., R_ij. This leads to the following estimating equation for IPWGEE1:

(5.13)

where

Here, is the estimated selection probability from the continuation ratio model. Because λ_ik is always smaller than 1, π_ij decreases as j increases. In other words, more weights are assigned to the later observations.

An alternative approach is to weight each subject by the inverse of the probability of dropping out at the observed time (Fitzmaurice et al., 1995). In this formulation, the estimating equation for IPWGEE2 becomes

(5.14)

where

Here, is the estimated dropout probability from the continuation ratio model, with m_i being the observed dropout time for subject i.

If the true value of λ_ik is used instead of being estimated from the continuation ratio model, the standard theory of estimating equations leads to the sandwich variance estimator for :

(5.15)

Here, U_i is the summand in (5.13) or (5.14) and I_N is the derivative of ∑_iU_i:

However, estimating the selection model contributes an additional term to the sandwich variance formula (5.15), and a rigorous proof is given in Robins et al. (1995). For simplicity, we just ignore the additional term, and the sandwich variance formula seems to work well in our simulation study.

5.6.3 A Simulation Study

We conduct a simulation study to compare the two IPWGEE methods with the naive GEE estimator that ignore the missing data. Consider N = 400 subjects, each with n_i = 5 observations. The true model is generated by a mixed effects model:

The Age variable follows the uniform distribution between 0 and 1, indicating the baseline age of the subject; the Time variable takes values from 1 up to 5, being the time of observation; the Treatment variable is equal to 0 for half of the subjects and 1 for the other half; b_i is the random intercept and ε_ij is the random error, both of which independently follow a normal distribution with variance 1.2² and 1, respectively. The marginal mean model is a probit model:

where Φ is the cumulative distribution function of a standard normal distribution. The dropout process is generated from a continuation ratio model:

The averaged parameter estimation and the 95% CI coverage are shown in Table 5.8. Both IPWGEE1 and IPWGEE2 are consistent and the CI coverage rates are close to the nominal level. The ordinary GEE is biased as we expected.

More extensive simulations that compare IPWGEE1 and IPWGEE2 can be found in Preisser et al. 2002). They found that IPWGEE2 is considerably less efficient than IPWGEE1, while both are consistent. When the dropout probability is low, IPWGEE2 could be quite inefficient and anticonservative (larger type 1 error rate than expected) for small samples.

5.6.4 Example: IMPACT Study

We next illustrate how to implement the two IPWGEE methods to analyze longitudinal data with dropouts (therefore, the missing data pattern is monotone).

With the Stata code presented in Appendix 5.A, we first estimate the probability that an outcome is observed at a specific time point, given the prior observed outcome variables and some selected baseline characteristics using the continuation ratio model. We then generate both observation-level weights and cluster (subject)-level weights based on the estimated probabilities. It is worth noting that the uncertainty in the estimated weights will not be accounted for in the IPWGEE analysis. One way to get around this is to apply a bootstrap method.

The results obtained from the two IPWGEE methods are listed in Tables 5.9 and 5.10. It can be seen that the estimated treatment effect (β₄ = − 0.3140) of the IPWGEE1 is close to what we had before. The estimated intervention effect (β₄ = − 0.2044) of the IPWGEE2 is much smaller with a much wider confidence interval, suggesting that the estimator using cluster-level weights may not be stable in this case.

5.8.1 Joint Imputation Model

Schafer (1997a 1997) proposed a Bayesian framework for MI using a multivariate linear mixed model. An important assumption he made was that all the variables subject to missingness follow a multivariate normal distribution. We will outline this method below.

We introduce the following notations that are different from previous sections. Let y_i be an n_i × r matrix, where each row of y_i corresponds to an observation time point for subject i, and each column corresponds to a variable subject to missingness. So this “outcome” matrix includes all the variables to be imputed: not only the outcome variables for the main analysis, but also the covariates with missing values. Assume that y_i follows a multivariate linear mixed model:

(5.17)

where X_i is an n_i × p matrix of covariates that are not subject to missingness, Z_i is an n_i × q design matrix for the random components, β is a p × r matrix of the fixed effects, and b_i is a q × r matrix of the random effects. We further assume that each row of the residual matrix, ε_i, independently follows N(0, Σ), while the random effects are distributed as

independently for i = 1, ..., N. The superscript V indicates vectorization of a matrix by stacking its columns. By integration over b_i, the marginal model becomes

where I_r is the identity matrix of r dimensions and ⊗ is the Kronecker product. We partition y_i into y_i(obs) and y_i(mis), denoting the observed and missing parts of y_i, respectively. Let and . Let and θ = (β, Σ, Ψ). The Gibbs sampler updates b_i, θ, and y_i(mis) iteratively, in the following three steps:

These three steps give the sequences {θ^(t)} and , which converge in distribution to P(θ|Y_obs) and P(Y_mis|Y_obs), respectively. The first step is to draw a sample from a multivariate normal distribution. The second step is a full Bayesian analysis pretending we have the complete data. With conjugate priors on θ, it can be shown that the posterior of β is from a multivariate normal distribution, whereas the variance components Σ⁻¹ and Ψ⁻¹ are from Wishart distributions. The last step is to draw a sample from the multivariate normal distribution, because of the normality assumption of the multivariate linear mixed model. The whole algorithm is implemented in the R library PAN.

The Gibbs sampler resembles the idea of data augmentation (Tanner and Wong, 1987) in drawing random sample from the posterior predictive distribution. The major difference is that the estimation of θ is not of high priority, as the multiple imputation model (5.17) itself is a nuisance. The interest here is only in drawing samples from the posterior predictive distribution P(Y_mis|Y_obs). In practice, after a suitable number of burn-in cycles, samples from every kth iteration are stored as the imputed data, in order to remove the autocorrelation within the samples. The advantage of the joint imputation model is that proper imputation (in the sense that the imputation comes from the posterior predictive distribution) is guaranteed, as long as the multivariate linear mixed model (5.17) holds. However, it cannot handle the missingness of categorical variables.

5.8.2 Imputation by Chained Equations

For a longitudinal study with a fixed-time design, the imputation could be performed on a “wide format” of the data set, meaning that each subject takes up one row. The outcome and time-varying covariates, which are observed at different but fixed time points, are treated as different variables (columns). In this case, the multiple imputation procedure is the same as that in cross-sectional studies. The outcome variable at one time point could predict the outcome at another time point in the imputation model, which reflects the within-subject dependence. If a longitudinal study does not have the fixed-time design, the data set can only be recorded in a “long format”; that is, each row corresponds to one observation of a subject and each subject contributes multiple rows. The NACC UDS data set is such an example, since the participants visited the clinics at different times. Then the imputation model should distinguish the subject-level and observation-level variables. The general framework of chained equations described in Section 2.3.2 still works. But for observational-level variables, the conditional distribution should specify the clustering structure, often by including random effects. For subject-level variables, the imputation should be performed only once for each subject. We will demonstrate these in detail with the analysis of NACC UDS data.

Beunckens et al. 2008) compared the MI and IPWGEE approaches in analyzing binary longitudinal data with dropout. They found that in finite sample settings, MI is less biased and more precise. In addition, MI is not very sensitive to the misspecification of the imputation model. The efficiency issue of IPWGEE could potentially be addressed by augmented IPW estimators, as we will introduce in the next section. However, the implementation of the augmented IPW estimators may be quite difficult, especially when the missingness involves both the outcome and the covariates. In these cases, the MI approach is generally recommended.

5.8.3 Example: NACC UDS Data

The NACC UDS example was introduced in Section 5.2, and we now demonstrate the use of R library mice to impute and analyze this example. Recall that our aim is to explore the risk factors that affect the decline of the MMSE score. Let Y_ij be the MMSE score for subject i at the jth time t_ij; let X_ij be the vector of covariates, including baseline age, gender, education, HIS, and APOE e4. Using generalized estimating equations, we can fit the following regression model:

(5.18)

The interaction term is of primary interest, which is interpreted as the increase in the average rate of MMSE decline per unit increase in X_ij.

Four variables are subject to missingness: the outcome, MMSE score, is a continuous-scale observation-level variable; HIS is an ordinal observation-level variable; APOE e4 is a binary subject-level variable; and education is an ordinal subject-level variable. Since the interaction terms are included in the analysis model (5.18), it is recommended that they should also be included in the imputation model.

We conduct the multiple imputation in several steps:

1. Prepare the data set for imputation by appending the transformation of variables, dummy variables, and interaction terms. Initialize the imputation parameters by “dry run” with the maximum number of iteration set to 0.
2. Specify the imputation methods. A two-level linear mixed effects model (2l.norm in the R package mice) is used to impute MMSE. MMSE is bounded between 0 and 30, whereas the 2l.norm method draws posterior samples from an unbounded normal distribution. Therefore, we transform the MMSE score as to force it unbounded and impute the transformed score. The back-transformed original score is generated by a “passive imputation.” The passive imputation means that the imputed values are computed from other imputed variables, as opposed to be generated from an imputation model. For the ordinal HIS, mice cannot handle the two-level clustering, so we can choose either the predictive mean matching (pmm) or the ordinal logit model (polr) method, both of which ignore the longitudinal structure. In our example, we use the pmm method. For imputing education and APOE e4, we choose the predictive mean matching at the subject level only (2lonly.pmm in the R package mice). The observation-level variables are aggregated within each cluster to serve as covariates in the imputation model. All the dummy variables and interaction terms that involve missing data were passively imputed.
3. Specify the prediction matrix of the imputation model. Each row of the prediction matrix corresponds to an imputation model for a variable, where “1” indicates the column variable is used as a predictor, whereas 0 means it is not used. In the two-level imputation methods, “−2” indicates the clustering variable and “2” indicates the random effects. Passive imputation overrules the predictor matrix elements, so the rows of passively imputed variables can be set as any value.
4. Specify the visit scheme, that is, the order of imputation. We need to keep the passive imputations synchronized with the imputed variables. In our case, the imputation of mmse2, education, apoee4, and hachin should be updated first, and then the transformations, dummy variables, and interactions are generated.
5. Call the mice function to impute the data set.

We also try to impute MMSE using the predictive mean matching method and see whether the result changes as we ignore the within-cluster correlation. Two imputation models are implemented for m = 20 times. We set m = 20 mainly for diagnostic purposes. The imputed values are plotted in Figure 5.2, which shows that the 20 imputations are well mixed indicating healthy convergence.

We can now conduct the analysis for the 20 imputed data sets. The combination is performed using Rubin's rule. The analysis results are shown in Table 5.11. CC refers to the complete-case analysis, imputation 1 uses 2l.norm for MMSE, and imputation 2 uses pmm for MMSE. We can see that both imputation methods give similar results, and the results of CC might be a bit biased with larger standard errors. Since baseline age is centered at 75 years, the coefficient for “year since aMCI” is interpreted as the average rate of decline per year (about 0.3 point decrease every year) for a female aged 75 at the baseline, with 0–12 years of education, 0 point HIS, and no APOE e4 allele. Imputation model 2 identifies all five risk factors to affect the level of MMSE; three of the risk factors significantly affect the rate of MMSE decline, namely, age, APOE e4, and education. For example, a subject with APOE e4 allele has about 0.44 point more decrease in MMSE every year compared with a subject without APOE e4 allele.

Although in this example ignoring the clustering in the imputation model does not seem to affect the final results, this is not warranted in other settings. Future research is needed to explore the effect of misspecifying the imputation model. The linear mixed effects model is the only available build-in function for the two-level imputation in mice, but the multivariate normality assumption is quite strong in practice. If more complicated imputation models are to be used in mice, the readers should refer to van Buuren and Groothuis-Oudshoorn 2011) for the user-defined imputation functions.

5.8.4 Example: IMPACT Study

5.8.4.1 Multiple Imputation with Monotone Missingness

With monotone missingness, the joint imputation method greatly simplifies in terms of the computation load. The basic idea is that missing data are imputed sequentially, from the variable with the least proportion of missingness to the one with the most missingness. While imputing for longitudinal data, the within-subject correlation needs to be taken into account as well. In general, we recommend multivariate imputation methods such as multivariate normal or chained equations for this type of imputation task. In the Stata code, given in Appendix 5.A, baseline covariates such as age, gender, study site, and so on are first used to impute the SCL-20 score at baseline using the linear regression model, and then baseline covariates plus the SCL-20 score at baseline are used to impute the SCL-20 score at month 3, and so on until missing SCL-20 scores at month 24 are imputed. In the following code, 10 imputed data sets are generated. Estimation is performed on each imputed data set, and the results are then pooled together to produce the final result. Because the analysis model examines the interaction between time and treatment, we conduct the imputation for the subjects in the intervention and control arms separately.

The results are listed in Table 5.12. The estimated intervention effect (group × period) is slightly smaller than that estimated based on available data analysis.

Parameter	Estimate	Robust SE	95% CI
β1	1.686	0.022	(1.644, 1.729)
β2	−0.335	0.022	(−0.379, −0.291)
β3	−0.003	0.030	(−0.056, 0.062)
β4	−0.288	0.031	(−0.350, −0.227)
	0.453	0.014	(0.426, 0.482)
	0.415	0.018	(0.382, 0.451)
Corr(b1i, b2i)	−0.202	0.048	(−0.293, −0.108)

5.8.4.2 Multiple Imputation Based on Multivariate Normal Distribution

For a continuous longitudinal outcome with missing values, another approach for missing data imputation is to first assume a multivariate normal distribution on the outcome vector and then to proceed with the imputation using the likelihood approach. The details have been provided in Schafer (1997b 1997). Also note that this imputation approach works for arbitrary missing data patterns. For categorical outcome with missing values, latent variable models could be employed with this imputation approach.

Table 5.13 shows the results of the multiple imputation method based on MVN. It can be seen that the results are close to those based on the multiple imputation with monotone missingness (Table 5.12).

Parameter	Estimate	Robust SE	95% CI
β1	1.686	0.022	(1.644, 1.729)
β2	−0.338	0.023	(−0.383, −0.293)
β3	−0.003	0.030	(−0.056, 0.062)
β4	−0.286	0.032	(−0.348, −0.224)
	0.454	0.014	(0.427, 0.482)
	0.422	0.019	(0.386, 0.462)
Corr(b1i, b2i)	−0.207	0.049	(−0.301, −0.109)

5.8.4.3 Multiple Imputation Based on Chained Equations

We next apply the multiple imputation with chained equations (MICE) approach to the IMPACT data set with monotone missingness. The results are shown in Table 5.14, from which it can be seen that the results are similar to the results based on the above two multiple imputation methods. It is worth noting that MICE applies to general missing data patterns. The results are very similar to those from both monotone MI and MVN MI.

Parameter	Estimate	Robust SE	95% CI
β1	1.686	0.022	(1.644, 1.729)
β2	−0.335	0.022	(−0.379, −0.291)
β3	−0.003	0.030	(−0.056, 0.062)
β4	−0.288	0.031	(−0.350, −0.227)
	0.453	0.014	(0.426, 0.482)
	0.415	0.018	(0.382, 0.451)
Corr(b1i, b2i)	−0.202	0.048	(−0.293, −0.108)

5.8.4.4 Multiple Imputation with Chained Equations for a General Missing Data Pattern

We now illustrate MICE when there are both the missing outcome and missing covariates with a general missing data pattern. In The IMPACT data, not only there are missing values in the SCL-20 score, there are also missing values in baseline number of chronic diseases (numdis1), preference on depression treatment (pref00), race (white), baseline general health status (ghlth00), and income (inc400). It is conceivable that we could do a better imputation of the missing outcome if we can gather more information on these important covariates. Specifically, we used logistic regression to impute race, ordinal logistic regression for treatment preference and general health status given they are multilevel categorical variables, Poisson regression model for a number of chronic conditions, predictive mean matching for income, and finally linear regression for SCL-20 scores. The results are shown in Table 5.15. It can be seen that for the IMPACT study, the results using all four multiple imputation methods generally agree.

Parameter	Estimate	Robust SE	95% CI
β1	1.672	0.020	(1.632, 1.712)
β2	−0.315	0.022	(−0.358, −0.273)
β3	−0.010	0.029	(−0.046, 0.066)
β4	−0.289	0.031	(−0.349, −0.227)
	0.448	0.014	(0.422, 0.476)
	0.419	0.019	(0.384, 0.457)
Corr(b1i, b2i)	−0.206	0.047	(−0.295, −0.112)

From Tables 5.5–5.7, 5.9, 5.10, and 5.12 –5.15, it is seen that for the IMPACT study, all the missing data methods show similar results, except the IPWGEE1. In general, we recommend the mixed effects model and multiple imputation methods for longitudinal data analysis with missing data. This is also the recommendation given by Schafer and Graham 2002).

5.10.1 Imputing Estimating Equations

Paik (1997) proposed an imputation approach that could handle the dropout of the outcome variable. The approach is based on generalized estimating equations, and the equations with missing observations are replaced by their conditional expectations. This idea is the same as the “mean score imputation” in the cross-sectional setting (Chapter 4). Two important assumptions are as follows: (a) The longitudinal observations are all scheduled at fixed time points, j = 1, ..., M. (b) The dropout process depends only on observed variables, that is, the MAR dropout process.

Let be the full vector of outcome variables for subject i and be the full matrix of corresponding covariates. The primary interest is the regression model of . The observed data for subject i are from the first n_i visits: and . Let H_ij = (X_i, Y_i1, ..., Y_ij) be the history of ith subject up to time j. Let R_ij be the missing indicator of Y_ij, similar as in the previous sections.

The full data estimating equation can be written as

where and is a working covariance matrix for . Paik's approach uses to replace those missing Y_ij (j > n_i). Then, the imputed estimating equation becomes

(5.20)

where . The expectation of equation (5.20) is 0, which leads to a consistent estimator of β. Now the crucial step is to estimate the conditional expectation .

Note that under MAR dropout,

(5.21)

So, a regression model can be fitted to the observed data, in order to estimate the first missing observation after dropout. For the second missing observation after dropout, we have

(5.22)

Hence, another regression model can be fitted to estimate the conditional expectation E(Y_ij|H_i,j−2, R_i,j−1 = 1). Now the regression model includes not only the observed data (those with R_i,j−1 = 1 and R_i,j = 1) but also the previously imputed observations (those with R_i,j−1 = 1 and R_i,j = 0). In other words, in fitting the regression models to impute the second missing observation, the imputed value of the first missing observation is also used as the outcome. If the previous imputation is consistent, the current regression model is also consistent. The same techniques can be repeated for the third potential observation, and so on, until all the missing data are replaced by the fitted conditional expectations. Eventually, the consistency of estimating is guaranteed in this procedure, if all the imputation models are correctly specified.

The whole procedure can be illustrated in Figure 5.3. The rows represent the missing patterns and the columns represent the observation times. The shaded cells are the missing observations that are to be imputed. We put each one of the subjects into a row according to their dropout times. The n_i observations for subject i enter the n_i cells in its row. The first diagonal of missing values is imputed in the first place, then the second diagonal, and so on. For the imputation of a cell c, the regression model uses all the cells in the same column but below cell c. For example, when imputing cell (3, 4) in the first diagonal, a regression model is fitted to observations in cells (4, 4) and (5, 4)—both are observed cells. When imputing cell (2, 4) in the second diagonal, not only cells (4, 4) and (5, 4) but also the imputed cell (3, 4) will enter the regression model as outcomes. The design matrix, however, is the history up to the second time point (H_i,j−2 in (5.44) is now H_i2), which does not involve any imputed cell or missing cell. The sequential imputation of the diagonals will eventually generate the complete data, and the estimation of the regression coefficients is followed by solving equation (5.20).

The explicit variance formula is given in Theorem 1 and Appendix B of Paik 1997). The variance indeed comes from two sources: one is from the estimating equation (5.20) while pretending all the imputed data were observed; the other one is from estimating the additional regression models for imputation.

In Paik's mean score imputation, the missing variables are replaced with the fitted values because the full data estimating equations are linear in the missing Y_ij. In the multiple imputation, as a comparison, the missing Y_ij are replaced with a random draw from the posterior predictive distribution. Under the correct imputation model, the multiply imputed values would be close on average to the fitted value in the regression model, that is, Paik's mean score:

when M gets larger. In a sense, Paik's mean score imputation is the limit situation of the multiple imputation. As shown in the simulation results of Paik 1997), the results of Paik's imputation and multiple imputation are similar in many settings.

5.A.4.1 Stata Code Used in Section 5.5.1.1

The following code conducts the complete-case analysis for the IMPACT data:

5.A.4.2 Stata Code Used in Section 5.5.1.2

The following code is to conduct the available data analysis for the IMPACT study:

5.A.4.3 Stata Code Used in Section 5.5.1.3

The following Stata code illustrates the LOCF analysis for the IMPACT study:

5.A.4.4 Stata Code Used in Section 5.6.4

Here we present the Stata code for the IMPACT study using IPWGEE1 and IPWGEE2. In the following code, we estimate the probability that an outcome is observed at a specific time point, given the prior observed outcome variables and some selected baseline characteristics using logistic regression, and then generate both observation-level weights and cluster (subject)-level weights based on the estimated probabilities.

The following code implements IPWGEE1 using observation-level weights. Because this approach is not currently implemented in Stata, a workaround using glm command is implemented.

The following code then performs IPWGEE2 using cluster-level weights:

5.A.4.5 R Code Used in Section 5.8.3

Here, we present the R code for implementing multiple imputation analysis for the NACC UDS data. The package mice is used. The following code appends the transformation of variables, dummy variables, and interaction terms.

> attach(NACCUDS)
> NACCUDS2=cbind(NACCUDS,cons=1,logcdr=log(cdr+1),
+ mmse2=logit((mmse+0.5)/31),edu2=(education==2)+0,
+ edu3=(education==3)+0,hachin2=(hachin==2)+0,
+ hachin3=(hachin==3)+0,age.year=age0*year,
+ gender.year=gender*year,edu.year2=(education==2)*year,
+ edu.year3=(education==3)*year,his.year2=(hachin==2)*year,
+ his.year3=(hachin==3)*year,ap.year=apoee4*year,
+ mmse.year=mmse*year)
> detach()

A “dry run” with the maximum number of iteration set to 0 is the first step to initialize the imputation parameters.

> ini=mice(NACCUDS2,maxit=0)

Then the imputation methods are specified.

> meth=ini $dollar $ meth
> meth[”mmse2”]=”2l.norm”
> meth[”mmse”]=”~I(round(1/(1+exp(-mmse2))*31-0.5))”
> meth[”hachin”]=”pmm”
> meth[”education”]=”2lonly.pmm”
> meth[”apoee4”]=”2lonly.pmm”
> meth[”edu2”]=”~I(education==2)”
> meth[”edu3”]=”~I(education==3)”
> meth[”hachin2”]=”~I(hachin==2)”
> meth[”hachin3”]=”~I(hachin==3)”
> meth[”edu.year2”]=”~I((education==2)*year)”
> meth[”edu.year3”]=”~I((education==3)*year)”
> meth[”his.year2”]=”~I((hachin==2)*year)”
> meth[”his.year3”]=”~I((hachin==3)*year)”
> meth[”ap.year”]=”~I(apoee4*year)”
> meth[”mmse.year”]=”~I(mmse*year)”

The next step is to specify the prediction matrix of the imputation model. The following is a print of our specified prediction matrix:

> pred[c(”mmse2”,”education”,”apoee4”,”hachin”),]
 mmse age0 gender education apoee4 year cdr hachin ID
mmse2 0 1 1 0 1 1 0 0 -2
education 1 1 1 0 1 0 0 0 -2
apoee4 1 1 1 0 0 0 0 0 -2
hachin 1 1 1 0 1 1 0 0 0
 cons logcdr mmse2 edu2 edu3 hachin2 hachin3 age.year
mmse2 2 1 0 1 1 1 1 1
education 0 1 0 0 0 1 1 0
apoee4 0 1 0 1 1 1 1 0
hachin 0 1 0 1 1 0 0 1
 gender.year edu.year2 edu.year3 his.year2 his.year3
mmse2 1 1 1 1 1
education 0 0 0 0 0
apoee4 0 0 0 0 0
hachin 0 1 1 0 0
 ap.year mmse.year
mmse2 1 0
education 0 0
apoee4 0 0
hachin 1 1

The visit scheme specifies the order of imputation. We need to keep the passive imputations synchronized with the imputed variables.

> vis=ini $dollar $ vis
> vis[”mmse2”]=1
> vis[”mmse”]=12
> vis
 mmse education apoee4 hachin mmse2 edu2
 12 4 5 8 1 13
 edu3 hachin2 hachin3 edu.year2 edu.year3 his.year2
 14 15 16 19 20 21
his.year3 ap.year mmse.year
 22 23 24

We also try imputing MMSE using the predictive mean matching method. The pmm does not make any distributional assumption, but only requires a mean model with identity link.

> meth2=meth
> meth2[”mmse2”]=”pmm”
> pred2=pred
> pred2[”mmse2”,”ID”]=0
> pred2[”mmse2”,”cons”]=0

The following code generates 20 complete data sets with missing values imputed by the two methods:

> imp2=mice(NACCUDS2,m=20,me=meth2,pred=pred2,vis=vis,maxit=20,
+ seed=10065)
> imp1=mice(NACCUDS2,m=20,me=meth,pred=pred,vis=vis,maxit=20,
+ seed=7523)
> plot(imp1,c(”hachin”,”education”,”apoee4”,”mmse”),
+ layout=c(2,4))

The analysis for the 20 imputed data sets is conducted by the following code.

> library(gee)
> fitC=gee(mmse~(I((age0-75)/10)+gender+apoee4+
+ as.factor(education)+as.factor(hachin))*year,
+ corstr=”independence”,id=ID,data=NACCUDS)
> fit1=with(imp1,gee(mmse~(I((age0-75)/10)+gender+apoee4+
+ as.factor(education)+as.factor(hachin))*year,
+ corstr=”independence”,id=ID))
> fit2=with(imp2,gee(mmse~(I((age0-75)/10)+gender+apoee4+
+ as.factor(education)+as.factor(hachin))*year,
+ corstr=”independence”,id=ID))

The code for combining fit1 using Rubin's rule is given below.

> PE1=VAR1=matrix(0,20,16)
> for(i in 1:20)
+ PE1[i,]=(fit1 $dollar $ analyses[[i]]) $dollar $ coef
+ VAR1[i,]=summary(fit1 $dollar $ analyses[[i]]) $dollar $ coef[,4]supstri 2
+
> COEF1=apply(PE1,2,mean)
> U1=apply(VAR1,2,mean)
> B1=apply(PE1,2,var)
> SE1=(U1+(1+1/20)*B1)supstri .5
> RES1=cbind(COEF1,SE1)

5.A.4.6 Stata Code Used in Section 5.8.4.1

The following Stata code is for the analysis of multiple imputation with monotone missingness:

5.A.4.7 Stata Code Used in Section 5.8.4.2

The following is the Stata code for multiple imputation based on the multivariate normal distribution for the IMPACT data:

5.A.4.8 Stata Code Used in Section 5.8.4.3

The multiple imputation with chained equations is applied to the IMPACT data and achieved by the following chunk of code:

5.A.4.9 Stata Code Used in Section 5.8.4.4

The following is the Stata code to conduct analysis of IMPACT data based on the multiple imputation with chained equations for a general missing data pattern. We recommend readers to start by following the steps listed in the following code chunk. One tip is to use “mi describe” often so that we can keep a close eye on our imputation.