Multivariate responses that are non-commensurate are common in the health sciences. For example, the multivariate response can consist of distinct continuous and categorical outcomes that are neither measured on the same scale, nor measure the same underlying variable (see, for example, [1]). On the other hand, the outcomes can be distinct but yet measured on the same scale. An example of the latter occurs in studies of AIDS, where CD4 and CD8 cell counts may be obtained longitudinally resulting in repeated measures of a bivariate outcome [2]. In this case, the two outcomes are both measured in terms of counts but are distinct markers of immune function (i.e., they are not commensurate). Another example where the outcomes are measured on the same scale, but are not commensurate, is repeated measures of systolic and diastolic blood pressures [3]. When the multiple outcomes are not commensurate, it is unlikely that the magnitudes of covariate effects on each outcome are similar, and thus separate regression models for each outcome will ordinarily be required.

With commensurate outcomes, we use the term “multiple source” to indicate that each response comes from a different source but provides a measure of the same underlying variable (i.e., each source represents a different component of the multivariate response). Longitudinal multiple source data arise in many areas of application, e.g., environmental studies where a pollutant is measured at varying depths in a lake through time [4], or in studies of bilateral organ systems. For example, Heitjan and Sharma [5] analyzed longitudinal bivariate data on intra-ocular pressure in both eyes. Studies of children are another common area where longitudinal multiple source outcomes arise. Information on children’s psychopathology may be gathered from multiple informants, or multiple psychiatric instruments, repeatedly through time [6]. When the multivariate responses are commensurate, certain covariates may have similar effects on two or more of the outcomes. Due to both the commensurate nature of the responses, and the positive correlation among the source outcomes, it is advantageous to analyze the outcomes jointly rather than separately. A joint analysis will often result in an efficiency gain, but more importantly it provides a formal basis for the comparison of covariate effects across sources.

One approach to the joint analysis of multivariate longitudinal data is to simply reduce the multivariate responses to a single outcome. When the responses are continuous, this could be achieved by taking the average of the multiple responses, resulting in a single response at each occasion. As Heitjan and Sharma [5] point out, however, some information will be lost, and this approach can be problematic when there are missing data. That is, ad hoc methods for calculating the mean when there are missing data effectively make strong assumptions about the missingness mechanism — namely that the missingness mechanism is missing completely at random (MCAR) [7].

In this paper, we focus on joint modelling of longitudinal multiple source Gaussian data via multivariate linear regression. The development of the methods in this paper was motivated by data arising from an interventional trial in psychiatry [8]. The aim of this study was to compare two interventional strategies to prevent the “social transmission” of affective disorders from a parent to other family members. Forty-nine families were randomized to an intervention consisting of lectures given to both parents to educate them on how to prevent such transmission from occurring. Sixty-four families randomized to the other interventional program attended clinician-facilitated counselling sessions. In order to assess whether the clinician-facilitated program was more effective than the lecture program, reports of family functioning were obtained from three sources — father, mother, and child. Family functioning was determined via a self-report measure, and was obtained from each source at five equally spaced times over a three-year period. Thus, each source provides a measure of the same underlying variable (family functioning) at each measurement occasion, and the analytic goal is to relate changes in family functioning to the intervention assignment. Much of the previous work on modelling multivariate longitudinal data, however, has focused on outcomes that are not commensurate (e.g., [1] and [2]). While there is some related work on longitudinal models for multiple source data, it has tended to make quite strong assumptions about the structure of the covariance among the outcomes (e.g., [3] and [5]).

Here, we present flexible regression models for longitudinal multiple source data that allow for more general covariance structures. We propose a joint model for the mean response vector. Jointly estimating the regression parameters of this model has the advantage of being more efficient than separately estimating the parameters for each source when there are common, or shared, effects across sources for one or more covariates [9]. This may often be the case with multiple source data since the outcomes are commensurate. An additional advantage of joint estimation, as mentioned earlier, is that it allows formal comparison of covariate effects across sources.

With a joint model, we also need to be concerned with specifying parsimonious models for the covariance matrix. While the model for the mean can have a large number of parameters, the proliferation of covariance parameters is of greater concern as the number of responses increases. In many longitudinal studies, the covariance parameters are considered a nuisance. Recognizing that the covariance parameters are often a nuisance, one possible approach is to utilize a “working independence” assumption that considers the observations to be independent for the purposes of estimation, but base inference on standard errors of the regression coefficients that are estimated via the empirical (or “sandwich”) covariance estimator [10]. This avoids explicit modelling of the covariance among outcomes. For many longitudinal designs, the loss of efficiency associated with the use of the working independence estimator of the regression parameters is modest. Use of the empirical covariance estimator is not desirable in all situations though. For example, Kauermann and Carroll [11] showed that the variability of the empirical covariance estimator is larger than that of a model-based covariance estimator in certain circumstances. In particular, this occurs when the sample size is modest, and when the covariate design matrices are not identical across subjects. They also showed that confidence intervals for the regression parameters generated using the empirical covariance estimator can result in undercoverage in such cases. Additionally, use of the empirical covariance estimator can be undesirable when there are many missing responses since it relies on replications across subjects to estimate the covariance non-parametrically [12]. In practice, many studies with longitudinal multiple source data have modest sample sizes, subject-varying design matrices, and missing data, making use of the empirical covariance estimator undesirable. The use of a suitable model-based covariance estimator can avoid such problems and is preferable.

Each family was randomly assigned to one of two preventive treatment strategies designed to improve family functioning. One strategy consisted of two 1-hour lectures given to the parents. The lectures were uniform and were given by a trained clinician. The other treatment consisted of clinician-facilitated psycho-educational sessions involving the entire family. These sessions were tailored to the individual families’ educational levels and consisted of 6 to 10 sessions. Sixty-four families were assigned to the clinician-facilitated group and 49 to the lecture group. Both treatment groups also received literature designed to educate them about affective disorders, and how parents a²icted with them can prevent their children from feeling the effects themselves. The families were assessed at baseline and at 4 approximately equally spaced times over the next 3 years. The aim of the study was to determine whether the clinician-facilitated treatment strategy was more efficacious in improving the functioning of the families over time.

In families with multiple children, one child was randomly selected for inclusion in the analysis. Thus, each family could have as many as 15 reports (3 sources reporting at 5 times). In all, approximately 30% of the responses were missing, but there was no reason to believe that the missingness was nonignorable. Throughout we assume that the missing responses are missing at random (MAR) [7]. The psychometric measure that was used to obtain family functioning information was the Family Relationships Index (FRI). The FRI is a self-report measure that uses three subscales — cohesion, expressiveness, and conflict — to obtain a final composite score. The conflict subscale is weighted negatively in the final score determination (see [18] and [19], for more information about the FRI). The FRI composite score is a continuous measure that was observed to be approximately normally distributed. Here, the unit of analysis is the family and the multiple “sources” are the father, mother, and child.

Note that if an unstructured covariance matrix was assumed, we would need to estimate 120 covariance parameters in addition to the regression coefficients. Given data on only 113 independent families, estimation may be unstable. Thus, reducing the number of covariance parameters that need to be estimated is necessary. We considered covariance structures resulting from both the Kronecker product covariances and the random effects models.

Figure 2(a) shows the mean observed FRI scores for each source through time for the clinician-facilitated group, and Figure 2(b) shows the mean observed FRI scores for the lecture group. The final model for the mean was obtained via a stepwise procedure utilizing likelihood ratio tests. The initial model that was fit to the mean was the saturated model with main effect terms for source, categorical time, treatment, and all possible two-way and three-way interactions. A quadratic time trend was found to provide an adequate fit to these data. A test of the null hypothesis of no treatment effect (i.e., a joint test of treatment-by-time and treatment-by-time-by-source interactions) was not significant (likelihood ratio = 6.1; p = 0.41). An additional test of collapsibility was conducted (i.e., a joint test of treatment main effect, treatment-by-time, treatment-by-source, and treatment-by-time-by-source interactions) and was also not significant (likelihood ratio = 7.79; p = 0.56). Thus the final model consists of a source main effect, a quadratic time trend, and both source-by-time interaction terms. That is,

Thus, no significant treatment effect was found to exist in these data, suggesting that there is no significant difference between the two interventions. The ML estimated means for each source are plotted in Figure 3. It appears that family functioning improved more rapidly at the beginning of the study, and then levelled off. The fathers’ and mothers’ reports tracked each other, while the children’s reports displayed a somewhat different trend. Although there were no differences in the effects of the interventions, family functioning improved over time.

The covariance structure first fit to these data was of the Kronecker product class. Note that fathers and mothers show a general increase in FRI scores through time. However, the mean response for children is more erratic. Analyses suggested that the three sources (i.e., father, mother, and child) were not exchangeable and that a Toeplitz correlation structure held over time. Due to these considerations, the Kronecker product structure that seemed most appropriate for these data considered the source marginal covariance as unstructured and the covariance matrix for the measurement occasions as Toeplitz with heterogeneous variances. This results in 5 parameters for the source covariance matrix, 8 parameters for the measurement occasions covariance matrix, and an overall scale factor for a total of 14 parameters.

A random-effects model was also assumed for these data. Recall that the exchangeability of informants assumption does not seem tenable; we also wanted to allow each source to have a random trajectory within family. Thus random source effects were drawn from a multivariate normal distribution, allowing each source to have a random intercept and trajectory, and also allowing correlation among the intercepts and trajectories for the different sources. The model can be written as,

where (η _i10, η _i11, η _i20, η _i21, η _i30, η _i31)′ ~ MV N(0, G) and Thus the multivariate normal distribution from which the random source effects were drawn was six dimensional. Note that G is assumed to be unstructured allowing the random source effects and random trajectories to be correlated. Since the psychometric reports obtained from children are often more variable than those obtained from adults [20], we also allowed the random measurement error to depend on source; for these data there was no a priori reason for doing so.

where g_i indicates the group assignment for subject i with i = 1, 2,…, N ; j = 1, 2, 3; k = 0, 1,…, T. The indicator function 1_c equals unity if condition c is true and zero otherwise. The coefficients for the source-specific random intercept and trajectory are given by (η_0ij, η_1ij)′, and the random measurement error is given by _ijk. These random coefficients are distributed as, (η_0i1, η_1i1, η_0i2, η_1i2, η_0i3, η_1i3) ~ MV N(0, G) and _ijk ~ N (0, σ²), with known G and σ². While several different covariance structures, for different choices of G and σ², were considered, we present results from a model that induces an overall association structure with moderate correlations, similar to that found in the family functioning data analyzed in Section 3. The inter-source correlations range from 0.15 to 0.40, the inter-source correlations range from 0.25 to 0.45, and the cross-correlations range from 0.15 to 0.35.

For each replication of the simulation, data were generated under this model and the fixed-effects (β) were estimated along with standard errors of the estimates obtained via both a model-based variance estimator and the empirical variance estimator under a correctly specified covariance model. This procedure was repeated 1000 times with samples of size N = 100 and N = 500. The medians of the model-based and empirical standard errors for the estimates of β₅ are given in Table 4 for sample sizes of 100 and 500, and for T = 3, 5, and 8 measurement occasions (giving 9, 15, and 24 repeated measures, respectively). The true values of the standard errors of β₅ are also given for each case. As can be seen, the medians are discernably different — particularly for the smaller sample size — with the model-based medians being closer to the true values of the standard errors. The discrepancy between the medians of the model-based and empirical standard error estimates tends to be larger as the number of repeated measures increases and decreases with an increase in sample size. In general, when the repeated measures-to-independent subjects ratio increases, there are discernable differences between the medians. In all cases, the center of the distribution of the empirical variance estimates is larger than the corresponding distribution of the model-based estimates. This was not observed for the other correlation models considered, however.

The distributions of the model-based standard errors were symmetric, however the distributions of the empirical standard errors were positively skewed. The skewness was more pronounced as the number of repeated measures per subject was increased. The top of Figure 4 illustrates the distribution of standard errors for model-based and empirical standard errors with three sources and three measurement occasions and a sample size of 100. The bottom of Figure 4 illustrates simulation results from the same mixed-effects model, but with a sample size of 500. It should be noted that data generated for these simulations consisted of almost independent repeated observations on each subject. It is apparent that the empirical standard errors have considerably more variability than the model-based, even when the number of subjects is relatively large compared to the number of repeated measures.

Table 5 gives the range of values for the standard errors between the 15^th percentile and the 85^th percentile. Since some distributions of the empirical standard errors were bimodal, using the interquartile range did not accurately reflect the variability of the standard error estimates in some cases. The empirical variance estimator shows poor performance, even as the number of measurements per subject increases. Increasing the sample size from 100 to 500 did not result in a discernable change in this performance.

There has been some previous work concerning the analysis of longitudinal multivariate data that has addressed the problem of the proliferation of covariance parameters. A model for multivariate longitudinal data was proposed by Carey and Rosner [3] that addressed this issue, achieving a parsimonious covariance structure through a set of somewhat restrictive assumptions about the relationships among outcomes. They assumed that the inter-source correlations and the variances are time invariant (however, they could differ by source). Additionally, they assumed that the intra-source and cross-correlations follow a damped autocorrelation structure, with a form that is similar to corr(Y (t), Y (t + s)) = γ^|s|θ; where γ is a time-invariant damping factor. Rochon [1] also considered models for multivariate longitudinal outcomes, with possibly incomplete data that are MCAR, using generalized estimating equations (GEEs); this approach could be extended to handle data that are MAR using the approaches of Robins et al. [21] and Paik [22]. The covariance structures Rochon [1] considered were of the vector autoregressive moving average (VARMA) form often used for multivariate time series data. Such structures can result in complex covariance patterns, but require that the intra-source and cross-correlations decay through time at a certain rate. Similar to the covariance structures considered by Carey and Rosner [3], these may not be useful to fit a general class of longitudinal multiple source models where the decay in the correlation levels off after a certain time lag.

Mixed-effects models have also been employed for the analysis of multivariate longitudinal data. Shah et al. [2] discussed models for non-commensurate outcomes in which the measurement errors for the same source at any two occasions are uncorrelated, but measurement errors between two different sources at the same occasion are correlated. In addition to correlated random measurement error, a random subject effect was included. The resulting structure assumed that outcomes at any given time were exchangeable (_jj′k = constant), and that the inter-source and cross-correlations were equal. Through the introduction of a random trajectory over time for each subject, these correlations would become parametric functions of time, but would still retain the basic constraints described. Reinsel [23] specified growth curve models for multivariate repeated measures data. The covariance structure specified by these growth models satisfied a more general multivariate sphericity condition described by Boik [24]. The structure described in Reinsel [23] would allow the variances and intra-source correlations to depend on the source under consideration, but are time invariant. The inter-source and cross-correlations would also depend on the two sources in question, but would also be time invariant. This assumption may not be plausible with longitudinal multiple source data. Finally, Heitjan and Sharma [5] extended the linear random effects/autoregressive linear model of Chi and Reinsel [25] to analyze longitudinal outcomes from two sources. Their model included a random subject effect and the resulting marginal covariance structure assumed the sources are exchangeable. The intra-source correlations have an autocorrelated structure, and the cross-source (τ_jj′kk′) correlations have a damped autocorrelation structure. In all of the previous work, the proposed covariance structures are plausible in a variety of situations, but make strong assumptions about the relationships among outcomes.

The linear multivariate regression model that we have presented addresses the issue of how to achieve a parsimonious covariance structure via two methods. The Kronecker product structures are useful in that they only require that the multiple source and longitudinal covariance structures be specified separately. They have the desirable property that the cross-correlations are no longer estimated separately since they are products of the marginal intra-source and inter-source correlations. The number of variance parameters that need to be estimated is also reduced. However, their appropriateness in any application will depend on whether the underlying conditional independence assumption is tenable. Additionally, the assumptions of the source invariance of the covariance among measurement occasions, and the time invariance of the covariance among sources must also be tenable. This has been found to hold in a variety of spatial and environmental applications (see [13], [26], and [27] for examples).

We have also discussed the use of mixed-effects models that induce relatively parsimonious structures on the covariance among outcomes. These models are remarkably flexible and can accommodate any degree of complexity. To be most useful, these models will often involve either random source effects nested within random subject effects, or correlated random source effects.