Methods for analyzing observational longitudinal prognosis studies for rheumatic diseases: a review & worked example using a clinic-based cohort of juvenile dermatomyositis patients

The GEE, which determines the mean population disease activity trajectory, is frequently used to answer this kind of question [7]. The GEE calculates the average DASm of the whole population at each visit. These population averages are then joined to make a “trajectory” of DASm for the whole population over time. The GEE assumes that the measurement schedule is unrelated to the outcome, i.e., both sick and well patients are presumed to have the same visit schedule on average. As sicker patients were likely seen more frequently than those who were well, this assumption was probably not supported in our clinic-based cohort. However, there have been extensions of the GEE (the weighted GEE) that allow GEE to be used even when the measurement schedule is related to the outcome [29].

The GEE is the most popular method used for longitudinal analysis for several reasons [30]. With the GEE, the mean population response at each occasion is modelled as the result of only the prognostic factors of interest and not of previous responses or random effects (individual heterogeneity) [9]. If the researcher’s interest is in the average population prognosis, then the GEE provides a simple interpretation [31].

We will briefly discuss the results when we used the GEE to analyze our study cohort. From the smoothed local regression plot of Fig. 1 [32], we can see that the average population DASm decreases rapidly and then plateaued to a lower degree of disease activity. This is clinically sensible as there are effective treatments for lowering disease activity in JDM, yet many patients remain chronically active.

In summary, the GEE is a good choice for analyzing studies where the overall population prognosis is the primary interest, e.g., when addressing questions about population trends or healthcare utilization.

The best way to address this question is to use the MRM, a so-called subject-specific model, that allows for potential inference at the level of the individual [1, 33]. For a continuous outcome like DASm, the MRM also provides us with the average population trajectory. The smoothed plot (Fig. 2) shows the DASm trajectories of both the individual patients (gray lines) and the mean population trajectory (bold black line) [32]. As the outcome is continuous, fitting the MRM resulted in a similar longitudinal trajectory to that derived from the GEE. However, if the outcome is not continuous, calculating the mean population trajectory from the MRM is very complex.

In the MRM, each individual’s trajectory is different from the mean population trajectory because of “random effects.” Random effects are a combination of unmeasured prognostic factors, confounders, environmental factors and genetic factors, which account for heterogeneity within the population [31]. Any difference between an individual’s trajectory and the average population trajectory is the result of that individual’s random effects; this forms the basis for individual level inference from the MRM. Random effects may be applied at intercepts, which represent between-individual differences. When random effects are applied to slopes, this allows individuals’ time trend to deviate from the average population trajectory. Although the random effects are taken into account in modeling, the results reported are that of the mean population trajectory in this case.

Although the results of the MRM may appear much like the GEE in continuous outcomes models, the interpretations are very different. The MRM models the average of individuals’ trajectories (subject specific) but the GEE models the trajectory from DASm averages at each time point (population average). Practically, in the context of the continuous outcomes models, the forms and the results from the 2 models are very similar. The differences between MRM and GEE are obvious when the outcomes are non-continuous, e.g., binary or ordinal [9, 34‐36]. Taking the example of a random intercepts (logistic) MRM, where the outcome could be disease remission (yes/no), and a time-invariant covariate could be calcinosis at baseline (yes/ no), the odds of remission would correspond to the effect of calcinosis plus an individual’s random effects. The effect of calcinosis therefore changes with different individuals. The odds ratios estimated by the logistic MRM are subject specific as these additionally adjust for heterogeneity between individuals. In the binary outcome GEE, the odds ratio corresponds to odds of an event among those with baseline calcinosis divided by the odds of an event in those without. As these are ratios of subpopulation risks, the GEE estimates are termed population average. From this, it should also be obvious that the results of non-continuous MRM will always be greater that than of non-continuous GEE due to inclusion of the random effects. The choice of model depends on the purposes of the study. If the mean prevalence of disease remission in a population over time predicted by baseline calcinosis is of interest, the GEE is suitable. If the investigators want to study individuals’ risk factors for personal predictions, then the MRM is the model of choice. As the estimates of the MRM are adjusted for random effects (unobserved individual characteristics), the results reflect the effect of baseline calcinosis status of an individual with a specific random effect (or individuals with the same random effect).

The models used in the preceding sections assumed that the shape of patient trajectories were homogenous, i.e., all patients follow the same disease course. If the investigator suspects that his study population contains heterogeneous patient trajectories, then latent class trajectory analysis (LCTA) (Additional file 1: Appendix) can be used to identify distinct subgroups with different disease trajectories (those with a clinically distinct prognosis) [10, 37].

When we applied LCTA to our cohort, we found three distinct subclasses. Each of these three subclasses has a class-specific average population disease activity trajectory, derived using individuals’ trajectories (with the MRM as in question 2a). From Fig. 4, we can see that Class 2 patients have high disease activity at diagnosis, and then rapidly improve. Class 1 patients have moderate disease activity at diagnosis that improves gradually to low disease activity over time. Class 3 patients have high disease activity at diagnosis that improves very gradually. For this kind of model, we determine a probability of belonging in each of the three classes but for each individual, but the probability of any individual of belonging to each class varies. When classified according to the highest probability of class membership for each individual, 42% are in class 1, 55% in class 2 and 3% in class 3. None of the patients was classified into 2 classes with similar probabilities, i.e., was ambiguous in the class membership probability (see Additional file 1: Appendix).

The LCTA is very flexible (Additional file 1: Appendix): 1) More than one and more than one kind of outcome trajectory can be modelled at a time (see next section); and 2) Trajectories of different subclasses can take on different shapes (reflecting different patterns of outcome evolutions) [37‐39].

The LCTA allows us to study the effects of time-invariant baseline factors, such as bDAS, in predicting membership in distinct subgroups. We use the bDAS to predict membership in the three subclasses. The higher the bDAS, the less likely is a patient to belong in class 1 (OR 0.11, 95% CI 0.02-0.63) or 2 (OR 0.27, 95% CI 0.05-1.41) compared to class 3. Furthermore, the higher the bDAS, the less likely is a patient to belong to class 1 (OR 0.41, 95% CI 0.25-0.68) compared to class 2. This means that those with highest bDAS are most likely to belong to class 3, with chronic high-grade disease activity. High bDAS patients are less likely to improve substantially or go into low disease activity states unlike those whose initial bDAS are low to moderate. This may appear different from the results of MRM where high bDAS predicted quicker resolution. In MRM, the whole population was studied together although there was significant heterogeneity within the population. LCTA allows us to group patients into more homogeneous subgroups and then more precisely clarify the effect of potential baseline membership predictors.

Time-varying covariates can also be studied in LCTA and can be formulated in different ways depending on the underlying question [40, 41]. As the computations are very complex and interpretations challenging, we chose to leave out time-varying covariates for this review.

In summary, if a researcher suspects that there are several prognostic subgroups within the study population, the LCTA can be used to identify these subpopulations. Baseline factors may identify patients’ memberships in different prognostic subpopulations, thus allowing more individualized management. Time-varying covariates can also be studied but interpretations are more complex (Additional file 1: Appendix).

In this case, we have two potential outcomes: the modified skin (SDAS) and musculoskeletal (MDAS) components of JDM. We are interested in how the disease courses evolve over time and the correlation of disease activity in the two components of JDM over time. The best way to study how two different outcomes evolve simultaneously over time is a joint model as shown in Fig. 5 (Additional file 1: Appendix) [12‐14, 42, 43]. The joint model refers to the concurrent modeling of two outcome trajectories (SDAS and MDAS) in one model. The basic model used here to model the trajectories is also the MRM.

The two disease components of JDM follow different trajectories. The MDAS improves more rapidly (within the first 10 months after diagnosis) to minimal activity. In contrast, the SDAS improves less rapidly and persists at a low activity level for almost 20 months (Fig. 5). The two disease components therefore follow discordant courses, as confirmed by the low correlation between the two trajectories (r = 0.32) [13]. This relationship between two or more outcome trajectories can only be studied using the joint model and not by using separate models for the individual components.

In our example above, both outcomes were treated as continuous outcomes. However, outcomes of different natures can be jointly modeled. We can model a continuous outcome (e.g., DASm), with a time-to-event outcome (e.g., calcinosis), a binary outcome (e.g., nailfold capillary abnormality) or a count (e.g., active joint count). Furthermore, as alluded to previously, more than one outcome can be jointly studied in the LCTA; in this case, members within the same subclass of joint outcomes would have similar trajectories or risk of events [38, 39, 44]. This way of examining outcomes more closely resembles the real world where several clinically relevant outcomes may be considered equally important in clinical decision making.

We evaluated the predictive effects of bDAS (time-invariant) on the joint MDAS and SDAS trajectories (Table 1). In this joint model, individuals with higher bDAS have a faster decline in the slope of the MDASm trajectory; but bDAS does not significantly affect the slope of the SDAS trajectory. Prior treatment with methotrexate or steroid (time-varying factors) does not significantly influence the MDAS and SDAS trajectories, perhaps because everyone received the same protocol of treatment. Thus, joint modeling can distinguish the prognostic effects of predictors for the major components of JDM disease activity.

In summary, if a researcher is interested in more than one (or more than one kind of) outcome and wants to evaluate the relationship (i.e., correlation) among these outcomes over time, the joint model can be used. The joint model can be used in the LCTA context so that latent classes of the different outcomes of interest can be delineated. The joint model also allows direct comparisons of the differing effects of the same predictors on multiple outcomes simultaneously.