Using joint modelling to assess the association between a time-varying biomarker and a survival outcome: an illustrative example in respiratory medicine

How does it work?

A schematic representation of the joint model [4, 5] is provided in figure 1a. At the heart of the joint model are the subject-specific biomarker trajectories m_i(t) that provide, for a given subject i, the value of the subject's time-varying biomarker at time t. Given m_i(t), the association between the time-varying biomarker and the risk for an event can be described through the proportional hazards modelwhere exp(β) is the HR for a unit increase in the value of the biomarker at time t. The hazard function h_i(t) may also depend on other covariates, such as age and gender, for which additional terms can be included.

In clinical studies, the value of a time-varying biomarker is not observed in between two measurement moments. In addition, a subject's observed biomarker measurement at a particular time point may be expected to fluctuate around the “true” value of the subject's biomarker trajectory at that time point due to short-term biological variation and variability in the measurement process (e.g. intra- and inter-assay variability). The joint model addresses this issue by approximating the true but unknown biomarker trajectories with linear regression equations. If we assume, for simplicity of exposition, that these trajectories change linearly with time, the model for m_i(t) can be specified aswhere a_i and b_i are the intercept and slope (modelled as random coefficients that are distributed according to a bivariate normal distribution) of the biomarker trajectory of the i-th subject. This approach is illustrated in figure 1b, where the red line represents the modelled biomarker trajectory (i.e. the fitted regression line) for a particular subject and the asterisks represent the observed biomarker measurements for that subject. The random fluctuation of the observed measurements around the modelled biomarker trajectory is represented by a normally distributed error term.

The different components of the joint model (i.e. the parameters of the hazard function, the parameters of the bivariate normal distribution of the intercepts and slopes of the subject-level biomarker trajectories, and the standard error of the normally distributed error term) are jointly estimated from the available longitudinal and survival data by maximising the joint likelihood of these observations. For a more detailed discussion of the construction of the joint likelihood function and the available estimation procedures, we refer to Rizopoulous [4].

Real example

Current and future application of joint modelling in respiratory medicine

By simultaneously modelling longitudinal and survival data, joint modelling can also be used to account for informative dropout when the longitudinal outcome is of main interest. This usage of joint modelling has already received some attention in respiratory medicine, especially in scleroderma-related interstitial lung disease (SSc-ILD) [9, 10]. However, joint modelling has seldom been used to examine the association between a time-varying biomarker and risk of experiencing a clinical event. Instead, such analyses were conducted using simpler methods. For example, in the SSc-ILD field, several studies identified change in FVC% as an independent predictor of mortality [11–16]. In these studies, change in FVC% was defined as the percentage change from baseline to a specific landmark time point (mostly 6 months to 12 months). Clinicians found that the prognostic value of change in FVC% over 12 months was highly variable across studies [16], possibly because more patients dropped out or died over a longer follow-up. Joint modelling would be a better choice here to study the association of the change in FVC% with survival by including all the available information and accounting for the measurement error of FVC.

Compared to the TDCM, the use of joint modelling has shown to result in less biased estimates of the association between a time-varying biomarker and a time-to-event outcome [2]. However, the approach does require the analyst to make additional assumptions that could adversely affect these estimates if left unchecked. For example, previous studies have shown that the estimated value of the hazard ratio could be severely biased when the shape of the biomarker trajectory is misspecified [2, 17]. This issue can be addressed by considering a more flexible function of time, e.g. polynomials or splines, for the subject-level biomarker trajectories. A similar bias has shown to occur when the baseline hazard is misspecified [2, 4]. The joint model is, however, relatively robust to misspecification of the distribution of the random coefficients of the subject-level biomarker trajectories [2, 18]. Another important aspect of joint modelling is the appropriate specification of the relationship between the biomarker trajectory and the risk for an event. Until now, we have assumed that the risk for an event depends on the current value of the biomarker trajectory. However, other specifications, such as where the hazard is modelled as a function of the slope of the biomarker trajectory or the cumulative area under the biomarker trajectory, are also possible [4]. Which specification of the relationship to use can be decided based upon statistical criteria like the Bayesian information criterion or can be based on reasoning regarding the underlying biological mechanisms.

While joint modelling may be more complicated to understand and utilise than some of the simpler methods, functionality to perform such analyses is nowadays implemented in most major statistical software packages, such as the JM [19] and joineR [20] packages in R, the %JM macro [21] in SAS, and the stjm command [22] in Stata. The availability of such software to fit a range of flexible models (e.g. competing risk, cure rates) and to account for epidemiological biases (e.g. late-entry, interval-censored) is likely to facilitate the application of joint modelling even further. Precision medicine, including the development of biomarkers, is increasingly influencing respiratory medicine, e.g. PDL1 in lung cancer, F_ENO in asthma, and fibrinogen in COPD [23]. Joint modelling is likely to gain more popularity because it can easily update a patient's prognosis after each visit in which new biomarker measurements are taken. This strategy more closely reflects contemporary trends in medicine: personalised and dynamic. Also, from the perspective of statistical design, joint modelling, by incorporating longitudinal biomarker information, may lead to a reduction of bias and increase in power in estimating treatment effects in randomised controlled trials [24].

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