2.2 Joint Models Using Longitudinal Quantile Regression

We then extend the regular joint models (consisting of a linear mixed sub-model for the longitudinal process and a Cox PHM sub-model for the survival process, referred to as LMJM), by replacing the linear mixed sub-model with an LQMM as in (3). Let Ti=min(Ti∗,Ci) be the observed event time for individual i, where Ti∗ is the true underlying event time and C_i is the censoring time. Let Δ_i be the event indicator (1 if the event is observed, and 0 otherwise). Let Y_i(t) be the continuous longitudinal outcome for individual i measured at time t. Note that Y_i(t) is only observed when t ≤ T_i, and the complete longitudinal measurements for individual i can be written as 𝒴_i(t) = {Y_i(s) : 0 ≤ s ≤ t}. We denote the true underlying longitudinal measurement at the τth quantile for individual i at time t with miτ(t) and his/her complete history of true longitudinal process as Miτ(t)={miτ(s):0≤s≤t}. The proposed quantile regression joint models (QRJM) can be written as a set of two sub-models:

where the first sub-model is the LQMM introduced in Section 2.1, in which X_i(t) are the fixed effect covariates and Z_i(t) are the covariates associated with k–dimensional random effects u_i. The second sub-model takes the format of PHM where h₀(·) is the baseline hazard function and W_i are the time-independent q–dimensional fixed effect covariates. These two sub-models are linked by incorporating miτ(t) (the true underlying longitudinal measurement at the τth quantile measured at time t) in the time-to-event process. The association parameter α_τ quantifies the strength of association between miτ(t) and the event hazard at the same time point, e.g., a positive α_τ indicates that the hazard rate will be exp(α_τ) times higher with a unit increase in the τth conditional quantile of the longitudinal outcome. Further extension of the JM in the functional form of the two processes is also possible, as discussed in Rizopoulos et al. ¹⁹

In the proposed QRJM (5), all parameters are functions of quantile τ. Thus, by choosing different quantiles, one can conduct a comprehensive analysis of the relationship between the outcome and the covariates. Depending on the research aims, we can take different strategies to utilize the flexibility of the QRJM. For example, to conduct a study over the entire conditional distribution of the longitudinal outcome, we can fit the QRJM through a set of pre-selected quantiles, collect and compare the resulting parameter estimations. Less varying values in the parameter estimates indicate a relatively stable covariate effect on the outcome, and vice versa. On the other hand, the interest may lie only in assessing the effect on some pre-specified quantiles (median, lower or higher quantile) of the longitudinal outcome and its association with the time-to-event process.