Biomarker Variability in Joint Model
Posted on
This note is for Wang, C., Shen, J., Charalambous, C., & Pan, J. (2024). Modeling biomarker variability in joint analysis of longitudinal and time-to-event data. Biostatistics, 25(2), 577–596. https://doi.org/10.1093/biostatistics/kxad009 and Wang, C., Shen, J., Charalambous, C., & Pan, J. (2024). Weighted biomarker variability in joint analysis of longitudinal and time-to-event data. The Annals of Applied Statistics, 18(3), 2576–2595. https://doi.org/10.1214/24-AOAS1896
Model biomarker variability in joint analysis of longitudinal and time-to-event data
role of visit-to-visit variability of a biomarker in predicting related disease
- existing measures of biological variability are criticized for being entangled with random variability resulted from measurement error or being unreliable due to limited measurement per individual
the paper proposed a new measure to quantify the biological variability of a biomarker by evaluating the fluctuation of each individual-specific trajectory behind longitudinal measurements
- given a mixed-effects model for longitudinal data with the mean function over time specified by cubic splines, the proposed variability measure can be mathematically expressed as a quadratic form of random effects.
- a cox model is assumed for time-to-event data by incorporating the defined variability as well as the current level of the underlying longitudinal trajectory as covariates, which, together with the longitudinal model, constitutes the joint modelling framework in this article.
- asymptotic properties of maximum likelihood estimators are established for the present joint model
- estimation is implemented via an EM algorithm with fully exponential Laplace approximation used in E-step to reduce the computation burden due to the increase of the random effects dimension.
- simulations compare
- two-stage method
- a simpler joint modeling approach which does not take into account biomarker variability
- finally, apply the model to investigate the effect of systolic blood pressure variability on cardiovascular events
Introduction
the article proposes a measure to characterize the inherent biological variability of a biomarker whose underlying trajectory possesses smooth and nonlinear shape.
the idea comes from smoothing splines on quantifying the roughness of a curve. They use the integrated squared second derivative, $\int_{t_0}^t {m’‘_i(s)}^2ds$, to capture the cumulative variability of a biomarker trajectory $m_i(\cdot)$ for the $i$-th individual from $t_0$ to current time $t$
Model
- let $T_i^\star$ and $C_i$ denote the event and censoring times, respectively, for the $i$-th individual, $i=1,\ldots,m$
- only observe $T_i=\min{T_i^\star, C_i}$ and $\delta_i = I{T_i^\star\le C_i}$
- $Y_i = (Y_{i1},\ldots, Y_{in_i})^\top$, longitudinal measurements for the $i$-th individual
- $Y_i^\star(t)$: the hypothetical trajectory behind $Y_i$
- assume that $C_i$ is independent of $T_i^\star$ given covariates
Longitudinal submodel
\[Y_i^\star(t) = x_i^\top\eta + m_i(t), i=1,\ldots, m\]allow $Y_{ij}$ to be affected by some other covariates $z_{ij}$ (e.g., in the MRC trial, is an indicator of whether the $j$-th SBP was measured by a doctor)
consider the following model
\[Y_{ij} = Y_i^\star(t_{ij}) + z_{ij}^\top\xi + \varepsilon_{ij}\]model $m_i(t)$ using regression splines with multidimensional random effects
\[m_i(t) = \sum_{k=1}^q (\beta_k + b_{ik}) B_k(t)\,,\]intuitively, ${m_i(t)}_{i=1}^m$ is a collection of random trajectories varying around a common mean trend $\sum_{k=1}^q \beta_kB_k(t)$ with random deviations specified by random effects.
Survival submodel
\[\lambda_i(t) = \lambda_0(t) \exp\left\{\gamma^\top w_i + \alpha_1(x_i^\top\eta + m_i(t)) + \alpha_2\left(\int_{t_0}^t(m_i''(s))^2 \right)^{1/2} \right\}, t > t_0\]How to do predictions, such as using C-index to evaluate the performance?
Weighted Biomarker Variability in Joint Analysis of Longitudinal and Time-to-event Data
the above paper restricted the study to the current biomarker level and cumulative biomarker variability
for a time-dependent covariate measured repeatedly during the follow-up, the relative importance of its preceding values in predicting the current risk may vary with the time lag between current time and the measurement time.
the concept of weighted cumulative exposure allows to investigate the combined effect of the value and timing of an exposure up to the current time.
one critical component in a weighted cumulative exposure is the weight function, whose shape determines the impact of the exposure at different previous times on the current hazard
Survival submodel
suppose the event time is associated with the longitudinal biomarker in terms of weighted cumulative level and variability
\[\lambda_i(t) = \lambda_0(t) \exp\left\{\gamma^\top w_i + \alpha_1\int_{t_0}^t w_{\sigma_1}(t-s)Y_i^\star(s)ds + \alpha_2\left[ \int_{t_0}^t w_{\sigma_2}(t-s) \{Y_i^{\star}{}''(s)\}^2ds \right]^{1/2} \right\}\,.\]