A joint logistic regression and covariate‐adjusted continuous‐time Markov chain model

Abstract

Summary The use of longitudinal measurements to predict a categorical outcome is an increasingly common goal in research studies. Joint models are commonly used to describe two or more models simultaneously by considering the correlated nature of their outcomes and the random error present in the longitudinal measurements. However, there is limited research on joint models with longitudinal predictors and categorical cross-sectional outcomes. Perhaps the most challenging task is how to model the longitudinal predictor process such that it represents the true biological mechanism that dictates the association with the categorical response. We propose a joint logistic regression and Markov chain model to describe a binary cross-sectional response, where the unobserved transition rates of a two-state continuous-time Markov chain are included as covariates. We use the method of maximum likelihood to estimate the parameters of our model. In a simulation study, coverage probabilities of about 95%, standard deviations close to standard errors, and low biases for the parameter values show that our estimation method is adequate. We apply the proposed joint model to a dataset of patients with traumatic brain injury to describe and predict a 6-month outcome based on physiological data collected post-injury and admission characteristics. Our analysis indicates that the information provided by physiological changes over time may help improve prediction of long-term functional status of these severely ill subjects.

Estimation of non‐monotonic transition rates in a semi‐Markov process with covariates adjustments and application to caregivers' stress data

“…Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) algorithm was exploited for the maximum likelihood estimators of parameters $$$ \boldsymbol{\gamma} $$$ and $$$ \boldsymbol{\delta} $$$ in transition‐specific models, while fixing $$$ \boldsymbol{\gamma} $$$ unchanged across transitions. BFGS algorithm is one of the most popular Quasi‐Newton methods for nonlinear optimization problems by approximating the inverse of Hessian matrix based on the history of gradients 20 . The estimation was performed using R package flexsurv 21 …”

“…BFGS algorithm is one of the most popular Quasi-Newton methods for nonlinear optimization problems by approximating the inverse of Hessian matrix based on the history of gradients. 20 The estimation was performed using R package flexsurv. 21 The likelihood function can be expressed as…”

Flexible cloglog links for binomial regression models as an alternative for imbalanced medical data

“…The RMSE was used to compare the methods given below: $$RMSE (\hat{\mbox{\boldmath {$\theta $}}_j})= \sqrt {\frac{1}{R}\sum _{r=1}^R(\hat{\mbox{\boldmath {$\theta $}}_j}^{(r)}-\mbox{\boldmath {$\theta $}}_j) ^2}$$, where R is the number of replicas in the simulation, $$\hat{\mbox{\boldmath {$\theta $}}_j}^{(r)}$$ is the posterior mean of the parameter $$\mbox{\boldmath {$\theta $}}_j$$ in the r th replica, and $$\mbox{\boldmath {$\theta $}}_j$$ is the j th component of $$\mbox{\boldmath {$\theta $}}= (\mbox{\boldmath {$\beta $}}^\top , \lambda )^\top$$. The 95% coverage probability for the credibility intervals of the MCMC chains was also calculated; one can think of this amount as the probability that the 95% credibility interval contains the true value of the parameter (Rubin et al., 2017; see Trikalinos et al., 2013). Another measure used to evaluate the recovery of the parameters was the computational time given in seconds, which was incorporated using the function system.time of the program R.…”

“…is the posterior mean of the parameter 𝜽 𝑗 in the 𝑟th replica, and 𝜽 𝑗 is the 𝑗th component of 𝜽 = (𝜷 ⊤ , 𝜆) ⊤ . The 95% coverage probability for the credibility intervals of the MCMC chains was also calculated; one can think of this amount as the probability that the 95% credibility interval contains the true value of the parameter (Rubin et al, 2017;see Trikalinos et al, 2013). Another measure used to evaluate the recovery of the parameters was the computational time given in seconds, which was incorporated using the function system.time of the program R. Codes for all models were developed and implemented in Stan through RStudio using the rstan package.…”

Variable Length Markov Chain with Exogenous Covariates

“…Several possible extensions are briefly discussed, which include the first order Markov chain with time-dependent covariates and the second order Markov chain with time-invariant covariates. After this pioneer work, there has been a growing interest in studying Markov chains with exogenous covariates (see for example Muenz and Rubinstein (1985), Azzalini (1994), Barrantes et al (1995), Cook and Ng (1997), Vermunt et al (1999), Aalen et al (2001), Heagerty (2002), Islam and Chowdhury (2006), Rubin et al (2017), Sirdari and Islam (2018)). On the other hand, formal inference regarding Markov Chains with exogenous covariates has not been carefully addressed.…”