jmodelMult {JSM}

Semiparametric Joint Models for Survival and Longitudinal Data with Nonparametric Multiplicative Random Effects

Description

This function applies a maximum likelihood approach to fit the semiparametric joint models of survival and normal longitudinal data. The survival model is assumed to come from a class of transformation models, including the Cox proportional hazards model and the proportional odds model as special cases. The longitudinal process is modeled by nonparametric multiplicative random effects (NMRE) model.

Usage

jmodelMult(fitLME, fitCOX, data, model = 1, rho = 0, timeVarY = NULL, 
           timeVarT = NULL, control = list(), ...)

Arguments

fitLME

an object inheriting from class lme representing a fitted nonparametric multiplicative random effects model. See Details and Note and Examples.

fitCOX

an object inheriting from class coxph representing a fitted Cox proportional hazards regression model. Specifying x = TRUE is required in the call to coxph() to include the design matrix in the object fit. See Note.

data

a data.frame containing all the variables included in the joint modeling. See Note.

model

an indicator specifying the dependency between the survival and longitudinal outcomes. Default is 1. See Details.

rho

a nonnegative real number specifying the transformation model you would like to fit. Default is 0, i.e. the Cox proportional hazards model. See Details.

timeVarY

a character string indicating the time variable in the NMRE model. See Examples.

timeVarT

a character string indicating the time variable in the coxph object. Normally it is NULL. See Note and Examples.

control

a list of control values for the estimation algorithm with components:

tol.P

tolerance value for convergence in the parameters with default value 1e-03. See Details.

tol.L

tolerance value for convergence in the log-likelihood with default value 1e-06. See Details.

max.iter

the maximum number of EM iterations with default value 250.

SE.method

a character string specifying the standard error estimation method. Default is "PRES". See Details and Note.

delta

a positive value used for numerical differentiation in the SE.method. Default is 1e-05 if "PRES" is used and 1e-03 otherwise. See Details.

nknot

the number of Gauss-Hermite quadrature knots used to approximate the integrals over the random effects. Under the nonparametric multiplicative random effects model, there are only one-dimensional integrations and the default for nknot is 11.

...

additional options to be passed to the control argument.

Details

The jmodelMult function fits joint models for survival and longitudinal data. Nonparametric multiplicative random effects models (NMRE) are assumed for the longitudinal processes. With the Cox proportional hazards model and the proportional odds model as special cases, a general class of transformation models are assumed for the survival processes. The baseline hazard functions are left unspecified, i.e. no parametric forms are assumed, thus leading to semiparametric models. For detailed model formulation, please refer to Xu, Hadjipantelis and Wang (2017).

The longitudinal model (NMRE) is written as

Y_i(t)=\mu_i(t) + \varepsilon_i(t) = b_i\times\mathbf{B}^\top(t)\boldsymbol\gamma + \varepsilon_i(t),

where \mathbf{B}(t)=(B_1(t), \cdots, B_L(t)) is a vector of B-spline basis functions and b_i is a random effect \sim\mathcal{N}(1, \sigma_b^2). Note that we also allow the inclusion of baseline covariates as columns of \mathbf{B}(t). If model = 1, then the linear predictor for the survival model is expressed as

\eta(t) = \mathbf{W}_i^\top(t)\boldsymbol\phi + \alpha\mu_i(t),

indicating that the entire longitudinal process (free of error) enters the survival model as a covariate. If other values are assigned to the model argument, the linear predictor for the surival model is then expressed as

\eta(t) = \mathbf{W}_i^\top(t)\boldsymbol\phi + \alpha b_i,

suggesting that the survival and longitudinal models only share the same random effect.

The survival model is written as

\Lambda(t|\eta(t)) = G\left[\int_0^t\exp\{\eta(s)\}d\Lambda_0(s)\right],

where G(x) = \log(1 + \rho x) / \rho with \rho \geq 0 is the class of logarithmic transfomrations. If rho = 0, then G(x) = x, yielding the Cox proportional hazards model. If rho = 1, then G(x) = \log(1 + x), yielding the proportional odds model. Users could assign any nonnegative real value to rho.

An expectation-maximization (EM) algorithm is implemented to obtain parameter estimates. The convergence criterion is either of (i) \max \{ | \boldsymbol\theta^{(t)} - \boldsymbol\theta^{(t - 1)} | / ( | \boldsymbol\theta^{(t - 1)} | + .Machine\$double.eps \times 2 ) \} < tol.P, or (ii) | L(\boldsymbol\theta^{(t)}) - L(\boldsymbol\theta^{(t - 1)})| / ( | L(\theta^{(t - 1)}) | + .Machine\$double.eps \times 2 ) < tol.L, is satisfied. Here \boldsymbol\theta^{(t)} and \boldsymbol\theta^{(t-1)} are the vector of parameter estimates at the t-th and (t-1)-th EM iterations, respectively; L(\boldsymbol\theta) is the value of the log-likelihood function evaluated at \boldsymbol\theta. Users could specify the tolerance values tol.P and tol.L through the control argument.

For standard error estimation for the parameter estimates, three methods are provided, namely "PRES", "PFDS" and "PLFD" (detailed information are referred to Xu, Baines and Wang (2014)). In the control argument, if SE.method = "PRES", numerically differentiating the profile Fisher score vector with Richardson extrapolation is applied; if SE.method = "PFDS", numerically differentiating the profile Fisher score vector with forward difference is applied; if SE.method = "PLFD", numerially (second) differentiating the profile likelihood with forward difference is applied. Generally, numerically differentiating a function f(x) (an arbitrary function) with forward difference is expressed as

f^\prime(x) = \frac{f(x + \delta) - f(x)}{\delta},

and that with Richardson extrapolation is expressed as

f^\prime(x) = \frac{f(x - 2\delta) - 8f(x - \delta) + 8f(x + \delta) - f(x + 2\delta)}{12\delta}.

Users could specify the value of \delta through the delta item in the control argument.

Value

See jmodelMultObject for the components of the fit.

Note

1. To fit a nonparametric multiplicative random effects model, the fixed effect in the fitLME object should be a matrix of B-spline basis functions (an object from the bs function) with the possibility of including baseline covariates and the random effect should only include a random intercept. In the bs function, it is a good practice to specify the boundary knots through the Boundary.knots argument, where the upper boundary knot is typically the longest follow-up time among all subjects. See Examples.

2. Currently, jmodelMult() could only handle the fitLME object with a simple random-effects structure (only the pdDiag() class). Moreover, the within-group correlation and heteroscedasticity structures in the fitLME object (i.e. the correlation and weights argument of lme()) are ignored.

3. The data argument in jmodelMult(), lme() and coxph() should be the same data frame.

4. For the fitCOX object, only the W_i(t) in the linear predictor \eta(t) for the survial model (see Details) should be involved in the formula argument of coxph{}. Since coxph() uses the same data frame as lme() does, a time-dependent Cox model must be fitted by coxph() although W_i(t) may only contain time-independent covariates. See Examples.

5. If W_i(t) in the linear predictor \eta(t) for the survial model (see Details) does involve time-dependent covariate, then timeVarT must specify the name of the time variable involved (see Examples).

6. The standard error estimates are obtained by numerical approximations which is naturally subject to numerical errors. Therefore, in extreme cases, there may be NA values for part of the standard error estimates.

Author(s)

Cong Xu [e-mail] Pantelis Z. Hadjipantelis [e-mail]

References

Dabrowska, D. M. and Doksun K. A. (1988) Partial Likelihood in Transformation Models with Censored Data. Scandinavian Journal of Statistics 15, 1–23.

Ding, J. and Wang, J. L. (2008) Modeling longitudinal data with nonparametric multiplicative random effects jointly with survival data. Biometrics 64, 546–556.

Tsiatis, A. A. and Davidian, M. (2004) Joint modeling of longitudinal and time-to-event data: an overview. Statistica Sinica 14, 809–834.

Xu, C., Baines, P. D. and Wang, J. L. (2014) Standard error estimation using the EM algorithm for the joint modeling of survival and longitudinal data. Biostatistics 15, 731–744

Xu, C., Hadjipantelis, P. Z. and Wang, J. L. (2020) Semiparametric joint modeling of survival and longitudinal data: the R package JSM. Journal of Statistical Software <doi:10.18637/jss.v093.i02>.

Zeng, D. and Lin, D. (2007) Maximum likelihood estimation in semiparametric regression models with censored data. Journal of the Royal Statistical Society: Series B 69, 507–564.

See Also

jmodelMultObject, lme, coxph, Surv, bs

Examples

# linear mixed-effects model fit where the fixed effect is modeled by 
# quadratic B-splie basis with no internal knots
fitLME <- lme(log(serBilir) ~ bs(obstime, degree = 2, Boundary.knots = c(0, 15)), 
              random = ~ 1 | ID, data = pbc)
# Cox proportional hazards model fit with a single time-independent covariate
fitCOX <- coxph(Surv(start, stop, event) ~ drug, data = pbc, x = TRUE)

# joint model fit which assumes the Cox proportional hazards model for the survival process 
# and NMRE for the longitudinal process. Use 'max.iter = 25', 'nknot = 3' and 
# the 'PFDS' method to calculate standard error estimates as a quick toy example 
fitJTMult.ph <- jmodelMult(fitLME, fitCOX, pbc, timeVarY = "obstime", 
                           control = list(SE.method = 'PFDS', max.iter = 25, nknot = 3))
summary(fitJTMult.ph)

## Not run: 
# joint model fit with the default control
fitJTMult.ph2 <- jmodelMult(fitLME, fitCOX, pbc, timeVarY = "obstime")
summary(fitJTMult.ph2)
# joint model fit where the survival and longitudinal processes only share 
# the same random effect
fitJTMult.ph3 <- jmodelMult(fitLME, fitCOX, pbc, model = 2, timeVarY = "obstime")
summary(fitJTMult.ph3)

# joint model fit which assumes the proportional odds model for the survival process 
# and NMRE model for the longitudinal process
fitJTMult.po <- jmodelMult(fitLME, fitCOX, pbc, rho = 1, timeVarY = "obstime")
summary(fitJTMult.po)
# joint model fit where the survival and longitudinal processes only share 
# the same random effect
fitJTMult.po2 <- jmodelMult(fitLME, fitCOX, pbc, model = 2, rho = 1, timeVarY = "obstime")
summary(fitJTMult.po2)

# allow baseline covariates in the NMRE model for the longitudinal process
fitLME2 <- lme(log(serBilir) ~ drug + bs(obstime, degree = 2, Boundary.knots = c(0, 15)), 
               random = ~1 | ID, data = pbc)
fitJTMult.ph4 <- jmodelMult(fitLME2, fitCOX, pbc, timeVarY = "obstime")
summary(fitJTMult.ph4)

# Cox proportional hazards model fit with a time-dependent covariate
fitCOX2 <- coxph(Surv(start, stop, event) ~ drug + as.numeric(drug) : obstime, 
                 data = pbc, x = TRUE)
# joint model fit in which \code{timeVarT} must be specified
fitJTMult.ph5 <- jmodelMult(fitLME, fitCOX2, pbc, timeVarY = "obstime", timeVarT = 'obstime', 
                            control = list(max.iter = 300))
summary(fitJTMult.ph5)

## End(Not run)