JM (version 1.5-2)

fitted: Fitted Values for Joint Models

Description

Calculates fitted values for joint models.

Usage

# S3 method for jointModel
fitted(object, process = c("Longitudinal", "Event"), 
type = c("Marginal", "Subject", "EventTime", "Slope"), scale = c("survival", 
"cumulative-Hazard", "log-cumulative-Hazard"), M = 200, ...)

Value

a numeric vector of fitted values.

Arguments

object

an object inheriting from class jointModel.

process

for which model (i.e., linear mixed model or survival model) to calculate the fitted values.

type

what type of fitted values to calculate for the survival outcome. See Details.

scale

in which scale to calculate; relevant only when process = "Event".

M

how many times to simulate random effects; see Details for more info.

...

additional arguments; currently none is used.

Author

Dimitris Rizopoulos

Details

For process = "Longitudinal", let \(X\) denote the design matrix for the fixed effects \(\beta\), and \(Z\) the design matrix for the random effects \(b\). Then for type = "Marginal" the fitted values are \(X \hat{\beta},\) whereas for type = "Subject" they are \(X \hat{\beta} + Z \hat{b}\). For type = "EventTime" is the same as type = "Subject" but evaluated at the observed event times. Finally, type == "Slope" returns \(Xs \hat{\beta} + Zs \hat{b}\) where \(Xs\) and \(Zs\) denote the fixed- and random-effects design matrices corresponding to the slope term which is specified in the derivForm argument of jointModel.

For process = "Event" and type = "Subject" the linear predictor conditional on the random effects estimates is calculated for each sample unit. Depending on the value of the scale argument the fitted survival function, cumulative hazard function or log cumulative hazard function is returned. For type = "Marginal", random effects values for each sample unit are simulated M times from a normal distribution with zero mean and covariance matrix the estimated covariance matrix for the random effects. The marginal survival function for the \(i\)th sample unit is approximated by $$S_i(t) = \int S_i(t | b_i) p(b_i) db_i \approx (1/M) \sum_{m = 1}^M S_i(t | b_{im}),$$ where \(p(b_i)\) denotes the normal probability density function, and \(b_{im}\) the \(m\)th simulated value for the random effect of the \(i\)th sample unit. The cumulative hazard and log cumulative hazard functions are calculated as \(H_i(t) = - \log S_i(t)\) and \(\log H_i(t) = \log \{ - \log S_i(t)\},\) respectively.

References

Rizopoulos, D. (2012) Joint Models for Longitudinal and Time-to-Event Data: with Applications in R. Boca Raton: Chapman and Hall/CRC.

Rizopoulos, D. (2010) JM: An R Package for the Joint Modelling of Longitudinal and Time-to-Event Data. Journal of Statistical Software 35 (9), 1--33. tools:::Rd_expr_doi("10.18637/jss.v035.i09")

See Also

residuals.jointModel

Examples

if (FALSE) {
# linear mixed model fit
fitLME <- lme(log(serBilir) ~ drug * year, 
    random = ~ 1 | id, data = pbc2)
# survival regression fit
fitSURV <- survreg(Surv(years, status2) ~ drug, 
    data = pbc2.id, x = TRUE)
# joint model fit, under the (default) Weibull model
fitJOINT <- jointModel(fitLME, fitSURV, timeVar = "year")

# fitted for the longitudinal process
head(cbind(
    "Marg" = fitted(fitJOINT), 
    "Subj" = fitted(fitJOINT, type = "Subject")
))

# fitted for the event process - survival function
head(cbind(
    "Marg" = fitted(fitJOINT, process = "Ev"), 
    "Subj" = fitted(fitJOINT, process = "Ev", type = "Subject")
))

# fitted for the event process - cumulative hazard function
head(cbind(
    "Marg" = fitted(fitJOINT, process = "Ev", 
        scale = "cumulative-Hazard"), 
    "Subj" = fitted(fitJOINT, process = "Ev", type = "Subject", 
        scale = "cumulative-Hazard")
))
}