jointmeta1: One stage joint meta function
In mesudell/joineRmeta: Joint Modelling for Meta-Analytic (Multi-Study) Data

Description Usage Arguments Value Details References Examples

Function to allow a one stage joint model (data from all studies analysed in one model) to be fitted to data from multiple studies. The function allows one longitudinal and one time-to-event outcome, and can accommodate baseline hazard stratified or not stratified by study, as well as random effects at the individual level and the study level. Currently only zero mean random effects only proportional association supported - see Wulfsohn and Tsiatis 1997

jointmeta1(data, long.formula, long.rand.ind, long.rand.stud = NULL,
  sharingstrct = c("randprop", "randsep", "value", "slope", "valandslope"),
  surv.formula, gpt, lgpt, max.it, tol, study.name, strat = F, longsep = F,
  survsep = F, bootrun = F, print.detail = F)

`data`	an object of class jointdata containing the variables named in the model formulae
`long.formula`	a formula object with the response varaible, and the covariates to include in the longitudinal sub-model
`long.rand.ind`	a vector of character strings to indicate what variables to assign individual level random effects to. A maximum of three individual level random effects can be assigned. To assign a random intercept include 'int' in the vector. To not include an individual level random intercept include 'noint' in the vector. For example to fit a model with individual level random intercept and random slope set `long.rand.ind = c('int', 'time')`, where `'time'` is the longitudinal time variable in the `data`.
`long.rand.stud`	a vector of character strings to indicate what variables to assign study level random effects to. If no study level random effects then this either not specified in function call or set to `NULL`. If a study level random intercept is required, include the name of the study membership variable for example `long.rand.stud = 'study'`.
`sharingstrct`	currently must be set to `'randprop'`. This gives a model that shares the zero mean random effects (at both individual and study level if specified) between the sub-models. Separate association parameters are calculated for the linear combination of random effects at each level. There are plans to expand to more sharing structures in the future.
`surv.formula`	a formula object with the survival time, censoring indicator and the covariates to include in the survival sub-model. The response must be a survival object as returned by the `Surv` function.
`gpt`	the number of quadrature points across which the integration with respect to the random effects will be performed. If random effects are specified at both the individual and the study level, the same number of quadrature points is used in both cases. Defaults to `gpt = 5`.
`lgpt`	the number of quadrature points which the log-likelihood is evaluated over following a model fit. This defaults to `lgpt = 7`.
`max.it`	the maximum number of iterations of the EM algorithm that the function will perform. Defaults to `max.it = 350` although more iterations could be required for large complex datasets.
`tol`	the tolerance level used to determine convergence in the EM algorithm. Defaults to `tol = 0.001`.
`study.name`	a character string denoting the name of the variable in the baseline dataset in `data` holding study membership, for example `study.name = 'study'`.
`strat`	logical value: if `TRUE` then the survival sub-model is calculated with a baseline stratified by study. Otherwise baseline is unstratified
`longsep`	logical value: if `TRUE` then parameter estimates, model fit and the log-likelihood from a separate linear mixed model analysis of the longitudinal data are returned (see the `lmer` function). The separate longitudinal model fit has the same specification as the longitudinal sub-model of the joint model.
`survsep`	logical value: if `TRUE` then parameter estimates, model fit and log-likelihood from a separate analysis of the survival data using the Cox Proportional Hazards model are returned (see `coxph` function for more details). This survival fit has the same specification (apart from the association structure) as the survival sub-model in the joint model.
`bootrun`	logical value: if `TRUE` then the log-likelihood for the model is not calculated. This option is available so that when bootstrapping to obtain standard errors, as the log-likelihood is not needed, it is not calculated, thus speeding up the bootstrapping process.
`print.detail`	logical value: if `TRUE` then details of the parameter estimates at each iteration of the EM algorithm are printed to the console.

An object of class jointmeta1 See jointmeta1.object

The jointmeta1 function fits a one stage joint model to survival and longitudinal data from multiple studies. This model is an extension of the model proposed by Wulfsohn and Tsiatis (1997). The model must contain at least one individual level random effect (specified using the long.rand.ind argument). The model can also contain study level random effects (specified using the long.rand.stud argument), which can differ from the individual level random effects. The maximum number of random effects that can be specified at each level is three. Note that the fitting and bootstrapping time increases as the number of included random effects increases. The model can also include a baseline hazard stratified by study, or can utilise a common baseline across the studies in the dataset. Interaction terms can be specified in either the longitudinal or the survival sub-model.

The longitudinal sub-model is a mixed effects model. If both individual level and study level random effects are included in the function call, then the sub-model has the following format:

Y_{kij} = X_{1kij}β_{1} + Z^{(2)}_{1kij}b^{(2)}_{ki} + Z^{(3)}_{1kij}b^{(3)}_{k} + ε_{kij}

Otherwise, if only individual level random effects are included in the function call, then the longitudinal sub-model has the following format:

Y_{kij} = X_{1kij}β_{1} + Z^{(2)}_{1kij}b^{(2)}_{ki} + ε_{kij}

In the above equation, Y represents the longitudinal outcome and X_1 represents the design matrix for the longitudinal fixed effects. The subscript 1 is used to distinguish between items from the longitudinal sub-model and items from the survival sub-model (which contain a subscript 2). The design matrices for random effects are represented using Z, fixed effect coefficients are represented by β, random effects by b and the measurement error by ε. Study membership is represented by the subscript k whilst individuals are identified by i and time points at which they are measured by j. The longitudinal outcome is assumed continuous.

Currently this function only supports one linking structure between the sub-models, namely a random effects only proportional sharing structure. In this structure, the zero mean random effects from the longitudinal sub-model are inserted into the survival sub-model, with a common association parameter for each level of random effects. Therefore the survival sub-model (for a case without baseline stratified by study) takes the following format:

λ_{ki}(t) = λ_{0}(t)exp(X_{2ki}β_{2} + α^{(2)}(Z^{(2)}_{1ki}b^{(2)}_{ki}) + α^{(3)}(Z^{(3)}_{1ki}b^{(3)}_{k}))

Otherwise, if only individual level random effects are included in the function call, this reduces to:

λ_{ki}(t) = λ_{0}(t)exp(X_{2ki}β_{2} + α^{(2)}(Z^{(2)}_{1ki}b^{(2)}_{ki})

In the above equation, λ_{ki}(t) represents the survival time of the individual i in study k, and λ_{0}(t) represents the baseline hazard. If a stratified baseline hazard were specified this would be replaced by λ_{0k}(t). The design matrix for the fixed effects in the survival sub-model is represented by X_{2ki}, with fixed effect coefficients represented by β_{2}. Association parameters quantifying the link between the sub-models are represented by α terms.

The model is fitted using an EM algorithm, starting values for which are extracted from initial separate longitudinal and survival fits. Pseudo adaptive Gauss - Hermite quadrature is used to evaluate functions of the random effects in the EM algorithm, see Rizopoulos 2012.

Wulfsohn, M.S. and A.A. Tsiatis, A Joint Model for Survival and Longitudinal Data Measured with Error. 1997, International Biometric Society. p. 330

Rizopoulos, D. (2012) Fast fitting of joint models for longitudinal and event time data using a pseudo-adaptive Gaussian quadrature rule. Computational Statistics & Data Analysis 56 (3) p.491-501

   #change example data to jointdata object
   jointdat2<-tojointdata(longitudinal = simdat2$longitudinal,
   survival = simdat2$survival, id = 'id',longoutcome = 'Y',
   timevarying = c('time','ltime'),
   survtime = 'survtime', cens = 'cens',time = 'time')

   #set variables to factors
   jointdat2$baseline$study <- as.factor(jointdat2$baseline$study)
   jointdat2$baseline$treat <- as.factor(jointdat2$baseline$treat)

   #fit multi-study joint model
   #note: for demonstration purposes only - max.it restricted to 5
   #model would need more iterations to truely converge
   onestagefit<-jointmeta1(data = jointdat2, long.formula = Y ~ 1 + time +
                           + treat + study, long.rand.ind = c('int', 'time'),
                           long.rand.stud = c('treat'),
                           sharingstrct = 'randprop',
                           surv.formula = Surv(survtime, cens) ~ treat,
                           study.name = 'study', strat = TRUE, max.it=5)