# dynSurv: Dynamic predictions for the time-to-event data sub-model In joineRML: Joint Modelling of Multivariate Longitudinal Data and Time-to-Event Outcomes

## Description

Calculates the conditional time-to-event distribution for a new subject from the last observation time given their longitudinal history data and a fitted mjoint object.

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 dynSurv( object, newdata, newSurvData = NULL, u = NULL, horizon = NULL, type = "first-order", M = 200, scale = 2, ci, progress = TRUE ) 

## Arguments

 object an object inheriting from class mjoint for a joint model of time-to-event and multivariate longitudinal data. newdata a list of data.frame objects for each longitudinal outcome for a single new patient in which to interpret the variables named in the formLongFixed and formLongRandom formulae of object. As per mjoint, the list structure enables one to include multiple longitudinal outcomes with different measurement protocols. If the multiple longitudinal outcomes are measured at the same time points for each patient, then a data.frame object can be given instead of a list. It is assumed that each data frame is in long format. newSurvData a data.frame in which to interpret the variables named in the formSurv formulae from the mjoint object. This is optional, and if omitted, the data will be searched for in newdata. Note that no event time or censoring indicator data are required for dynamic prediction. Defaults to newSurvData=NULL. u an optional time that must be greater than the last observed measurement time. If omitted (default is u=NULL), then conditional failure probabilities are reported for all observed failure times in the mjoint object data from the last known follow-up time of the subject. horizon an optional horizon time. Instead of specifying a specific time u relative to the time origin, one can specify a horizon time that is relative to the last known follow-up time. The prediction time is essentially equivalent to horizon + t_{obs}, where t_{obs} is the last known follow-up time where the patient had not yet experienced the event. Default is horizon=NULL. If horizon is non-NULL, then the output will be reported still in terms of absolute time (from origin), u. type a character string for whether a first-order (type="first-order") or Monte Carlo simulation approach (type="simulated") should be used for the dynamic prediction. Defaults to the computationally faster first-order prediction method. M for type="simulated", the number of simulations to performs. Default is M=200. scale a numeric scalar that scales the variance parameter of the proposal distribution for the Metropolis-Hastings algorithm, which therefore controls the acceptance rate of the sampling algorithm. ci a numeric value with value in the interval (0, 1) specifying the confidence interval level for predictions of type='simulated'. If missing, defaults to ci=0.95 for a 95% confidence interval. If type='first-order' is used, then this argument is ignored. progress logical: should a progress bar be shown on the console to indicate the percentage of simulations completed? Default is progress=TRUE.

## Details

Dynamic predictions for the time-to-event data sub-model based on an observed measurement history for the longitudinal outcomes of a new subject are based on either a first-order approximation or Monte Carlo simulation approach, both of which are described in Rizopoulos (2011). Namely, given that the subject was last observed at time t, we calculate the conditional survival probability at time u > t as

P[T ≥ u | T ≥ t; y, θ] \approx \frac{S(u | \hat{b}; θ)}{S(t | \hat{b}; θ)},

where T is the failure time for the new subject, y is the stacked-vector of longitudinal measurements up to time t and S(u | \hat{b}; θ) is the survival function.

First order predictions

For type="first-order", \hat{b} is the mode of the posterior distribution of the random effects given by

\hat{b} = {\arg \max}_b f(b | y, T ≥ t; θ).

The predictions are based on plugging in θ = \hat{θ}, which is extracted from the mjoint object.

Monte Carlo simulation predictions

For type="simulated", θ is drawn from a multivariate normal distribution with means \hat{θ} and variance-covariance matrix both extracted from the fitted mjoint object via the coef() and vcov() functions. \hat{b} is drawn from the the posterior distribution of the random effects

f(b | y, T ≥ t; θ)

by means of a Metropolis-Hasting algorithm with independent multivariate non-central t-distribution proposal distributions with non-centrality parameter \hat{b} from the first-order prediction and variance-covariance matrix equal to scale \times the inverse of the negative Hessian of the posterior distribution. The choice os scale can be used to tune the acceptance rate of the Metropolis-Hastings sampler. This simulation algorithm is iterated M times, at each time calculating the conditional survival probability.

## Value

A list object inheriting from class dynSurv. The list returns the arguments of the function and a data.frame with first column (named u) denoting times and the subsequent columns returning summary statistics for the conditional failure probabilities For type="first-order", a single column named surv is appended. For type="simulated", four columns named mean, median, lower and upper are appended, denoting the mean, median and lower and upper confidence intervals from the Monte Carlo draws. Additional objects are returned that are used in the intermediate calculations.

## Author(s)

Graeme L. Hickey (graemeleehickey@gmail.com)

## References

Rizopoulos D. Dynamic predictions and prospective accuracy in joint models for longitudinal and time-to-event data. Biometrics. 2011; 67: 819–829.

Taylor JMG, Park Y, Ankerst DP, Proust-Lima C, Williams S, Kestin L, et al. Real-time individual predictions of prostate cancer recurrence using joint models. Biometrics. 2013; 69: 206–13.

mjoint, dynLong, and plot.dynSurv.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ## Not run: # Fit a joint model with bivariate longitudinal outcomes data(heart.valve) hvd <- heart.valve[!is.na(heart.valve$log.grad) & !is.na(heart.valve$log.lvmi), ] fit2 <- mjoint( formLongFixed = list("grad" = log.grad ~ time + sex + hs, "lvmi" = log.lvmi ~ time + sex), formLongRandom = list("grad" = ~ 1 | num, "lvmi" = ~ time | num), formSurv = Surv(fuyrs, status) ~ age, data = list(hvd, hvd), inits = list("gamma" = c(0.11, 1.51, 0.80)), timeVar = "time", verbose = TRUE) hvd2 <- droplevels(hvd[hvd\$num == 1, ]) dynSurv(fit2, hvd2) dynSurv(fit2, hvd2, u = 7) # survival at 7-years only out <- dynSurv(fit2, hvd2, type = "simulated") out ## End(Not run)