Fit Flexible Additive Joint Models

Family object to fit a flexible additive joint model for longitudinal and survival data under a Bayesian approach as presented in Koehler et al. (2017a, b). All parts of the joint model can be specified as structured additive predictors. See the details and examples.

## JM family object.
jm_bamlss(...)

## "bamlss.frame" transformer function
## to set up joint models.
jm_transform(x, y, data, terms, knots, formula, family, subdivisions = 25,
  timedependent = c("lambda", "mu", "alpha", "dalpha"), timevar = NULL, 
  idvar = NULL, alpha = .Machine$double.eps, mu = NULL, sigma = NULL, 
  sparse = TRUE, nonlinear = FALSE, edf_alt = FALSE, start_mu = NULL, 
  k_mu = 6, ...)
                    
## Posterior mode optimizing engine.     
opt_JM(x, y, start = NULL, weights = NULL, offset = NULL, 
  criterion = c("AICc", "BIC", "AIC"), maxit = c(100, 1), 
  nu = c("lambda" = 0.1, "gamma" = 0.1, "mu" = 1, "sigma" = 1,
    "alpha" = 1, "dalpha" = 1),
  update.nu = FALSE, eps = 0.0001, alpha.eps = 0.001, ic.eps = 1e-08, 
  nback = 40, verbose = TRUE, digits = 4, ...)

jm_mode(x, y, start = NULL, weights = NULL, offset = NULL, 
  criterion = c("AICc", "BIC", "AIC"), maxit = c(100, 1), 
  nu = c("lambda" = 0.1, "gamma" = 0.1, "mu" = 1, "sigma" = 1,
    "alpha" = 1, "dalpha" = 1),
  update.nu = FALSE, eps = 0.0001, alpha.eps = 0.001, ic.eps = 1e-08, 
  nback = 40, verbose = TRUE, digits = 4, ...)

## Sampler function.
sam_JM(x, y, family, start = NULL, weights = NULL, offset = NULL, 
  n.iter = 1200, burnin = 200, thin = 1, verbose = TRUE, digits = 4, 
  step = 20, ...)

jm_mcmc(x, y, family, start = NULL, weights = NULL, offset = NULL, 
  n.iter = 1200, burnin = 200, thin = 1, verbose = TRUE, digits = 4, 
  step = 20, ...)
  
## Predict function, set to default in jm_bamlss().
jm_predict(object, newdata,
  type = c("link", "parameter", "probabilities", "cumhaz", "loglik"),
  dt, steps, id, FUN = function(x) { mean(x, na.rm = TRUE) }, 
  subdivisions = 100, cores = NULL, chunks = 1,
  verbose = FALSE,  ...)
  
## Survival plot.
jm_survplot(object, id = 1, dt = NULL, steps = 10, 
  points = TRUE, rug = !points)

Arguments

x: The x list, as returned from function bamlss.frame (and transformed by function jm_transform()), holding all model matrices and other information that is used for fitting the model.
y: The model response, as returned from function bamlss.frame.
data: A data.frame or list containing the model response variable(s) and covariates specified in the formula in long format. By default the variables are taken from environment(formula): typically the environment from which bamlss is called.
terms: The corresponding terms.bamlss object needed for processing.
knots: An optional list containing user specified knots, see the documentation of function gam.
formula: The corresponding bamlss.formula.
family: The bamlss.family object.
subdivisions: How many time points should be created for each individual.
timedependent: A character vector specifying the names of parameters in x that are time-dependent. Time grid design matrices are only computed for these parameters.
timevar: A character specifying the name of the survival time variable in the data set.
idvar: Depending on the type of data set, this is the name of the variable specifying identifier of individuals.
alpha: Numeric, a starting value for the intercept of the association parameter alpha.
mu: Numeric, a starting value for the intercept of the mu parameter.
sigma: Numeric, a starting value for the intercept of the sigma parameter.
sparse: Logical, indicating if sparse matrix structures are used for updating and sampling of mu parameter model terms.
nonlinear: Logical, indicating if association is nonlinear in mu. See Details on the different model specifications.
edf_alt: Logical, indicating if an alternative computation of estimated degrees of freedom for penalized model terms should be used.
start_mu: Starting values for the computation of mu. For estimating associations which are nonlinear in mu, knot placement is based on these starting values which can improve stability.
k_mu: Number of knots for spline basis of association nonlinear in mu. Reducing this number improves stability of the estimation.
start: A named numeric vector containing possible starting values, the names are based on function parameters.
weights: Currently not supported.
offset: Currently not supported.
criterion: Information criterion to be used, e.g., for smoothing variance selection. Options are the corrected AIC "AICc" (see Details), the "BIC" and "AIC". Defaults to "AICc"?
maxit: Vector containing the maximum number of iterations for the backfitting algorithm with maxit[1] defining the iterations for the full model and maxit[2] the iterations within each predictor. maxit[2] defaults to 1 if only one value is specified.
nu: Vector of step lengths for parameter updates of one Newton-Raphson update for each predictor of the joint model.
update.nu: Should the updating step length be optimized in each iteration of the backfitting algorithm? Uses nu as starting value if set to TRUE.
eps: The relative convergence tolerance of the backfitting algorithm.
alpha.eps: The relative convergence tolerance of the backfitting algorithm for predictor alpha.
ic.eps: The relative convergence tolerance of the information criterion used, e.g., for smoothing variance selection.
nback: For computing ic.eps, how many iterations back should be included when computing relative convergence tolerance of the information criterion.
verbose: Print information during runtime of the algorithm.
digits: Set the digits for printing when verbose = TRUE.
n.iter: the number of MCMC iterations.
burnin: the burn-in phase of the sampler, i.e., the number of starting samples that should be removed.
thin: the thinning parameter for MCMC simulation. E.g., thin = 10 means, that only every 10th sampled parameter will be stored.
step: How many times should algorithm runtime information be printed, divides n.iter.
object: A "bamlss" object processed with the JM optimizer function opt_JM() ans/or sampler function sam_JM() for which the survival plot should be created.
newdata: Dataset for which to create predictions. Not needed for conditional survival probabilities.
type: Character string indicating which type of predictions to compute. link returns estimates for all predictors with the respective link functions applied, "parameter" returns the estimates for all pedictors, "probabilities" returns the survival probabilities conditional on the survival up to the last longitudinal measurement, and "cumhaz" return the cumulative hazard up to the survival time or for a time window after the last longitudinal measurement. If type is set to "loglik", the log-likelihood of the joint model is returned.
id: Integer or character, that specifies the individual for which the plot should be created.
dt: The time window after the last observed measurement for which predictions should be computed. The default is 0.4 * max(obstime) and obstime are the individual's longitudinal measurement times.
steps: Integer, the number of steps for which to evaluate the conditional survival probability up to dt.
FUN: A function that should be applied on the samples of predictors or parameters, depending on argument type.
cores: Specifies the number of cores that should be used for prediction. Note that this functionality is based on the parallel package.
chunks: Should computations be split into chunks? Prediction is then processed sequentially.
points: Should longitudinal observations be added to the plot.
rug: Should longitudinal observed time points be added on the x-axis to the plot.
...: Currently not used.

Details

We refer to the papers of Koehler et al. (2017a, b) for details on the flexible additive joint model. In short, we model the hazard of subject $i$ an event at time $t$ as $$h_{i}(t)= \exp [\eta_{\lambda i}(t)+ \eta_{\gamma i}+\eta_{\alpha i}(\eta_{\mu i}(t), t) ]$$ with predictor $\eta_{\lambda}$ for all survival covariates that are time-varying or have a time-varying coefficient (including the log baseline hazard), predictor $\eta_{\gamma}$ for baseline survival covariates, predictor $\eta_{\alpha}$ representing the potentially time-varying or nonlinear association between the longitudinal marker $\eta_{\mu}$ and the hazard. The longitudinal response $y_{ij}$ at time points $t_{ij}$ is modeled as $$y_{ij}=\eta_{\mu i}(t_{ij})+e_{ij}$$ with independent normal errors $N(0, \exp[\eta_{\sigma i}(t_{ij})]^2)$.

Each predictor $\eta_{ki}$ is a structured additive predictor, i.e. a sum of functions of covariates $\eta_{ki} = \sum_{m=1}^{M_k} f_{km}(\bm{x}_{ki})$. Each of these functions can be modeled parametrically or using basis function evaluations from the smooth constructors in mgcv such as s, te and ti and can include smooth time-varying, random or spatial effects. For the Bayesian estimation of these effects we specify corresponding priors: For linear or parametric terms we use vague normal priors, smooth and random effect terms are regularized by placing generic multivariate normal priors on the coefficients and for anisotropic smooths, when multiple smoothing variance parameters are involved, more complex prior are in place (cf. Koehler et al., 2017a). We use inverse Gamma hyper-priors, i.e. IG(0.001, 0.001) to obtain an inverse Gamma full conditional for the variance parameters. We estimate the posterior mode by maximizing the log-posterior of the model using a Newton-Raphson procedure, the posterior mean is obtained via derivative-based Metropolis-Hastings sampling. We recommend to use posterior mode estimates for a quick model assessment. In order to draw correct inferences from the model, posterior mean estimates should be computed. We approximate integration in the survival part of the likelihood using trapezoidal rule. For posterior mode estimation.

A variety specifications of the association $\eta_{\alpha i}(\eta_{\mu i}(t), t)$ are possible with an important distinction between associations which are nonlinear in $\eta_{\mu}$ for nonlinear = TRUE (Koehler et al. 2017b) or linear where $\eta_{\alpha i}(\eta_{\mu i}(t), t) = \eta_{\alpha i}(t)\eta_{\mu i}(t)$ for nonlinear = FALSE (Koehler et al. 2017a).

Note

The indicator nonlinear for associations with are linear or nonlinear in $\eta_{\mu}$ was named interaction] in earlier versions stages of the development.

References

Koehler M, Umlauf N, Beyerlein, A., Winkler, C. Ziegler, A.-G., Greven S (2017). Flexible Bayesian Additive Joint Models with an Application to Type 1 Diabetes Research. Biometrical Journal. doi:10.1002/bimj.201600224

Meike Koehler, Nikolaus Umlauf, and Sonja Greven (2018). Nonlinear association structures in flexible Bayesian additive joint models. Statistics in Medicine. doi:10.1002/sim.7967

Examples

if (FALSE) {

set.seed(123)
## Simulate survival data
## with random intercepts/slopes and a linear effect of time,
## constant association alpha and no effect of the derivative
d <- simJM(nsub = 200, long_setting = "linear", 
  alpha_setting = "constant",
  dalpha_setting = "zero", full = FALSE)

## Formula of the according joint model
f <- list(
  Surv2(survtime, event, obs = y) ~ s(survtime, bs = "ps"),
  gamma ~ s(x1, bs = "ps"),
  mu ~ obstime + s(id, bs = "re") + 
    s(id, obstime, bs = "re"),
  sigma ~ 1,
  alpha ~ 1,
  dalpha ~ -1
)

## Joint model estimation
## jm_bamlss() sets the default optimizer and sampler function.
## First, posterior mode estimates are computed using function
## opt_JM(), afterwards the sampler sam_JM() is started.
b <- bamlss(f, data = d, family = "jm",
           timevar = "obstime", idvar = "id")

## Plot estimated effects.
plot(b)

## Predict event probabilities for two individuals
## at 12 time units after their last longitudinal measurement.
## The event probability is conditional on their survival
## up to their last observed measurement.
p <- predict(b, type = "probabilities", id = c(1, 2), dt = 12, FUN = c95)
print(p)

## Plot of survival probabilities and
## corresponding longitudinal effects
## for individual id.
jm_survplot(b, id = 3)
jm_survplot(b, id = 30)


## Simulate survival data
## with functional random intercepts and a nonlinear effect 
## of time, time-varying association alpha and no effect 
## of the derivative.
## Note: This specification is the simJM default.
d <- simJM(nsub = 200, full = FALSE)

## Formula of the according joint model
## number of knots for the smooth nonlinear effect of time
klong <- 8
f <- list(
  Surv2(survtime, event, obs = y) ~ s(survtime, bs = "ps"),
  gamma ~ s(x1, bs = "ps"),
  mu ~ ti(id, bs = "re") + 
    ti(obstime, bs = "ps", k = klong) + 
    ti(id, obstime, bs = c("re", "ps"), 
      k = c(nlevels(d$id), klong)) + 
    s(x2, bs = "ps"),
  sigma ~ 1,
  alpha ~ s(survtime, bs = "ps"),
  dalpha ~ -1
)

## Estimating posterior mode only using opt_JM()
b_mode <- bamlss(f, data = d, family = "jm",
                 timevar = "obstime", idvar = "id", 
                 sampler = FALSE)

## Estimating posterior means using sam_JM() 
##  with starting values generated from posterior mode 
b_mean <- bamlss(f, data = d, family = "jm",
                 timevar = "obstime", idvar = "id", optimizer = FALSE,
                 start = parameters(b_mode), results = FALSE)

## Plot effects.
plot(b_mean, model = "alpha")



## Simulate survival data
## with functional random intercepts and an association nonlinear in mu
set.seed(234)
d <- simJM(nsub = 300, long_setting = "functional", alpha_setting = "nonlinear", 
           nonlinear = TRUE, full = FALSE, probmiss = 0.9)

## Calculate longitudinal model to obtain starting values for mu
long_df <- 7
f_start <- y ~ ti(id, bs = "re") + ti(obstime, bs = "ps", k = long_df) + 
               ti(id, obstime, bs = c("re", "ps"), k = c(nlevels(d$id), long_df)) + 
               s(x2, bs = "ps")
b_start <- bamlss(f_start, data = d, sampler = FALSE)
mu <- predict(b_start)$mu

## Fit joint model with nonlinear association (nonlinear = TRUE)
f <- list(
  Surv2(survtime, event, obs = y) ~ s(survtime, bs = "ps"),
  gamma ~ x1,
  mu ~ ti(id, bs = "re") + ti(obstime, bs = "ps", k = long_df) + 
       ti(id, obstime, bs = c("re", "ps"), k = c(nlevels(d$id), long_df)) + 
       s(x2, bs = "ps"),
  sigma ~ 1,
  alpha ~ 1,
  dalpha ~ -1
)
b <- bamlss(f, data = d, family = "jm", timevar = "obstime", idvar = "id", 
            nonlinear = TRUE, start_mu = mu, 
            n.iter = 6000, burnin = 2000, thin = 2)
plot(b)
samplestats(b$samples)
}

Arguments

Details

Note

References

See also

Examples