BayesPIM: Bayesian Prevalence-Incidence Mixture Model

Description

Models time-to-event data from interval-censored screening studies. It accounts for latent prevalence at baseline and incorporates misclassification due to imperfect test sensitivity. For usage details, see the package vignette ("BayesPIM_intro"). Further details can be found in T. Klausch, B. I. Lissenberg-Witte, and V. M. Coupe (2024), "A Bayesian prevalence-incidence mixture model for screening outcomes with misclassification", doi:10.48550/arXiv.2412.16065.

Author(s)

Maintainer: Thomas Klausch t.klausch@amsterdamumc.nl

See Also

Useful links:

• https://github.com/thomasklausch2/bayespim

• Report bugs at https://github.com/thomasklausch2/BayesPIM/issues

Fitting Bayesian Prevalence-Incidence Mixture Model

Description

Estimates the Pattern Mixture model of Klausch et al. (2025) using a Bayesian Gibbs sampler. The model is formulated as an interval-censored survival model over successive intervals, with the possibility of missed events due to imperfect test sensitivity. In addition, baseline tests at time zero may fail to detect pre-study events (prevalence).

Usage

bayes.2S(
  Vobs,
  Z.X = NULL,
  Z.W = NULL,
  r = NULL,
  dist.X = "weibull",
  kappa = 0.5,
  update.kappa = FALSE,
  kappa.prior = NULL,
  ndraws = 1000,
  prop.sd.X = NULL,
  chains = 3,
  thining = 1,
  parallel = TRUE,
  update.till.converge = FALSE,
  maxit = Inf,
  conv.crit = "upper",
  min_effss = chains * 10,
  beta.prior = "norm",
  beta.prior.X = 1,
  sig.prior.X = 1,
  tau.w = 1,
  fix.sigma.X = FALSE,
  prev.run = NULL,
  update.burnin = TRUE,
  ndraws.update = NULL,
  prev = TRUE,
  vanilla = FALSE,
  ndraws.naive = 10000,
  naive.run.prop.sd.X = prop.sd.X,
  par.exp = FALSE,
  collapsed.g = TRUE,
  k.prior = 1,
  fix.k = FALSE
)

Arguments

Details

This Bayesian prevalence-incidence mixture model (PIM) characterizes the time to incidence using an accelerated failure time (AFT) model of the form:

\log(x_i) = \bm{z}_{xi}' \bm{\beta}_x + \sigma \epsilon_i

where \epsilon_i is chosen such that x_i follows a weibull, lognormal, or loglog (log-logistic) distribution, as specified by the dist argument. The covariate vector \bm{z}_{xi} for individual i is provided in the Z.X matrix.

Baseline prevalence is modeled using a probit formulation Pr(g_i=1 | \bm{z}_{wi}) = Pr(w_i > 0 | \bm{z}_{wi}) with

w_i = \bm{z}_{wi}' \bm{\beta}_w + \psi_i

where \psi_i follows a standard normal distribution, and the covariate vector \bm{z}_{wi} is given in the Z.W matrix. The latent variable w_i determines prevalence status, such that g_i = 1 if w_i > 0 and g_i = 0 otherwise.

The argument Vobs provides the observed testing times for all individuals. It is a list of numeric vectors, where each vector starts with 0 (representing the baseline time) and is followed by one or more screening times. The final entry is Inf in the case of right censoring or indicates the time of a positive test if an event is observed. Specifically:

• If the baseline test is positive, the vector consists solely of c(0).

• If the baseline test is negative and right censoring occurs before the first regular screening, the vector is c(0, Inf).

• Otherwise, the vector ends with Inf in the case of right censoring (e.g., c(0, 1, 3, 6, Inf)) or ends at the event time (e.g., c(0, 1, 3, 6) for an event detected at time 6).

By convention, every vector in Vobs starts with 0. However, the binary vector r of length n indicates whether the baseline test was conducted (r[i] = 1) or missing (r[i] = 0) for each individual i in Vobs. For further details on coding, see Section 2 of the main paper.

Test sensitivity can be fixed to a value kappa by setting update.kappa = FALSE, or it can be estimated if update.kappa = TRUE. When estimated, a Beta prior is used, centered on the first element of kappa.prior, with a standard deviation equal to its second element. An internal optimization process finds the Beta prior hyperparameters that best match this choice. If the chosen prior is not feasible, unexpected behavior may occur. If kappa.prior is not specified (the default), an uninformative uniform(0,1) prior is used. In general, we advise against using an uninformative prior, but this default avoids favoring any specific informative prior.

The Gibbs sampler runs for ndraws iterations for each of chains total chains. The Metropolis step used for sampling the parameters of the incidence model applies a normal proposal (jumping) distribution with a standard deviation prop.sd.X, which must be selected by trial and error. An optimal acceptance rate is approximately 23%, which can be computed per MCMC run from the model output. Alternatively, the function search.prop.sd provides a heuristic for selecting an effective proposal distribution standard deviation.

If parallel = TRUE, the Gibbs sampler runs in parallel with one chain per CPU (if possible), using the foreach package. If this package causes issues on some operating systems, set parallel = FALSE or use the bayes.2S_seq function, which iterates over 1:chains using a for loop. This sequential function may also be useful in Monte Carlo simulations that parallelize experimental replications using foreach.

We recommend running at least two chains in parallel, and preferably more, to facilitate standard MCMC diagnostics such as the Gelman-Rubin R statistic. Additionally, we suggest first running the sampler for a moderate number of iterations to assess its behavior before using the updating functionality in prev.run to extend sampling (see below).

The option update.till.convergence = TRUE allows bayes.2S to run until convergence. Convergence is achieved when R < 1.1 for all parameters and the minimum effective sample size min_effs is reached. The sampler continues updating until convergence is attained or maxit is reached.

The priors for the regression coefficients in the prevalence and incidence models can be controlled using beta.prior, beta.prior.X, sig.prior.X, and tau.w. Specifically:

• beta.prior determines the prior type for \beta_{xj} (either normal or Student-t t).

• beta.prior.X specifies either the standard deviation (for normal priors) or degrees of freedom (for Student-t priors). The default is a standard normal prior.

• A half-normal prior is used for \sigma, with sig.prior.X controlling the standard deviation.

• A zero-centered normal prior is assigned to \beta_{wj}, with tau.w controlling its standard deviation (default: standard normal).

Sometimes model fitting can be improved by fixing the \sigma parameter to a value, which is achieved through setting fix.sigma.X = TRUE. Then, the value specified as sig.prior.X is regarded as the correct value for \sigma, akin to a point prior on this value. The functionality can also be used to obtain the exponential distribution, aking to a Markov model. For this choose dist='weibull', sig.prior.X = 1, and fix.sigma.X=TRUE.

The prev.run argument allows updating a previous run with additional MCMC draws. The MCMC chain resumes from the last draws, continues, and merges with the original run. If an initial model was fit using mod <- bayes.2S(...), it can be updated using mod_update <- bayes.2S(prev.run = mod). By default, ndraws additional iterations are added unless otherwise specified via ndraws.update. When updating, the number of discarded burn-in draws can be adjusted to half of the total draws (update.burnin = TRUE) or remain at the initial number (update.burnin = FALSE).

The Gibbs sampler requires starting values, which are obtained from an initial Bayesian interval-censored survival model using the specified dist distribution. The jumping distribution variance and the number of MCMC draws for this initialization are controlled via ndraws.naive and naive.run.prop.sd.X. The default values typically suffice but may need adjustment if initialization fails (e.g., increasing ndraws.naive or tuning naive.run.prop.sd.X). If starting values are found but still lead to an infinite posterior at initialization, the error "Bad starting values" is returned. Then it usually sufficces to re-run bayes.2S with an increased ndraws.naive value.

Value

A list containing the following elements:

Additionally, most input arguments are returned as part of the output for reference.

References

T. Klausch, B. I. Lissenberg-Witte, and V. M. Coupe (2024). "A Bayesian prevalence-incidence mixture model for screening outcomes with misclassification.", doi:10.48550/arXiv.2412.16065.

J. S. Liu and Y. N. Wu, “Parameter Expansion for Data Augmentation,” Journal of the American Statistical Association, vol. 94, no. 448, pp. 1264–1274, 1999, doi:10.2307/2669940.

Examples

library(BayesPIM)

# Generate data according to the Klausch et al. (2024) PIM
set.seed(2025)
dat <- gen.dat(kappa = 0.7, n = 1e3, theta = 0.2,
               p = 1, p.discrete = 1,
               beta.X = c(0.2, 0.2), beta.W = c(0.2, 0.2),
               v.min = 20, v.max = 30, mean.rc = 80,
               sigma.X = 0.2, mu.X = 5, dist.X = "weibull",
               prob.r = 1)

# Initial model fit with fixed test sensitivity kappa (approx. 1-3 minutes runtime)
mod <- bayes.2S(Vobs = dat$Vobs,
                Z.X = dat$Z,
                Z.W = dat$Z,
                r = dat$r,
                kappa = 0.7,
                update.kappa = FALSE,
                ndraws = 1e4,
                chains = 2,
                prop.sd.X = 0.008,
                parallel = TRUE,
                dist.X = "weibull")


# Inspect results
mod$runtime  # Runtime of the Gibbs sampler
plot(trim.mcmc(mod$par.X.all, thining = 10))  # MCMC chains including burn-in, also see ?trim.mcmc
plot(trim.mcmc(mod$par.X.bi, thining = 10))   # MCMC chains excluding burn-in
apply(mod$ac.X, 2, mean)  # Acceptance rates per chain
gelman.diag(mod$par.X.bi)  # Gelman convergence diagnostics

# Model updating
mod_update <- bayes.2S(prev.run = mod)      # Adds ndraws additional MCMC draws
mod_update <- bayes.2S(prev.run = mod, 
                       ndraws.update = 1e3) # Adds ndraws.update additional MCMC draws

# Example with kappa estimated/updated
mod2 <- bayes.2S(Vobs = dat$Vobs,
                 Z.X = dat$Z,
                 Z.W = dat$Z,
                 r = dat$r,
                 kappa = 0.7,
                 update.kappa = TRUE,
                 kappa.prior = c(0.7, 0.1), # Beta prior, mean = 0.7, s.d. = 0.1
                 ndraws = 1e4,
                 chains = 2,
                 prop.sd.X = 0.008,
                 parallel = TRUE,
                 dist.X = "weibull")

# Inspect results
mod2$runtime # runtime of Gibbs sampler
plot( trim.mcmc( mod2$par.X.all, thining = 10) ) # kappa returned as part of the mcmc.list

bayes.2S_seq: Bayesian Prevalence-Incidence Mixture Model (sequential processing)

Description

Estimates the Pattern Mixture model of Klausch et al. (2025) using a Bayesian Gibbs sampler. Identical to bayes.2S but uses a for loop over MCMC chains instead of foreach.

Usage

bayes.2S_seq(
  Vobs,
  Z.X = NULL,
  Z.W = NULL,
  r = NULL,
  dist.X = "weibull",
  kappa = 0.5,
  update.kappa = FALSE,
  kappa.prior = NULL,
  ndraws = 1000,
  prop.sd.X = NULL,
  chains = 3,
  thining = 1,
  parallel = TRUE,
  update.till.converge = FALSE,
  maxit = Inf,
  conv.crit = "upper",
  min_effss = chains * 10,
  beta.prior = "norm",
  beta.prior.X = 1,
  sig.prior.X = 1,
  tau.w = 1,
  fix.sigma.X = FALSE,
  prev.run = NULL,
  update.burnin = TRUE,
  ndraws.update = NULL,
  prev = TRUE,
  vanilla = FALSE,
  ndraws.naive = 5000,
  naive.run.prop.sd.X = prop.sd.X,
  par.exp = FALSE,
  collapsed.g = TRUE,
  k.prior = 1,
  fix.k = FALSE
)

Arguments

Details

This function is dentical to bayes.2S with the only difference being that the chains MCMC chains are run in sequence using a for loop instead of parallel processing. This can be useful if operating systems do not support foreach or for simulation studies that parallize replication of experiments using foreach and/or need a worker that does not apply foreach internally.

Value

A list containing the following elements:

Additionally, most input arguments are returned as part of the output for reference.

References

T. Klausch, B. I. Lissenberg-Witte, and V. M. Coupe (2024). "A Bayesian prevalence-incidence mixture model for screening outcomes with misclassification.", doi:10.48550/arXiv.2412.16065.

J. S. Liu and Y. N. Wu, “Parameter Expansion for Data Augmentation,” Journal of the American Statistical Association, vol. 94, no. 448, pp. 1264–1274, 1999, doi: 10.2307/2669940.

Examples

library(BayesPIM)

# Generate data according to the Klausch et al. (2025) PIM
set.seed(2025)
dat <- gen.dat(kappa = 0.7, n = 1e3, theta = 0.2,
               p = 1, p.discrete = 1,
               beta.X = c(0.2, 0.2), beta.W = c(0.2, 0.2),
               v.min = 20, v.max = 30, mean.rc = 80,
               sigma.X = 0.2, mu.X = 5, dist.X = "weibull",
               prob.r = 1)

# Initial model fit with fixed test sensitivity kappa (approx. 4-12 minutes runtime)
mod <- bayes.2S_seq(Vobs = dat$Vobs,
                Z.X = dat$Z,
                Z.W = dat$Z,
                r = dat$r,
                kappa = 0.7,
                update.kappa = FALSE,
                ndraws = 1e4,
                chains = 4,
                prop.sd.X = 0.008,
                parallel = TRUE,
                dist.X = "weibull")

# Inspect results
mod$runtime  # Runtime of the Gibbs sampler
plot(trim.mcmc(mod$par.X.all, thining = 10))  # MCMC chains including burn-in, also see ?trim.mcmc
plot(trim.mcmc(mod$par.X.bi, thining = 10))   # MCMC chains excluding burn-in
apply(mod$ac.X, 2, mean)  # Acceptance rates per chain
gelman.diag(mod$par.X.bi)  # Gelman convergence diagnostics

# Model updating
mod_update <- bayes.2S(prev.run = mod)      # Adds ndraws additional MCMC draws
mod_update <- bayes.2S(prev.run = mod, 
                       ndraws.update = 1e3) # Adds ndraws.update additional MCMC draws

# Example with kappa estimated/updated
mod2 <- bayes.2S_seq(Vobs = dat$Vobs,
                 Z.X = dat$Z,
                 Z.W = dat$Z,
                 r = dat$r,
                 kappa = 0.7,
                 update.kappa = TRUE,
                 kappa.prior = c(0.7, 0.1), # Beta prior, mean = 0.7, s.d. = 0.1
                 ndraws = 1e4,
                 chains = 4,
                 prop.sd.X = 0.008,
                 parallel = TRUE,
                 dist.X = "weibull")

# Inspect results
mod2$runtime # runtime of Gibbs sampler
plot( trim.mcmc( mod2$par.X.all, thining = 10) ) # kappa returned as part of the mcmc.list

gen.dat: Simulate Screening Data for a Prevalence-Incidence Mixture Model

Description

Generates synthetic data according to the Bayesian prevalence-incidence mixture (PIM) framework of Klausch et al. (2025) with interval-censored screening outcomes. The function simulates continuous or discrete baseline covariates, event times from one of several parametric families, and irregular screening schedules, yielding interval-censored observations suitable for testing or demonstrating PIM-based or other interval-censored survival methods.

Usage

gen.dat(
  kappa = 0.7,
  n = 1000,
  p = 2,
  p.discrete = 0,
  r = 0,
  s = 1,
  sigma.X = 1/2,
  mu.X = 4,
  beta.X = NULL,
  beta.W = NULL,
  theta = 0.15,
  v.min = 1,
  v.max = 6,
  mean.rc = 40,
  dist.X = "weibull",
  k = 1,
  sel.mod = "probit",
  prob.r = 0
)

Arguments

Details

The data-generating process includes:

1. Covariates Z: Continuous covariates are simulated using a correlation structure specified by r and a common standard deviation s. If p.discrete = 1, a single discrete covariate is added, drawn from \mathrm{Bernoulli}(0.5).

2. Event Times X: An Accelerated Failure Time (AFT) model is used:

   \log(x_i) = \beta_{x0} + \beta_{x}^\top z_{xi} + \sigma_X \,\epsilon_i,

   where \beta_{x0} is the intercept (set by mu.X) and \beta_{x} are the other regression coefficients (provided via beta.X). The error term \epsilon_i is drawn from the distribution chosen by dist.X: "weibull", "lognormal", "loglog" (log-logistic), or "gengamma" (generalized gamma). For "gengamma", the shape parameter k is additionally used.

3. Irregular Screening Schedules V_i: Each individual has multiple screening times generated randomly between v.min and v.max, ending in right censoring or the time of detection. These screening times (including a 0 for baseline and Inf for censoring) are returned in Vobs.

4. Prevalence Indicator g_i: Baseline prevalence is modeled via either a probit or logit link, consistent with:

   w_i = \beta_{w0} + \beta_{w}^\top z_{wi} + \psi_i,

   where \beta_{w0} is determined by theta, and \beta_{w} by beta.W. Specifically:

   • If sel.mod = "probit", then \beta_{w0} = \mathrm{qnorm}(\theta).

   • If sel.mod = "logit", then \beta_{w0} = \log(\theta / (1-\theta)).

   We set g_i = 1 if w_i > 0, and g_i = 0 otherwise.

5. Baseline Test Missingness r_i: A baseline test indicator r_i \in \{0,1\} is generated via \mathrm{Bernoulli}(\text{prob.r}), so r_i = 1 means the baseline test is performed and r_i = 0 means it is missing.

6. Test Sensitivity \kappa: A misclassification parameter \kappa (test sensitivity) can be specified via kappa. If \kappa < 1, some truly positive cases are missed.

Value

A list with the following elements:

Vobs

A list of length n, each entry containing screening times. The first element is 0 (baseline), and Inf may indicate right censoring.

X.true

Numeric vector of length n giving the true (latent) event times x_i.

Z

Numeric matrix of dimension n \times p (plus an extra column if p.discrete = 1) containing the covariates.

C

Binary vector of length n, indicating whether an individual is truly positive at baseline (g_i = 1).

r

Binary vector of length n, indicating whether the baseline test was performed (r_i = 1) or missing (r_i = 0).

p.W

Numeric vector of length n giving the true prevalence probabilities, P(g_i = 1).

References

T. Klausch, B. I. Lissenberg-Witte, and V. M. Coupé, “A Bayesian prevalence-incidence mixture model for screening outcomes with misclassification,” arXiv:2412.16065.

Examples

# Generate a small dataset for testing
set.seed(2025)
sim_data <- gen.dat(n = 100, p = 1, p.discrete = 1,
                    sigma.X = 0.5, mu.X = 2,
                    beta.X = c(0.2, 0.2), beta.W = c(0.5, -0.2),
                    theta = 0.2,
                    dist.X = "weibull", sel.mod = "probit")
str(sim_data)

Compute Information Criteria for a Bayesian Prevalence-Incidence Mixture Model

Description

Computes and returns information criteria for a fitted Bayesian prevalence-incidence mixture model, including the Widely Applicable Information Criterion 1 (WAIC-1), WAIC-2, and the Deviance Information Criterion (DIC). These criteria are commonly used for model comparison and evaluation in Bayesian analysis. See Gelman et al. (2014) for further details on these criteria.

Usage

get.IC_2S(mod, samples = nrow(mod$par.X.bi[[1]]), cores = NULL)

Arguments

Details

This function calculates information criteria for a fitted Bayesian prevalence-incidence mixture model (bayes.2S). The information criteria include:

• WAIC-1: Based on the sum of posterior variances of log-likelihood contributions.

• WAIC-2: Similar to WAIC-1 but incorporates an alternative variance estimate.

• DIC: Measures model fit by penalizing complexity via the effective number of parameters.

The computation is performed by evaluating log-likelihood values for MCMC samples. By default, all MCMC samples after burn-in are used, though a subset can be specified via the samples argument.

Parallelization is available via the foreach package, utilizing multiple cores if cores is set accordingly. If cores = NULL, all available cores will be used.

Value

A matrix containing WAIC-1, WAIC-2, and DIC values for the model.

References

Gelman, A., Hwang, J., & Vehtari, A. (2014). Understanding predictive information criteria for Bayesian models. Stat Comput, 24(6), 997–1016.

Examples

# Generate data according to the Klausch et al. (2024) PIM
set.seed(2025)
dat = gen.dat(kappa = 0.7, n= 1e3, theta = 0.2,
              p = 1, p.discrete = 1,
              beta.X = c(0.2,0.2), beta.W = c(0.2,0.2),
              v.min = 20, v.max = 30, mean.rc = 80,
              sigma.X = 0.2, mu.X=5, dist.X = "weibull",
              prob.r  = 1)

# An initial model fit with fixed test sensitivity kappa (approx. 1-3 minutes, depending on machine)
mod = bayes.2S( Vobs = dat$Vobs,
                Z.X = dat$Z,
                Z.W = dat$Z,
                r= dat$r,
                kappa = 0.7,
                update.kappa = FALSE,
                ndraws= 1e4,
                chains = 2,
                prop.sd.X = 0.008,
                parallel = TRUE,
                dist.X = 'weibull'
)

# Get information criteria
get.IC_2S(mod, samples = 1e3, cores = 2)

Posterior Predictive Cumulative Incidence Function

Description

Computes the posterior predictive cumulative incidence function (CIF) from a bayes.2S prevalence-incidence mixture model. The function can return quantiles corresponding to user-specified percentiles (i.e., time points at which the cumulative probability reaches certain thresholds), or vice versa (percentiles at user-specified quantiles). Additionally, it allows for marginal or conditional CIFs of either the mixture population (including prevalence as a point-mass at time zero) or the non-prevalent (healthy) subpopulation.

Usage

get.ppd.2S(
  mod,
  fix_Z.X = NULL,
  fix_Z.W = NULL,
  pst.samples = 1000,
  perc = seq(0, 1, 0.01),
  type = "x",
  ppd.type = "percentiles",
  quant = NULL
)

Arguments

Details

For a prevalence-incidence mixture model, some fraction of the population may already have experienced the event (prevalent cases) at baseline, while the remaining (healthy) fraction has not. This function estimates the CIF in two ways:

• type = "xstar" (mixture CIF): Includes a point-mass at time zero representing baseline prevalence, with incidence beginning thereafter.

• type = "x" (non-prevalent CIF): Excludes prevalent cases, so it only shows incidence among the initially healthy subpopulation.

You may request a marginal CIF by setting both fix_Z.X = NULL and fix_Z.W = NULL, thus integrating over all covariates. Alternatively, a conditional CIF can be obtained by partially or fully specifying fixed covariate values in fix_Z.X (and optionally fix_Z.W) while integrating out the unspecified covariates (NA entries).

The function operates in two main modes:

• ppd.type = "quantiles": Given a set of perc (cumulative probabilities), returns corresponding quantiles (time points).

• ppd.type = "percentiles": Given a set of quant (time points), returns corresponding percentiles (cumulative probabilities).

Value

A list with some or all of the following elements:

med.cdf

• If ppd.type = "quantiles", med.cdf contains the median quantiles (time points) across posterior samples for each percentile in perc.

• If ppd.type = "percentiles", med.cdf contains the median cumulative probabilities across posterior samples for each time point in quant.

med.cdf.ci

A 2-row matrix with the 2.5% and 97.5% posterior quantiles for the estimated med.cdf, reflecting uncertainty.

quant

If ppd.type = "percentiles", this is a copy of the input quant (time points).

perc

If ppd.type = "quantiles", this is a copy of the input perc (cumulative probabilities).

Examples

# Generate data according to the Klausch et al. (2024) PIM
set.seed(2025)
dat <- gen.dat(kappa = 0.7, n = 1e3, theta = 0.2,
               p = 1, p.discrete = 1,
               beta.X = c(0.2, 0.2), beta.W = c(0.2, 0.2),
               v.min = 20, v.max = 30, mean.rc = 80,
               sigma.X = 0.2, mu.X = 5, dist.X = "weibull",
               prob.r  = 1)

# Fit a bayes.2S model (example with moderate ndraws = 2e4)
mod <- bayes.2S(Vobs = dat$Vobs, Z.X = dat$Z, Z.W = dat$Z, r = dat$r,
                kappa = 0.7, update.kappa = FALSE, ndraws = 1e4,
                chains = 2, prop.sd.X = 0.008, parallel = TRUE,
                dist.X = "weibull")

###################
# (1) Provide percentiles, get back quantiles (times)
###################
cif_nonprev <- get.ppd.2S(mod, pst.samples = 1e3, type = "x",
                          ppd.type = "quantiles", perc = seq(0, 1, 0.01))
cif_mix     <- get.ppd.2S(mod, pst.samples = 1e3, type = "xstar",
                          ppd.type = "quantiles", perc = seq(0, 1, 0.01))

# Plot: Non-prevalent stratum CIF vs. mixture CIF (marginal)
oldpar <- par(no.readonly = TRUE)
par(mfrow = c(1,2))

plot(cif_nonprev$med.cdf, cif_nonprev$perc, type = "l", xlim = c(0,300), ylim = c(0,1),
     xlab = "Time", ylab = "Cumulative Incidence")
lines(cif_nonprev$med.cdf.ci[1,], cif_nonprev$perc, lty = 2)
lines(cif_nonprev$med.cdf.ci[2,], cif_nonprev$perc, lty = 2)

plot(cif_mix$med.cdf, cif_mix$perc, type = "l", xlim = c(0,300), ylim = c(0,1),
     xlab = "Time", ylab = "Cumulative Incidence")
lines(cif_mix$med.cdf.ci[1,], cif_mix$perc, lty = 2)
lines(cif_mix$med.cdf.ci[2,], cif_mix$perc, lty = 2)

###################
# (2) Provide quantiles (times), get back percentiles (cumulative probabilities)
###################
cif2_nonprev <- get.ppd.2S(mod, pst.samples = 1e3, type = "x",
                           ppd.type = "percentiles", quant = 1:300)
cif2_mix     <- get.ppd.2S(mod, pst.samples = 1e3, type = "xstar",
                           ppd.type = "percentiles", quant = 1:300)

# Plot: Non-prevalent vs. mixture CIF using times on the x-axis
plot(cif2_nonprev$quant, cif2_nonprev$med.cdf, type = "l", xlim = c(0,300), ylim = c(0,1),
     xlab = "Time", ylab = "Cumulative Incidence")
lines(cif2_nonprev$quant, cif2_nonprev$med.cdf.ci[1,], lty = 2)
lines(cif2_nonprev$quant, cif2_nonprev$med.cdf.ci[2,], lty = 2)

plot(cif2_mix$quant, cif2_mix$med.cdf, type = "l", xlim = c(0,300), ylim = c(0,1),
     xlab = "Time", ylab = "Cumulative Incidence")
lines(cif2_mix$quant, cif2_mix$med.cdf.ci[1,], lty = 2)
lines(cif2_mix$quant, cif2_mix$med.cdf.ci[2,], lty = 2)

###################
# (3) Conditional CIFs by fixing some covariates
###################
cif_mix_m1 <- get.ppd.2S(mod, fix_Z.X = c(-1, NA), pst.samples = 1e3,
                         type = "xstar", ppd.type = "quantiles", perc = seq(0,1,0.01))
cif_mix_0  <- get.ppd.2S(mod, fix_Z.X = c(0, NA),  pst.samples = 1e3,
                         type = "xstar", ppd.type = "quantiles", perc = seq(0,1,0.01))
cif_mix_p1 <- get.ppd.2S(mod, fix_Z.X = c(1, NA),  pst.samples = 1e3,
                         type = "xstar", ppd.type = "quantiles", perc = seq(0,1,0.01))

# Plot: mixture CIF for three different values of the first covariate
par(mfrow = c(1,1))
plot(cif_mix_m1$med.cdf, cif_mix_m1$perc, type = "l", xlim = c(0,300), ylim = c(0,1),
     xlab = "Time", ylab = "Cumulative Incidence", col=1)
lines(cif_mix_0$med.cdf,  cif_mix_m1$perc, col=2)
lines(cif_mix_p1$med.cdf, cif_mix_m1$perc, col=3)

par(oldpar)

Automated Heuristic Search of a Proposal Standard Deviation for bayes.2S

Description

The bayes.2S Gibbs sampler uses a Metropolis step for sampling the incidence model parameters and requires specifying a standard deviation for the normal proposal (jumping) distribution. This function uses a heuristic algorithm to find a proposal distribution standard deviation such that the Metropolis sampler accepts proposed draws at an acceptance rate within the user-defined interval (by default around 20–25%).

Usage

search.prop.sd(m, ndraws = 1000, succ.min = 3, acc.bounds.X = c(0.2, 0.25))

Arguments

Details

Starting from an initial bayes.2S model object m, the function attempts to calibrate the standard deviation of the proposal distribution. Specifically, it:

1. Runs an initial update of ndraws iterations and computes an acceptance rate.

2. If the acceptance rate lies within acc.bounds.X, the number of MCMC draws ndraws is doubled, and the process repeats.

3. Otherwise, the proposal standard deviation \sigma is adjusted based on whether the acceptance rate p is below the lower bound a or above the upper bound b of acc.bounds.X.

4. The formula for adjustment is:

   \sigma \leftarrow \sigma \times (1 - \frac{ (a-p)}{a}) \quad\text{if } p < a, \quad \sigma \leftarrow \sigma \times (1 + \frac{ (p-b)}{b}) \quad\text{if } p > b.

By default, if the acceptance rate falls within [0.2, 0.25], that \sigma is considered acceptable, and the process continues until succ.min consecutive successes (doubles) are achieved.

Value

A list with the following elements:

prop.sd.X

The final (adjusted) proposal standard deviation.

ac.X

The acceptance rate in the last iteration.

Examples

# Generate data according to Klausch et al. (2025) PIM
set.seed(2025)
dat = gen.dat(kappa = 0.7, n = 1e3, theta = 0.2,
              p = 1, p.discrete = 1,
              beta.X = c(0.2, 0.2), beta.W = c(0.2, 0.2),
              v.min = 20, v.max = 30, mean.rc = 80,
              sigma.X = 0.2, mu.X = 5, dist.X = "weibull",
              prob.r  = 1)

# An initial model fit with a moderate number of ndraws (e.g., 1e3)
mod = bayes.2S(
  Vobs = dat$Vobs, Z.X = dat$Z, Z.W = dat$Z, r = dat$r,
  kappa = 0.7, update.kappa = FALSE, ndraws = 1e3, chains = 2,
  prop.sd.X = 0.005, parallel = TRUE, dist.X = "weibull"
)

# Running the automated search
search.sd <- search.prop.sd(m = mod)
print(search.sd)

Automated Heuristic Search of a Proposal Standard Deviation for bayes.2S (sequential processing)

Description

The bayes.2S Gibbs sampler uses a Metropolis step for sampling the incidence model parameters and requires specifying a standard deviation for the normal proposal (jumping) distribution. This function uses a heuristic algorithm to find a proposal distribution standard deviation such that the Metropolis sampler accepts proposed draws at an acceptance rate within the user-defined interval (by default around 20–25%). This is the sequential processing analogue to search.prop.sd which does parallel processing by default.

Usage

search.prop.sd_seq(m, ndraws = 1000, succ.min = 3, acc.bounds.X = c(0.2, 0.25))

Arguments

Details

Starting from an initial bayes.2S model object m, the function attempts to calibrate the standard deviation of the proposal distribution. Specifically, it:

1. Runs an initial update of ndraws iterations and computes an acceptance rate.

2. If the acceptance rate lies within acc.bounds.X, the number of MCMC draws ndraws is doubled, and the process repeats.

3. Otherwise, the proposal standard deviation \sigma is adjusted based on whether the acceptance rate p is below the lower bound a or above the upper bound b of acc.bounds.X.

4. The formula for adjustment is:

   \sigma \leftarrow \sigma \times (1 - \frac{ (a-p)}{a}) \quad\text{if } p < a, \quad \sigma \leftarrow \sigma \times (1 + \frac{ (p-b)}{b}) \quad\text{if } p > b.

By default, if the acceptance rate falls within [0.2, 0.25], that \sigma is considered acceptable, and the process continues until succ.min consecutive successes (doubles) are achieved.

Value

A list with the following elements:

prop.sd.X

The final (adjusted) proposal standard deviation.

ac.X

The acceptance rate in the last iteration.

Examples

# Generate data according to Klausch et al. (2025) PIM
set.seed(2025)
dat = gen.dat(kappa = 0.7, n = 1e3, theta = 0.2,
              p = 1, p.discrete = 1,
              beta.X = c(0.2, 0.2), beta.W = c(0.2, 0.2),
              v.min = 20, v.max = 30, mean.rc = 80,
              sigma.X = 0.2, mu.X = 5, dist.X = "weibull",
              prob.r  = 1)

# An initial model fit with a moderate number of ndraws (e.g., 1e3)
mod = bayes.2S(
  Vobs = dat$Vobs, Z.X = dat$Z, Z.W = dat$Z, r = dat$r,
  kappa = 0.7, update.kappa = FALSE, ndraws = 1e3, chains = 2,
  prop.sd.X = 0.005, parallel = TRUE, dist.X = "weibull"
)

# Running the automated search
search.sd <- search.prop.sd_seq(m = mod)
print(search.sd)

Subset MCMC draws (burn-in and thinning)

Description

Takes an mcmc.list object (or a list of MCMC chains) and returns a new mcmc.list containing only the specified subset of iterations (from burnin to end) with the specified thinning interval.

Usage

trim.mcmc(obj, burnin = 1, end = nrow(as.matrix(obj[[1]])), thining = 1)

Arguments

Details

This function subsets each chain of the input obj to the specified iteration indices and creates a new mcmc.list. If you have multiple MCMC chains, each chain is trimmed in the same way.

Value

An object of class mcmc.list, representing the trimmed subset of the original MCMC draws.

Examples

# Example with a toy mcmc.list
set.seed(123)
x1 <- matrix(rnorm(2000), ncol = 2)
x2 <- matrix(rnorm(2000), ncol = 2)
mcmc_list <- mcmc.list(mcmc(x1), mcmc(x2))

# Trim and thin the chains
trimmed_mcmc <- trim.mcmc(mcmc_list, burnin = 100, end = 800, thining = 5)
summary(trimmed_mcmc)