R/prevalence.R
In chjackson/msm: Multi-State Markov and Hidden Markov Models in Continuous Time
Defines functions
plot.prevalence.msm
print.msm.prevalence
prevalence.msm
expected.msm
observed.msm
get.covhist
Documented in
plot.prevalence.msm
prevalence.msm
## Enumerate the distinct subject covariate histories in the data
## Works for time homogeneous and inhomogeneous models
## Used to calculate expected prevalences for a population with those covariates
get.covhist <- function(x, subset=NULL) {
## Keep only times where the covariate changes, or first or last obs
mf <- x$data$mf
if (x$qcmodel$ncovs > 0) {
if (!is.null(subset)) {
subs <- mf$"(subject)" %in% subset
mf <- mf[subs,,drop=FALSE]
}
subj <- match(mf$"(subject)", unique(mf$"(subject)"))
n <- length(subj)
apaste <- do.call("paste", mf[,attr(mf,"covnames"),drop=FALSE])
first <- !duplicated(subj); last <- rev(!duplicated(rev(subj)))
keep <- (c(0, apaste[1:(n-1)]) != apaste) | first | last
## Keep and tabulate unique covariate series
covseries <- split(apaste[keep], subj[keep]) # as a list of char vectors
covseries <- sapply(covseries, paste, collapse=" , ") # as one char vector, one series per pt.
## also need p matrices for different times as well as different covs.
## but only interested in cov change times if there's more than one
## transition (at least one times change point)
change.times <- mf$"(time)"; change.times[first] <- change.times[last] <- 0
change.times <- split(change.times[keep & (!(first|last))], subj[keep & (!(first|last))])
change.times <- sapply(change.times, paste, collapse= " , ")
covseries.t <- paste(covseries, change.times, sep="; ")
ids <- unique(subj)[!duplicated(covseries.t)] # subj ids, one with each distinct series
ncombs <- table(covseries.t)[unique(covseries.t)]# how many per series
covmat <- cbind(subject=subj, time=mf$"(time)", mf[,attr(mf,"covnames"),drop=FALSE])
covmat <- covmat[(subj %in% ids) & keep,]
list(example=covmat, # rows of the original data sufficient to define the distinct histories
hist=covseries.t) # one per subject listing their covariate history as a string
}
else NULL
}
### Estimate observed state occupancies in the data at a series of times
### Assume previous observed state is retained until next observation time
### Assumes times are sorted within patient (they are in data in msm objects)
observed.msm <- function(x, times=NULL, interp=c("start","midpoint"), censtime=Inf, subset=NULL)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
## For general HMMs use the Viterbi estimate of the observed state.
if (!is.null(x$pci)) {
state <- x$data$mf$"(state)"[!x$data$mf$"(pci.imp)"]
time <- x$data$mf$"(time)"[!x$data$mf$"(pci.imp)"]
subject <- x$data$mf$"(subject)"
subject <- subject[!x$data$mf$"(pci.imp)"]
} else {
if ((x$hmodel$hidden && !x$emodel$misc) ||
(!x$emodel$misc && x$cmodel$ncens>0) )
state <- viterbi.msm(x)$fitted
else if (x$emodel$misc && x$cmodel$ncens>0) {
vit <- viterbi.msm(x)$fitted
state <- x$data$mf$"(state)"
state[state %in% x$cmodel$censor] <- vit[state %in% x$cmodel$censor]
## TODO for misc models with censoring, impute only censored obs states from viterbi
} else
state <- x$data$mf$"(state)"
time <- x$data$mf$"(time)"; subject <- x$data$mf$"(subject)"
}
if (is.null(subset)) subset <- unique(subject)
time <- time[subject %in% subset]
state <- state[subject %in% subset]
subject <- subject[subject %in% subset]
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
states.expand <- matrix(nrow=length(unique(subject)), ncol=length(times))
pts <- unique(subject)
absorb <- absorbing.msm(x)
interp <- match.arg(interp)
if (!is.numeric(censtime)) stop("censtime should be numeric")
if (length(censtime)==1) censtime <- rep(censtime, length(pts))
else if (length(censtime)!=length(pts)) stop("censtime of length ", length(censtime), ", should be 1 or ", length(pts))
for (i in seq_along(pts)){
state.i <- state[(subject==pts[i])]
time.i <- time[(subject==pts[i])]
j <- 1
while(j <= length(times)) {
if (times[j] < time.i[1]) {
mtime <- max(which(times-time.i[1] < 0))
states.expand[i, j:mtime] <- NA
j <- mtime + 1
next;
} else if (times[j] > time.i[length(time.i)]) {
if (state.i[length(time.i)] %in% absorb && (times[j] <= censtime[i])) {
states.expand[i, j:(length(times))] <- state.i[length(time.i)]
} else states.expand[i, j:(length(times))] <- NA
break;
} else {
prevtime.ind <- max(which(time.i <= times[j]))
prevtime <- time.i[prevtime.ind]
if (interp=="midpoint") {
nexttime.ind <- min(which(time.i >= times[j]))
nexttime <- time.i[nexttime.ind]
midpoint <- (prevtime + nexttime) / 2
states.expand[i,j] <- state.i[if (times[j] <= midpoint) prevtime.ind else nexttime.ind]
} else
states.expand[i,j] <- state.i[prevtime.ind]
}
j <- j+1
}
}
obstab <- t(apply(states.expand, 2, function(y) table(factor(y, levels=seq(length.out=x$qmodel$nstates)))))
obsperc <- 100*obstab / rep(rowSums(obstab), ncol(obstab))
dimnames(obstab) <- dimnames(obsperc) <- list(times, paste("State", 1:x$qmodel$nstates))
obstab <- cbind(obstab, Total=rowSums(obstab))
covhist <- get.covhist(x, subset)
covcat <- ## distinct covariate history group each subject falls into (ordinal)
if (is.null(covhist)) rep(1, length(unique(subject)))
else match(covhist$hist, unique(covhist$hist))
risk <- matrix(nrow=length(times), ncol=length(unique(covcat)), dimnames = list(times, unique(covhist$hist)))
for (i in seq_along(unique(covcat))) {
obst <- t(apply(states.expand[covcat==unique(covcat)[i],,drop=FALSE], 2,
function(y) table(factor(y, levels=seq(length.out=x$qmodel$nstates)))))
risk[,i] <- rowSums(obst)
}
list(obstab=obstab, obsperc=obsperc, risk=risk)
}
expected.msm <- function(x,
times=NULL,
timezero=NULL,
initstates=NULL,
covariates="population",
misccovariates="mean",
piecewise.times=NULL,
piecewise.covariates=NULL,
risk=NULL,
subset=NULL,
ci=c("none","normal","bootstrap"),
cl = 0.95,
B = 1000,
cores = NULL
)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
time <- model.extract(x$data$mf, "time")
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
if (is.null(timezero)) timezero <- min(time)
if (is.null(risk)) risk <- observed.msm(x, times=times, subset=subset)$risk
exptab <- matrix(0, nrow=length(times), ncol=x$qmodel$nstates)
start <- min(which(times - timezero >= 0))
if (x$emodel$misc)
initprobs <- x$emodel$initprobs
else {
if (is.null(initstates))
initstates <- observed.msm(x, times=timezero)$obstab[1:x$qmodel$nstates]
initprobs <- initstates / sum(initstates)
}
if (length(times) >= start) {
for (j in start:length(times)) {
if (x$qcmodel$ncovs>0 && isTRUE(covariates=="population")) {
covmat <- get.covhist(x, subset=subset)$example
for (i in 1:length(unique(covmat$subject))) {
## sum expected prevalences for each covariate history observed in the data
subji <- unique(covmat$subject)[i]
ni <- sum(covmat$subject==subji)
ctimes <- covmat$time[covmat$subject==subji][-c(1,ni)]
covs <- covmat[covmat$subject==subji, attr(x$data$mf,"covnames"),drop=FALSE][-ni,,drop=FALSE]
ccovs <- list()
for (k in 1:nrow(covs)) ccovs[[k]] <- as.list(covs[k,,drop=FALSE])
pmat <- pmatrix.piecewise.msm(x, t1=timezero, t2=times[j], times=ctimes, covariates=ccovs)
expji <- risk[j,i] * initprobs %*% pmat
if (x$emodel$misc) { # return expected prev of obs (not true) states
if (x$ecmodel$ncovs==0) emat <- ematrix.msm(x, ci="none")
else {
ecovs <- if(length(ctimes)==0) ccovs else ccovs[[length(ccovs)]]
emat <- ematrix.msm(x, covariates=ecovs, ci="none")
}
expji <- expji %*% emat
}
exptab[j,] <- exptab[j,] + expji
}
}
else {
pmat <-
if (is.null(piecewise.times))
pmatrix.msm(x, t=times[j] - timezero, t1=timezero, covariates=covariates)
else
pmatrix.piecewise.msm(x, timezero, times[j], piecewise.times, piecewise.covariates)
expj <- rowSums(risk)[j] * initprobs %*% pmat
if (x$emodel$misc) # return expected prev of obs (not true) states
expj <- expj %*% ematrix.msm(x, covariates=misccovariates, ci="none")
exptab[j,] <- expj
}
}
}
exptab <- cbind(exptab, apply(exptab, 1, sum))
dimnames(exptab) <- list(times, c(rownames(x$qmodel$qmatrix),"Total"))
expperc <- 100*exptab[,1:x$qmodel$nstates] / exptab[, x$qmodel$nstates+1]
ci <- match.arg(ci)
e.ci <- switch(ci,
bootstrap = expected.ci.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, risk, cl, B, cores),
normal = expected.normci.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, risk, cl, B),
none = NULL)
res <-
if (ci=="none") list(exptab=exptab, expperc=expperc)
else list(exptab=list(estimates=exptab, ci=e.ci[[1]]),
expperc=list(estimates=expperc, ci=e.ci[[2]]))
names(res) <- c("Expected","Expected percentages")
res
}
### Table of observed and expected prevalences (works for misclassification and non-misclassification models)
#' Tables of observed and expected prevalences
#'
#' This provides a rough indication of the goodness of fit of a multi-state
#' model, by estimating the observed numbers of individuals occupying each
#' state at a series of times, and comparing these with forecasts from the
#' fitted model.
#'
#' The fitted transition probability matrix is used to forecast expected
#' prevalences from the state occupancy at the initial time. To produce the
#' expected number in state \eqn{j} at time \eqn{t} after the start, the number
#' of individuals under observation at time \eqn{t} (including those who have
#' died, but not those lost to follow-up) is multiplied by the product of the
#' proportion of individuals in each state at the initial time and the
#' transition probability matrix in the time interval \eqn{t}. The proportion
#' of individuals in each state at the "initial" time is estimated, if
#' necessary, in the same way as the observed prevalences.
#'
#' For misclassification models (fitted using an \code{ematrix}), this aims to
#' assess the fit of the full model for the \emph{observed} states. That is,
#' the combined Markov progression model for the true states and the
#' misclassification model. Thus, expected prevalences of \emph{true} states
#' are estimated from the assumed proportion occupying each state at the
#' initial time using the fitted transition probabiliy matrix. The vector of
#' expected prevalences of true states is then multiplied by the fitted
#' misclassification probability matrix to obtain the expected prevalences of
#' \emph{observed} states.
#'
#' For general hidden Markov models, the observed state is taken to be the
#' predicted underlying state from the Viterbi algorithm
#' (\code{\link{viterbi.msm}}). The goodness of fit of these states to the
#' underlying Markov model is tested.
#'
#' In any model, if there are censored states, then these are replaced by
#' imputed values of highest probability from the Viterbi algorithm in order to
#' calculate the observed state prevalences.
#'
#' For an example of this approach, see Gentleman \emph{et al.} (1994).
#'
#' @param x A fitted multi-state model produced by \code{\link{msm}}.
#' @param times Series of times at which to compute the observed and expected
#' prevalences of states.
#' @param timezero Initial time of the Markov process. Expected values are
#' forecasted from here. Defaults to the minimum of the observation times given
#' in the data.
#' @param initstates Optional vector of the same length as the number of
#' states. Gives the numbers of individuals occupying each state at the initial
#' time, to be used for forecasting expected prevalences. The default is those
#' observed in the data. These should add up to the actual number of people in
#' the study at the start.
#' @param covariates Covariate values for which to forecast expected state
#' occupancy. With the default \code{covariates="population"}, expected
#' prevalences are produced by summing model predictions over the covariates
#' observed in the original data, for a fair comparison with the observed
#' prevalences. This may be slow, particularly with continuous covariates.
#'
#' Predictions for fixed covariates can be obtained by supplying covariate
#' values in the standard way, as in \code{\link{qmatrix.msm}}. Therefore if
#' \code{covariates="population"} is too slow, using the mean observed values
#' through \code{covariates="mean"} may give a reasonable approximation.
#'
#' This argument is ignored if \code{piecewise.times} is specified. If there
#' are a mixture of time-constant and time-dependent covariates, then the
#' values for all covariates should be supplied in \code{piecewise.covariates}.
#' @param misccovariates (Misclassification models only) Values of covariates
#' on the misclassification probability matrix for converting expected true to
#' expected misclassified states. Ignored if \code{covariates="population"},
#' otherwise defaults to the mean values of the covariates in the data set.
#' @param piecewise.times Times at which piecewise-constant intensities change.
#' See \code{\link{pmatrix.piecewise.msm}} for how to specify this. Ignored if
#' \code{covariates="population"}. This is only required for
#' time-inhomogeneous models specified using explicit time-dependent
#' covariates, and should not be used for models specified using "pci".
#' @param piecewise.covariates Covariates on which the piecewise-constant
#' intensities depend. See \code{\link{pmatrix.piecewise.msm}} for how to
#' specify this. Ignored if \code{covariates="population"}.
#' @param ci If \code{"normal"}, then calculate a confidence interval for the
#' expected prevalences by simulating \code{B} random vectors from the
#' asymptotic multivariate normal distribution implied by the maximum
#' likelihood estimates (and covariance matrix) of the log transition
#' intensities and covariate effects, then calculating the expected prevalences
#' for each replicate.
#'
#' If \code{"bootstrap"} then calculate a confidence interval by non-parametric
#' bootstrap refitting. This is 1-2 orders of magnitude slower than the
#' \code{"normal"} method, but is expected to be more accurate. See
#' \code{\link{boot.msm}} for more details of bootstrapping in \pkg{msm}.
#'
#' If \code{"none"} (the default) then no confidence interval is calculated.
#' @param cl Width of the symmetric confidence interval, relative to 1
#' @param B Number of bootstrap replicates
#' @param cores Number of cores to use for bootstrapping using parallel
#' processing. See \code{\link{boot.msm}} for more details.
#' @param interp Suppose an individual was observed in states \eqn{S_{r-1}} and
#' \eqn{S_r} at two consecutive times \eqn{t_{r-1}} and \eqn{t_r}, and we want
#' to estimate 'observed' prevalences at a time \eqn{t} between \eqn{t_{r-1}}
#' and \eqn{t_r}.
#'
#' If \code{interp="start"}, then individuals are assumed to be in state
#' \eqn{S_{r-1}} at time \eqn{t}, the same state as they were at \eqn{t_{r-1}}.
#'
#' If \code{interp="midpoint"} then if \eqn{t <= (t_{r-1} + t_r) / 2}, the
#' midpoint of \eqn{t_{r-1}} and \eqn{t_r}, the state at \eqn{t} is assumed to
#' be \eqn{S_{r-1}}, otherwise \eqn{S_{r}}. This is generally more reasonable
#' for "progressive" models.
#' @param censtime Adjustment to the observed prevalences to account for
#' limited follow-up in the data.
#'
#' If the time is greater than \code{censtime} and the patient has reached an
#' absorbing state, then that subject will be removed from the risk set. For
#' example, if patients have died but would only have been observed up to this
#' time, then this avoids overestimating the proportion of people who are dead
#' at later times.
#'
#' This can be supplied as a single value, or as a vector with one element per
#' subject (after any \code{subset} has been taken), in the same order as the
#' original data. This vector also only includes subjects with complete data,
#' thus it excludes for example subjects with only one observation (thus no
#' observed transitions), and subjects for whom every observation has missing
#' values. (Note, to help construct this, the complete data used for the model
#' fit can be accessed with \code{model.frame(x)}, where \code{x} is the fitted
#' model object)
#'
#' This is ignored if it is less than the subject's maximum observation time.
#' @param subset Subset of subjects to calculate observed prevalences for.
#' @param plot Generate a plot of observed against expected prevalences. See
#' \code{\link{plot.prevalence.msm}}
#' @param ... Further arguments to pass to \code{\link{plot.prevalence.msm}}.
#' @return A list of matrices, with components:
#'
#' \item{Observed}{Table of observed numbers of individuals in each state at
#' each time}
#'
#' \item{Observed percentages}{Corresponding percentage of the individuals at
#' risk at each time.}
#'
#' \item{Expected}{Table of corresponding expected numbers.}
#'
#' \item{Expected percentages}{Corresponding percentage of the individuals at
#' risk at each time.}
#'
#' Or if \code{ci.boot = TRUE}, the component \code{Expected} is a list with
#' components \code{estimates} and \code{ci}.\cr \code{estimates} is a matrix
#' of the expected prevalences, and \code{ci} is a list of two matrices,
#' containing the confidence limits. The component \code{Expected percentages}
#' has a similar format.
#' @author C. H. Jackson \email{chris.jackson@@mrc-bsu.cam.ac.uk}
#' @seealso \code{\link{msm}}, \code{\link{summary.msm}}
#' @references Gentleman, R.C., Lawless, J.F., Lindsey, J.C. and Yan, P.
#' Multi-state Markov models for analysing incomplete disease history data with
#' illustrations for HIV disease. \emph{Statistics in Medicine} (1994) 13(3):
#' 805--821.
#'
#' Titman, A.C., Sharples, L. D. Model diagnostics for multi-state models.
#' \emph{Statistical Methods in Medical Research} (2010) 19(6):621-651.
#' @keywords models
#' @export prevalence.msm
prevalence.msm <- function(x,
times=NULL,
timezero=NULL,
initstates=NULL,
covariates="population",
misccovariates="mean",
piecewise.times=NULL,
piecewise.covariates=NULL,
ci=c("none","normal","bootstrap"),
cl = 0.95,
B = 1000,
cores = NULL,
interp=c("start","midpoint"),
censtime=Inf,
subset=NULL,
plot = FALSE, ...
)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
## Estimate observed state occupancies in the data at a series of times
time <- model.extract(x$data$mf, "time")
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
obs <- observed.msm(x, times, interp, censtime, subset)
## Work out expected state occupancies by forecasting from transition probabilities
expec <- expected.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, obs$risk, subset, ci, cl, B, cores)
res <- list(observed=obs$obstab, expected=expec[[1]], obsperc=obs$obsperc, expperc=expec[[2]])
names(res) <- c("Observed", "Expected", "Observed percentages", "Expected percentages")
class(res) <- "msm.prevalence"
if (plot) plot.prevalence.msm(x, mintime=min(times), maxtime=max(times), timezero=timezero, initstates=initstates,
interp=interp, censtime=censtime, covariates=covariates, misccovariates=misccovariates,
piecewise.times=piecewise.times, piecewise.covariates=piecewise.covariates, ...)
res
}
#' @export
print.msm.prevalence <- function(x,...){
print(unclass(x))
}
#' Plot of observed and expected prevalences
#'
#' Provides a rough indication of goodness of fit of a multi-state model, by
#' estimating the observed numbers of individuals occupying a state at a series
#' of times, and plotting these against forecasts from the fitted model, for
#' each state. Observed prevalences are indicated as solid lines, expected
#' prevalences as dashed lines.
#'
#' See \code{\link{prevalence.msm}} for details of the assumptions underlying
#' this method.
#'
#' Observed prevalences are plotted with a solid line, and expected prevalences
#' with a dotted line.
#'
#' @param x A fitted multi-state model produced by \code{\link{msm}}.
#' @param mintime Minimum time at which to compute the observed and expected
#' prevalences of states.
#' @param maxtime Maximum time at which to compute the observed and expected
#' prevalences of states.
#' @param timezero Initial time of the Markov process. Expected values are
#' forecasted from here. Defaults to the minimum of the observation times given
#' in the data.
#' @param initstates Optional vector of the same length as the number of
#' states. Gives the numbers of individuals occupying each state at the initial
#' time, to be used for forecasting expected prevalences. The default is those
#' observed in the data. These should add up to the actual number of people in
#' the study at the start.
#' @param interp Interpolation method for observed states, see
#' \code{\link{prevalence.msm}}.
#' @param censtime Subject-specific maximum follow-up times, see
#' \code{\link{prevalence.msm}}.
#' @param subset Vector of the subject identifiers to calculated observed
#' prevalences for.
#' @param covariates Covariate values for which to forecast expected state
#' occupancy. See \code{\link{prevalence.msm}} --- if this function runs too
#' slowly, as it may if there are continuous covariates, replace
#' \code{covariates="population"} with \code{covariates="mean"}.
#' @param misccovariates (Misclassification models only) Values of covariates
#' on the misclassification probability matrix. See
#' \code{\link{prevalence.msm}}.
#' @param piecewise.times Times at which piecewise-constant intensities change.
#' See \code{\link{prevalence.msm}}.
#' @param piecewise.covariates Covariates on which the piecewise-constant
#' intensities depend. See \code{\link{prevalence.msm}}.
#' @param xlab x axis label.
#' @param ylab y axis label.
#' @param lwd.obs Line width for observed prevalences. See \code{\link{par}}.
#' @param lwd.exp Line width for expected prevalences. See \code{\link{par}}.
#' @param lty.obs Line type for observed prevalences. See \code{\link{par}}.
#' @param lty.exp Line type for expected prevalences. See \code{\link{par}}.
#' @param col.obs Line colour for observed prevalences. See \code{\link{par}}.
#' @param col.exp Line colour for expected prevalences. See \code{\link{par}}.
#' @param legend.pos Vector of the \eqn{x} and \eqn{y} position, respectively,
#' of the legend.
#' @param ... Further arguments to be passed to the generic \code{\link{plot}}
#' function.
#' @seealso \code{\link{prevalence.msm}}
#' @references Gentleman, R.C., Lawless, J.F., Lindsey, J.C. and Yan, P.
#' Multi-state Markov models for analysing incomplete disease history data with
#' illustrations for HIV disease. \emph{Statistics in Medicine} (1994) 13(3):
#' 805--821.
#' @keywords models
#' @export plot.prevalence.msm
#' @export
plot.prevalence.msm <- function(x, mintime=NULL, maxtime=NULL, timezero=NULL, initstates=NULL,
interp=c("start","midpoint"), censtime=Inf, subset=NULL,
covariates="population", misccovariates="mean",
piecewise.times=NULL, piecewise.covariates=NULL, xlab="Times",ylab="Prevalence (%)",
lwd.obs=1, lwd.exp=1, lty.obs=1, lty.exp=2,
col.obs="blue", col.exp="red", legend.pos=NULL,...){
if (!inherits(x, "msm")) stop("expected x to be a msm model")
time <- model.extract(x$data$mf, "time")
if (is.null(mintime)) mintime <- min(time)
if (is.null(maxtime)) maxtime <- max(time)
t <- seq(mintime, maxtime, length.out=100)
obs <- observed.msm(x, t, interp, censtime, subset)
expec <- expected.msm(x, t, timezero=timezero, initstates=initstates, covariates=covariates, misccovariates=misccovariates,
piecewise.times=piecewise.times, piecewise.covariates=piecewise.covariates, risk=obs$risk, subset=subset, ci="none")[[2]]
states <- seq(length.out=x$qmodel$nstates)
S <- length(states)
ncols <- ceiling(sqrt(S))
nrows <- if (floor(sqrt(S))^2 < S && S <= floor(sqrt(S))*ceiling(sqrt(S))) floor(sqrt(S)) else ceiling(sqrt(S))
par(mfrow=c(nrows, ncols))
for (i in states) {
plot(t, obs$obsperc[,i], type="l", ylim=c(0, 100), xlab=xlab, ylab=ylab, lwd=lwd.obs, lty=lty.obs, col=col.obs,
main=rownames(x$qmodel$qmatrix)[i],...)
lines(t, expec[,i], lwd=lwd.exp, lty=lty.exp, col=col.exp)
}
if (!is.numeric(legend.pos) || length(legend.pos) != 2)
legend.pos <- c(0.4*maxtime, 40)
legend(x=legend.pos[1], y=legend.pos[2], legend=c("Observed","Expected"), lty=c(lty.obs,lty.exp), lwd=c(lwd.obs,lwd.exp), col=c(col.obs,col.exp))
invisible()
}
chjackson/msm documentation built on March 3, 2024, 1:05 a.m.
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Defines functions plot.prevalence.msm print.msm.prevalence prevalence.msm expected.msm observed.msm get.covhist
Documented in plot.prevalence.msm prevalence.msm
## Enumerate the distinct subject covariate histories in the data
## Works for time homogeneous and inhomogeneous models
## Used to calculate expected prevalences for a population with those covariates
get.covhist <- function(x, subset=NULL) {
## Keep only times where the covariate changes, or first or last obs
mf <- x$data$mf
if (x$qcmodel$ncovs > 0) {
if (!is.null(subset)) {
subs <- mf$"(subject)" %in% subset
mf <- mf[subs,,drop=FALSE]
}
subj <- match(mf$"(subject)", unique(mf$"(subject)"))
n <- length(subj)
apaste <- do.call("paste", mf[,attr(mf,"covnames"),drop=FALSE])
first <- !duplicated(subj); last <- rev(!duplicated(rev(subj)))
keep <- (c(0, apaste[1:(n-1)]) != apaste) | first | last
## Keep and tabulate unique covariate series
covseries <- split(apaste[keep], subj[keep]) # as a list of char vectors
covseries <- sapply(covseries, paste, collapse=" , ") # as one char vector, one series per pt.
## also need p matrices for different times as well as different covs.
## but only interested in cov change times if there's more than one
## transition (at least one times change point)
change.times <- mf$"(time)"; change.times[first] <- change.times[last] <- 0
change.times <- split(change.times[keep & (!(first|last))], subj[keep & (!(first|last))])
change.times <- sapply(change.times, paste, collapse= " , ")
covseries.t <- paste(covseries, change.times, sep="; ")
ids <- unique(subj)[!duplicated(covseries.t)] # subj ids, one with each distinct series
ncombs <- table(covseries.t)[unique(covseries.t)]# how many per series
covmat <- cbind(subject=subj, time=mf$"(time)", mf[,attr(mf,"covnames"),drop=FALSE])
covmat <- covmat[(subj %in% ids) & keep,]
list(example=covmat, # rows of the original data sufficient to define the distinct histories
hist=covseries.t) # one per subject listing their covariate history as a string
}
else NULL
}
### Estimate observed state occupancies in the data at a series of times
### Assume previous observed state is retained until next observation time
### Assumes times are sorted within patient (they are in data in msm objects)
observed.msm <- function(x, times=NULL, interp=c("start","midpoint"), censtime=Inf, subset=NULL)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
## For general HMMs use the Viterbi estimate of the observed state.
if (!is.null(x$pci)) {
state <- x$data$mf$"(state)"[!x$data$mf$"(pci.imp)"]
time <- x$data$mf$"(time)"[!x$data$mf$"(pci.imp)"]
subject <- x$data$mf$"(subject)"
subject <- subject[!x$data$mf$"(pci.imp)"]
} else {
if ((x$hmodel$hidden && !x$emodel$misc) ||
(!x$emodel$misc && x$cmodel$ncens>0) )
state <- viterbi.msm(x)$fitted
else if (x$emodel$misc && x$cmodel$ncens>0) {
vit <- viterbi.msm(x)$fitted
state <- x$data$mf$"(state)"
state[state %in% x$cmodel$censor] <- vit[state %in% x$cmodel$censor]
## TODO for misc models with censoring, impute only censored obs states from viterbi
} else
state <- x$data$mf$"(state)"
time <- x$data$mf$"(time)"; subject <- x$data$mf$"(subject)"
}
if (is.null(subset)) subset <- unique(subject)
time <- time[subject %in% subset]
state <- state[subject %in% subset]
subject <- subject[subject %in% subset]
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
states.expand <- matrix(nrow=length(unique(subject)), ncol=length(times))
pts <- unique(subject)
absorb <- absorbing.msm(x)
interp <- match.arg(interp)
if (!is.numeric(censtime)) stop("censtime should be numeric")
if (length(censtime)==1) censtime <- rep(censtime, length(pts))
else if (length(censtime)!=length(pts)) stop("censtime of length ", length(censtime), ", should be 1 or ", length(pts))
for (i in seq_along(pts)){
state.i <- state[(subject==pts[i])]
time.i <- time[(subject==pts[i])]
j <- 1
while(j <= length(times)) {
if (times[j] < time.i[1]) {
mtime <- max(which(times-time.i[1] < 0))
states.expand[i, j:mtime] <- NA
j <- mtime + 1
next;
} else if (times[j] > time.i[length(time.i)]) {
if (state.i[length(time.i)] %in% absorb && (times[j] <= censtime[i])) {
states.expand[i, j:(length(times))] <- state.i[length(time.i)]
} else states.expand[i, j:(length(times))] <- NA
break;
} else {
prevtime.ind <- max(which(time.i <= times[j]))
prevtime <- time.i[prevtime.ind]
if (interp=="midpoint") {
nexttime.ind <- min(which(time.i >= times[j]))
nexttime <- time.i[nexttime.ind]
midpoint <- (prevtime + nexttime) / 2
states.expand[i,j] <- state.i[if (times[j] <= midpoint) prevtime.ind else nexttime.ind]
} else
states.expand[i,j] <- state.i[prevtime.ind]
}
j <- j+1
}
}
obstab <- t(apply(states.expand, 2, function(y) table(factor(y, levels=seq(length.out=x$qmodel$nstates)))))
obsperc <- 100*obstab / rep(rowSums(obstab), ncol(obstab))
dimnames(obstab) <- dimnames(obsperc) <- list(times, paste("State", 1:x$qmodel$nstates))
obstab <- cbind(obstab, Total=rowSums(obstab))
covhist <- get.covhist(x, subset)
covcat <- ## distinct covariate history group each subject falls into (ordinal)
if (is.null(covhist)) rep(1, length(unique(subject)))
else match(covhist$hist, unique(covhist$hist))
risk <- matrix(nrow=length(times), ncol=length(unique(covcat)), dimnames = list(times, unique(covhist$hist)))
for (i in seq_along(unique(covcat))) {
obst <- t(apply(states.expand[covcat==unique(covcat)[i],,drop=FALSE], 2,
function(y) table(factor(y, levels=seq(length.out=x$qmodel$nstates)))))
risk[,i] <- rowSums(obst)
}
list(obstab=obstab, obsperc=obsperc, risk=risk)
}
expected.msm <- function(x,
times=NULL,
timezero=NULL,
initstates=NULL,
covariates="population",
misccovariates="mean",
piecewise.times=NULL,
piecewise.covariates=NULL,
risk=NULL,
subset=NULL,
ci=c("none","normal","bootstrap"),
cl = 0.95,
B = 1000,
cores = NULL
)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
time <- model.extract(x$data$mf, "time")
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
if (is.null(timezero)) timezero <- min(time)
if (is.null(risk)) risk <- observed.msm(x, times=times, subset=subset)$risk
exptab <- matrix(0, nrow=length(times), ncol=x$qmodel$nstates)
start <- min(which(times - timezero >= 0))
if (x$emodel$misc)
initprobs <- x$emodel$initprobs
else {
if (is.null(initstates))
initstates <- observed.msm(x, times=timezero)$obstab[1:x$qmodel$nstates]
initprobs <- initstates / sum(initstates)
}
if (length(times) >= start) {
for (j in start:length(times)) {
if (x$qcmodel$ncovs>0 && isTRUE(covariates=="population")) {
covmat <- get.covhist(x, subset=subset)$example
for (i in 1:length(unique(covmat$subject))) {
## sum expected prevalences for each covariate history observed in the data
subji <- unique(covmat$subject)[i]
ni <- sum(covmat$subject==subji)
ctimes <- covmat$time[covmat$subject==subji][-c(1,ni)]
covs <- covmat[covmat$subject==subji, attr(x$data$mf,"covnames"),drop=FALSE][-ni,,drop=FALSE]
ccovs <- list()
for (k in 1:nrow(covs)) ccovs[[k]] <- as.list(covs[k,,drop=FALSE])
pmat <- pmatrix.piecewise.msm(x, t1=timezero, t2=times[j], times=ctimes, covariates=ccovs)
expji <- risk[j,i] * initprobs %*% pmat
if (x$emodel$misc) { # return expected prev of obs (not true) states
if (x$ecmodel$ncovs==0) emat <- ematrix.msm(x, ci="none")
else {
ecovs <- if(length(ctimes)==0) ccovs else ccovs[[length(ccovs)]]
emat <- ematrix.msm(x, covariates=ecovs, ci="none")
}
expji <- expji %*% emat
}
exptab[j,] <- exptab[j,] + expji
}
}
else {
pmat <-
if (is.null(piecewise.times))
pmatrix.msm(x, t=times[j] - timezero, t1=timezero, covariates=covariates)
else
pmatrix.piecewise.msm(x, timezero, times[j], piecewise.times, piecewise.covariates)
expj <- rowSums(risk)[j] * initprobs %*% pmat
if (x$emodel$misc) # return expected prev of obs (not true) states
expj <- expj %*% ematrix.msm(x, covariates=misccovariates, ci="none")
exptab[j,] <- expj
}
}
}
exptab <- cbind(exptab, apply(exptab, 1, sum))
dimnames(exptab) <- list(times, c(rownames(x$qmodel$qmatrix),"Total"))
expperc <- 100*exptab[,1:x$qmodel$nstates] / exptab[, x$qmodel$nstates+1]
ci <- match.arg(ci)
e.ci <- switch(ci,
bootstrap = expected.ci.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, risk, cl, B, cores),
normal = expected.normci.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, risk, cl, B),
none = NULL)
res <-
if (ci=="none") list(exptab=exptab, expperc=expperc)
else list(exptab=list(estimates=exptab, ci=e.ci[[1]]),
expperc=list(estimates=expperc, ci=e.ci[[2]]))
names(res) <- c("Expected","Expected percentages")
res
}
### Table of observed and expected prevalences (works for misclassification and non-misclassification models)
#' Tables of observed and expected prevalences
#'
#' This provides a rough indication of the goodness of fit of a multi-state
#' model, by estimating the observed numbers of individuals occupying each
#' state at a series of times, and comparing these with forecasts from the
#' fitted model.
#'
#' The fitted transition probability matrix is used to forecast expected
#' prevalences from the state occupancy at the initial time. To produce the
#' expected number in state \eqn{j} at time \eqn{t} after the start, the number
#' of individuals under observation at time \eqn{t} (including those who have
#' died, but not those lost to follow-up) is multiplied by the product of the
#' proportion of individuals in each state at the initial time and the
#' transition probability matrix in the time interval \eqn{t}. The proportion
#' of individuals in each state at the "initial" time is estimated, if
#' necessary, in the same way as the observed prevalences.
#'
#' For misclassification models (fitted using an \code{ematrix}), this aims to
#' assess the fit of the full model for the \emph{observed} states. That is,
#' the combined Markov progression model for the true states and the
#' misclassification model. Thus, expected prevalences of \emph{true} states
#' are estimated from the assumed proportion occupying each state at the
#' initial time using the fitted transition probabiliy matrix. The vector of
#' expected prevalences of true states is then multiplied by the fitted
#' misclassification probability matrix to obtain the expected prevalences of
#' \emph{observed} states.
#'
#' For general hidden Markov models, the observed state is taken to be the
#' predicted underlying state from the Viterbi algorithm
#' (\code{\link{viterbi.msm}}). The goodness of fit of these states to the
#' underlying Markov model is tested.
#'
#' In any model, if there are censored states, then these are replaced by
#' imputed values of highest probability from the Viterbi algorithm in order to
#' calculate the observed state prevalences.
#'
#' For an example of this approach, see Gentleman \emph{et al.} (1994).
#'
#' @param x A fitted multi-state model produced by \code{\link{msm}}.
#' @param times Series of times at which to compute the observed and expected
#' prevalences of states.
#' @param timezero Initial time of the Markov process. Expected values are
#' forecasted from here. Defaults to the minimum of the observation times given
#' in the data.
#' @param initstates Optional vector of the same length as the number of
#' states. Gives the numbers of individuals occupying each state at the initial
#' time, to be used for forecasting expected prevalences. The default is those
#' observed in the data. These should add up to the actual number of people in
#' the study at the start.
#' @param covariates Covariate values for which to forecast expected state
#' occupancy. With the default \code{covariates="population"}, expected
#' prevalences are produced by summing model predictions over the covariates
#' observed in the original data, for a fair comparison with the observed
#' prevalences. This may be slow, particularly with continuous covariates.
#'
#' Predictions for fixed covariates can be obtained by supplying covariate
#' values in the standard way, as in \code{\link{qmatrix.msm}}. Therefore if
#' \code{covariates="population"} is too slow, using the mean observed values
#' through \code{covariates="mean"} may give a reasonable approximation.
#'
#' This argument is ignored if \code{piecewise.times} is specified. If there
#' are a mixture of time-constant and time-dependent covariates, then the
#' values for all covariates should be supplied in \code{piecewise.covariates}.
#' @param misccovariates (Misclassification models only) Values of covariates
#' on the misclassification probability matrix for converting expected true to
#' expected misclassified states. Ignored if \code{covariates="population"},
#' otherwise defaults to the mean values of the covariates in the data set.
#' @param piecewise.times Times at which piecewise-constant intensities change.
#' See \code{\link{pmatrix.piecewise.msm}} for how to specify this. Ignored if
#' \code{covariates="population"}. This is only required for
#' time-inhomogeneous models specified using explicit time-dependent
#' covariates, and should not be used for models specified using "pci".
#' @param piecewise.covariates Covariates on which the piecewise-constant
#' intensities depend. See \code{\link{pmatrix.piecewise.msm}} for how to
#' specify this. Ignored if \code{covariates="population"}.
#' @param ci If \code{"normal"}, then calculate a confidence interval for the
#' expected prevalences by simulating \code{B} random vectors from the
#' asymptotic multivariate normal distribution implied by the maximum
#' likelihood estimates (and covariance matrix) of the log transition
#' intensities and covariate effects, then calculating the expected prevalences
#' for each replicate.
#'
#' If \code{"bootstrap"} then calculate a confidence interval by non-parametric
#' bootstrap refitting. This is 1-2 orders of magnitude slower than the
#' \code{"normal"} method, but is expected to be more accurate. See
#' \code{\link{boot.msm}} for more details of bootstrapping in \pkg{msm}.
#'
#' If \code{"none"} (the default) then no confidence interval is calculated.
#' @param cl Width of the symmetric confidence interval, relative to 1
#' @param B Number of bootstrap replicates
#' @param cores Number of cores to use for bootstrapping using parallel
#' processing. See \code{\link{boot.msm}} for more details.
#' @param interp Suppose an individual was observed in states \eqn{S_{r-1}} and
#' \eqn{S_r} at two consecutive times \eqn{t_{r-1}} and \eqn{t_r}, and we want
#' to estimate 'observed' prevalences at a time \eqn{t} between \eqn{t_{r-1}}
#' and \eqn{t_r}.
#'
#' If \code{interp="start"}, then individuals are assumed to be in state
#' \eqn{S_{r-1}} at time \eqn{t}, the same state as they were at \eqn{t_{r-1}}.
#'
#' If \code{interp="midpoint"} then if \eqn{t <= (t_{r-1} + t_r) / 2}, the
#' midpoint of \eqn{t_{r-1}} and \eqn{t_r}, the state at \eqn{t} is assumed to
#' be \eqn{S_{r-1}}, otherwise \eqn{S_{r}}. This is generally more reasonable
#' for "progressive" models.
#' @param censtime Adjustment to the observed prevalences to account for
#' limited follow-up in the data.
#'
#' If the time is greater than \code{censtime} and the patient has reached an
#' absorbing state, then that subject will be removed from the risk set. For
#' example, if patients have died but would only have been observed up to this
#' time, then this avoids overestimating the proportion of people who are dead
#' at later times.
#'
#' This can be supplied as a single value, or as a vector with one element per
#' subject (after any \code{subset} has been taken), in the same order as the
#' original data. This vector also only includes subjects with complete data,
#' thus it excludes for example subjects with only one observation (thus no
#' observed transitions), and subjects for whom every observation has missing
#' values. (Note, to help construct this, the complete data used for the model
#' fit can be accessed with \code{model.frame(x)}, where \code{x} is the fitted
#' model object)
#'
#' This is ignored if it is less than the subject's maximum observation time.
#' @param subset Subset of subjects to calculate observed prevalences for.
#' @param plot Generate a plot of observed against expected prevalences. See
#' \code{\link{plot.prevalence.msm}}
#' @param ... Further arguments to pass to \code{\link{plot.prevalence.msm}}.
#' @return A list of matrices, with components:
#'
#' \item{Observed}{Table of observed numbers of individuals in each state at
#' each time}
#'
#' \item{Observed percentages}{Corresponding percentage of the individuals at
#' risk at each time.}
#'
#' \item{Expected}{Table of corresponding expected numbers.}
#'
#' \item{Expected percentages}{Corresponding percentage of the individuals at
#' risk at each time.}
#'
#' Or if \code{ci.boot = TRUE}, the component \code{Expected} is a list with
#' components \code{estimates} and \code{ci}.\cr \code{estimates} is a matrix
#' of the expected prevalences, and \code{ci} is a list of two matrices,
#' containing the confidence limits. The component \code{Expected percentages}
#' has a similar format.
#' @author C. H. Jackson \email{chris.jackson@@mrc-bsu.cam.ac.uk}
#' @seealso \code{\link{msm}}, \code{\link{summary.msm}}
#' @references Gentleman, R.C., Lawless, J.F., Lindsey, J.C. and Yan, P.
#' Multi-state Markov models for analysing incomplete disease history data with
#' illustrations for HIV disease. \emph{Statistics in Medicine} (1994) 13(3):
#' 805--821.
#'
#' Titman, A.C., Sharples, L. D. Model diagnostics for multi-state models.
#' \emph{Statistical Methods in Medical Research} (2010) 19(6):621-651.
#' @keywords models
#' @export prevalence.msm
prevalence.msm <- function(x,
times=NULL,
timezero=NULL,
initstates=NULL,
covariates="population",
misccovariates="mean",
piecewise.times=NULL,
piecewise.covariates=NULL,
ci=c("none","normal","bootstrap"),
cl = 0.95,
B = 1000,
cores = NULL,
interp=c("start","midpoint"),
censtime=Inf,
subset=NULL,
plot = FALSE, ...
)
{
if (!inherits(x, "msm")) stop("expected x to be a msm model")
## Estimate observed state occupancies in the data at a series of times
time <- model.extract(x$data$mf, "time")
if (is.null(times))
times <- seq(min(time), max(time), (max(time) - min(time))/10)
obs <- observed.msm(x, times, interp, censtime, subset)
## Work out expected state occupancies by forecasting from transition probabilities
expec <- expected.msm(x, times, timezero, initstates, covariates, misccovariates,
piecewise.times, piecewise.covariates, obs$risk, subset, ci, cl, B, cores)
res <- list(observed=obs$obstab, expected=expec[[1]], obsperc=obs$obsperc, expperc=expec[[2]])
names(res) <- c("Observed", "Expected", "Observed percentages", "Expected percentages")
class(res) <- "msm.prevalence"
if (plot) plot.prevalence.msm(x, mintime=min(times), maxtime=max(times), timezero=timezero, initstates=initstates,
interp=interp, censtime=censtime, covariates=covariates, misccovariates=misccovariates,
piecewise.times=piecewise.times, piecewise.covariates=piecewise.covariates, ...)
res
}
#' @export
print.msm.prevalence <- function(x,...){
print(unclass(x))
}
#' Plot of observed and expected prevalences
#'
#' Provides a rough indication of goodness of fit of a multi-state model, by
#' estimating the observed numbers of individuals occupying a state at a series
#' of times, and plotting these against forecasts from the fitted model, for
#' each state. Observed prevalences are indicated as solid lines, expected
#' prevalences as dashed lines.
#'
#' See \code{\link{prevalence.msm}} for details of the assumptions underlying
#' this method.
#'
#' Observed prevalences are plotted with a solid line, and expected prevalences
#' with a dotted line.
#'
#' @param x A fitted multi-state model produced by \code{\link{msm}}.
#' @param mintime Minimum time at which to compute the observed and expected
#' prevalences of states.
#' @param maxtime Maximum time at which to compute the observed and expected
#' prevalences of states.
#' @param timezero Initial time of the Markov process. Expected values are
#' forecasted from here. Defaults to the minimum of the observation times given
#' in the data.
#' @param initstates Optional vector of the same length as the number of
#' states. Gives the numbers of individuals occupying each state at the initial
#' time, to be used for forecasting expected prevalences. The default is those
#' observed in the data. These should add up to the actual number of people in
#' the study at the start.
#' @param interp Interpolation method for observed states, see
#' \code{\link{prevalence.msm}}.
#' @param censtime Subject-specific maximum follow-up times, see
#' \code{\link{prevalence.msm}}.
#' @param subset Vector of the subject identifiers to calculated observed
#' prevalences for.
#' @param covariates Covariate values for which to forecast expected state
#' occupancy. See \code{\link{prevalence.msm}} --- if this function runs too
#' slowly, as it may if there are continuous covariates, replace
#' \code{covariates="population"} with \code{covariates="mean"}.
#' @param misccovariates (Misclassification models only) Values of covariates
#' on the misclassification probability matrix. See
#' \code{\link{prevalence.msm}}.
#' @param piecewise.times Times at which piecewise-constant intensities change.
#' See \code{\link{prevalence.msm}}.
#' @param piecewise.covariates Covariates on which the piecewise-constant
#' intensities depend. See \code{\link{prevalence.msm}}.
#' @param xlab x axis label.
#' @param ylab y axis label.
#' @param lwd.obs Line width for observed prevalences. See \code{\link{par}}.
#' @param lwd.exp Line width for expected prevalences. See \code{\link{par}}.
#' @param lty.obs Line type for observed prevalences. See \code{\link{par}}.
#' @param lty.exp Line type for expected prevalences. See \code{\link{par}}.
#' @param col.obs Line colour for observed prevalences. See \code{\link{par}}.
#' @param col.exp Line colour for expected prevalences. See \code{\link{par}}.
#' @param legend.pos Vector of the \eqn{x} and \eqn{y} position, respectively,
#' of the legend.
#' @param ... Further arguments to be passed to the generic \code{\link{plot}}
#' function.
#' @seealso \code{\link{prevalence.msm}}
#' @references Gentleman, R.C., Lawless, J.F., Lindsey, J.C. and Yan, P.
#' Multi-state Markov models for analysing incomplete disease history data with
#' illustrations for HIV disease. \emph{Statistics in Medicine} (1994) 13(3):
#' 805--821.
#' @keywords models
#' @export plot.prevalence.msm
#' @export
plot.prevalence.msm <- function(x, mintime=NULL, maxtime=NULL, timezero=NULL, initstates=NULL,
interp=c("start","midpoint"), censtime=Inf, subset=NULL,
covariates="population", misccovariates="mean",
piecewise.times=NULL, piecewise.covariates=NULL, xlab="Times",ylab="Prevalence (%)",
lwd.obs=1, lwd.exp=1, lty.obs=1, lty.exp=2,
col.obs="blue", col.exp="red", legend.pos=NULL,...){
if (!inherits(x, "msm")) stop("expected x to be a msm model")
time <- model.extract(x$data$mf, "time")
if (is.null(mintime)) mintime <- min(time)
if (is.null(maxtime)) maxtime <- max(time)
t <- seq(mintime, maxtime, length.out=100)
obs <- observed.msm(x, t, interp, censtime, subset)
expec <- expected.msm(x, t, timezero=timezero, initstates=initstates, covariates=covariates, misccovariates=misccovariates,
piecewise.times=piecewise.times, piecewise.covariates=piecewise.covariates, risk=obs$risk, subset=subset, ci="none")[[2]]
states <- seq(length.out=x$qmodel$nstates)
S <- length(states)
ncols <- ceiling(sqrt(S))
nrows <- if (floor(sqrt(S))^2 < S && S <= floor(sqrt(S))*ceiling(sqrt(S))) floor(sqrt(S)) else ceiling(sqrt(S))
par(mfrow=c(nrows, ncols))
for (i in states) {
plot(t, obs$obsperc[,i], type="l", ylim=c(0, 100), xlab=xlab, ylab=ylab, lwd=lwd.obs, lty=lty.obs, col=col.obs,
main=rownames(x$qmodel$qmatrix)[i],...)
lines(t, expec[,i], lwd=lwd.exp, lty=lty.exp, col=col.exp)
}
if (!is.numeric(legend.pos) || length(legend.pos) != 2)
legend.pos <- c(0.4*maxtime, 40)
legend(x=legend.pos[1], y=legend.pos[2], legend=c("Observed","Expected"), lty=c(lty.obs,lty.exp), lwd=c(lwd.obs,lwd.exp), col=c(col.obs,col.exp))
invisible()
}
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Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.