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.
prevalence.msm(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, ...)A list of matrices, with components:
Table of observed numbers of individuals in each state at each time
Corresponding percentage of the individuals at risk at each time.
Table of corresponding expected numbers.
Corresponding percentage of the individuals at risk at each time.
Or if ci.boot = TRUE, the component Expected is a list
with components estimates and ci.
estimates is a matrix of the expected prevalences, and
ci is a list of two matrices, containing the confidence limits.
The component
Expected percentages has a similar format.
A fitted multi-state model produced by msm.
Series of times at which to compute the observed and expected prevalences of states.
Initial time of the Markov process. Expected values are forecasted from here. Defaults to the minimum of the observation times given in the data.
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.
Covariate values for which to forecast expected
state occupancy. With the default 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
qmatrix.msm. Therefore if
covariates="population" is too slow, using the mean observed
values through covariates="mean" may give a reasonable
approximation.
This argument is ignored if 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
piecewise.covariates.
(Misclassification models only)
Values of covariates on the misclassification probability matrix
for converting expected true to expected misclassified states.
Ignored if covariates="population", otherwise
defaults to the mean values of the covariates in the data set.
Times at which piecewise-constant intensities
change. See pmatrix.piecewise.msm for how
to specify this. Ignored if 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".
Covariates on which the piecewise-constant
intensities depend. See pmatrix.piecewise.msm for how
to specify this. Ignored if covariates="population".
If "normal", then calculate a confidence interval for
the expected prevalences by simulating 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 "bootstrap" then calculate a confidence interval by
non-parametric bootstrap refitting. This is 1-2 orders of magnitude
slower than the "normal" method, but is expected to be more
accurate. See boot.msm for more details of
bootstrapping in msm.
If "none" (the default) then no confidence interval is
calculated.
Width of the symmetric confidence interval, relative to 1
Number of bootstrap replicates
Number of cores to use for bootstrapping using parallel
processing. See boot.msm for more details.
Suppose an individual was observed in states \(S_{r-1}\) and \(S_r\) at two consecutive times \(t_{r-1}\) and \(t_r\), and we want to estimate 'observed' prevalences at a time \(t\) between \(t_{r-1}\) and \(t_r\).
If interp="start", then individuals are assumed to be in
state \(S_{r-1}\) at time \(t\), the same state as they were at \(t_{r-1}\).
If interp="midpoint" then if \(t <= (t_{r-1} + t_r) / 2\),
the midpoint of \(t_{r-1}\) and \(t_r\), the state at \(t\)
is assumed to be \(S_{r-1}\), otherwise \(S_{r}\). This is
generally more reasonable for "progressive" models.
Adjustment to the observed prevalences to account for limited follow-up in the data.
If the time is greater than 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 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 model.frame(x), where x is the
fitted model object)
This is ignored if it is less than the subject's maximum observation time.
Subset of subjects to calculate observed prevalences for.
Generate a plot of observed against expected
prevalences. See plot.prevalence.msm
Further arguments to pass to plot.prevalence.msm.
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
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 \(j\) at time \(t\) after the start, the number of individuals under observation at time \(t\) (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 \(t\). 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 ematrix), this
aims to assess the fit of the full model for the observed
states. That is, the combined Markov progression model for the true
states and the misclassification model. Thus, expected prevalences of 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 observed states.
For general hidden Markov models, the observed state is taken to be the
predicted underlying state from the Viterbi algorithm
(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 et al. (1994).
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. Statistics in Medicine (1994) 13(3): 805--821.
Titman, A.C., Sharples, L. D. Model diagnostics for multi-state models. Statistical Methods in Medical Research (2010) 19(6):621-651.
msm, summary.msm