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