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.

## Usage

```
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,
...
)
```

## Arguments

- x
A fitted multi-state model produced by

`msm`

.- times
Series of times at which to compute the observed and expected prevalences of states.

- 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.

- 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.

- covariates
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`

.- misccovariates
(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.- piecewise.times
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".- piecewise.covariates
Covariates on which the piecewise-constant intensities depend. See

`pmatrix.piecewise.msm`

for how to specify this. Ignored if`covariates="population"`

.- ci
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.- cl
Width of the symmetric confidence interval, relative to 1

- B
Number of bootstrap replicates

- cores
Number of cores to use for bootstrapping using parallel processing. See

`boot.msm`

for more details.- interp
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.- censtime
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
Subset of subjects to calculate observed prevalences for.

- plot
Generate a plot of observed against expected prevalences. See

`plot.prevalence.msm`

- ...
Further arguments to pass to

`plot.prevalence.msm`

.

## Value

A list of matrices, with components:

- Observed
Table of observed numbers of individuals in each state at each time

- Observed percentages
Corresponding percentage of the individuals at risk at each time.

- Expected
Table of corresponding expected numbers.

- Expected percentages
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.

## Details

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).

## 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. *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.

## Author

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk