# Bootstrap resampling for multi-state models

### Description

Draw a number of bootstrap resamples, refit a `msm`

model to the
resamples, and calculate statistics on the refitted models.

### Usage

1 | ```
boot.msm(x, stat=pmatrix.msm, B=1000, file=NULL, cores=NULL)
``` |

### Arguments

`x` |
A fitted msm model, as output by |

`stat` |
A function to call on each refitted msm model. By default
this is |

`B` |
Number of bootstrap resamples. |

`file` |
Name of a file in which to save partial results after each
replicate. This is saved using |

`cores` |
Number of processor cores to use for parallel processing.
Requires the doParallel package to be installed. If not
specified, parallel processing is not used. If |

### Details

The bootstrap datasets are computed by resampling independent transitions between pairs of states (for non-hidden models without censoring), or independent individual series (for hidden models or models with censoring). Therefore this approach doesn't work if, for example, the data for a HMM consist of a series of observations from just one individual, and is inaccurate for small numbers of independent transitions or individuals.

Confidence intervals or standard errors for the corresponding
statistic can be calculated by summarising the returned list of
`B`

replicated outputs. This is currently implemented for most
the output functions `qmatrix.msm`

, `ematrix.msm`

,
`qratio.msm`

,
`pmatrix.msm`

, `pmatrix.piecewise.msm`

,
`totlos.msm`

and `prevalence.msm`

.
For other outputs, users will have to write their own code to
summarise the output of `boot.msm`

.

Most of msm's output functions present confidence intervals
based on asymptotic standard errors calculated from the
Hessian. These are expected to be underestimates of the true standard
errors (Cramer-Rao lower bound). Some of these functions use a
further approximation, the delta method (see
`deltamethod`

) to obtain standard errors of transformed
parameters. Bootstrapping should give a more
accurate estimate of the uncertainty.

An alternative method which is less accurate though faster than
bootstrapping, but more accurate than the delta method, is to draw a
sample from the asymptotic multivariate normal distribution implied by
the maximum likelihood estimates (and covariance matrix), and
summarise the transformed estimates. See `pmatrix.msm`

.

All objects used in the original call to `msm`

which
produced `x`

, such as the `qmatrix`

, should be in the
working environment, or else `boot.msm`

will produce an
“object not found” error. This enables `boot.msm`

to
refit the original model to the replicate datasets. However there is
currently a limitation. In the original call to `msm`

, the
`"formula"`

argument should be specified directly, as, for
example,

`msm(state ~ time, data = ...)`

and not, for example,

`form = data$state ~ data$time`

`msm(formula=form, data = ...)`

otherwise `boot.msm`

will be unable to draw the replicate
datasets.

`boot.msm`

will also fail with an incomprehensible error if the
original call to msm used a used-defined object whose name is the same
as a built-in R object, or an object in any other loaded package. For
example, if you have called a Q matrix `q`

, when `q()`

is
the built-in function for quitting R.

If `stat`

is `NULL`

, then `B`

different `msm`

model
objects will be stored in memory. This is unadvisable, as `msm`

objects tend to be large, since they contain the original data used for
the `msm`

fit, so this will be wasteful of memory.

To specify more than one statistic, write a function consisting of a list of different function calls, for example,

```
stat = function(x) list (pmatrix.msm(x, t=1), pmatrix.msm(x,
t=2))
```

### Value

A list with `B`

components, containing the result of calling
function `stat`

on each of the refitted models. If `stat`

is `NULL`

, then each component just contains the refitted
model. If one of the `B`

model fits was unsuccessful and
resulted in an error, then the corresponding list component will
contain the error message.

### Author(s)

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

### References

Efron, B. and Tibshirani, R.J. (1993) *An Introduction to the
Bootstrap*, Chapman and Hall.

### See Also

`qmatrix.msm`

, `qratio.msm`

,
`sojourn.msm`

, `ematrix.msm`

, `pmatrix.msm`

, `pmatrix.piecewise.msm`

, `totlos.msm`

, `prevalence.msm`

.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
## Not run:
## Psoriatic arthritis example
data(psor)
psor.q <- rbind(c(0,0.1,0,0),c(0,0,0.1,0),c(0,0,0,0.1),c(0,0,0,0))
psor.msm <- msm(state ~ months, subject=ptnum, data=psor, qmatrix =
psor.q, covariates = ~ollwsdrt+hieffusn,
constraint = list(hieffusn=c(1,1,1),ollwsdrt=c(1,1,2)),
control = list(REPORT=1,trace=2), method="BFGS")
## Bootstrap the baseline transition intensity matrix. This will take a long time.
q.list <- boot.msm(psor.msm, function(x)x$Qmatrices$baseline)
## Manipulate the resulting list of matrices to calculate bootstrap standard errors.
apply(array(unlist(q.list), dim=c(4,4,5)), c(1,2), sd)
## Similarly calculate a bootstrap 95% confidence interval
apply(array(unlist(q.list), dim=c(4,4,5)), c(1,2),
function(x)quantile(x, c(0.025, 0.975)))
## Bootstrap standard errors are larger than the asymptotic standard
## errors calculated from the Hessian
psor.msm$QmatricesSE$baseline
## End(Not run)
``` |