Compute the estimate and approximate standard error of the ratio of two estimated transition intensities from a fitted multi-state model at a given set of covariate values.

## Usage

```
qratio.msm(
x,
ind1,
ind2,
covariates = "mean",
ci = c("delta", "normal", "bootstrap", "none"),
cl = 0.95,
B = 1000,
cores = NULL
)
```

## Arguments

- x
A fitted multi-state model, as returned by

`msm`

.- ind1
Pair of numbers giving the indices in the intensity matrix of the numerator of the ratio, for example,

`c(1,2)`

.- ind2
Pair of numbers giving the indices in the intensity matrix of the denominator of the ratio, for example,

`c(2,1)`

.- covariates
The covariate values at which to estimate the intensities. This can either be:

the string

`"mean"`

, denoting the means of the covariates in the data (this is the default),the number

`0`

, indicating that all the covariates should be set to zero,or a list of values, with optional names. For example

`list (60, 1)`

where the order of the list follows the order of the covariates originally given in the model formula, or a named list,

`list (age = 60, sex = 1)`

- ci
If

`"delta"`

(the default) then confidence intervals are calculated by the delta method.If

`"normal"`

, then calculate a confidence interval 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 transforming.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.- cl
Width of the symmetric confidence interval to present. Defaults to 0.95.

- B
Number of bootstrap replicates, or number of normal simulations from the distribution of the MLEs

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

`boot.msm`

for more details.

## Value

A named vector with elements `estimate`

, `se`

, `L`

and `U`

containing the estimate, standard error, lower and upper
confidence limits, respectively, of the ratio of intensities.

## Details

For example, we might want to compute the ratio of the progression rate and
recovery rate for a fitted model `disease.msm`

with a health state
(state 1) and a disease state (state 2). In this case, the progression rate
is the (1,2) entry of the intensity matrix, and the recovery rate is the
(2,1) entry. Thus to compute this ratio with covariates set to their means,
we call

`qratio.msm(disease.msm, c(1,2), c(2,1))`

.

Standard errors are estimated by the delta method. Confidence limits are estimated by assuming normality on the log scale.

## Author

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