# Criteria for comparing two multi-state models with nested state spaces

### Description

A modification of Akaike's information criterion, and a leave-one-out likelihood cross-validation criterion, for comparing the predictive ability of two Markov multi-state models with nested state spaces. This is evaluated based on the restricted or aggregated data which the models have in common.

Note that standard AIC can be computed for one or more fitted
`msm`

models `x,y,...`

using `AIC(x,y,...)`

, and
this can be used to compare models fitted to the same
data. `draic.msm`

and `drlcv.msm`

are designed for models
fitted to data with differently-aggregated state spaces.

### Usage

1 2 3 4 |

### Arguments

`msm.full` |
Model on the bigger state space. |

`msm.coarse` |
Model on the smaller state space. The two models must both be non-hidden Markov models without censored states. The two models must be fitted to the same datasets, except that the
state space of the coarse model must be an aggregated version of the
state space of the full model. That is, every state in the full
dataset must correspond to a unique state in the coarse dataset.
For example, for the full state variable The structure of allowed transitions in the coarse model must also be a collapsed version of the big model structure, but no check is currently made for this in the code. To use these functions, all objects
which were used in the calls to fit |

`likelihood.only` |
Don't calculate Hessians and trace term (DRAIC). |

`information` |
Use observed or expected information in the DRAIC
trace term. Expected is the default, and much faster, though is only
available for models fitted to pure panel data (all |

`tl` |
Width of symmetric tracking interval, by default 0.95 for a 95% interval. |

`cores` |
Number of processor cores to use in |

`verbose` |
Print intermediate results of each iteration of cross-validation to the console while running. May not work with parallel processing. |

`outfile` |
Output file to print intermediate results of cross-validation. Useful to track execution speed when using parallel processing, where output to the console may not work. |

### Details

The difference in restricted AIC (Liquet and Commenges, 2011), as computed by this function, is defined as

*D_RAIC = l(gamma_n |x'' ) - l(theta_n |x'' ) + trace ( J(theta_n |x'')J(theta_n |x)^{-1} - J(gamma_n |x'' )J(gamma_n |x' )^{-1})*

where *gamma* and *theta* are the maximum likelihood estimates of the smaller and bigger models, fitted to the smaller and bigger data, respectively.

*l(gamma_n |x'')* represents the likelihood of the simpler model evaluated on the restricted data.

*l(theta_n |x'')* represents the likelihood of
the complex model evaluated on the restricted data. This is a hidden
Markov model, with a misclassification matrix and initial state
occupancy probabilities as described by Thom et al (2014).

*J()* are the corresponding (expected or observed, as specified by the user) information matrices.

*x* is the expanded data, to which the bigger model was originally fitted, and
*x'* is the data to which the smaller model was originally fitted.
*x''* is the restricted data which the two models
have in common. *x'' = x* in this implementation, so the models are nested.

The difference in likelihood cross-validatory criteria (Liquet and Commenges, 2011) is defined as

*D_{RLCV} = 1/n ∑_{i=1}^n log( h_{X''}(x_i'' | gamma_{-i}) / g_{X''}(x_i''| theta_{-i}))*

where *γ_{-i}* and *θ_{-i}* are the maximum likelihood estimates from the smaller and bigger models fitted to datasets with subject *i* left out, *g()* and *h()* are the densities of the corresponding models, and *x_i''* is the restricted data from subject *i*.

Tracking intervals are analogous to confidence intervals, but not strictly the same, since the quantity which D_RAIC aims to estimate, the difference in expected Kullback-Leibler discrepancy for predicting a replicate dataset, depends on the sample size. See the references.

Positive values for these criteria indicate the coarse model is preferred, while negative values indicate the full model is preferred.

### Value

A list containing *D_RAIC* (`draic.msm`

) or
*D_RLCV* (`drlcv.msm`

), its component terms, and tracking intervals.

### Author(s)

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk, H. H. Z. Thom howard.thom@bristol.ac.uk

### References

Thom, H. and Jackson, C. and Commenges, D. and Sharples, L. (2015) State selection in multistate models with application to quality of life in psoriatic arthritis. Statistics In Medicine 34(16) 2381 - 2480.

Liquet, B. and Commenges D. (2011) Choice of estimators based on different observations: Modified AIC and LCV criteria. Scandinavian Journal of Statistics; 38:268-287.

### See Also

`logLik.msm`