Multi-state Markov and hidden Markov models in continuous time

Description

Fit a continuous-time Markov or hidden Markov multi-state model by maximum likelihood. Observations of the process can be made at arbitrary times, or the exact times of transition between states can be known. Covariates can be fitted to the Markov chain transition intensities or to the hidden Markov observation process.

Usage

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msm ( formula, subject=NULL, data = list(), qmatrix, gen.inits = FALSE,
      ematrix=NULL, hmodel=NULL, obstype=NULL, obstrue=NULL,
      covariates = NULL, covinits = NULL, constraint = NULL,
      misccovariates = NULL, misccovinits = NULL, miscconstraint = NULL,
      hcovariates = NULL, hcovinits = NULL, hconstraint = NULL, hranges=NULL,
      qconstraint=NULL, econstraint=NULL, initprobs = NULL,
      est.initprobs=FALSE, initcovariates = NULL, initcovinits = NULL,
      deathexact = NULL, death=NULL, exacttimes = FALSE, censor=NULL,
      censor.states=NULL, pci=NULL, phase.states=NULL, phase.inits=NULL,
      cl = 0.95, fixedpars = NULL, center=TRUE,
      opt.method="optim", hessian=NULL, use.deriv=TRUE,
      use.expm=TRUE, analyticp=TRUE, na.action=na.omit, ... )

Arguments

formula

A formula giving the vectors containing the observed states and the corresponding observation times. For example,

state ~ time

Observed states should be numeric variables in the set 1, ..., n, where n is the number of states. Factors are allowed only if their levels are called "1", ..., "n".

The times can indicate different types of observation scheme, so be careful to choose the correct obstype.

For hidden Markov models, state refers to the outcome variable, which need not be a discrete state. It may also be a matrix, giving multiple observations at each time (see hmmMV).

subject

Vector of subject identification numbers for the data specified by formula. If missing, then all observations are assumed to be on the same subject. These must be sorted so that all observations on the same subject are adjacent.

data

Optional data frame in which to interpret the variables supplied in formula, subject, covariates, misccovariates, hcovariates, obstype and obstrue.

qmatrix

Matrix which indicates the allowed transitions in the continuous-time Markov chain, and optionally also the initial values of those transitions. If an instantaneous transition is not allowed from state r to state s, then qmatrix should have (r,s) entry 0, otherwise it should be non-zero.

If supplying initial values yourself, then the non-zero entries should be those values. If using gen.inits=TRUE then the non-zero entries can be anything you like (conventionally 1). Any diagonal entry of qmatrix is ignored, as it is constrained to be equal to minus the sum of the rest of the row.

For example,

rbind( c( 0, 0.1, 0.01 ), c( 0.1, 0, 0.2 ), c( 0, 0, 0 ) )

represents a 'health - disease - death' model, with initial transition intensities 0.1 from health to disease, 0.01 from health to death, 0.1 from disease to health, and 0.2 from disease to death.

If the states represent ordered levels of severity of a disease, then this matrix should usually only allow transitions between adjacent states. For example, if someone was observed in state 1 ("mild") at their first observation, followed by state 3 ("severe") at their second observation, they are assumed to have passed through state 2 ("moderate") in between, and the 1,3 entry of qmatrix should be zero.

The initial intensities given here are with any covariates set to their means in the data (or set to zero, if center = FALSE). If any intensities are constrained to be equal using qconstraint, then the initial value is taken from the first of these (reading across rows).

gen.inits

If TRUE, then initial values for the transition intensities are generated automatically using the method in crudeinits.msm. The non-zero entries of the supplied qmatrix are assumed to indicate the allowed transitions of the model. This is not available for hidden Markov models, including models with misclassified states.

ematrix

If misclassification between states is to be modelled, this should be a matrix of initial values for the misclassification probabilities. The rows represent underlying states, and the columns represent observed states. If an observation of state s is not possible when the subject occupies underlying state r, then ematrix should have (r,s) entry 0. Otherwise ematrix should have (r,s) entry corresponding to the probability of observing s conditionally on occupying true state r. The diagonal of ematrix is ignored, as rows are constrained to sum to 1. For example,

rbind( c( 0, 0.1, 0 ), c( 0.1, 0, 0.1 ), c( 0, 0.1, 0 ) )

represents a model in which misclassifications are only permitted between adjacent states.

If any probabilities are constrained to be equal using econstraint, then the initial value is taken from the first of these (reading across rows).

For an alternative way of specifying misclassification models, see hmodel.

hmodel

Specification of the hidden Markov model (HMM). This should be a list of return values from HMM constructor functions. Each element of the list corresponds to the outcome model conditionally on the corresponding underlying state. Univariate constructors are described in thehmm-dists help page. These may also be grouped together to specify a multivariate HMM with a set of conditionally independent univariate outcomes at each time, as described in hmmMV.

For example, consider a three-state hidden Markov model. Suppose the observations in underlying state 1 are generated from a Normal distribution with mean 100 and standard deviation 16, while observations in underlying state 2 are Normal with mean 54 and standard deviation 18. Observations in state 3, representing death, are exactly observed, and coded as 999 in the data. This model is specified as

hmodel = list(hmmNorm(mean=100, sd=16), hmmNorm(mean=54, sd=18), hmmIdent(999))

The mean and standard deviation parameters are estimated starting from these initial values. If multiple parameters are constrained to be equal using hconstraint, then the initial value is taken from the value given on the first occasion that parameter appears in hmodel.

See the hmm-dists help page for details of the constructor functions for each univariate distribution.

A misclassification model, that is, a hidden Markov model where the outcomes are misclassified observations of the underlying states, can either be specified using a list of hmmCat or hmmIdent objects, or by using an ematrix.

For example,

ematrix = rbind( c( 0, 0.1, 0, 0 ), c( 0.1, 0, 0.1, 0 ), c( 0, 0.1, 0, 0), c( 0, 0, 0, 0) )

is equivalent to

hmodel = list( hmmCat(prob=c(0.9, 0.1, 0, 0)), hmmCat(prob=c(0.1, 0.8, 0.1, 0)), hmmCat(prob=c(0, 0.1, 0.9, 0)), hmmIdent())

obstype

A vector specifying the observation scheme for each row of the data. This can be included in the data frame data along with the state, time, subject IDs and covariates. Its elements should be either 1, 2 or 3, meaning as follows:

1

An observation of the process at an arbitrary time (a "snapshot" of the process, or "panel-observed" data). The states are unknown between observation times.

2

An exact transition time, with the state at the previous observation retained until the current observation. An observation may represent a transition to a different state or a repeated observation of the same state (e.g. at the end of follow-up). Note that if all transition times are known, more flexible models could be fitted with packages other than msm - see the note under exacttimes.

Note also that if the previous state was censored using censor, for example known only to be state 1 or state 2, then obstype 2 means that either state 1 is retained or state 2 is retained until the current observation - this does not allow for a change of state in the middle of the observation interval.

3

An exact transition time, but the state at the instant before entering this state is unknown. A common example is death times in studies of chronic diseases.

If obstype is not specified, this defaults to all 1. If obstype is a single number, all observations are assumed to be of this type. The obstype value for the first observation from each subject is not used.

This is a generalisation of the deathexact and exacttimes arguments to allow different schemes per observation. obstype overrides both deathexact and exacttimes.

exacttimes=TRUE specifies that all observations are of obstype 2.

deathexact = death.states specifies that all observations of death.states are of type 3. deathexact = TRUE specifies that all observations in the final absorbing state are of type 3.

obstrue

In misclassification models specified with ematrix, obstrue is a vector of logicals (TRUE or FALSE) or numerics (1 or 0) specifying which observations (TRUE, 1) are observations of the underlying state without error, and which (FALSE, 0) are realisations of a hidden Markov model.

In HMMs specified with hmodel, where the hidden state is known at some times, if obstrue is supplied it is assumed to contain the actual true state data. Elements of obstrue at times when the hidden state is unknown are set to NA. This allows the information from HMM outcomes generated conditionally on the known state to be included in the model, thus improving the estimation of the HMM outcome distributions.

In HMMs where there are also censored states, obstrue should be set to 1 for observed states which are censored but not misclassified.

covariates

A formula or a list of formulae representing the covariates on the transition intensities via a log-linear model. If a single formula is supplied, like

covariates = ~ age + sex + treatment

then these covariates are assumed to apply to all intensities. If a named list is supplied, then this defines a potentially different model for each named intensity. For example,

covariates = list("1-2" = ~ age, "2-3" = ~ age + treatment)

specifies an age effect on the state 1 - state 2 transition, additive age and treatment effects on the state 2 - state 3 transition, but no covariates on any other transitions that are allowed by the qmatrix.

If covariates are time dependent, they are assumed to be constant in between the times they are observed, and the transition probability between a pair of times (t1, t2) is assumed to depend on the covariate value at t1.

covinits

Initial values for log-linear effects of covariates on the transition intensities. This should be a named list with each element corresponding to a covariate. A single element contains the initial values for that covariate on each transition intensity, reading across the rows in order. For a pair of effects constrained to be equal, the initial value for the first of the two effects is used.

For example, for a model with the above qmatrix and age and sex covariates, the following initialises all covariate effects to zero apart from the age effect on the 2-1 transition, and the sex effect on the 1-3 transition. covinits = list(sex=c(0, 0, 0.1, 0), age=c(0, 0.1, 0, 0))

For factor covariates, name each level by concatenating the name of the covariate with the level name, quoting if necessary. For example, for a covariate agegroup with three levels 0-15, 15-60, 60-, use something like

covinits = list("agegroup15-60"=c(0, 0.1, 0, 0), "agegroup60-"=c(0.1, 0.1, 0, 0))

If not specified or wrongly specified, initial values are assumed to be zero.

constraint

A list of one numeric vector for each named covariate. The vector indicates which covariate effects on intensities are constrained to be equal. Take, for example, a model with five transition intensities and two covariates. Specifying

constraint = list (age = c(1,1,1,2,2), treatment = c(1,2,3,4,5))

constrains the effect of age to be equal for the first three intensities, and equal for the fourth and fifth. The effect of treatment is assumed to be different for each intensity. Any vector of increasing numbers can be used as indicators. The intensity parameters are assumed to be ordered by reading across the rows of the transition matrix, starting at the first row, ignoring the diagonals.

Negative elements of the vector can be used to indicate that particular covariate effects are constrained to be equal to minus some other effects. For example:

constraint = list (age = c(-1,1,1,2,-2), treatment = c(1,2,3,4,5))

constrains the second and third age effects to be equal, the first effect to be minus the second, and the fifth age effect to be minus the fourth. For example, it may be realisitic that the effect of a covariate on the "reverse" transition rate from state 2 to state 1 is minus the effect on the "forward" transition rate, state 1 to state 2. Note that it is not possible to specify exactly which of the covariate effects are constrained to be positive and which negative. The maximum likelihood estimation chooses the combination of signs which has the higher likelihood.

For categorical covariates, defined using factor(covname), specify constraints as follows:

list(..., covnameVALUE1 = c(...), covnameVALUE2 = c(...), ...)

where covname is the name of the factor, and VALUE1, VALUE2, ... are the labels of the factor levels (usually excluding the baseline, if using the default contrasts).

Make sure the contrasts option is set appropriately, for example, the default

options(contrasts=c(contr.treatment, contr.poly))

sets the first (baseline) level of unordered factors to zero, then the baseline level is ignored in this specification.

To assume no covariate effect on a certain transition, use the fixedpars argument to fix it at its initial value (which is zero by default) during the optimisation.

misccovariates

A formula representing the covariates on the misclassification probabilities, analogously to covariates, via multinomial logistic regression. Only used if the model is specified using ematrix, rather than hmodel.

This must be a single formula - lists are not supported, unlike covariates. If a different model on each probability is required, include all covariates in this formula, and use fixedpars to fix some of their effects (for particular probabilities) at their default initial values of zero.

misccovinits

Initial values for the covariates on the misclassification probabilities, defined in the same way as covinits. Only used if the model is specified using ematrix.

miscconstraint

A list of one vector for each named covariate on misclassification probabilities. The vector indicates which covariate effects on misclassification probabilities are constrained to be equal, analogously to constraint. Only used if the model is specified using ematrix.

hcovariates

List of formulae the same length as hmodel, defining any covariates governing the hidden Markov outcome models. The covariates operate on a suitably link-transformed linear scale, for example, log scale for a Poisson outcome model. If there are no covariates for a certain hidden state, then insert a NULL in the corresponding place in the list. For example, hcovariates = list(~acute + age, ~acute, NULL).

hcovinits

Initial values for the hidden Markov model covariate effects. A list of the same length as hcovariates. Each element is a vector with initial values for the effect of each covariate on that state. For example, the above hcovariates can be initialised with hcovariates = list(c(-8, 0), -8, NULL). Initial values must be given for all or no covariates, if none are given these are all set to zero. The initial value given in the hmodel constructor function for the corresponding baseline parameter is interpreted as the value of that parameter with any covariates fixed to their means in the data. If multiple effects are constrained to be equal using hconstraint, then the initial value is taken from the first of the multiple initial values supplied.

hconstraint

A named list. Each element is a vector of constraints on the named hidden Markov model parameter. The vector has length equal to the number of times that class of parameter appears in the whole model.

For example consider the three-state hidden Markov model described above, with normally-distributed outcomes for states 1 and 2. To constrain the outcome variance to be equal for states 1 and 2, and to also constrain the effect of acute on the outcome mean to be equal for states 1 and 2, specify

hconstraint = list(sd = c(1,1), acute=c(1,1))

Note this excludes initial state occupancy probabilities and covariate effects on those probabilities, which cannot be constrained.

hranges

Range constraints for hidden Markov model parameters. Supplied as a named list, with each element corresponding to the named hidden Markov model parameter. This element is itself a list with two elements, vectors named "lower" and "upper". These vectors each have length equal to the number of times that class of parameter appears in the whole model, and give the corresponding mininum amd maximum allowable values for that parameter. Maximum likelihood estimation is performed with these parameters constrained in these ranges (through a log or logit-type transformation). Lower bounds of -Inf and upper bounds of Inf can be given if the parameter is unbounded above or below.

For example, in the three-state model above, to constrain the mean for state 1 to be between 0 and 6, and the mean of state 2 to be between 7 and 12, supply

hranges=list(mean=list(lower=c(0, 7), upper=c(6, 12)))

These default to the natural ranges, e.g. the positive real line for variance parameters, and [0,1] for probabilities. Therefore hranges need not be specified for such parameters unless an even stricter constraint is desired. If only one limit is supplied for a parameter, only the first occurrence of that parameter is constrained.

Initial values should be strictly within any ranges, and not on the range boundary, otherwise optimisation will fail with a "non-finite value" error.

qconstraint

A vector of indicators specifying which baseline transition intensities are equal. For example,

qconstraint = c(1,2,3,3)

constrains the third and fourth intensities to be equal, in a model with four allowed instantaneous transitions. When there are covariates on the intensities and center=TRUE (the default), qconstraint is applied to the intensities with covariates taking the values of the means in the data. When center=FALSE, qconstraint is applied to the intensities with covariates set to zero.

econstraint

A similar vector of indicators specifying which baseline misclassification probabilities are constrained to be equal. Only used if the model is specified using ematrix, rather than hmodel.

initprobs

Only used in hidden Markov models. Underlying state occupancy probabilities at each subject's first observation. Can either be a vector of nstates elements with common probabilities to all subjects, or a nsubjects by nstates matrix of subject-specific probabilities. This refers to observations after missing data and subjects with only one observation have been excluded.

If these are estimated (see est.initprobs), then this represents an initial value, and defaults to equal probability for each state. Otherwise this defaults to c(1, rep(0, nstates-1)), that is, in state 1 with a probability of 1. Scaled to sum to 1 if necessary. The state 1 occupancy probability should be non-zero.

est.initprobs

Only used in hidden Markov models. If TRUE, then the underlying state occupancy probabilities at the first observation will be estimated, starting from a vector of initial values supplied in the initprobs argument. Structural zeroes are allowed: if any of these initial values are zero they will be fixed at zero during optimisation, even if est.initprobs=TRUE, and no covariate effects on them are estimated. The exception is state 1, which should have non-zero occupancy probability.

Note that the free parameters during this estimation exclude the state 1 occupancy probability, which is fixed at one minus the sum of the other probabilities.

initcovariates

Formula representing covariates on the initial state occupancy probabilities, via multinomial logistic regression. The linear effects of these covariates, observed at the individual's first observation time, operate on the log ratio of the state r occupancy probability to the state 1 occupancy probability, for each r = 2 to the number of states. Thus the state 1 occupancy probability should be non-zero. If est.initprobs is TRUE, these effects are estimated starting from their initial values. If est.initprobs is FALSE, these effects are fixed at theit initial values.

initcovinits

Initial values for the covariate effects initcovariates. A named list with each element corresponding to a covariate, as in covinits. Each element is a vector with (1 - number of states) elements, containing the initial values for the linear effect of that covariate on the log odds of that state relative to state 1, from state 2 to the final state. If initcovinits is not specified, all covariate effects are initialised to zero.

deathexact

Vector of indices of absorbing states whose time of entry is known exactly, but the individual is assumed to be in an unknown transient state ("alive") at the previous instant. This is the usual situation for times of death in chronic disease monitoring data. For example, if you specify deathexact = c(4, 5) then states 4 and 5 are assumed to be exactly-observed death states.

See the obstype argument. States of this kind correspond to obstype=3. deathexact = TRUE indicates that the final absorbing state is of this kind, and deathexact = FALSE or deathexact = NULL (the default) indicates that there is no state of this kind.

The deathexact argument is overridden by obstype or exacttimes.

Note that you do not always supply a deathexact argument, even if there are states that correspond to deaths, because they do not necessarily have obstype=3. If the state is known between the time of death and the previous observation, then you should specify obstype=2 for the death times, or exacttimes=TRUE if the state is known at all times, and the deathexact argument is ignored.

death

Old name for the deathexact argument. Overridden by deathexact if both are supplied. Deprecated.

censor

A state, or vector of states, which indicates censoring. Censoring means that the observed state is known only to be one of a particular set of states. For example, censor=999 indicates that all observations of 999 in the vector of observed states are censored states. By default, this means that the true state could have been any of the transient (non-absorbing) states. To specify corresponding true states explicitly, use a censor.states argument.

Note that in contrast to the usual terminology of survival analysis, here it is the state which is considered to be censored, rather than the event time. If at the end of a study, an individual has not died, but their true state is known, then censor is unnecessary, since the standard multi-state model likelihood is applicable. Also a "censored" state here can be at any time, not just at the end.

Note in particular that general time-inhomogeneous Markov models with piecewise constant transition intensities can be constructed using the censor facility. If the true state is unknown on occasions when a piecewise constant covariate is known to change, then censored states can be inserted in the data on those occasions. The covariate may represent time itself, in which case the pci option to msm can be used to perform this trick automatically, or some other time-dependent variable.

censor.states

Specifies the underlying states which censored observations can represent. If censor is a single number (the default) this can be a vector, or a list with one element. If censor is a vector with more than one element, this should be a list, with each element a vector corresponding to the equivalent element of censor. For example

censor = c(99, 999), censor.states = list(c(2,3), c(3,4))

means that observations coded 99 represent either state 2 or state 3, while observations coded 999 are really either state 3 or state 4.

pci

Model for piecewise-constant intensities. Vector of cut points defining the times, since the start of the process, at which intensities change for all subjects. For example

pci = c(5, 10)

specifies that the intensity changes at time points 5 and 10. This will automatically construct a model with a categorical (factor) covariate called timeperiod, with levels "[-Inf,5)", "[5,10)" and "[10,Inf)", where the first level is the baseline. This covariate defines the time period in which the observation was made. Initial values and constraints on covariate effects are specified the same way as for a model with a covariate of this name, for example,

covinits = list("timeperiod[5,10)"=c(0.1,0.1), "timeperiod[10,Inf)"=c(0.1,0.1))

Thus if pci is supplied, you cannot have a previously-existing variable called timeperiod as a covariate in any part of a msm model.

Internally, this works by inserting censored observations in the data at times when the intensity changes but the state is not observed.

If the supplied times are outside the range of the time variable in the data, pci is ignored and a time-homogeneous model is fitted.

After fitting a time-inhomogeneous model, qmatrix.msm can be used to obtain the fitted intensity matrices for each time period, for example,

qmatrix.msm(example.msm, covariates=list(timeperiod="[5,Inf)"))

This facility does not support interactions between time and other covariates. Such models need to be specified "by hand", using a state variable with censored observations inserted. Note that the data component of the msm object returned from a call to msm with pci supplied contains the states with inserted censored observations and time period indicators. These can be used to construct such models.

Note that you do not need to use pci in order to model the effect of a time-dependent covariate in the data. msm will automatically assume that covariates are piecewise-constant and change at the times when they are observed. pci is for when you want all intensities to change at the same pre-specified times for all subjects.

phase.states

Indices of states which have a two-phase sojourn distribution. This defines a semi-Markov model, in which the hazard of an onward transition depends on the time spent in the state.

This uses the technique described by Titman and Sharples (2009). A hidden Markov model is automatically constructed on an expanded state space, where the phases correspond to the hidden states. The "tau" proportionality constraint described in this paper is currently not supported.

Covariates, constraints, deathexact and censor are expressed with respect to the expanded state space. If not supplied by hand, initprobs is defined automatically so that subjects are assumed to begin in the first of the two phases.

Hidden Markov models can additionally be given phased states. The user supplies an outcome distribution for each original state using hmodel, which is expanded internally so that it is assumed to be the same within each of the phased states. initprobs is interpreted on the expanded state space. Misclassification models defined using ematrix are not supported, and these must be defined using hmmCat or hmmIdent constructors, as described in the hmodel section of this help page. Or the HMM on the expanded state space can be defined by hand.

Output functions are presented as it were a hidden Markov model on the expanded state space, for example, transition probabilities between states, covariate effects on transition rates, or prevalence counts, are not aggregated over the hidden phases.

Numerical estimation will be unstable when there is weak evidence for a two-phase sojourn distribution, that is, if the model is close to Markov.

See d2phase for the definition of the two-phase distribution and the interpretation of its parameters.

This is an experimental feature, and some functions are not implemented. Please report any experiences of using this feature to the author!

phase.inits

Initial values for phase-type models. A list with one component for each "two-phased" state. Each component is itself a list of two elements. The first of these elements is a scalar defining the transition intensity from phase 1 to phase 2. The second element is a matrix, with one row for each potential destination state from the two-phased state, and two columns. The first column is the transition rate from phase 1 to the destination state, and the second column is the transition rate from phase 2 to the destination state. If there is only one destination state, then this may be supplied as a vector.

In phase type models, the initial values for transition rates out of non-phased states are taken from the qmatrix supplied to msm, and entries of this matrix corresponding to transitions out of phased states are ignored.

exacttimes

By default, the transitions of the Markov process are assumed to take place at unknown occasions in between the observation times. If exacttimes is set to TRUE, then the observation times are assumed to represent the exact times of transition of the process. The subject is assumed to be in the same state between these times. An observation may represent a transition to a different state or a repeated observation of the same state (e.g. at the end of follow-up). This is equivalent to every row of the data having obstype = 2. See the obstype argument. If both obstype and exacttimes are specified then exacttimes is ignored.

Note that the complete history of the multi-state process is known with this type of data. The models which msm fits have the strong assumption of constant (or piecewise-constant) transition rates. Knowing the exact transition times allows more realistic models to be fitted with other packages. For example parametric models with sojourn distributions more flexible than the exponential can be fitted with the flexsurv package, or semi-parametric models can be implemented with survival in conjunction with mstate.

cl

Width of symmetric confidence intervals for maximum likelihood estimates, by default 0.95.

fixedpars

Vector of indices of parameters whose values will be fixed at their initial values during the optimisation. These are given in the order: transition intensities (reading across rows of the transition matrix), covariates on intensities (ordered by intensities within covariates), hidden Markov model parameters, including misclassification probabilities (ordered by parameters within states), hidden Markov model covariate parameters (ordered by covariates within parameters within states), initial state occupancy probabilities (excluding the first probability, which is fixed at one minus the sum of the others).

If there are equality constraints on certain parameters, then fixedpars indexes the set of unique parameters, excluding those which are constrained to be equal to previous parameters.

To fix all parameters, specify fixedpars = TRUE.

This can be useful for profiling likelihoods, and building complex models stage by stage.

center

If TRUE (the default, unless fixedpars=TRUE) then covariates are centered at their means during the maximum likelihood estimation. This usually improves stability of the numerical optimisation.

opt.method

If "optim", "nlm" or "bobyqa", then the corresponding R function will be used for maximum likelihood estimation. optim is the default. "bobyqa" requires the package minqa to be installed. See the help of these functions for further details. Advanced users can also add their own optimisation methods, see the source for optim.R in msm for some examples.

If "fisher", then a specialised Fisher scoring method is used (Kalbfleisch and Lawless, 1985) which can be faster than the generic methods, though less robust. This is only available for Markov models with panel data (obstype=1), that is, not for models with censored states, hidden Markov models, exact observation or exact death times (obstype=2,3).

hessian

If TRUE then standard errors and confidence intervals are obtained from a numerical estimate of the Hessian (the observed information matrix). This is the default when maximum likelihood estimation is performed. If all parameters are fixed at their initial values and no optimisation is performed, then this defaults to FALSE. If requested, the actual Hessian is returned in x$paramdata$opt$hessian, where x is the fitted model object.

If hessian is set to FALSE, then standard errors and confidence intervals are obtained from the Fisher (expected) information matrix, if this is available. This may be preferable if the numerical estimation of the Hessian is computationally intensive, or if the resulting estimate is non-invertible or not positive definite.

use.deriv

If TRUE then analytic first derivatives are used in the optimisation of the likelihood, where available and an appropriate quasi-Newton optimisation method, such as BFGS, is being used. Analytic derivatives are not available for all models.

use.expm

If TRUE then any matrix exponentiation needed to calculate the likelihood is done using the expm package. Otherwise the original routines used in msm 1.2.4 and earlier are used. Set to FALSE for backward compatibility, and let the package maintainer know if this gives any substantive differences.

analyticp

By default, the likelihood for certain simpler 3, 4 and 5 state models is calculated using an analytic expression for the transition probability (P) matrix. For all other models, matrix exponentiation is used to obtain P. To revert to the original method of using the matrix exponential for all models, specify analyticp=FALSE. See the PDF manual for a list of the models for which analytic P matrices are implemented.

na.action

What to do with missing data: either na.omit to drop it and carry on, or na.fail to stop with an error. Missing data includes all NAs in the states, times, subject or obstrue, all NAs at the first observation for a subject for covariates in initcovariates, all NAs in other covariates (excluding the last observation for a subject), all NAs in obstype (excluding the first observation for a subject), and any subjects with only one observation (thus no observed transitions).

...

Optional arguments to the general-purpose R optimisation routine, optim by default. For example method="Nelder-Mead" to change the optimisation algorithm from the "BFGS" method that msm calls by default.

It is often worthwhile to normalize the optimisation using control=list(fnscale = a), where a is the a number of the order of magnitude of the -2 log likelihood.

If 'false' convergence is reported and the standard errors cannot be calculated due to a non-positive-definite Hessian, then consider tightening the tolerance criteria for convergence. If the optimisation takes a long time, intermediate steps can be printed using the trace argument of the control list. See optim for details.

For the Fisher scoring method, a control list can be supplied in the same way, but the only supported options are reltol, trace and damp. The first two are used in the same way as for optim. If the algorithm fails with a singular information matrix, adjust damp from the default of zero (to, e.g. 1). This adds a constant identity matrix multiplied by damp to the information matrix during optimisation.

Details

For full details about the methodology behind the msm package, refer to the PDF manual ‘msm-manual.pdf’ in the ‘doc’ subdirectory of the package. This includes a tutorial in the typical use of msm. The paper by Jackson (2011) in Journal of Statistical Software presents the material in this manual in a more concise form.

msm was designed for fitting continuous-time Markov models, processes where transitions can occur at any time. These models are defined by intensities, which govern both the time spent in the current state and the probabilities of the next state. In discrete-time models, transitions are known in advance to only occur at multiples of some time unit, and the model is purely governed by the probability distributions of the state at the next time point, conditionally on the state at the current time. These can also be fitted in msm, assuming that there is a continuous-time process underlying the data. Then the fitted transition probability matrix over one time period, as returned by pmatrix.msm(...,t=1) is equivalent to the matrix that governs the discrete-time model. However, these can be fitted more efficiently using multinomial logistic regression, for example, using multinom from the R package nnet (Venables and Ripley, 2002).

For simple continuous-time multi-state Markov models, the likelihood is calculated in terms of the transition intensity matrix Q. When the data consist of observations of the Markov process at arbitrary times, the exact transition times are not known. Then the likelihood is calculated using the transition probability matrix P(t) = exp(tQ), where exp is the matrix exponential. If state i is observed at time t and state j is observed at time u, then the contribution to the likelihood from this pair of observations is the i,j element of P(u - t). See, for example, Kalbfleisch and Lawless (1985), Kay (1986), or Gentleman et al. (1994).

For hidden Markov models, the likelihood for an individual with k observations is calculated directly by summing over the unknown state at each time, producing a product of k matrices. The calculation is a generalisation of the method described by Satten and Longini (1996), and also by Jackson and Sharples (2002), and Jackson et al. (2003).

There must be enough information in the data on each state to estimate each transition rate, otherwise the likelihood will be flat and the maximum will not be found. It may be appropriate to reduce the number of states in the model, the number of allowed transitions, or the number of covariate effects, to ensure convergence. Hidden Markov models, and situations where the value of the process is only known at a series of snapshots, are particularly susceptible to non-identifiability, especially when combined with a complex transition matrix. Choosing an appropriate set of initial values for the optimisation can also be important. For flat likelihoods, 'informative' initial values will often be required. See the PDF manual for other tips.

Value

To obtain summary information from models fitted by the msm function, it is recommended to use extractor functions such as qmatrix.msm, pmatrix.msm, sojourn.msm, msm.form.qoutput. These provide estimates and confidence intervals for quantities such as transition probabilities for given covariate values.

For advanced use, it may be necessary to directly use information stored in the object returned by msm. This is documented in the help page msm.object.

Printing a msm object by typing the object's name at the command line implicitly invokes print.msm. This formats and prints the important information in the model fit, and also returns that information in an R object. This includes estimates and confidence intervals for the transition intensities and (log) hazard ratios for the corresponding covariates. When there is a hidden Markov model, the chief information in the hmodel component is also formatted and printed. This includes estimates and confidence intervals for each parameter.

Author(s)

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

References

Jackson, C.H. (2011). Multi-State Models for Panel Data: The msm Package for R., Journal of Statistical Software, 38(8), 1-29. URL http://www.jstatsoft.org/v38/i08/.

Kalbfleisch, J., Lawless, J.F., The analysis of panel data under a Markov assumption Journal of the Americal Statistical Association (1985) 80(392): 863–871.

Kay, R. A Markov model for analysing cancer markers and disease states in survival studies. Biometrics (1986) 42: 855–865.

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.

Satten, G.A. and Longini, I.M. Markov chains with measurement error: estimating the 'true' course of a marker of the progression of human immunodeficiency virus disease (with discussion) Applied Statistics 45(3): 275-309 (1996)

Jackson, C.H. and Sharples, L.D. Hidden Markov models for the onset and progression of bronchiolitis obliterans syndrome in lung transplant recipients Statistics in Medicine, 21(1): 113–128 (2002).

Jackson, C.H., Sharples, L.D., Thompson, S.G. and Duffy, S.W. and Couto, E. Multi-state Markov models for disease progression with classification error. The Statistician, 52(2): 193–209 (2003)

Titman, A.C. and Sharples, L.D. Semi-Markov models with phase-type sojourn distributions. Biometrics 66, 742-752 (2009).

Venables, W.N. and Ripley, B.D. (2002) Modern Applied Statistics with S, second edition. Springer.

See Also

simmulti.msm, plot.msm, summary.msm, qmatrix.msm, pmatrix.msm, sojourn.msm.

Examples

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### Heart transplant data
### For further details and background to this example, see
### Jackson (2011) or the PDF manual in the doc directory.
print(cav[1:10,])
twoway4.q <- rbind(c(-0.5, 0.25, 0, 0.25), c(0.166, -0.498, 0.166, 0.166),
c(0, 0.25, -0.5, 0.25), c(0, 0, 0, 0))
statetable.msm(state, PTNUM, data=cav)
crudeinits.msm(state ~ years, PTNUM, data=cav, qmatrix=twoway4.q)
cav.msm <- msm( state ~ years, subject=PTNUM, data = cav,
                 qmatrix = twoway4.q, deathexact = 4, 
                 control = list ( trace = 2, REPORT = 1 )  )
cav.msm
qmatrix.msm(cav.msm)
pmatrix.msm(cav.msm, t=10)
sojourn.msm(cav.msm)

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