These functions are used to specify the distribution of the response
conditionally on the underlying state in a hidden Markov model. A list
of these function calls, with one component for each state, should be
used for the hmodel argument to msm. The initial values
for the parameters of the distribution should be given as arguments.
Note the initial values should be supplied as literal values - supplying
them as variables is currently not supported.
hmmCat(prob, basecat)
hmmIdent(x)
hmmUnif(lower, upper)
hmmNorm(mean, sd)
hmmLNorm(meanlog, sdlog)
hmmExp(rate)
hmmGamma(shape, rate)
hmmWeibull(shape, scale)
hmmPois(rate)
hmmBinom(size, prob)
hmmTNorm(mean, sd, lower, upper)
hmmMETNorm(mean, sd, lower, upper, sderr, meanerr=0)
hmmMEUnif(lower, upper, sderr, meanerr=0)
hmmNBinom(disp, prob)
hmmBetaBinom(size, meanp, sdp)
hmmBeta(shape1,shape2)
hmmT(mean,scale,df)(hmmCat) Vector of probabilities of observing
category 1, 2, …, length(prob) respectively. Or
the probability governing a binomial or negative binomial
distribution.
(hmmCat) Category which is considered to be the "baseline",
so that during estimation, the probabilities are parameterised as
probabilities relative to this baseline category. By default, the
category with the greatest probability is
used as the baseline.
(hmmIdent) Code in the data which denotes the exactly-observed state.
(hmmNorm,hmmLNorm,hmmTNorm) Mean defining a Normal, or truncated Normal
distribution.
(hmmNorm,hmmLNorm,hmmTNorm) Standard deviation defining a
Normal, or truncated Normal distribution.
(hmmNorm,hmmLNorm,hmmTNorm) Mean on the log
scale, for a log Normal distribution.
(hmmNorm,hmmLNorm,hmmTNorm) Standard deviation on
the log scale, for a log Normal distribution.
First and second parameters of a beta
distribution (see dbeta).
(hmmGamma) Scale parameter of a Gamma
distribution (see dgamma), or unstandardised Student t
distribution.
Degrees of freedom of the Student t distribution.
Order of a Binomial distribution (see dbinom).
Dispersion parameter of a negative binomial distribution,
also called size or order. (see
dnbinom).
Mean outcome probability in a beta-binomial distribution
Standard deviation describing the overdispersion of the outcome probability in a beta-binomial distribution
(hmmUnif,hmmTNorm,hmmMEUnif) Lower limit for an Uniform or truncated Normal distribution.
(hmmUnif,hmmTNorm,hmmMEUnif) Upper limit for an Uniform or truncated Normal
distribution.
(hmmMETNorm,hmmUnif) Standard deviation of the Normal measurement error
distribution.
(hmmMETNorm,hmmUnif) Additional shift in the
measurement error, fixed to 0 by default. This may
be modelled in terms of covariates.
Each function returns an object of class hmodel, which is a
list containing information about the model. The only component
which may be useful to end users is r, a function of one
argument n which returns a random sample of size n from
the given distribution.
hmmCat represents a categorical response distribution on the
set 1, 2, …, length(prob). The
Markov model with misclassification is an example of this type of model. The
categories in this case are (some subset of) the underlying states.
The hmmIdent distribution is used for underlying states which are
observed exactly without error. For hidden Markov models with multiple outcomes,
(see hmmMV), the outcome in the data which takes the
special hmmIdent value must be the first of the multiple outcomes.
hmmUnif, hmmNorm, hmmLNorm, hmmExp,
hmmGamma, hmmWeibull, hmmPois, hmmBinom,
hmmTNorm, hmmNBinom and hmmBeta represent Uniform, Normal,
log-Normal, exponential, Gamma, Weibull, Poisson, Binomial, truncated Normal,
negative binomial and beta distributions, respectively, with parameterisations
the same as the default parameterisations in the corresponding base R
distribution functions.
hmmT is the Student t distribution with general mean
\(\mu\), scale \(\sigma\) and degrees of freedom
df.
The variance is \(\sigma^2 df/(df + 2)\).
Note the t distribution in base R dt is a standardised one with
mean 0 and scale 1. These allow any positive (integer or non-integer)
df. By default, all three
parameters, including df, are estimated when fitting a hidden
Markov model, but in practice, df might need to be fixed for identifiability - this can be done
using the fixedpars argument to msm.
The hmmMETNorm and hmmMEUnif distributions are
truncated Normal and Uniform distributions, but with additional Normal measurement error on the
response. These are generalisations of the distributions proposed by
Satten and Longini (1996) for modelling the progression of CD4 cell
counts in monitoring HIV disease. See medists for
density, distribution, quantile and random generation functions for
these distributions. See also tnorm for
density, distribution, quantile and random generation functions for
the truncated Normal distribution.
See the PDF manual msm-manual.pdf in the doc
subdirectory for algebraic definitions of all these distributions.
New hidden Markov model response distributions can be added to
msm by following the instructions in Section 2.17.1.
Parameters which can be modelled in terms of covariates, on the scale of a link function, are as follows.
| PARAMETER NAME | LINK FUNCTION |
mean |
identity |
meanlog |
identity |
rate |
log |
scale |
log |
meanerr |
identity |
meanp |
logit |
Parameters basecat, lower, upper, size, meanerr are fixed at
their initial values. All other parameters are estimated while fitting
the hidden Markov model, unless the appropriate fixedpars
argument is supplied to msm.
For categorical response distributions (hmmCat) the
outcome probabilities initialized to zero are fixed at zero, and the
probability corresponding to basecat is fixed to one minus the
sum of the remaining probabilities. These remaining probabilities are
estimated, and can be modelled in terms of covariates via multinomial
logistic regression (relative to basecat).
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 progresison 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).