est_LM_cat_multinom: Estimation of Constrained Hidden Markov Models

In LMest: Generalized Latent Markov Models

Estimation of Constrained Hidden Markov Models

Description

Function that estimates a hidden Markov model to a univariate categorical longitudinal response. The observed response is generated by an unobserved first-order Markov chain with k latent states. Conditional on the latent state, responses are independent over time. Covariates may affect both the initial latent-state probabilities and the transition probabilities, and linear equality constraints may be imposed separately on transition intercepts and covariate effects.

Usage

est_LM_cat_multinom(resp_name, formula, k, data, reltol = 10^-10,
                    rand.start = FALSE,
                    nstart = 1, baseline = c("initial", "central"),
                    model_int = c("all", "const", "dist1", "dist2",
                                  "dist3", "two", "symm", "rsymm", "diff"),
                    model_cov = c("all", "const", "dist1", "dist2", "dist3",
                                  "two", "symm", "rsymm", "diff"),
                    formula_init = NULL,
                    par.init = list(la = NULL, psi = NULL, Pr = NULL),
                    output = FALSE, fort = TRUE)

Arguments

resp_name	Character string giving the name of the observed response variable in data. The response must be numeric and coded with consecutive integer categories 1, ..., c, where c is the number of response categories, with no missing categories.
formula	A one-sided formula specifying the covariates in the transition model, for example ~ gender + race + education + age_c. An intercept should be included. If formula_init is NULL, the same covariates are used for the initial latent-state probabilities.
k	Positive integer giving the number of latent states. A one-state model is allowed; in that case only the response probabilities and fit criteria are estimated, and the arguments formula, formula_init, baseline, model_int and model_cov have no effect.
data	Data frame in long format. The first column is used as the subject identifier and the second as the measurement occasion, irrespective of their names. Subject identifiers must be consecutive integers, records must be ordered by subject and occasion, and occasions for each subject must be consecutive and start at 1. Individual trajectories may have different lengths. Rows with missing values in the response or model variables should be removed before fitting.
reltol	Positive numeric value giving the relative log-likelihood tolerance used to stop the EM algorithm. Iteration stops when the relative increment in log-likelihood is no greater than reltol.
rand.start	Logical value controlling initialization when nstart = 1. If FALSE, the initial latent allocation is obtained by k-means clustering of the response. If TRUE, a random latent allocation is used. Components supplied in par.init override the corresponding internally generated starting values.
nstart	Positive integer giving the number of fits from different starting values. If greater than 1, the first fit uses the k-means initialization and the remaining fits use random initial allocations; the fit with the largest log-likelihood is returned.
baseline	Character string specifying the reference category in the multinomial logit parameterization for the transition probabilities. With "initial", the transition towards the first state is used as the reference category for each row of the transition matrix. With "central", the self-transition is used as the reference category for each row of the transition matrix.
model_int	Character string specifying the equality-constraint structure for the transition-logit intercepts. Available choices are "all", "const", "dist1", "dist2", "dist3", "two", "symm", "rsymm", and "diff". See Details and design_matrices.
model_cov	Character string specifying the equality-constraint structure for the transition-logit regression coefficients. The available choices are the same as for model_int. See Details and design_matrices.
formula_init	Optional one-sided formula specifying the covariates for the initial latent-state probabilities. Only measurements at occasion 1 are used. If NULL, formula is also used for this submodel. Use ~ 1 for an intercept-only initial-probability model.
par.init	Named list of optional starting values. Component la contains the coefficients for the initial-probability logits; psi contains the reduced parameter vector for the constrained transition logits; and Pr is a k by c matrix of conditional response probabilities. Any component may be left NULL to use an internally generated starting value.
output	Logical value. If TRUE, the fitted subject-specific initial probabilities Piv and transition probabilities PI are returned in addition to the main estimation results.
fort	Logical value selecting the numerical implementation. If TRUE, the compiled Fortran routines are used and must be available and loaded. If FALSE, the corresponding R routines are used.

Details

Let U denote the latent state for subject i at occasion t. The measurement model is P(Y = y \mid U = u), represented by the returned matrix Pr. The arguments model_int and model_cov apply the same family of constraint structures to different parameter blocks: model_int acts on the transition intercepts, whereas model_cov acts componentwise on the transition covariate coefficients. The two structures can be selected independently. With baseline = "central", let a_{uv} denote a transition parameter from origin state u to destination state v, with v \ne u. The available structures are:

"all"

No equality constraints: every transition has its own parameter.

"const"

A common parameter is used for all off-diagonal transitions. For model_cov, each covariate therefore has one common effect across all transitions.

"dist1"

Parameters are grouped by signed displacement v-u. Transitions with the same direction and the same size of movement share a parameter.

"dist2"

Parameters are grouped by absolute displacement |v-u|; movements of the same size share a parameter in both directions.

"dist3"

Parameters depend on |v-u|, with opposite signs for upward and downward movements of the same size.

"two"

Two parameters are used: one for transitions with v>u and one for transitions with v<u, irrespective of movement size.

"symm"

Opposite transitions share a parameter, so that a_{uv}=a_{vu}.

"rsymm"

Opposite transitions have parameters with reversed signs, so that a_{uv}=-a_{vu}.

"diff"

Transition parameters are differences between destination- and origin-state parameters, a_{uv}=a_v-a_u, with the state-1 parameter fixed at zero for identifiability.

The distance-based structures "dist1", "dist2", "dist3", and "two" are intended for substantively ordered latent states. In particular, model_cov = "all" is the unconstrained transition-covariate model, model_cov = "dist1" pools effects by signed displacement, and model_cov = "const" imposes one effect per covariate across all transitions. The argument model_int operates analogously on the transition intercepts.

When nstart > 1, the returned object is the solution with the largest log-likelihood and component lkv records the log-likelihood obtained in each fit.

Value

An object of class "LM_cat_multinom" containing:

Pr

A k by c matrix of estimated conditional response probabilities, with latent states in rows and observed categories in columns.

lk

The maximized observed-data log-likelihood.

np

The number of free model parameters.

aic

The Akaike information criterion.

bic

The Bayesian information criterion, using the number of subjects as the sample size.

call

The matched call used to fit the model.

For k > 1, the object also contains:

la

Estimated regression coefficients for the initial latent-state probabilities.

psi

Estimated reduced parameter vector for the constrained transition logits.

eta

Expanded transition-logit parameter vector obtained by applying the design matrix constraints to psi.

W

Array of posterior latent-state probabilities produced by the final E-step, with dimensions n by k by TT, where n is the number of subjects.

CL

Matrix of modal posterior latent-state classifications by subject and occasion, with dimensions n by TT.

If output = TRUE and k > 1, it additionally contains:

Piv

Matrix of subject-specific fitted initial probabilities, with dimensions n by k.

PI

Array of subject- and occasion-specific fitted transition probabilities, with dimensions k by k by n by TT, where TT is the maximum number of occasions.

If nstart > 1, component lkv contains the log-likelihood from each attempted start.

Author(s)

Francesco Bartolucci and Silvia Pandolfi, University of Perugia (Italy); Luca Brusa and Fulvia Pennoni, University of Milano-Bicocca (Italy).

References

Bartolucci, F., Pandolfi, S., and Pennoni, F. (2026). Parsimonious parametrizations of transition matrices of Markov chain and hidden Markov models. Annals of Operations Research, 363, 233–266.

See Also

design_matrices

Examples

## Not run: 

# Load and prepare data

data("data_SRHS_long")
srhs <- data_SRHS_long[,c(2,1,3:7)]

# 1 = poor, ..., 5 = excellent
srhs$srhs1 <- 6L - as.integer(srhs$srhs)
srhs$gender <- factor(srhs$gender)
srhs$race <- factor(srhs$race)
srhs$education <- factor(srhs$education)
srhs$age_c <- srhs$age - mean(srhs$age)
srhs$id <- as.integer(factor(srhs$id))
srhs <- srhs[order(srhs$id, srhs$t), ]

# Reduce the sample size for fast example
set.seed(123)

all_ids <- unique(srhs$id)
n_example <- min(1000L, length(all_ids))
example_ids <- sample(all_ids, size = n_example, replace = FALSE)
srhs_example <- srhs[srhs$id %in% example_ids,,drop = FALSE]

# Consecutive identifiers required by the estimation function
srhs_example$id <- match(srhs_example$id, example_ids)
srhs_example <- srhs_example[order(srhs_example$id, srhs_example$t), ]

# Estimate a HM-ALL-COV: benchmark without constraints on covariate effects

hm_all <- est_LM_cat_multinom("srhs1", ~ gender + race + education + age_c,
                              k = 3, data = srhs_example, baseline = "central",
                              model_cov = "all", model_int = "all",
                              output = TRUE, fort = TRUE)
print(hm_all)
summary(hm_all)

# Model with a common covariate effect across transitions
hm_const <- est_LM_cat_multinom("srhs1", ~ gender + race + education + age_c,
                                k = 3, data = srhs_example, baseline = "central",
                                model_cov = "const", model_int = "all",
                                output = TRUE, fort = TRUE)
print(hm_const)
summary(hm_const)

## End(Not run)

LMest documentation built on Oct. 8, 2026, 5:08 p.m.