mc_multinom: Estimate MC model with constraints

In LMest: Generalized Latent Markov Models

Estimate MC model with constraints

Description

Function that estimates Markov chain model with suitable constraints on the transition matrix.

Usage

mc_multinom(formula, data, 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, reltol = 10^-10, output = FALSE)

Arguments

formula	A model formula specifying the response variable and the covariates affecting the transition probabilities. The response variable must be numeric, coded consecutively from 1 to k, where k is the number of observed states. See the ‘Details’ section for the specification of the covariates
data	A balanced data frame in long format. The first two required variables are id, identifying subjects, and t, identifying time occasions. Subject identifiers must be consecutive integers from 1 to the number of subjects; for each subject the time occasions must be the consecutive integers from 1 to the number of occasions available for that subject, which must be at least two; the rows must be ordered by subject and, within each subject, by time occasion. Missing values are not allowed in the variables entering formula and formula_init
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 constraint structure for the intercepts of the transition logits. See design_matrices for the available options and their interpretation.
model_cov	Character string specifying the constraint structure for the regression coefficients associated with the transition covariates. See design_matrices for the available options and their interpretation
formula_init	Optional formula specifying the covariates affecting the initial probabilities. If NULL, the covariates specified in formula are also used for the initial probabilities
reltol	Positive numeric value specifying the relative convergence tolerance used in the Newton–Raphson iterations for the initial and transition components of the log-likelihood
output	Logical value. If TRUE, subject-specific initial and transition probability arrays are returned in addition to the parameter estimates

Details

The covariates are included in the model through model.matrix, so that factors are automatically coded by means of dummy variables and transformations of the covariates may be specified directly in the formula, as in I(age - 50) or I((age - 50)^2/100). The model always includes an intercept and only one response variable is allowed. Covariates affecting the initial probabilities are specified through the separate argument formula_init; when this is NULL, the covariates in formula are used for both the initial and the transition probabilities, whereas formula_init = y ~ 1 gives initial probabilities that do not depend on covariates.

Value

An object of class "mc_multinom" containing the following components:

la	Estimated regression coefficients for the initial probabilities
psi	Estimated reduced vector of regression coefficients for the constrained transition logits
eta	Expanded vector of transition-logit parameters obtained by applying the design-matrix constraints to psi
lk	Total log-likelihood, equal to the sum of the initial-probability and transition-probability log-likelihoods
lk1	Log-likelihood contribution of the initial probabilities
lk2	Log-likelihood contribution of the transition probabilities
np	Number of free model parameters
aic	Akaike Information Criterion
bic	Bayesian Information Criterion
sela	Standard errors of the estimated coefficients la
sepsi	Standard errors of the estimated coefficients psi
call	Matched call used to fit the model
Piv	Matrix of subject-specific initial probabilities, with dimensions n x k. Returned only when output = TRUE
PI	Array of subject-specific transition probabilities, with dimensions k x k x n x TT, where TT is the maximum number of time occasions. Returned only when output = TRUE

Author(s)

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

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.

Examples

## Not run: 
data("data_SRHS_long")
srhs <- data_SRHS_long

# Covariates
srhs$gender <- factor(
  srhs$gender,
  levels = c(1, 2),
  labels = c("male", "female")
)

srhs$race <- factor(
  srhs$race,
  levels = c(1, 2, 3),
  labels = c("white", "black", "others")
)

srhs$education <- factor(
  srhs$education,
  levels = 1:5,
  labels = c(
    "high school",
    "general educational diploma",
    "high school graduate",
    "some college",
    "college and above"
  )
)

srhs$age_c <- srhs$age - mean(srhs$age)

# 1 = poor, ..., 5 = excellent

srhs$srhs1 <- 6L - as.integer(srhs$srhs)

# Consecutive numeric identifiers and ordering by subject and time

srhs$id <- as.integer(factor(srhs$id))
srhs <- srhs[order(srhs$id, srhs$t), ]

# Select a small reproducible balanced sample for a fast example

set.seed(2026)

n_example <- 2000L
example_ids <- sample(unique(srhs$id), n_example)

srhs_example <- srhs[srhs$id %in% example_ids, , drop = FALSE]

srhs_example$id <- match(srhs_example$id, example_ids)
srhs_example <- srhs_example[order(srhs_example$id, srhs_example$t), ]


# Example 1: gender and age affect both initial and transition probabilities

mod_cov <- mc_multinom(
  srhs1 ~ gender + age_c,
  data = srhs_example,
  baseline = "central",
  model_int = "dist1",
  model_cov = "dist1",
  output = TRUE)

print(mod_cov)
summary(mod_cov)


# Average initial and transition probabilities

colMeans(mod_cov$Piv)
round(apply(mod_cov$PI[, , , 2:max(srhs_example$t)], c(1, 2), mean),3)

# Example 2: covariates affect transitions only

mod_trans <- mc_multinom(
  srhs1 ~ gender + age_c,
  data = srhs_example,
  baseline = "central",
  model_int = "dist1",
  model_cov = "dist1",
  formula_init = srhs1 ~ 1)

print(mod_trans)
summary(mod_trans)

## End(Not run)

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