Type: Package
Title: Saturated Model Log-Likelihood for Multinomial Outcomes
Version: 0.0.0.6
Date: 2025-07-09
Author: Jorge Villalba ORCID iD [aut, cre], Humberto Llinas ORCID iD [aut], Jorge Borja ORCID iD [aut], Jorge Tilano ORCID iD [aut]
Maintainer: Jorge Villalba <jvillalba@utb.edu.co>
Description: When the response variable Y takes one of R > 1 values, the function 'glsm()' computes the maximum likelihood estimates (MLEs) of the parameters under four models: null, complete, saturated, and logistic. It also calculates the log-likelihood values for each model. This method assumes independent, non-identically distributed variables. For grouped data with a multinomial outcome, where observations are divided into J populations, the function 'glsm()' provides estimation for any number K of explanatory variables.
Depends: R (≥ 3.5.0)
Imports: stats, dplyr (≥ 1.0.0), ggplot2 (≥ 1.0.0), VGAM (≥ 1.0.0), plyr
License: MIT + file LICENSE
Encoding: UTF-8
RoxygenNote: 7.3.2
LazyData: true
NeedsCompilation: no
Packaged: 2025-07-09 14:14:12 UTC; jvillalba
Repository: CRAN
Date/Publication: 2025-07-14 17:10:02 UTC

Confidence Intervals for Coefficients in glsm Objects

Description

Calculates confidence intervals for the coefficients in a fitted glsm model. Includes exponentiated intervals (Odds Ratios) for easier interpretation.

Usage

## S3 method for class 'glsm'
confint(object, parm, level = 0.95, ...)

Arguments

object

The type of prediction required. The default is on the scale of the linear predictors. The alternative response gives the predicted probabilities.

parm

calculate confidence intervals for the coefficients

level

It gives the desired confidence level for the confidence interval. For example, a default value is level = 0.95, which will generate a 95% confidence interval." The alternative response gives the predicted probabilities.

...

further arguments passed to or from other methods.

Details

Confint Method for 'glsm'

The saturated model is characterized by the assumptions 1 and 2 presented in section 2.3 by Llinas (2006, ISSN:2389-8976).

Value

An object of class "confint.glsm", which is a list containing:

object

a glsm object

parm

calculate confidence intervals for the coefficients.

level

confidence levels

Author(s)

Humberto Llinás (Universidad del Norte, Barranquilla-Colombia; author), Jorge Villalba (Universidad Tecnológica de Bolívar, Cartagena-Colombia; author and creator), Jorge Borja (Universidad del Norte, Barranquilla-Colombia; author and creator), Jorge Tilano (Universidad del Norte, Barranquilla-Colombia; author)

References

Hosmer, D., Lemeshow, S., & Sturdivant, R. (2013). Applied Logistic Regression (3rd ed.). New York: Wiley. ISBN: 978-0-470-58247-3 Llinás, H. (2006). Precisiones en la teoría de los modelos logísticos. Revista Colombiana de Estadística, 29(2), 239–265. Llinás, H., & Carreño, C. (2012). The Multinomial Logistic Model for the Case in Which the Response Variable Can Assume One of Three Levels and Related Models. Revista Colombiana de Estadística, 35(1), 131–138. Orozco, E., Llinás, H., & Fonseca, J. (2020). Convergence theorems in multinomial saturated and logistic models. Revista Colombiana de Estadística, 43(2), 211–231. Llinás, H., Arteta, M., & Tilano, J. (2016). El modelo de regresión logística para el caso en que la variable de respuesta puede asumir uno de tres niveles: estimaciones, pruebas de hipótesis y selección de modelos. Revista de Matemática: Teoría y Aplicaciones, 23(1), 173–197.

Examples

# Load the glsm package and example dataset
library(glsm)
data("hsbdemo", package = "glsm")

# Fit a multinomial logistic regression model using glsm()
model <- glsm(prog ~ ses + gender, data = hsbdemo)

# Get confidence intervals for all model coefficients (default 95% level)
confint(model)

# Get confidence intervals for a specific coefficient
params <- names(model$coefficients)

results <- lapply(params, function(p) {
  cat("\nConfidence interval for:", p, "\n")
  print(confint(model, parm = p, level = 0.95))
})

Saturated Model Log-Likelihood for Multinomial Outcomes

Description

When the response variable Y takes one of R > 1 values, the function "glsm()" computes the maximum likelihood estimates (MLEs) of the parameters under four models: null, complete, saturated, and logistic. It also calculates the log-likelihood values for each model.

The method assumes independent, non-identically distributed variables. For grouped data with a multinomial outcome variable, where the observations are divided into J populations, the function '"glsm()" offers reliable estimation for any number K of explanatory variables.

Usage

glsm(formula, data, ref = NaN)

Arguments

formula

An object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. See 'Details' for more information on model specification.

data

An optional data frame, list, or environment (or object coercible via as.data.frame) containing the variables in the model. If variables are not found in data, they are taken from environment(formula), typically the environment from which glsm() is called.

ref

Optional character string indicating the reference level of the response variable. If not specified, the first level is used by default.

Details

glsm.R

An expression of the form y ~ model is interpreted as a specification that the response variable y is modeled by a linear predictor, symbolically defined by model (the systematic component). The model consists of terms separated by + operators. Each term can include variable or factor names, and interactions between variables are denoted by :. Such a term represents the interaction of all included variables and factors. In this context, y is the outcome variable, which may be binary or polychotomous.

Value

An object of class "glsm", which is a list containing at least the following components:

coefficients

Vector of estimated coefficients, including intercepts and slopes.

coef

Alias for coefficients. Returns the same vector of estimated intercepts and slopes.

Std.Error

Vector of standard errors for the estimated coefficients (intercepts and slopes).

ExpB

Vector containing the exponentiated coefficients (i.e., exp(beta)) for interpretation as odds ratios.

Wald

Wald test statistic used to assess the significance of each coefficient (assumed to follow a chi-squared distribution).

DF

Degrees of freedom associated with the Wald test's chi-squared distribution.

P.value

P-values corresponding to the Wald test statistics.

Log_Lik_Complete

Log-likelihood value of the complete model.

Log_Lik_Null

Log-likelihood value of the null model.

Log_Lik_Logit

Log-likelihood value of the logistic model.

Log_Lik_Saturate

Log-likelihood value of the saturated model.

Populations

Number of populations considered in the saturated model.

Dev_Null_vs_Logit

Deviance statistic comparing the null and logistic models.

Dev_Logit_vs_Complete

Deviance statistic comparing the logistic and complete models.

Dev_Logit_vs_Saturate

Deviance statistic comparing the logistic and saturated models.

Df_Null_vs_Logit

Degrees of freedom for the deviance test comparing the null and logistic models.

Df_Logit_vs_Complete

Degrees of freedom for the deviance test comparing the logistic and complete models.

Df_Logit_vs_Saturate

Degrees of freedom for the deviance test comparing the logistic and saturated models.

P.v_Null_vs_Logit

P-value for the hypothesis test comparing the null and logistic models.

P.v_Logit_vs_Complete

P-value for the hypothesis test comparing the logistic and complete models.

P.v_Logit_vs_Saturate

P-value for the hypothesis test comparing the logistic and saturated models.

Logit_r

Matrix of log-odds values, with respect to the reference category r of the outcome variable Y.

p_hat_complete

Vector of probabilities that the outcome variable takes the value 1, given the jth population (estimated from the complete model, excluding the logistic model).

p_hat_null

Vector of probabilities that the outcome variable takes the value 1, given the jth population (estimated from the null model, excluding the logistic model).

p_rj

Matrix containing the estimated values of each prj, the probability that the outcome variable takes the value r, given the jth population (estimated using the logistic model).

odd

Vector containing the odds for each jth population.

OR

Vector containing the odds ratios for each variable's coefficient.

z_rj

Vector containing the values of each Zrj, defined as the sum of observations in the jth population.

nj

Vector containing the number of observations (nj) in each jth population.

p_rj_tilde

Vector containing the estimated values of each prj, the probability that the outcome variable takes the value r, given the jth population (estimated under the saturated model, without estimating logistic parameters).

v_rj

Vector of variances of the Bernoulli variables in the jth population and category r.

m_rj

Vector of expected values of Zj in the jth population and category r.

V_rj

Vector of variances of Zj in the jth population and category r.

V

Variance–covariance matrix of Z, the vector containing all Zj values.

S_p

Score vector computed under the saturated model.

I_p

Fisher information matrix under the saturated model.

Zast_j

Vector of standardized values for the variable Zj.

mcov

Variance–covariance matrix of the coefficient estimates.

mcor

Correlation matrix of the coefficient estimates.

Esm

Estimated Saturated Matrix. A data frame containing estimates from the saturated model. For each population j, it includes the values of the explanatory variables, nj, Zrj, prj_tilde, and the log-likelihood Lp_tilde.

Elm

Estimated Logit Matrix. A data frame containing estimates from the logistic model. For each population j, it includes the values of the explanatory variables, nj, Zrj, prj, the logit transformation Logit_rj, and the variance of the logit (var_logit_rj).

call

The original function call used to fit the glsm model.

Author(s)

Humberto Llinás (Universidad del Norte, Barranquilla-Colombia; author), Jorge Villalba (Universidad Tecnológica de Bolívar, Cartagena-Colombia; author and creator), Jorge Borja (Universidad del Norte, Barranquilla-Colombia; author and creator), Jorge Tilano (Universidad del Norte, Barranquilla-Colombia; author)

References

Hosmer, D., Lemeshow, S., & Sturdivant, R. (2013). Applied Logistic Regression (3rd ed.). New York: Wiley. ISBN: 978-0-470-58247-3 Llinás, H. (2006). Precisiones en la teoría de los modelos logísticos. Revista Colombiana de Estadística, 29(2), 239–265. Llinás, H., & Carreño, C. (2012). The Multinomial Logistic Model for the Case in Which the Response Variable Can Assume One of Three Levels and Related Models. Revista Colombiana de Estadística, 35(1), 131–138. Orozco, E., Llinás, H., & Fonseca, J. (2020). Convergence theorems in multinomial saturated and logistic models. Revista Colombiana de Estadística, 43(2), 211–231. Llinás, H., Arteta, M., & Tilano, J. (2016). El modelo de regresión logística para el caso en que la variable de respuesta puede asumir uno de tres niveles: estimaciones, pruebas de hipótesis y selección de modelos. Revista de Matemática: Teoría y Aplicaciones, 23(1), 173–197.

Examples

library(glsm)
data("hsbdemo", package = "glsm")
model <- glsm(prog ~ ses + gender, data = hsbdemo, ref = "academic")
model

hsbdemo: School data for testing.

Description

Entering high school students make program choices among general program, vocational program and academic program. Their choice might be modeled using their writing score and their social economic status.The data set contains variables on 200 students. The outcome variable is prog, program type. The predictor variables are social economic status, ses, a three-level categorical variable and writing score, write, a continuous variable.

Usage

hsbdemo

Format

A data frame with 200 rows and 17 columns:

Student

Categorical. Student identification code.

id

Categorical. Unique identifier for each student.

gender

Categorical. Student gender: "female" or "male".

ses

Categorical. Socioeconomic status: "low", "middle", "high".

schtyp

Categorical. Type of school: "private" or "public".

# corregido

prog

Categorical. Program of study chosen: 0 = General, 1 = Vocational, 2 = Academic.

read

Continuous. Reading test score.

write

Continuous. Writing test score.

math

Continuous. Math test score.

science

Continuous. Science test score.

socst

Continuous. Social studies test score.

honors

Categorical. Honors enrollment status: "enrolled" or "not enrolled".

awards

Integer. Number of awards received, ranging from 0 to 9.

cid

Categorical. Unspecified score, ranging from 0 to 20.

prog0

Binary. 1 if prog = General, 0 otherwise.

prog1

Binary. 1 if prog = Vocational, 0 otherwise.

prog2

Binary. 1 if prog = Academic, 0 otherwise.

Source

Simulated dataset inspired by high school program choices.

Summary Method for in glsm Objects

Description

Summarizes a fitted glsm model, including coefficients, standard errors, odds ratios, Wald tests, and likelihood-ratio comparisons with nested models.

Usage

## S3 method for class 'glsm'
summary(object, ...)

Arguments

object

The glsm model to summarize. The details of the model specification are provided under Details.

...

Other arguments passed to or from other methods.

Details

Summary Method for 'glsm'

Value

"summary.glsm" returns an object of class summary.glsm, a list with components:

Call

The original call used to fit the model.

coeff

A matrix of coefficients with columns for the estimated coefficients (Coef(B)), standard errors (Std.Error), exponentiated coefficients (Exp(B)), Wald test statistics (Wald), degrees of freedom (DF), and the corresponding p-values (P.value).

comparison test

A matrix with comparison tests of the logistic model against the following models: Null, Complete, and Saturated. It includes the test statistic (Deviance), degrees of freedom (DF), and p-values (P.value).

#' @details The glsm function estimates a multinomial logistic regression model when the response variable takes more than two levels. The model compares the logistic specification against nested models (null, complete, and saturated), and provides maximum likelihood estimates, asymptotic inference for coefficients, and goodness-of-fit measures. This summary method presents the key components of the model in a structured format.

Author(s)

Humberto Llinás (Universidad del Norte, Barranquilla-Colombia; author), Jorge Villalba (Universidad Tecnológica de Bolívar, Cartagena-Colombia; author and creator), Jorge Borja (Universidad del Norte, Barranquilla-Colombia; author and creator), Jorge Tilano (Universidad del Norte, Barranquilla-Colombia; author)

References

Hosmer, D., Lemeshow, S., & Sturdivant, R. (2013). Applied Logistic Regression (3rd ed.). New York: Wiley. ISBN: 978-0-470-58247-3 Llinás, H. (2006). Precisiones en la teoría de los modelos logísticos. Revista Colombiana de Estadística, 29(2), 239–265. Llinás, H., & Carreño, C. (2012). The Multinomial Logistic Model for the Case in Which the Response Variable Can Assume One of Three Levels and Related Models. Revista Colombiana de Estadística, 35(1), 131–138. Orozco, E., Llinás, H., & Fonseca, J. (2020). Convergence theorems in multinomial saturated and logistic models. Revista Colombiana de Estadística, 43(2), 211–231. Llinás, H., Arteta, M., & Tilano, J. (2016). El modelo de regresión logística para el caso en que la variable de respuesta puede asumir uno de tres niveles: estimaciones, pruebas de hipótesis y selección de modelos. Revista de Matemática: Teoría y Aplicaciones, 23(1), 173–197.

Examples

data("hsbdemo", package = "glsm")
model <- glsm(prog ~ ses + gender, data = hsbdemo)
summary(model)