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Introduction to GMLTM

Overview

The GMLTM package implements three Bayesian Item Response Theory models for cognitive diagnosis:

All models decompose item difficulty into cognitive operations (rules), making them suitable for studying the cognitive components underlying test performance. Estimation uses Hamiltonian Monte Carlo via rstan.

Installation

install.packages("GMLTM")

The Q matrix and components

All models require a Q matrix (items × rules) indicating which cognitive rules each item involves. The MLTM-D and GMLTM-D additionally require a components list grouping rules into higher-level dimensions.

# Example Q matrix: 5 items, 3 rules
Q <- matrix(c(1,0,1,
              0,1,1,
              1,1,0,
              1,0,0,
              0,1,0), nrow = 5, byrow = TRUE)

# Group rules into 2 components
components <- list(comp1 = c(1, 2), comp2 = 3)

Basic usage — LLTM

library(GMLTM)
data(analogy)

Q <- matrix(...)  # define your Q matrix (items x rules)

fit <- LLTM(analogy, Q,
            iters = 2000, iter_warmup = 500, chains = 2)

fit$EAP$eta      # rule difficulty estimates
fit$EAP$beta     # item difficulty estimates
reliability(fit) # marginal reliability
ppchecks(fit)    # posterior predictive check plot

Basic usage — GMLTM-D

components <- list(global = c(1, 2, 3), local = c(4, 5))

fit <- GMLTM(analogy, Q, components,
             iters = 2000, iter_warmup = 500, chains = 2)

fit$EAP$eta      # rule difficulties per component
fit$EAP$alpha    # item discriminations per component
fit$EAP$guessing # item guessing parameters
reliability(fit) # marginal reliability per component

Customising prior distributions

All model functions accept a priors argument. Only the elements to change need to be specified; unspecified elements retain their defaults.

# More diffuse prior on rule difficulties
fit_diffuse <- LLTM(analogy, Q,
                    priors = list(eta = list(sigma = 3)))

# Moderately informative prior for guessing in GMLTM
fit_gmltm <- GMLTM(analogy, Q, components,
                   priors = list(c = list(shape1 = 2, shape2 = 10)))

# Uniform prior for guessing (least informative)
fit_uniform_c <- GMLTM(analogy, Q, components,
                       priors = list(c = list(shape1 = 1, shape2 = 1)))

Available priors per model

Parameter Distribution Models
theta (ability) Normal(mu, sigma) All
eta (rule difficulty) Normal(mu, sigma) All
alpha (discrimination) half-Normal(sigma) MLTM, GMLTM
c (guessing) Beta(shape1, shape2) GMLTM

Model comparison with LOO-CV

# Fit two models with different priors
fit1 <- GMLTM(analogy, Q, components, iters = 2000, iter_warmup = 500)
fit2 <- GMLTM(analogy, Q, components,
              priors = list(c = list(shape1 = 1, shape2 = 1)),
              iters = 2000, iter_warmup = 500)

# Compare with LOO
result <- compute_model_validation(list(fit1, fit2))
print(result$Summary)

Customizing prior distributions

All models in GMLTM support user-defined prior distributions via the priors argument. This is useful for prior sensitivity analysis — refitting models with different priors to assess how much the posteriors change.

How priors work in GMLTM

Priors are passed as a named list where each element corresponds to a model parameter. You only need to specify the parameters you want to change; unspecified ones keep their defaults.

The general structure is:

priors = list(
  parameter_name = list(hyperparameter1 = value, hyperparameter2 = value)
)

LLTM priors

The LLTM has two parameters with customizable priors:

Parameter Distribution Hyperparameters Default
theta (ability) Normal mu, sigma mu=0, sigma=1
eta (rule difficulty) Normal mu, sigma mu=0, sigma=1 
library(GMLTM)
data(analogy)

# Default priors (weakly informative)
fit_default <- LLTM(analogy, Q,
                    iters = 2000, iter_warmup = 500, chains = 2)

# More diffuse prior on rule difficulties (eta)
fit_diffuse <- LLTM(analogy, Q,
                    iters = 2000, iter_warmup = 500, chains = 2,
                    priors = list(eta = list(mu = 0, sigma = 3)))

# Informative prior centered on positive difficulty
fit_informed <- LLTM(analogy, Q,
                     iters = 2000, iter_warmup = 500, chains = 2,
                     priors = list(eta = list(mu = 1, sigma = 0.5)))

MLTM priors

The MLTM adds discrimination parameters (alpha):

Parameter Distribution Hyperparameters Default
theta (ability) Normal mu, sigma mu=0, sigma=1
eta (rule difficulty) Normal mu, sigma mu=0, sigma=1
alpha (discrimination) Half-Normal sigma sigma=1

Note: alpha uses a half-Normal prior (truncated at 0) to enforce positive discrimination. Only sigma is meaningful; mu is ignored.

components <- list(global = c(1, 2, 3), local = c(4, 5))

# Wider prior for discrimination
fit_mltm <- MLTM(analogy, Q, components,
                 iters = 2000, iter_warmup = 500, chains = 2,
                 priors = list(
                   theta = list(mu = 0, sigma = 1),
                   alpha = list(sigma = 2)
                 ))

GMLTM-D priors

The GMLTM-D adds a guessing parameter (c):

Parameter Distribution Hyperparameters Default
theta (ability) Normal mu, sigma mu=0, sigma=1
eta (rule difficulty) Normal mu, sigma mu=0, sigma=1
alpha (discrimination) Half-Normal sigma sigma=1
c (guessing) Beta shape1, shape2 shape1=3, shape2=20

The default Beta(3, 20) prior for guessing concentrates probability below 0.20, consistent with typical multiple-choice guessing rates.

# Default: Beta(3,20) -- conservative guessing prior
fit_gmltm <- GMLTM(analogy, Q, components,
                   iters = 2000, iter_warmup = 500, chains = 2)

# Uniform prior for guessing -- no prior assumption
fit_uniform_c <- GMLTM(analogy, Q, components,
                       iters = 2000, iter_warmup = 500, chains = 2,
                       priors = list(c = list(shape1 = 1, shape2 = 1)))

# Moderately informative prior
fit_moderate_c <- GMLTM(analogy, Q, components,
                        iters = 2000, iter_warmup = 500, chains = 2,
                        priors = list(c = list(shape1 = 2, shape2 = 10)))

Prior sensitivity analysis

A good practice is to refit the model with at least two different prior specifications and compare the posterior means:

# Conservative priors
fit_conservative <- LLTM(analogy, Q, chains = 2, iters = 2000,
                         priors = list(eta = list(sigma = 1)))

# Diffuse priors
fit_diffuse <- LLTM(analogy, Q, chains = 2, iters = 2000,
                    priors = list(eta = list(sigma = 5)))

# Compare eta estimates
cbind(
  conservative = fit_conservative$EAP$eta,
  diffuse      = fit_diffuse$EAP$eta
)

If the estimates are similar, your results are robust to prior choice. Large differences indicate the data are not very informative and results should be interpreted with caution.

GMLTM default priors

The default priors in GMLTM() are weakly informative and suitable for most applications:

Parameter Distribution Default
theta (ability) Normal mu=0, sigma=1
eta (rule difficulty) Normal mu=0, sigma=1
alpha (discrimination) Half-Normal sigma=1
c (guessing) Beta shape1=3, shape2=20

To replicate moderately diffuse priors (formerly GMLTM1):

fit_gmltm1_style <- GMLTM(analogy, Q, components,
  priors = list(
    theta = list(mu = 0, sigma = 2),
    eta   = list(mu = 0, sigma = 2),
    c     = list(shape1 = 2, shape2 = 5)
  ))

To replicate diffuse priors (formerly GMLTM2):

fit_gmltm2_style <- GMLTM(analogy, Q, components,
  priors = list(
    theta = list(mu = 0, sigma = 5),
    eta   = list(mu = 0, sigma = 5),
    c     = list(shape1 = 1, shape2 = 1)
  ))

References

Fischer, G. H. (1973). The linear logistic test model as an instrument in educational research. Acta Psychologica, 37(6), 359–374.

Embretson, S. E., & Yang, X. (2013). A multicomponent latent trait model for diagnosis. Psychometrika, 78, 14–36.

Ramírez, E. S., Jiménez, M., Franco, V. R., & Alvarado, J. M. (2024). Delving into the complexity of analogical reasoning: A detailed exploration with the Generalized Multicomponent Latent Trait Model for Diagnosis. Journal of Intelligence, 12, 67. https://doi.org/10.3390/jintelligence12070067

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