Spelling data and dichotomous IRT

The R code below first loads edstan, which implicitly loads rstan. Then the two commands that follow set options related to rstan, which are generally recommended. The first causes compiled Stan models to be saved to disc, which allows models to run more quickly after the first time. The second causes Stan to run multiple MCMC chains in parallel.

# Load packages and set options
library(edstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())

Next we get summary information for the spelling data.

# Summarize the spelling data
str(spelling)

The dataset is a response matrix indicating whether the 658 respondents spelled four words correctly. It also includes a dummy variable for whether the respondent is male. Data is fed into Stan in list form, and the next block of code demonstrates how a suitable data list may be constructed using the irt_data() function. The response matrix (that is, all columns but the first) is provided as the response_matrix option.

# Make a data list
simple_list <- irt_data(response_matrix = spelling[, -1])
str(simple_list)

This data list may be fed to the irt_stan() function to fit a Rasch or 2PL model, but for this example we will add a person covariate to the model using the optional arguments covariates and formula. The covariates option takes a data frame of person-related covariates and the formula option takes a formula to be applied to that data frame. The left side of the formula should be left empty, though implicitly it is latent ability. The choice of what to include in the latent regression (if anything) is made when calling irt_data() to assemble a data list.

In the next block a data list is made that includes male as a covariate. Note that drop = FALSE is specified when subsetting the data frame below because R will simplify the result to a vector otherwise. This complication is default R behavior and not specific to edstan.

# Make a data list with person covariates
latent_reg_list <- irt_data(response_matrix = spelling[, -1],
                            covariates = spelling[, "male", drop = FALSE],
                            formula = ~ 1 + male)
str(latent_reg_list)

The only difference between the two lists are K and W. The first list has W with \(K=1\) columns, which is a constant for the model intercept. The second list has W with \(K=2\) columns, which correspond to the constant for the model intercept and the male indicator variable. In either list, W has one row per person.

Next we fit the model using the irt_stan() function, which is a wrapper for stan() from rstan. latent_reg_list is provided as the data, and the name of one of the .stan files to use is provided as the model argument. We also choose the number of chains and number of iterations per chain. By default the first half of each chain is discarded as warm up.

# Fit the Rasch model
fit_rasch <- irt_stan(latent_reg_list, model = "rasch_latent_reg.stan",
                      iter = 300, chains = 4)

The stan_columns_plot() shows convergence statistics for the fitted model. As an aside, it may also be used to show other statistics such as posterior means or numbers of effective samples.

# View convergence statistics
stan_columns_plot(fit_rasch)

print_irt_stan() provides a summary of parameter posteriors. It is a wrapper for the print() method for fitted Stan models, selecting and organizing the most interesting parameters. Labels for the items are automatically taken from the column names of the response matrix. Regarding the parameters in the output: beta refers to item difficulties, lambda refers to latent regression coefficients, and sigma is the standard deviation of the ability distribution.

# View a summary of parameter posteriors                      
print_irt_stan(fit_rasch, latent_reg_list)

If we prefer to fit the 2PL model with latent regression, we could call irt_stan() and select that .stan file.

# Fit the Rasch model
fit_rasch <- irt_stan(latent_reg_list, model = "2pl_latent_reg.stan",
                      iter = 300, chains = 4)

Verbal aggression data and polytomous IRT

The second example will use the verbal aggression data to fit a polytomous item response theory model.

# Describe the data
str(aggression)

The data consists of 316 persons responding to 24 items with responses scored as 0, 1, or 2. The variable person is a person ID, item is an item ID, and poly is the scored response. In this way, the data may be said to be in “long form” as there is one row per person-item combination. The data includes two person covariates: male is a dummy variable for whether the respondent is male, and anger is the person’s trait anger raw score, a separate measure.

The labelled_integer() function may help prepare the data list when the data are in long form. Stan models generally require that ID variables be consecutive integers starting at one. In other words, the first item must have an ID value of 1, the second 2, and so on. The same is true of person IDs. But sometimes the raw data may have alphanumerics or arbitrary numbers for person or item IDs, and if so we can use labelled_integer() to create an appropriate vector of consecutive integers. The verbal aggression data includes the variable description, which is a brief text description of the item. Below, labelled_integer() is applied to the first few values of this variable.

# Show an example of using labelled_integer()
labelled_integer(aggression$description[1:5])

The result is integer values that are labeled with the original text descriptions. For the verbal aggression data, we could simply use the item variable for an item index rather than apply labelled_integer() to the descriptions, but a benefit of using labelled_integer() is that the table of posterior summaries (shown shortly) will include the item labels.

Next we make the data list using irt_data(). However, we’ll use a different set of arguments because the data are in long form: y takes the responses in vector form, ii takes a vector of item IDs, and jj takes a vector of person IDs. The use of covariates and formula are the same as before.

# Make the data list
agg_list <- irt_data(y = aggression$poly,
                     ii = labelled_integer(aggression$description),
                     jj = aggression$person,
                     covariates = aggression[, c("male", "anger")],
                     formula = ~ 1 + male*anger)
str(agg_list)

The model is fit using the generalized partial credit model.

# Fit the generalized partial credit model
fit_gpcm <- irt_stan(agg_list, model = "gpcm_latent_reg.stan",
                     iter = 300, chains = 4)

As before, a plot of convergence statistics is made.

# View convergence statistics
stan_columns_plot(fit_gpcm)

And lastly we obtain a table of posterior summaries. Here, alpha refers to discriminations, beta refers to step difficulties, and lambda refers to latent regression coefficients.

# View a summary of parameter posteriors    
print_irt_stan(fit_gpcm, agg_list)

Notation

Variables and parameters are similar across edstan models. The variables used are:

• \(i = 1 \ldots I\) indexes items.
• \(j = 1 \ldots J\) indexes persons.
• \(m_i\) is simultaneously the maximum score and the number of step difficulty parameters for item \(i\) for partial credit models. Alternatively, \(m\) is the same across all items for rating scale models.
• \(s = 1 \ldots m_i\) or \(s = 1 \ldots m\) indexes steps within items.
• \(y_{ij}\) is the scored response of person \(j\) to item \(i\). The lowest score for items must be zero (except for rating scale models).
• \(w_{j}\) is the vector of covariates for person \(j\), the first element of which must equal one for a model intercept. \(w_{j}\) may be assembled into a \(J\)-by-\(K\) covariate matrix \(W\), where \(K\) is number of elements in \(w_j\).

The parameters used are:

• For the Rasch and 2PL models, \(\beta_i\) is the difficulty for item \(i\). For the rating scale models, \(\beta_i\) is the mean difficulty for item \(i\). For partial credit models, \(\beta_{is}\) is the difficulty for step \(s\) of item \(i\).
• \(\kappa_s\) is a step difficulty for the (generalized) rating scale model.
• \(\alpha_i\) is the discrimination parameter for item \(i\) (when applicable).
• \(\theta_j\) is the ability for person \(j\).
• \(\lambda\) is the vector of latent regression parameters of length \(K\) (when applicable).
• \(\sigma\) is standard deviation for the ability distribution (when applicable).

The .stan files and the notation for the models below closely adhere to these conventions.

Rasch family models

Models featuring discrimination parameters

Priors

For Rasch family models, the prior distributions for the person-related parameters are \[\theta_j \sim \mathrm{N}(w_j' \lambda, \sigma^2)\] \[\lambda \sim t_3(0, 1)\] \[\sigma \sim \mathrm{Exp}(.1)\]

For models with discrimination parameters, the priors are \[\theta_j \sim \mathrm{N}(w_j' \lambda, 1)\] \[\lambda \sim t_3(0, 1)\]

In either case, the prior for \(\lambda\) is applied to centered and scaled versions of the person covariates. Specifically: (1) continuous covariates are mean-centered and then divided by two times their standard deviations, (2) binary covariates are mean-centered and divided their maximum minus minimum values, and (3) no change is made to the constant, set to one, for the model intercept. \(t_3\) is the Student’s \(t\) distribution with three degrees of freedom. Though the model is estimated using adjusted covariates, the Stan models report \(\lambda\) for the original values of the covariates.

The priors for the item parameters are \[\alpha \sim \mathrm{lognormal}(1, 1)\] \[\beta \sim \mathrm{N}(0, 9)\] \[\kappa \sim \mathrm{N}(0, 9)\] No prior is placed on the last \(\beta_i\), as it is constrained such that the vector \(\beta\) has a mean of zero. The same is true regarding the vector \(\kappa\).