---
title: "Introduction to irtQ"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Introduction to irtQ}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 6.5,
  fig.height = 4
)
```

## Overview

The **irtQ** package fits unidimensional item response theory (IRT) models to
test data that may contain both dichotomous and polytomous items. The
dichotomous models are the one-, two-, and three-parameter logistic models
(1PLM, 2PLM, and 3PLM), and the polytomous models are the graded response
model (GRM) and the generalized partial credit model (GPCM). A single test can
mix these models.

The package covers the steps of a typical IRT analysis: estimating item
parameters by marginal maximum likelihood with the EM algorithm
(`est_irt()`, `est_mg()`), calibrating pretest items with fixed item
parameters or fixed abilities (`est_irt()`, `est_item()`), estimating
examinees' abilities (`est_score()`), evaluating model-data fit (`irtfit()`,
`sx2_fit()`), and detecting differential item functioning (`rdif()`,
`crdif()`, `grdif()`, `catsib()`). It also provides functions for item and
test information (`info()`), item response curves (`traceline()`),
classification accuracy and consistency (`cac_lee()`, `cac_rud()`), multistage
test panels (`panel_info()`, `reval_mst()`, `run_mst()`), and classical test
theory (`ctt()`).

This vignette walks through a short workflow: read item parameters, simulate
responses, calibrate the items, estimate abilities, check fit, and compute test
information. Every example uses the scaling constant `D = 1`.

## Item metadata

Most irtQ functions take a data frame of item metadata as their first
argument, or a fitted object that contains one. Each row describes one item.

| Column | Meaning |
|---|---|
| `id` | Item identifier |
| `cats` | Number of score categories (2 for a dichotomous item) |
| `model` | IRT model of the item: `"1PLM"`, `"2PLM"`, `"3PLM"`, `"GRM"`, or `"GPCM"` |
| `par.1` | Slope (discrimination) parameter |
| `par.2` | Difficulty for a dichotomous item, or the first threshold for a polytomous item |
| `par.3` | Guessing parameter for a 3PLM item, or the second threshold for a polytomous item; `NA` for 1PLM and 2PLM items |
| `par.4`, ... | Further thresholds of a polytomous item; `NA` for dichotomous items |

Item metadata can be read from the output of other IRT software with the
`bring.*()` functions (`bring.flexmirt()`, `bring.bilog()`,
`bring.parscale()`, and `bring.mirt()`, which requires the suggested **mirt**
package) or created with `shape_df()`. The example below reads the flexMIRT
parameter file that ships with the package and keeps the first 25 items, all of
which follow the 3PLM.

```{r metadata}
library(irtQ)

flex_file <- system.file("extdata", "flexmirt_sample-prm.txt", package = "irtQ")
x <- bring.flexmirt(file = flex_file, type = "par")$Group1$full_df
x <- x[1:25, ]
head(x)
```

A small metadata frame can also be written directly with `shape_df()`.

```{r shape-df}
shape_df(
  par.drm = list(a = c(1.0, 1.2), b = c(-0.5, 0.5), g = c(0.2, 0.2)),
  cats = 2, model = "3PLM"
)
```

## Simulating item responses

`simdat()` generates item responses from item metadata and a vector of
abilities. Here 500 examinees are drawn from a standard normal distribution.

```{r simulate}
set.seed(2026)
theta <- rnorm(500, mean = 0, sd = 1)
resp <- simdat(x = x, theta = theta, D = 1)
dim(resp)
```

## Estimating item parameters

`est_irt()` calibrates the items. The model and the number of score categories
are given for all items at once. A Beta(5, 16) prior on the guessing parameters,
which is the default, stabilizes the 3PLM estimates; it is written out here for
clarity.

```{r calibrate}
fit <- est_irt(
  data = resp, D = 1, model = "3PLM", cats = 2, item.id = x$id,
  use.gprior = TRUE, gprior = list(dist = "beta", params = c(5, 16)),
  verbose = FALSE
)
fit
```

Printing the fitted object reports the number of EM cycles, the number of
quadrature points, the first- and second-order convergence checks, and the
log-likelihood. A fuller estimation summary is available from `summary(fit)`.

`getirt()` extracts parts of the fitted object, such as the item parameter
estimates.

```{r par-est}
par_est <- getirt(fit, what = "par.est")
head(par_est)
```

## Estimating abilities

`est_score()` estimates the abilities of the examinees. A fitted `est_irt`
object can be passed directly, and the response data stored in the object are
used. The code below uses maximum likelihood estimation (ML) and compares the
estimates with the abilities used to simulate the data. The `range` argument
keeps the ML estimates within [-4, 4], which matters for examinees who answer
all items correctly or all incorrectly.

```{r score}
scores <- est_score(fit, method = "ML", range = c(-4, 4))
head(scores$est.theta)
cor(scores$est.theta, theta)
```

## Model-data fit

`sx2_fit()` computes the S-X2 item fit statistic, and `irtfit()` computes
chi-square and likelihood-ratio statistics, infit and outfit, and the
contingency tables behind the residual plots.

```{r sx2}
sx2 <- sx2_fit(fit)
head(sx2$fit_stat)
```

`irtfit()` groups the examinees by the ML estimates from the previous section,
and the plot shows the raw and standardized residuals of the first item.

```{r irtfit-plot}
fit_irt <- irtfit(
  x = fit, score = scores$est.theta, group.method = "equal.width",
  n.width = 10, loc.theta = "middle"
)
plot(fit_irt, item.loc = 1, type = "both", ci.method = "wald",
     show.table = FALSE, ylim.sr.adjust = TRUE)
```

## Test information

`info()` computes item and test information functions on a grid of ability
values, and its `plot()` method draws them. Without `item.loc`, the plot shows
the test information function.

```{r info}
theta_grid <- seq(-4, 4, 0.1)
tif <- info(x = fit, theta = theta_grid, tif = TRUE)
plot(tif)
```

## Where to go next

The package website at <https://hwangQ.github.io/irtQ/> has articles that
cover each topic in more detail:

- [Getting Started: A Detailed Guide](https://hwangQ.github.io/irtQ/articles/getting-started.html)
- [Item Parameter Estimation](https://hwangQ.github.io/irtQ/articles/item-parameter-estimation.html)
- [Ability Estimation](https://hwangQ.github.io/irtQ/articles/ability-estimation.html)
- [Model-Data Fit Evaluation](https://hwangQ.github.io/irtQ/articles/model-fit-evaluation.html)
- [DIF Detection](https://hwangQ.github.io/irtQ/articles/dif-detection.html)
- [Classification Accuracy and Consistency](https://hwangQ.github.io/irtQ/articles/classification-analysis.html)
- [Utility Functions](https://hwangQ.github.io/irtQ/articles/utilities.html)
- [Classical Test Theory (CTT) Analysis](https://hwangQ.github.io/irtQ/articles/ctt-analysis.html)
- [MST Panel Evaluation and Simulation](https://hwangQ.github.io/irtQ/articles/mst-panel-evaluation.html)