--- title: "Rasch Partial Credit Model with easyRaschBayes" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Rasch Partial Credit Model} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ## Overview This vignette demonstrates a Partial Credit Model (PCM) Rasch analysis workflow using `easyRaschBayes`. The analysis uses the `eRm::pcmdat2` dataset, a small polytomous item response data set included in the `eRm` package. All functions in this package work with a Bayesian `brms` model object fitted with the `acat` (adjacent categories) family, which parameterises the PCM. Dichotomous Rasch models can also be fit using `brms` and analyzed with the functions in this package. A code example is available [here](https://pgmj.github.io/reliability.html#rasch-dichotomous-model), and more detail is available in [Bürkner, 2021](https://doi.org/10.18637/jss.v100.i05). Below, there is some brief texts explaining the output and interpretation of key functions. For a more extensive treatment of various Rasch analysis aspects, please see the [`easyRasch` vignette](https://pgmj.github.io/raschrvignette/RaschRvign.html). Also, each function includes documentation and example code, use `?function` in your console. ## Data Preparation ``` r library(easyRaschBayes) library(brms) library(dplyr) library(tidyr) library(tibble) library(ggplot2) ``` `eRm::pcmdat2` is in wide format with item responses coded 0, 1, 2, …. The `brms` `acat` family expects response categories starting at **1**, so we add 1 to every response before reshaping to long format. ``` r df_pcm <- eRm::pcmdat2 %>% mutate(across(everything(), ~ .x + 1)) %>% rownames_to_column("id") %>% pivot_longer(!id, # don't include the id variable in the wide->long transformation names_to = "item", values_to = "response") head(df_pcm) #> # A tibble: 6 × 3 #> id item response #> #> 1 1 I1 2 #> 2 1 I2 2 #> 3 1 I3 2 #> 4 1 I4 2 #> 5 2 I1 1 #> 6 2 I2 1 ``` ## Visualizing response data Before analyzing data, it is good to review the response distributions. There are three simple plotting functions included in the package. All make use of wide format datasets, where items are separate columns and respondents are rows. ``` r plot_stackedbars(eRm::pcmdat2) ```

Stacked bars

The other two visualization functions are `plot_tile()` and `plot_bars()` ## Fitting the PCM The model is fitted once and saved to disk. The code chunk below shows the fitting call (not evaluated during `R CMD check`). A pre-fitted model stored at `fits/fit_pcm.rds` is loaded instead. ``` r prior_pcm <- prior("normal(0, 3)", class = "Intercept") + prior("normal(0, 3)", class = "sd", group = "id") ``` ``` r fit_pcm <- brm( response | thres(gr = item) ~ 1 + (1 | id), data = df_pcm, family = acat, prior = prior_pcm, chains = 4, cores = 4, iter = 2000 ) saveRDS(fit_pcm, "fits/fit_pcm.rds") ``` ``` r fit_pcm <- readRDS("fits/fit_pcm.rds") ``` ## Item Fit: Conditional Infit Statistics `infit_statistic()` computes posterior predictive infit (and outfit) statistics for each item. Values near 1.0 indicate good fit; values above 1 suggest underfit (unexpected responses, often indicating multidimensionality), values below 1 suggest overfit (too predictable, often coinciding with local dependence issues). The `ndraws_use` argument limits the number of posterior draws used, which speeds up computation during exploration. For final reporting, use all draws (set `ndraws_use = NULL` or omit it). ``` r fit_stats <- infit_statistic(fit_pcm, ndraws_use = 500) # Post-process Infit infit_results <- infit_post(fit_stats) infit_results$summary #> # A tibble: 4 × 4 #> item infit_obs infit_rep infit_ppp #> #> 1 I1 1.04 1.00 0.292 #> 2 I2 1.08 1 0.11 #> 3 I3 0.917 1.00 0.89 #> 4 I4 1.03 0.999 0.364 infit_results$plot ```

Conditional infit figure

`infit_obs` indicates the observed conditional infit, which can be compared to `infit_rep`, which is akin to the model expected value. Posterior predictive p-values (`*_ppp`) close to 0.5 indicate that the observed statistic falls near the centre of the posterior predictive distribution, implying good fit. Values near 0 or 1 warrant further investigation. ## Item–Rest Score Association `item_restscore_statistic()` computes Goodman-Kruskal's gamma between each item's observed responses and the rest score (total score minus the focal item). In a well-fitting Rasch model, gamma should be positive and moderate and of similar magnitude for all items; high values may indicate redundancy, low values suggest the item does not relate well to the latent trait. ``` r rest_stats <- item_restscore_statistic(fit_pcm, ndraws_use = 500) rest_results <- item_restscore_post(rest_stats) rest_results$summary #> # A tibble: 4 × 5 #> item gamma_obs gamma_rep gamma_diff ppp #> #> 1 I1 0.473 0.541 -0.068 0.118 #> 2 I2 0.441 0.543 -0.102 0.056 #> 3 I3 0.643 0.53 0.112 0.962 #> 4 I4 0.535 0.537 -0.002 0.492 rest_results$plot ```

Item-restscore figure

## Conditional ICC Plot Shows item fit across the latent continuum, dividing the sample into n class intervals (default is 5). ``` r plot_icc(fit_pcm) ```

Conditional Item Characteristic Curves figure

## Dimensionality: Residual PCA `plot_residual_pca()` performs a principal components analysis on the person-item residuals and plots the standardized loadings on the first residual contrast factor together with item locations and the uncertainty of both. ``` r pca <- plot_residual_pca(fit_pcm, ndraws_use = 500) pca$plot ```

1st residual contrast loadings & locations

If the observed largest eigenvalue is smaller than the replicated, unidimensionality is supported. The ppp should not be close to 1. ## Local Dependence: Q3 Residual Correlations `q3_statistic()` computes Yen's Q3 statistic — the correlation between person-item residuals for every item pair. After conditioning on the latent trait, residuals should be uncorrelated; elevated Q3 values indicate local dependence (LD). Our primary metric here is the ppp, that should not be close to 1. ``` r q3_stats <- q3_statistic(fit_pcm, ndraws_use = 500) q3_results <- q3_post(q3_stats) q3_results$summary #> # A tibble: 6 × 7 #> item_pair item_1 item_2 q3_obs q3_rep q3_diff q3_ppp #> #> 1 I3 : I4 I3 I4 0.34 0.001 0.339 1 #> 2 I1 : I2 I1 I2 0.102 0.002 0.1 0.994 #> 3 I1 : I3 I1 I3 -0.072 -0.004 -0.068 0.016 #> 4 I2 : I3 I2 I3 -0.088 -0.003 -0.085 0 #> 5 I1 : I4 I1 I4 -0.132 0 -0.132 0 #> 6 I2 : I4 I2 I4 -0.163 -0.002 -0.161 0 q3_results$plot ```

Q3 residual correlations

## Item Category Probabilities This plot shows the probability of using a response category on the y axis and the latent score on the x axis. The crossover points, where lines meet, are the item category threshold locations. Uncertainty is shown with the shaded area around each line. ``` r plot_ipf(fit_pcm, theta_range = c(-6,5)) ```

Item Probability Functions

## Person–Item Targeting `plot_targeting()` produces a Wright map (person–item targeting plot) showing the distribution of person locations alongside the item threshold locations on the same logit scale. Good targeting occurs when person and item distributions overlap substantially. ``` r plot_targeting(fit_pcm) ```

Targeting figure

## Reliability: Relative Measurement Uncertainty `RMUreliability()` provides a Bayesian reliability estimate via Relative Measurement Uncertainty (RMU, see Bignardi et al., 2025). It requires a matrix of person location draws with dimensions \[persons × draws\]. The output is a point estimate and lower/upper 95% highest density continuous intervals (HDCI). ``` r person_draws <- fit_pcm %>% as_draws_df() %>% as_tibble() %>% select(starts_with("r_id")) %>% t() rmu <- RMUreliability(person_draws) rmu #> rmu_estimate hdci_lowerbound hdci_upperbound .width .point .interval #> 1 0.6727799 0.6137229 0.7280385 0.95 mean hdci ``` RMU values range from 0 to 1, with higher values indicating higher reliability, similarly to traditional reliability metrics such as Cronbach's alpha. ## Item Parameters ``` r ipar <- item_parameters(fit_pcm) knitr::kable(ipar$summary) ``` |item | threshold| location| se| hdci_lower| hdci_upper| n_eff| |:----|---------:|--------:|------:|----------:|----------:|-----:| |I1 | 1| -0.4792| 0.1819| -0.8505| -0.1457| 3734| |I1 | 2| 1.8135| 0.1840| 1.4532| 2.1753| 3503| |I2 | 1| 0.2591| 0.1751| -0.0743| 0.6055| 3423| |I2 | 2| 1.6465| 0.1908| 1.3032| 2.0377| 3350| |I3 | 1| -2.5798| 0.3347| -3.2377| -1.9333| 4000| |I3 | 2| 0.2413| 0.1569| -0.0609| 0.5475| 3255| |I4 | 1| -1.4880| 0.2529| -1.9889| -1.0082| 4000| |I4 | 2| 0.5866| 0.1608| 0.2573| 0.8902| 4000| ``` r knitr::kable(ipar$locations_wide) ``` |item | t1| t2| location| |:----|-------:|------:|--------:| |I3 | -2.5798| 0.2413| -1.1693| |I4 | -1.4880| 0.5866| -0.4507| |I1 | -0.4792| 1.8135| 0.6672| |I2 | 0.2591| 1.6465| 0.9528| ## Person Parameters This estimates latent scores. ``` r ppar <- person_parameters(fit_pcm) knitr::kable(ppar$score_table) ``` | sum_score| n| eap| eap_se| wle| wle_se| |---------:|--:|-------:|------:|-------:|------:| | 0| 7| -1.9948| 0.8723| -7.0000| NaN| | 1| 4| -1.3413| 0.7962| -3.0165| 1.3597| | 2| 23| -0.7713| 0.7451| -1.5860| 0.9883| | 3| 35| -0.2446| 0.7225| -0.6911| 0.8665| | 4| 80| 0.2582| 0.7107| 0.0616| 0.8151| | 5| 35| 0.7609| 0.7140| 0.7901| 0.8208| | 6| 51| 1.2865| 0.7428| 1.6205| 0.9128| | 7| 28| 1.8658| 0.8013| 2.9849| 1.3554| | 8| 37| 2.5625| 0.8963| 7.0000| NaN| ``` r hist(ppar$person_estimates$eap, col = "lightblue", main = "Histogram of EAP scores") ```

EAP score histogram

## References - Bürkner, P.-C. (2021). Bayesian Item Response Modeling in R with brms and Stan. *Journal of Statistical Software*, *100*, 1–54. - Bignardi, G., Kievit, R., & Bürkner, P.-C. (2025). A general method for estimating reliability using Bayesian Measurement Uncertainty. OSF. - Levy, R., Mislevy, R. J., & Sinharay, S. (2009). Posterior Predictive Model Checking for Multidimensionality in Item Response Theory. Applied Psychological Measurement, 33(7), 519–537. - Sinharay, S. (2006). Bayesian item fit analysis for unidimensional item response theory models. British Journal of Mathematical and Statistical Psychology, 59(2), 429–449. - Müller, M. (2020). Item fit statistics for Rasch analysis: Can we trust them? Journal of Statistical Distributions and Applications, 7(1), 5. - Kreiner, S. (2011). A Note on Item–Restscore Association in Rasch Models. Applied Psychological Measurement, 35(7), 557–561.