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Rasch Partial Credit Model with easyRaschBayes
In easyRaschBayes: Bayesian Rasch Analysis Using 'brms'
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, and more
detail is available in Bürkner, 2021.
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.
Also, each function includes documentation and example code, use ?function in your
console.
Data Preparation
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.
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
#> <chr> <chr> <dbl>
#> 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.
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.
prior_pcm <- prior("normal(0, 3)", class = "Intercept") +
prior("normal(0, 3)", class = "sd", group = "id")
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")
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).
fit_stats <- infit_statistic(fit_pcm, ndraws_use = 500)
# Post-process Infit
infit_results <- infit_post(fit_stats)
infit_results$summary
#> # A tibble: 4 × 4
#> item infit_obs infit_rep infit_ppp
#> <chr> <dbl> <dbl> <dbl>
#> 1 I1 1.04 1.00 0.292
#> 2 I2 1.08 1 0.11
#> 3 I3 0.917 1.00 0.89
#> 4 I4 1.03 0.999 0.364
infit_results$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.
rest_stats <- item_restscore_statistic(fit_pcm, ndraws_use = 500)
rest_results <- item_restscore_post(rest_stats)
rest_results$summary
#> # A tibble: 4 × 5
#> item gamma_obs gamma_rep gamma_diff ppp
#> <chr> <dbl> <dbl> <dbl> <dbl>
#> 1 I1 0.473 0.541 -0.068 0.118
#> 2 I2 0.441 0.543 -0.102 0.056
#> 3 I3 0.643 0.53 0.112 0.962
#> 4 I4 0.535 0.537 -0.002 0.492
rest_results$plot
Item-restscore figure
Conditional ICC Plot
Shows item fit across the latent continuum, dividing the sample into n class
intervals (default is 5).
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.
pca <- plot_residual_pca(fit_pcm, ndraws_use = 500)
pca$plot
1st residual contrast loadings & locations
If the observed largest eigenvalue is smaller than the replicated, unidimensionality
is supported. The ppp should not be close to 1.
Local Dependence: Q3 Residual Correlations
q3_statistic() computes Yen's Q3 statistic — the correlation between
person-item residuals for every item pair. After conditioning on the latent
trait, residuals should be uncorrelated; elevated Q3 values indicate local
dependence (LD). Our primary metric here is the ppp, that should not be close to 1.
q3_stats <- q3_statistic(fit_pcm, ndraws_use = 500)
q3_results <- q3_post(q3_stats)
q3_results$summary
#> # A tibble: 6 × 7
#> item_pair item_1 item_2 q3_obs q3_rep q3_diff q3_ppp
#> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl>
#> 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$plot
Q3 residual correlations
Item Category Probabilities
This plot shows the probability of using a response category on the y axis and
the latent score on the x axis. The crossover points, where lines meet, are the
item category threshold locations. Uncertainty is shown with the shaded area around
each line.
plot_ipf(fit_pcm, theta_range = c(-6,5))
Item Probability Functions
Person–Item Targeting
plot_targeting() produces a Wright map (person–item targeting plot) showing
the distribution of person locations alongside the item threshold locations on
the same logit scale. Good targeting occurs when person and item distributions
overlap substantially.
plot_targeting(fit_pcm)
Targeting figure
Reliability: Relative Measurement Uncertainty
RMUreliability() provides a Bayesian reliability estimate via Relative
Measurement Uncertainty (RMU, see Bignardi et al., 2025). It requires a matrix
of person location draws with dimensions [persons × draws]. The output is a
point estimate and lower/upper 95% highest density continuous intervals (HDCI).
person_draws <- fit_pcm %>%
as_draws_df() %>%
as_tibble() %>%
select(starts_with("r_id")) %>%
t()
rmu <- RMUreliability(person_draws)
rmu
#> rmu_estimate hdci_lowerbound hdci_upperbound .width .point .interval
#> 1 0.6727799 0.6137229 0.7280385 0.95 mean hdci
RMU values range from 0 to 1, with higher values indicating higher reliability,
similarly to traditional reliability metrics such as Cronbach's alpha.
Item Parameters
ipar <- item_parameters(fit_pcm)
knitr::kable(ipar$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|
knitr::kable(ipar$locations_wide)
|item | t1| t2| location|
|:----|-------:|------:|--------:|
|I3 | -2.5798| 0.2413| -1.1693|
|I4 | -1.4880| 0.5866| -0.4507|
|I1 | -0.4792| 1.8135| 0.6672|
|I2 | 0.2591| 1.6465| 0.9528|
Person Parameters
This estimates latent scores.
ppar <- person_parameters(fit_pcm)
knitr::kable(ppar$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|
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.
https://doi.org/10.18637/jss.v100.i05
-
Bignardi, G., Kievit, R., & Bürkner, P.-C. (2025). A general method for
estimating reliability using Bayesian Measurement Uncertainty. OSF.
https://doi.org/10.31234/osf.io/h54k8_v1
-
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. https://doi.org/10.1177/0146621608329504
-
Sinharay, S. (2006). Bayesian item fit analysis for unidimensional item
response theory models. British Journal of Mathematical and Statistical
Psychology, 59(2), 429–449. https://doi.org/10.1348/000711005X66888
-
Müller, M. (2020). Item fit statistics for Rasch analysis: Can we trust them?
Journal of Statistical Distributions and Applications, 7(1), 5.
https://doi.org/10.1186/s40488-020-00108-7
-
Kreiner, S. (2011). A Note on Item–Restscore Association in Rasch Models.
Applied Psychological Measurement, 35(7), 557–561.
https://doi.org/10.1177/0146621611410227
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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, and more
detail is available in Bürkner, 2021.
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.
Also, each function includes documentation and example code, use ?function in your
console.
Data Preparation
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.
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 #> <chr> <chr> <dbl> #> 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.
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.
prior_pcm <- prior("normal(0, 3)", class = "Intercept") + prior("normal(0, 3)", class = "sd", group = "id")
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")
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).
fit_stats <- infit_statistic(fit_pcm, ndraws_use = 500) # Post-process Infit infit_results <- infit_post(fit_stats) infit_results$summary #> # A tibble: 4 × 4 #> item infit_obs infit_rep infit_ppp #> <chr> <dbl> <dbl> <dbl> #> 1 I1 1.04 1.00 0.292 #> 2 I2 1.08 1 0.11 #> 3 I3 0.917 1.00 0.89 #> 4 I4 1.03 0.999 0.364 infit_results$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.
rest_stats <- item_restscore_statistic(fit_pcm, ndraws_use = 500) rest_results <- item_restscore_post(rest_stats) rest_results$summary #> # A tibble: 4 × 5 #> item gamma_obs gamma_rep gamma_diff ppp #> <chr> <dbl> <dbl> <dbl> <dbl> #> 1 I1 0.473 0.541 -0.068 0.118 #> 2 I2 0.441 0.543 -0.102 0.056 #> 3 I3 0.643 0.53 0.112 0.962 #> 4 I4 0.535 0.537 -0.002 0.492 rest_results$plot
Item-restscore figure
Conditional ICC Plot
Shows item fit across the latent continuum, dividing the sample into n class intervals (default is 5).
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.
pca <- plot_residual_pca(fit_pcm, ndraws_use = 500) pca$plot
1st residual contrast loadings & locations
If the observed largest eigenvalue is smaller than the replicated, unidimensionality is supported. The ppp should not be close to 1.
Local Dependence: Q3 Residual Correlations
q3_statistic() computes Yen's Q3 statistic — the correlation between
person-item residuals for every item pair. After conditioning on the latent
trait, residuals should be uncorrelated; elevated Q3 values indicate local
dependence (LD). Our primary metric here is the ppp, that should not be close to 1.
q3_stats <- q3_statistic(fit_pcm, ndraws_use = 500) q3_results <- q3_post(q3_stats) q3_results$summary #> # A tibble: 6 × 7 #> item_pair item_1 item_2 q3_obs q3_rep q3_diff q3_ppp #> <chr> <chr> <chr> <dbl> <dbl> <dbl> <dbl> #> 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$plot
Q3 residual correlations
Item Category Probabilities
This plot shows the probability of using a response category on the y axis and the latent score on the x axis. The crossover points, where lines meet, are the item category threshold locations. Uncertainty is shown with the shaded area around each line.
plot_ipf(fit_pcm, theta_range = c(-6,5))
Item Probability Functions
Person–Item Targeting
plot_targeting() produces a Wright map (person–item targeting plot) showing
the distribution of person locations alongside the item threshold locations on
the same logit scale. Good targeting occurs when person and item distributions
overlap substantially.
plot_targeting(fit_pcm)
Targeting figure
Reliability: Relative Measurement Uncertainty
RMUreliability() provides a Bayesian reliability estimate via Relative
Measurement Uncertainty (RMU, see Bignardi et al., 2025). It requires a matrix
of person location draws with dimensions [persons × draws]. The output is a
point estimate and lower/upper 95% highest density continuous intervals (HDCI).
person_draws <- fit_pcm %>% as_draws_df() %>% as_tibble() %>% select(starts_with("r_id")) %>% t() rmu <- RMUreliability(person_draws) rmu #> rmu_estimate hdci_lowerbound hdci_upperbound .width .point .interval #> 1 0.6727799 0.6137229 0.7280385 0.95 mean hdci
RMU values range from 0 to 1, with higher values indicating higher reliability, similarly to traditional reliability metrics such as Cronbach's alpha.
Item Parameters
ipar <- item_parameters(fit_pcm) knitr::kable(ipar$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|
knitr::kable(ipar$locations_wide)
|item | t1| t2| location| |:----|-------:|------:|--------:| |I3 | -2.5798| 0.2413| -1.1693| |I4 | -1.4880| 0.5866| -0.4507| |I1 | -0.4792| 1.8135| 0.6672| |I2 | 0.2591| 1.6465| 0.9528|
Person Parameters
This estimates latent scores.
ppar <- person_parameters(fit_pcm) knitr::kable(ppar$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|
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. https://doi.org/10.18637/jss.v100.i05
-
Bignardi, G., Kievit, R., & Bürkner, P.-C. (2025). A general method for estimating reliability using Bayesian Measurement Uncertainty. OSF. https://doi.org/10.31234/osf.io/h54k8_v1
-
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. https://doi.org/10.1177/0146621608329504
-
Sinharay, S. (2006). Bayesian item fit analysis for unidimensional item response theory models. British Journal of Mathematical and Statistical Psychology, 59(2), 429–449. https://doi.org/10.1348/000711005X66888
-
Müller, M. (2020). Item fit statistics for Rasch analysis: Can we trust them? Journal of Statistical Distributions and Applications, 7(1), 5. https://doi.org/10.1186/s40488-020-00108-7
-
Kreiner, S. (2011). A Note on Item–Restscore Association in Rasch Models. Applied Psychological Measurement, 35(7), 557–561. https://doi.org/10.1177/0146621611410227
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