Rasch analysis with Bayesian and frequentist methods

A comparison using NHANES data for PHQ-9

Author
Affiliation

Magnus Johansson, PhD

Karolinska Institutet

Published

2026-03-31

Modified

2026-06-22

library(easyRasch2)
library(iarm)
library(eRm)
library(mirt)
library(tidyverse)
library(brms)
library(loo)

options(mc.cores = 8)
set.seed(92756246)
library(priorsense)
library(tidybayes)
library(easyRaschBayes) # current CRAN version 0.2.0.1
library(haven)
library(knitr)

select <- dplyr::select
count <- dplyr::count
rename <- dplyr::rename

1 Introduction

This is an example Rasch analysis using frequentist methods and their Bayesian counterparts, as implemented in the easyRaschBayes R package. The frequentist methods are used through the easyRasch2 package, which is mostly a wrapper package except for some bootstrap/simulation functions (see also Johansson 2025). As such, the sections above will highlight which R package contains the frequentist function. For a more extensive discussion on the metrics and their interpretation, see the old easyRasch vignette.

2 Importing data

Data files manually downloaded from September 2024 NHANES data website. As the dataset is quite large with 6337 observations, we will select a subsample of n = 750 with complete responses to all 9 items of the PHQ-9 and a reasonable amount (10%) of respondents with all zeroes.

d_all <- read_xpt("data/DPQ_L_2024_nhanes.xpt")

d_sub <- d_all[2:10] %>% 
  mutate(across(everything(), ~ car::recode(.x, "4:9=NA"))) %>% #only responses 0-3 are valid
  na.omit() %>% 
  set_names(paste0("q",1:9)) %>% 
  mutate(sumscore = rowSums(.))

d_sub %>% 
  summarise(mean(sumscore == 0)) # 24.3%

d_zero <- d_sub %>% 
  filter(sumscore == 0) %>% 
  slice_sample(n = 75)

d <- d_sub %>% 
  filter(sumscore > 0) %>% 
  slice_sample(n = 675) %>% 
  bind_rows(d_zero) %>% 
  select(!sumscore)

#saveRDS(d, "data/phq9_nhanes2024_subsample.rds")
d <- readRDS("data/phq9_nhanes2024_subsample.rds")
head(d)
# A tibble: 6 × 9
     q1    q2    q3    q4    q5    q6    q7    q8    q9
  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1     1     0     0     1     2     1     0     0     0
2     1     2     3     1     0     1     1     0     1
3     0     1     0     1     0     3     0     0     0
4     0     0     1     1     1     0     0     0     0
5     1     2     2     1     3     3     1     0     0
6     0     0     2     1     0     0     0     0     0

All PHQ-9 questions/items are introduced with the phrase

Over the last 2 weeks, how often have you been bothered by the following problems

All items have the same ordinal response categories:

  • “Not at all”
  • “Several days”
  • “More than half the days”
  • “Nearly every day”

We’ll set up vectors for item and response category labels for later use in plots, etc.

item_desc <- c(
  "Little interest or pleasure in doing things",
  "Feeling down, depressed, or hopeless",
  "Trouble falling or staying asleep, or sleeping too much",
  "Feeling tired or having little energy", "Poor appetite or overeating",
  "Feeling bad about yourself - or that you are a failure or have let yourself or your family down",
  "Trouble concentrating on things, such as reading the newspaper or watching television",
  "Moving or speaking so slowly that other people could have noticed?",
  "Thoughts that you would be better off dead or of hurting yourself in some way"
)

item_resp <- c("Not at all","Several days","More than \nhalf the days","Nearly every day")

3 Response distributions

Using functions from easyRaschBayes with the wide format dataset.

plot_bars(d, text_y = 10, category_labels = item_resp)

plot_stackedbars(d, text_size = 4.5, category_labels = item_resp) + 
  theme_minimal(base_size = 14)

plot_tile(d, category_labels = item_resp, 
          item_labels = str_wrap(item_desc,24)) + 
  theme_minimal(base_size = 13)

Low frequency use of upper categories of item 9, which is expected in this normal population sample, even if we manipulated the sampling a bit to minimize the floor effects.

4 Fit brms model

For brms, we need data in long format and with the lowest category coded as 1.

dl <- d %>% 
  mutate(across(everything(), ~ .x + 1)) %>% 
  rownames_to_column("id") %>% 
  pivot_longer(!id, 
               names_to = "item", 
               values_to = "response")

Now we can fit a Rasch partial credit model (PCM) with a weakly informative prior.

prior_pcm <- prior(normal(0, 3), class = "Intercept") +
  prior(normal(0, 3), class = "sd", group = "id")

fit <- brm( 
  response | thres(gr = item) ~ 1 + (1 | id),
  data   = dl,
  family = acat("logit"),
  prior = prior_pcm,
  chains = 4,
  cores  = 4, 
  iter   = 4000,
  warmup = 1500,
  save_pars = save_pars(all = TRUE)
)

saveRDS(fit,"data/phq9fit.rds")

The model took about 175s to fit on my Macbook Pro with M4 Max chip.

fit <- readRDS("data/phq9fit.rds")

5 brms model fit

First, we’ll run some basic diagnostics for the model.

summary(fit) # review Rhat and ESS metrics
 Family: acat 
  Links: mu = logit 
Formula: response | thres(gr = item) ~ 1 + (1 | id) 
   Data: dl (Number of observations: 6750) 
  Draws: 4 chains, each with iter = 4000; warmup = 1500; thin = 1;
         total post-warmup draws = 10000

Multilevel Hyperparameters:
~id (Number of levels: 750) 
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept)     1.22      0.05     1.12     1.32 1.00     2065     3621

Regression Coefficients:
                Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept[q1,1]     0.82      0.10     0.62     1.02 1.00     4323     6517
Intercept[q1,2]     1.94      0.16     1.63     2.26 1.00     5810     7128
Intercept[q1,3]     1.77      0.22     1.35     2.19 1.00     7731     7474
Intercept[q2,1]     0.78      0.10     0.57     0.98 1.00     4524     6036
Intercept[q2,2]     2.06      0.16     1.74     2.39 1.00     6146     7100
Intercept[q2,3]     2.05      0.23     1.61     2.51 1.00     7793     7516
Intercept[q3,1]    -0.09      0.11    -0.30     0.11 1.00     4710     6190
Intercept[q3,2]     1.51      0.14     1.23     1.78 1.00     5170     7020
Intercept[q3,3]     0.99      0.17     0.66     1.32 1.00     6108     7198
Intercept[q4,1]    -0.80      0.11    -1.00    -0.59 1.00     5021     6854
Intercept[q4,2]     1.61      0.13     1.36     1.88 1.00     5418     5959
Intercept[q4,3]     1.33      0.17     1.00     1.66 1.00     5835     6915
Intercept[q5,1]     0.87      0.11     0.66     1.07 1.00     4818     6244
Intercept[q5,2]     1.81      0.16     1.50     2.12 1.00     5973     7352
Intercept[q5,3]     1.63      0.21     1.23     2.04 1.00     7603     7620
Intercept[q6,1]     1.10      0.11     0.89     1.31 1.00     4923     6316
Intercept[q6,2]     2.10      0.18     1.75     2.46 1.00     6420     7535
Intercept[q6,3]     1.52      0.23     1.07     1.96 1.00     7760     7072
Intercept[q7,1]     1.19      0.11     0.98     1.40 1.00     4709     5983
Intercept[q7,2]     1.97      0.17     1.64     2.31 1.00     6764     6924
Intercept[q7,3]     2.25      0.24     1.78     2.74 1.00     7957     6948
Intercept[q8,1]     2.47      0.14     2.21     2.74 1.00     5012     6183
Intercept[q8,2]     2.60      0.27     2.09     3.13 1.00     7457     7318
Intercept[q8,3]     1.94      0.33     1.30     2.61 1.00     7689     7316
Intercept[q9,1]     3.68      0.20     3.31     4.09 1.00     7203     7379
Intercept[q9,2]     2.59      0.36     1.91     3.32 1.00     7234     7018
Intercept[q9,3]     2.88      0.54     1.88     3.97 1.00     8877     7277

Further Distributional Parameters:
     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
disc     1.00      0.00     1.00     1.00   NA       NA       NA

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
fit <- add_criterion(fit, criterion = "loo")
powerscale_sensitivity(fit, get_variables(fit)[1:28])
Sensitivity based on cjs_dist
Prior selection: all priors
Likelihood selection: all data

          variable prior likelihood diagnosis
 b_Intercept[q1,1] 0.014      0.103         -
 b_Intercept[q1,2] 0.016      0.332         -
 b_Intercept[q1,3] 0.018      0.413         -
 b_Intercept[q2,1] 0.014      0.078         -
 b_Intercept[q2,2] 0.017      0.360         -
 b_Intercept[q2,3] 0.019      0.274         -
 b_Intercept[q3,1] 0.009      0.164         -
 b_Intercept[q3,2] 0.016      0.225         -
 b_Intercept[q3,3] 0.019      0.446         -
 b_Intercept[q4,1] 0.007      0.311         -
 b_Intercept[q4,2] 0.016      0.216         -
 b_Intercept[q4,3] 0.018      0.427         -
 b_Intercept[q5,1] 0.013      0.097         -
 b_Intercept[q5,2] 0.016      0.362         -
 b_Intercept[q5,3] 0.019      0.378         -
 b_Intercept[q6,1] 0.015      0.078         -
 b_Intercept[q6,2] 0.018      0.301         -
 b_Intercept[q6,3] 0.016      0.369         -
 b_Intercept[q7,1] 0.014      0.145         -
 b_Intercept[q7,2] 0.017      0.333         -
 b_Intercept[q7,3] 0.020      0.381         -
 b_Intercept[q8,1] 0.016      0.255         -
 b_Intercept[q8,2] 0.017      0.313         -
 b_Intercept[q8,3] 0.017      0.290         -
 b_Intercept[q9,1] 0.018      0.201         -
 b_Intercept[q9,2] 0.012      0.245         -
 b_Intercept[q9,3] 0.024      0.279         -
  sd_id__Intercept 0.023      1.053         -

There are additional posterior predictive checks that could be run as well, using pp_check(). A recent preprint presents some new types of pp-checks that seem highly relevant but AFAIK has not been implemented in the bayesplot package yet. The same paper also critiques the usefulness of pp_check(type = "bars_grouped") (Säilynoja et al. 2025).

6 Rasch analysis

For this example, I have opted to use 2000 draws available from the brmsfit object. It takes quite a bit of time to extract and compute a large number of draws for the various functions. I would suggest to use 1000 draws for a casual first analysis, and probably 4000+ for a publication ready analysis. This is just based on small scale experimenting, so feel free to run a simulation study if you want hard evidence, or rely on Kruschke’s recommendation to have an ESS (effective sample size) of at least 10 000 draws (Kruschke 2018) to be on the safe side.

6.1 Conditional item infit

See Müller (2020) and Johansson (2025) for more details on this metric and particularly Müller on why the traditional unconditional item fit should not be used with sample sizes above 200. The easyRasch2 package uses parametric bootstrap to determine adaptive cutoff values for every item.

The Bayesian model doesn’t need bootstrap, since we can get expected and observed values directly from the model fit object.

simfit <- RMitemInfitCutoff(d, iterations = 400)
RMitemInfit(d, simfit)
MSQ values based on conditional estimation (n = 750 complete cases). Cutoff values based on 341 simulation iterations (99.9% HDCI). Flagged: overfit = infit below range (more predictable); underfit = above range (noisier).
Item Infit MSQ Infit low Infit high Flagged Relative location
q1 0.942 0.884 1.111 1.60
q2 0.713 0.908 1.106 overfit 1.72
q3 1.384 0.888 1.123 underfit 0.89
q4 1.018 0.869 1.134 0.81
q5 1.159 0.889 1.127 underfit 1.53
q6 0.848 0.864 1.144 overfit 1.66
q7 0.962 0.872 1.118 1.89
q8 1.241 0.862 1.130 underfit 2.42
q9 1.002 0.883 1.108 3.12

Item 2 is strongly overfit. - 2: “Feeling down, depressed, or hopeless”

Item 3 is strongly underfit, followed by item 5. - 3: “Trouble falling or staying asleep, or sleeping too much” - 5: “Poor appetite or overeating”

The two psychosomatic symptoms (item 3 and 5) deviate from other items, likely due to multidimensionality, while item 2 is a very general statement that is strongly related to the latent construct.

inf1 <- infit_statistic(fit, ndraws_use = 2000)
inf1post <- infit_post(inf1)
inf1post$summary
# A tibble: 9 × 4
  item  infit_obs infit_rep infit_ppp
  <chr>     <dbl>     <dbl>     <dbl>
1 q1        0.96      1         0.813
2 q2        0.781     1.00      1    
3 q3        1.27      1         0    
4 q4        1.01      0.997     0.382
5 q5        1.12      1         0.003
6 q6        0.887     1.00      0.98 
7 q7        0.975     1         0.695
8 q8        1.19      1.00      0.009
9 q9        0.98      0.998     0.59 
inf1post$hdi
# A tibble: 9 × 3
  item  underfit overfit
  <chr>    <dbl>   <dbl>
1 q1       0.003   0.17 
2 q2       0       0.994
3 q3       1       0    
4 q4       0.076   0.017
5 q5       0.743   0    
6 q6       0       0.649
7 q7       0.016   0.101
8 q8       0.672   0    
9 q9       0.072   0.066
inf1post$plot

Based on ppp-values:

  • item 2 is overfit
  • item 3, 5, and 8 are underfit
    • 8: “Moving or speaking so slowly that other people could have noticed?”

6.2 Item-restscore

Using Goodman-Kruskal’s gamma (Kreiner 2011).

Item-restscore associations (n = 750 complete cases). Flagged (adj. p < .05): overfit = observed above expected (over-discrimination, often local dependence); underfit = below (under-discrimination, often multidimensionality/noise).
Item Observed Expected Difference Adj. p-value (BH) Flagged Rel. location
q1 0.66 0.60 0.06 0.080 1.60
q2 0.74 0.59 0.15 0.000 overfit 1.72
q3 0.46 0.60 -0.14 0.000 underfit 0.89
q4 0.58 0.58 0.00 0.828 0.81
q5 0.54 0.61 -0.07 0.080 1.53
q6 0.69 0.61 0.08 0.007 overfit 1.66
q7 0.63 0.60 0.03 0.547 1.89
q8 0.64 0.64 0.00 0.883 2.42
q9 0.87 0.67 0.20 0.000 overfit 3.12
  • Items 2, 6 and 9 are overfit
  • Item 3 is underfit.
    • 3: “Trouble falling or staying asleep, or sleeping too much”
irest1 <- item_restscore_statistic(fit, ndraws_use = 2000)
irest1post <- item_restscore_post(irest1)
irest1post$summary
# A tibble: 9 × 5
  item  gamma_obs gamma_rep gamma_diff   ppp
  <chr>     <dbl>     <dbl>      <dbl> <dbl>
1 q1        0.659     0.597      0.062 0.986
2 q2        0.742     0.59       0.152 1    
3 q3        0.457     0.6       -0.143 0    
4 q4        0.575     0.586     -0.011 0.342
5 q5        0.543     0.605     -0.062 0.018
6 q6        0.691     0.609      0.083 0.997
7 q7        0.629     0.603      0.027 0.805
8 q8        0.644     0.636      0.009 0.583
9 q9        0.872     0.667      0.204 1    
irest1post$plot

Since the observed item-restscore is non-parametric, there is no posterior distribution, only a point estimate.

Based on ppp-values:

  • items 2, 6 and 9 are overfit
  • items 3 and 5 underfit
    • 3: “Trouble falling or staying asleep, or sleeping too much”
    • 5: “Poor appetite or overeating”

6.3 Posterior Predictive Item Fit

Based on code in Bürkner (2020) and adapted for polytomous data.

# Categorical log-likelihood criterion (for polytomous models)
ll_categorical <- function(y, p) log(p)

ll_fit <- fit_statistic_pcm(fit, 
                  criterion = ll_categorical, 
                  ndraws_use = 1000,
                  group = item)

# Summarise: posterior predictive p-values per item
ll_fit %>%
  group_by(item) %>%
  summarise(
    observed   = mean(crit),
    replicated = mean(crit_rep),
    ppp        = mean(crit_rep > crit)
  )
# A tibble: 9 × 4
  item  observed replicated   ppp
  <chr>    <dbl>      <dbl> <dbl>
1 q1       -582.      -595. 0.287
2 q2       -529.      -589. 0.001
3 q3       -795.      -714. 1    
4 q4       -731.      -723. 0.64 
5 q5       -634.      -600. 0.92 
6 q6       -525.      -554. 0.102
7 q7       -532.      -539. 0.391
8 q8       -342.      -323. 0.788
9 q9       -151.      -168. 0.249
# Use ggplot2 to make a histogram
ll_fit %>%
  ggplot(aes(crit_diff)) +
  geom_histogram(aes(fill = ifelse(crit_diff > 0, "above","below"))) +
  facet_wrap("item") +
  theme_bw() +
  theme(legend.position = "none")

Based on ppp:

  • item 2 is overfit
  • item 3 is underfit, maybe also item 5

6.4 Conditional ICC

6.5 Local dependency

Using Yen’s \(Q_3\) statistic (Christensen et al. 2017), again with parametric bootstrap to determine a critical value.

simcor <- RMlocdepQ3Cutoff(d, iterations = 1000)
q3ld <- RMlocdepQ3(d, simcor)
q3ld$matrix
Dynamic cut-off: 0.093 (mean Q3 -0.099 + 0.192). Global simulation cutoff (99th pctl of max-mean Q3) from 840 iterations. Correlations exceeding the cut-off may indicate local dependence; see the per-pair table for detail.
q1 q2 q3 q4 q5 q6 q7 q8 q9 above_cutoff
q1
q2 0.19 *
q3 -0.25 -0.23
q4 -0.17 -0.23 -0.06
q5 -0.23 -0.18 -0.14 -0.09
q6 -0.03 0.19 -0.25 -0.16 -0.16 *
q7 -0.06 -0.1 -0.22 -0.12 -0.07 -0.13
q8 -0.14 -0.12 -0.13 -0.07 -0.09 -0.15 -0.01
q9 -0.05 0.06 -0.09 -0.1 -0.14 0.13 -0.06 -0.09 * 
RMlocdepQ3Plot(simcor, d, n_pairs = 10)

Based on 1000 simulations and the 99th percentile as a global cutoff for the matrix table.

Correlated item pairs, sorted on strongest correlations

  • 1 & 2
    • 1: “Little interest or pleasure in doing things”
    • 2: “Feeling down, depressed, or hopeless”
  • 2 & 6
    • 6: “Feeling bad about yourself - or that you are a failure or have let yourself or your family down”
  • 6 & 9
    • 9: “Thoughts that you would be better off dead or of hurting yourself in some way”
q3_1 <- q3_statistic(fit, ndraws_use = 2000)
q3_1post <- q3_post(q3_1, n_pairs = 8)
q3_1post$summary %>% head(8)
# A tibble: 8 × 7
  item_pair item_1 item_2 q3_obs q3_rep q3_diff q3_ppp
  <chr>     <chr>  <chr>   <dbl>  <dbl>   <dbl>  <dbl>
1 q2 : q6   q2     q6      0.297  0.002   0.295  1    
2 q1 : q2   q1     q2      0.29  -0.002   0.292  1    
3 q6 : q9   q6     q9      0.174  0.001   0.173  1    
4 q2 : q9   q2     q9      0.111  0       0.111  1    
5 q1 : q6   q1     q6      0.097  0       0.097  1    
6 q3 : q4   q3     q4      0.065  0       0.065  1    
7 q7 : q8   q7     q8      0.06  -0.001   0.061  0.998
8 q1 : q7   q1     q7      0.058  0       0.058  0.999
q3_1post$hdi %>% head(8)
# A tibble: 8 × 5
  item_pair item_1 item_2    ld    lr
  <chr>     <chr>  <chr>  <dbl> <dbl>
1 q1 : q2   q1     q2     1         0
2 q2 : q6   q2     q6     1         0
3 q6 : q9   q6     q9     1         0
4 q2 : q9   q2     q9     0.974     0
5 q1 : q6   q1     q6     0.97      0
6 q3 : q4   q3     q4     0.736     0
7 q1 : q7   q1     q7     0.414     0
8 q7 : q8   q7     q8     0.246     0

Item pairs most strongly correlated:

  • 2 & 6
  • 1 & 2
  • 6 & 9

A bit less strong but still likely too much:

  • 2 & 9
  • 1 & 6

These seem to be clustered together.

q3_1post$plot

6.6 PCA of residuals

This is primarily a way to evaluate patterns in the standardized model residuals.

pca1 <- plot_residual_pca(fit, ndraws_use = 2000)
pca1
$plot


$data
# A tibble: 9 × 7
  item  location location_lower location_upper  loading loading_lower
  <chr>    <dbl>          <dbl>          <dbl>    <dbl>         <dbl>
1 q1     -0.140        -0.264          -0.0161 -0.143         -0.626 
2 q2     -0.0218       -0.156           0.108  -0.0838        -0.405 
3 q3     -0.849        -0.955          -0.742   0.0179        -0.854 
4 q4     -0.933        -1.05           -0.825   0.0310        -0.226 
5 q5     -0.216        -0.335          -0.0946  0.136         -0.0266
6 q6     -0.0797       -0.203           0.0462 -0.0622        -0.350 
7 q7      0.153         0.00839         0.294   0.00229       -0.280 
8 q8      0.688         0.518           0.867   0.394         -0.234 
9 q9      1.40          1.14            1.71   -0.0208        -0.111 
# ℹ 1 more variable: loading_upper <dbl>

$eigenvalue
# A tibble: 1 × 6
  eigenvalue_obs eigenvalue_rep eigenvalue_diff   ppp var_explained_obs
           <dbl>          <dbl>           <dbl> <dbl>             <dbl>
1           1.57           1.31           0.263 0.896             0.176
# ℹ 1 more variable: var_explained_rep <dbl>
simpca <- RMdimResidualPCACutoff(d, iterations = 500)
RMdimResidualPCA(d, simpca)
Partial Credit Model (750 complete cases, 9 items). Total observed variance: 49.6% explained by measures, 50.4% unexplained (basis for PCA; n = 750 non-extreme cases). First-contrast cutoff = 1.411 based on 422 simulation iterations (99th percentile).
Component Eigenvalue Proportion of variance Flagged
PC1 1.452 0.191 TRUE
PC2 1.300 0.171 FALSE
PC3 1.201 0.158 FALSE
PC4 1.098 0.144 FALSE
PC5 0.927 0.122 FALSE 
RMdimResidualPCA(d, simpca, output = "loadings")

6.7 Response category probabilities

A.k.a. Item Probability Functions

RMitemCatProb(d, category_labels = item_resp)

plot_ipf(fit, theta_range = c(-3,5),
         category_labels = item_resp,
         ribbon_alpha = 0.25)

6.8 Targeting

6.9 DIF

For the case of demonstration, we’ll generate a random “gender” variable.

d2 <- d %>% 
  mutate(gender = factor(sample(c("M", "F"), n(), TRUE)))

dl2 <- d2 %>% 
  mutate(across(where(is.numeric), ~ .x + 1)) %>% 
  rownames_to_column("id") %>% 
  pivot_longer(!c(id,gender), 
               names_to = "item", 
               values_to = "response")
#saveRDS(list(d2,dl2), file = "data/phq9difdata.rds")
difdata <- readRDS("data/phq9difdata.rds")
d2 <- difdata[[1]]
dl2 <- difdata[[2]]

First, we need to check the response distribution for both gender categories.

RMplotTile(d, group = d2$gender)

No empty cells, which is good, particularly in the lower categories.

RMdifLR(d, dif_var = d2$gender, output = "kable")
Partial Credit Model split by DIF variable (2 groups). Andersen LR chi^2 = 17.58, df = 26, p = 0.891. n = 750 complete cases, 9 items. Bold = MaxDiff > 0.5 logits.
Item F M All MaxDiff Flagged SE_F SE_M SE_All
q1 0.015 -0.058 -0.019 0.073 no 0.212 0.220 0.152
q2 0.089 0.116 0.099 0.027 no 0.211 0.236 0.157
q3 -0.696 -0.769 -0.728 0.073 no 0.180 0.187 0.130
q4 -0.880 -0.750 -0.814 0.130 no 0.174 0.190 0.128
q5 -0.093 -0.101 -0.092 0.008 no 0.205 0.218 0.149
q6 0.006 0.102 0.045 0.097 no 0.226 0.235 0.161
q7 0.258 0.291 0.274 0.033 no 0.223 0.247 0.165
q8 0.845 0.750 0.801 0.095 no 0.324 0.360 0.239
q9 1.401 1.626 1.496 0.225 no 0.474 0.551 0.357 
RMdifLR(d, dif_var = d2$gender, output = "ggplot")

RMdifLR(d, dif_var = d2$gender, output = "ggplot", level = "threshold")

RMitemICCPlot(d, dif_var = d2$gender)

Does not require refitting the model.

plot_icc(fit, 
         dif_var = "gender", 
         dif_data = dl2)

This does require refitting the model with the DIF variable included, which will take quite a bit of time. I’ll preload the one below, since it took almost 30 minutes on 4 cores.

phq9_dif_gender <- dif_statistic(fit, group_var = "gender", data = dl2)

saveRDS(phq9_dif_gender, "data/phq9_dif_gender_fit.rds")

You can also use dif_statistic() to just output a formula for brms and refit the model manually.

dif_statistic(fit, group_var = "gender", data = dl2, refit = FALSE)$dif_formula
response | thres(gr = item) ~ 1 + (1 | id) + gender:item
NoteNote

For polytomous models, it is also of interest to review the potential effect of a DIF variable on individual threshold locations, which can be done with dif_type = "non-uniform". This type of model needs to be fit using dif_statistic(), since it does some pre-processing under the hood.

phq9_dif_gender <- readRDS("data/phq9_dif_gender_fit.rds")

phq9_dif_gender$summary %>% 
  kable()
item dif_estimate dif_sd dif_lower dif_upper pd rope_percentage flag
q1 -1.4846677 1.718012 -4.862511 1.9213643 0.8092 0.1533 Inconclusive
q2 -1.6785010 1.750830 -5.104834 1.7283809 0.8322 0.1426 Inconclusive
q3 -0.8570800 1.759725 -4.356726 2.5547662 0.6852 0.2047 Inconclusive
q4 -0.8698661 1.748307 -4.268959 2.4530867 0.6894 0.2055 Inconclusive
q5 -1.4727613 1.755595 -4.883501 1.9926160 0.8026 0.1593 Inconclusive
q6 -1.5902585 1.772406 -5.015850 1.8983073 0.8165 0.1468 Inconclusive
q7 -1.8656254 1.719208 -5.242316 1.4421569 0.8607 0.1319 Inconclusive
q8 -2.3197795 1.709899 -5.660178 1.0559836 0.9087 0.0921 Inconclusive
q9 -3.3026390 1.753691 -6.725325 0.1993221 0.9678 0.0405 Inconclusive 
phq9_dif_gender$plot

7 Summary

These PHQ-9 data clearly show several significant psychometric weaknesses. Several items show strong misfit to the measurement model, residual correlations, and disordered item category threshold locations. A next step in the analysis process would be to remove one or two misfitting items, in this data items 3 (underfit) and 2 (overfit and having residual correlations), and run a new model. Since this is intended primarily as a demonstration of the functions available in easyRaschBayes, I’ll leave the continued analysis to the curious reader.

However, there are some functions left to demonstrate that would normally only be used when having arrived at a set of items with adequate psychometric properties. These include reliability and item and person parameters.

For this last part, we’ll use a model including only PHQ items 1, 4, 6, and 7, since these make up a well-functioning item set regarding item fit and LD. Item threshold locations are not optimally separated, as will be shown after we look at the recently proposed reliability metric Relative Measurement Uncertainty (RMU, Bignardi et al. 2025).

7.1 Reliability

First, we refit the model, modified to include only four items.

dl2 <- dl %>% 
  filter(item %in% paste0("q",c(1,4,6,7)))
fit_4item <- update(fit, newdata = dl2)
saveRDS(fit_4item,"data/phq9fit4items.rds")

Next, the draws of person parameters are extracted from the model.

fit_4item <- readRDS("data/phq9fit4items.rds")

posterior_draws <- fit_4item %>%
  as_draws_df() %>%
  as_tibble() %>% 
  select(starts_with("r_id")) %>%
  t()

RMUreliability(posterior_draws)
  rmu_estimate hdci_lowerbound hdci_upperbound .width .point .interval
1    0.6918987       0.6600473       0.7247605   0.95   mean      hdci

7.2 Item category probabilities

plot_ipf(fit_4item)

itemps <- item_parameters(fit_4item)
itemps$threshold_order
# A tibble: 4 × 4
  item  n_thresholds ordered prob_disordered
  <chr>        <int> <lgl>             <dbl>
1 q1               3 FALSE             0.598
2 q4               3 FALSE             0.764
3 q6               3 FALSE             0.933
4 q7               3 TRUE              0.133
itemps$locations_wide %>% 
  arrange(item) %>% 
  kable()
item t1 t2 t3 location
q1 -0.6869 0.5615 0.4776 0.1174
q4 -2.3742 0.1711 -0.0075 -0.7369
q6 -0.3956 0.7256 0.2317 0.1872
q7 -0.2984 0.6093 0.9857 0.4322

Note that this function is not yet available in the CRAN package version 0.8.0, but only in the development version on GitHub.

RMitemParameters(d[,c(1,4,6,7)], format = "wide")
Item thresholds via CML (Andrich thresholds, logit scale). SE and 95% Wald CI shown.
item t1 t2 t3 location se_t1 se_t2 se_t3
q4 -2.397 0.201 -0.010 -0.735 0.119 0.124 0.165
q1 -0.681 0.589 0.450 0.119 0.098 0.156 0.216
q6 -0.387 0.749 0.209 0.190 0.099 0.172 0.225
q7 -0.292 0.642 0.928 0.426 0.099 0.165 0.246

7.3 Person locations

We get EAP, expected a-posteriori, from brms, but Weighted Likelihood Estimation (WLE, Warm 1989) is also available in the function output. WLE has been shown to have slightly less bias (see also Kreiner 2025). Standard errors of measurement are also included in the output object.

personps <- person_parameters(fit_4item, theta_range = c(-4,4))
hist(personps$person_estimates$eap, breaks = 20)

hist(personps$person_estimates$wle, breaks = 20)

cor.test(personps$person_estimates$eap,
         personps$person_estimates$wle)

    Pearson's product-moment correlation

data:  personps$person_estimates$eap and personps$person_estimates$wle
t = 277.04, df = 748, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.9944184 0.9958074
sample estimates:
      cor 
0.9951623

For comparison, here are the WLE person parameters from easyRasch2. Note that this function is not yet available in the CRAN package version 0.8.0, but only in the development version on GitHub.

ppar <- RMpersonParameters(d[,c(1,4,6,7)], output = "dataframe")
hist(ppar$theta, breaks = 20)

7.4 Transformation table

When data fits the Rasch model, the ordinal sum score is a sufficient statistic for the latent variable. However, the ordinal sum score is not an interval scale variable and it is likely beneficial to transform the sum score to EAP or WLE prior to further statistical analysis.

personps$score_table %>% 
  kable()
sum_score n eap eap_se wle wle_se
0 153 -2.9762 0.9632 -4.0000 NaN
1 179 -2.1802 0.8304 -2.9004 1.3733
2 119 -1.5796 0.7267 -1.6893 0.9143
3 89 -1.1102 0.6506 -1.0450 0.7224
4 58 -0.7281 0.5929 -0.5855 0.6139
5 37 -0.4058 0.5545 -0.2163 0.5530
6 31 -0.1199 0.5291 0.1093 0.5237
7 28 0.1470 0.5193 0.4220 0.5204
8 20 0.4072 0.5213 0.7503 0.5457
9 10 0.6767 0.5338 1.1349 0.6151
10 9 0.9646 0.5685 1.6696 0.7868
11 14 1.3009 0.6252 2.7795 1.4514
12 3 1.7252 0.7232 4.0000 NaN 
RMscoreSE(d[,c(1,4,6,7)])
Person locations via Warm’s WLE (CML item parameters from eRm).
Ordinal sum score Logit score Logit std.error
0 -3.571 1.746
1 -1.974 1.014
2 -1.177 0.763
3 -0.703 0.641
4 -0.362 0.574
5 -0.084 0.537
6 0.163 0.520
7 0.399 0.517
8 0.637 0.531
9 0.897 0.564
10 1.207 0.630
11 1.638 0.773
12 2.526 1.268

8 References

Bignardi, Giacomo, Rogier Kievit, and Paul-Christian Bürkner. 2025. A General Method for Estimating Reliability Using Bayesian Measurement Uncertainty. OSF. https://doi.org/10.31234/osf.io/h54k8_v1.
Bürkner, Paul-Christian. 2020. “Analysing Standard Progressive Matrices (SPM-LS) with Bayesian Item Response Models.” Journal of Intelligence 8 (1). https://doi.org/10.3390/jintelligence8010005.
Christensen, Karl Bang, Guido Makransky, and Mike Horton. 2017. “Critical Values for Yen’s Q3: Identification of Local Dependence in the Rasch Model Using Residual Correlations.” Applied Psychological Measurement 41 (3): 178–94. https://doi.org/10.1177/0146621616677520.
Johansson, Magnus. 2025. “Detecting Item Misfit in Rasch Models.” Educational Methods & Psychometrics 3 (18). https://doi.org/10.61186/emp.2025.5.
Kreiner, Svend. 2011. “A Note on Item–Restscore Association in Rasch Models.” Applied Psychological Measurement 35 (7): 557–61. https://doi.org/10.1177/0146621611410227.
Kreiner, Svend. 2025. “On Specific Objectivity and Measurement by Rasch Models: A Statistical Viewpoint.” Educational Methods & Psychometrics 3. https://doi.org/10.61186/emp.2025.7.
Kruschke, John K. 2018. “Rejecting or Accepting Parameter Values in Bayesian Estimation.” Advances in Methods and Practices in Psychological Science 1 (2): 270–80. https://doi.org/10.1177/2515245918771304.
Müller, Marianne. 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.
Säilynoja, Teemu, Andrew R. Johnson, Osvaldo A. Martin, and Aki Vehtari. 2025. Recommendations for Visual Predictive Checks in Bayesian Workflow. arXiv. https://doi.org/10.48550/arXiv.2503.01509.
Warm, Thomas A. 1989. “Weighted Likelihood Estimation of Ability in Item Response Theory.” Psychometrika 54 (3): 427–50. https://doi.org/10.1007/BF02294627.

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Citation

BibTeX citation:
@online{johansson2026,
  author = {Johansson, Magnus},
  title = {Rasch Analysis with {Bayesian} and Frequentist Methods},
  date = {2026-03-31},
  url = {https://pgmj.github.io/rasch_bayes_comp.html},
  langid = {en}
}
For attribution, please cite this work as:
Johansson, Magnus. 2026. “Rasch Analysis with Bayesian and Frequentist Methods.” March 31. https://pgmj.github.io/rasch_bayes_comp.html.