Explanatory Rasch models

The linear logistic test model (LLTM; Fischer, 1973) explains dichotomous item locations using observed item characteristics. The linear partial credit model (LPCM; Fischer and Ponocny, 1994) extends the same formulation to polytomous thresholds. Both retain the common discrimination and conditional Rasch structure.

\[ \log\frac{P(X_{ni}=k)}{P(X_{ni}=k-1)} =\theta_n-\delta_{ik}, \qquad \delta_{ik}=\mathbf z_{ik}^{\mathsf T}\gamma . \]

The vector \(\mathbf z_{ik}\) contains the nominated item or threshold characteristics. An interaction permits the effect of one characteristic to differ according to another. The scale origin is fixed at mean item location zero, as in rasch(); the formula intercept is therefore absorbed by the origin constraint.

Design columns are centred and rescaled internally for numerical stability. Coefficients and standard errors are reported in the supplied predictor units, for both Rasch and comparative judgement models.

Item predictors

Item-level metadata contain one row per item. This example explains eight dichotomous item locations by the operation required and the response format.

set.seed(1)
item_design <- data.frame(
  item = paste0("I", 1:8),
  operation = rep(c("recall", "inference"), each = 4),
  format = rep(c("selected", "constructed"), 4)
)

difficulty <- 0.8 * (item_design$operation == "inference") +
  0.4 * (item_design$format == "constructed")
group <- factor(rep(c("A", "B"), each = 250))
theta <- rnorm(500) + ifelse(group == "B", 0.3, 0)
X <- sapply(difficulty, function(delta)
  rbinom(length(theta), 1, plogis(theta - delta)))
colnames(X) <- item_design$item

fit <- rasch_explanatory(
  X,
  predictors = item_design,
  formula = ~ operation + format,
  level = "item",
  factors = data.frame(group = group)
)
fit$est$coefficients
#>             term estimate    se       t  df       p   p_adj
#>  operationrecall   -0.851 0.069 -12.294 461 < 0.001 < 0.001
#>   formatselected   -0.327 0.071  -4.606 461 < 0.001 < 0.001

If the same person contributes more than one response row, the coefficient covariance is clustered by person. Supported repeated-person fits use a linearised delete-one-person covariance correction. Every supported fit takes a \(t\) reference with degrees of freedom equal to the number of independent person units contributing conditional information minus one, whether or not identifiers repeat: one response row per person still leaves a finite count of sampling units, and reading that count as a limiting normal rejects a true null more often than the printed probability states. Coefficient inference is withheld when the calibration lacks enough independent information. Holm adjustment covers the coefficient family. With few persons and unequal numbers of response rows, these approximate tests can still be mildly liberal. The correction does not guarantee nominal small-sample coverage.

Predictors may be continuous, categorical or ordinal. Numeric vectors are continuous and unordered factors are categorical. An ordered factor is scored with successive contrasts: each coefficient is the change between two adjacent levels in the declared order. Character vectors are converted to unordered factors; the first factor level is the reference category. For example:

item_design$demand <- as.numeric(item_design$demand)
item_design$format <- factor(item_design$format)
item_design$complexity <- ordered(
  item_design$complexity,
  levels = c("low", "moderate", "high")
)

This coding uses the order without assuming equal spacing and does not fit polynomial contrasts. Use an unordered factor for comparisons with a single reference category, or a numeric predictor when the spacing has a substantive scale. Selected interactions can be added, for example ~ demand + format + complexity + format:complexity. Formula offsets (offset()) are not supported in explanatory Rasch or CJ models.

Threshold predictors

For an LPCM, item metadata may still be supplied once per item. The reserved factor threshold identifies the within-item threshold and can be used in the formula:

lpcm <- rasch_explanatory(
  responses,
  predictors = item_design,
  formula = ~ operation + format + threshold + format:threshold,
  level = "item"
)

Use level = "threshold" when a predictor varies within an item. The metadata must then contain item, threshold, and one row for every observed item threshold. threshold_number is also available as an integer predictor.

Comparing the restriction with a free calibration

rasch_explanatory() retains a free partial credit calibration of the same responses. explanatory_test() compares it with the active explanatory model. Pairwise conditional estimation produces a composite likelihood, so the reported inferential result is the first-order Kent-adjusted chi-square rather than an ordinary likelihood-ratio test. The same table reports calibration \(R^2\), the proportion of variation in the free threshold calibration reproduced by the explanatory thresholds. Comparative judgement uses the corresponding free object calibration. The Kent comparison is asymptotic and does not use the finite-person-cluster correction applied to individual coefficients.

explanatory_test(fit)
#>  model parameters free_parameters r_squared r_squared_adj              r2_basis
#>   LLTM          2               7     0.974         0.964 threshold calibration
#>   chisq df p_naive chisq_kent     p p_kent
#>  16.526  5   0.005      4.245 0.515  0.515

A non-significant result does not establish that the explanatory structure is true. It indicates that the free calibration has not detected a departure at the available precision. Coefficients should be interpreted together with their standard errors, item fit, targeting, and the study design.

Fixed departures

rasch_explanatory() begins with the exact explanatory restriction. Its error formulation uses nominated fixed departures rather than a random item effect, so it does not introduce an item population distribution. An explanatory model can be too restrictive for a small number of items. explanatory_diagnostics() adds each available departure separately and adjusts the complete family of probabilities by Holm’s method.

departures <- explanatory_diagnostics(fit)
head(departures)
#>  item     component parameters_added departure deviance_reduction df     p weak
#>    I1 Item location                1     0.184              8.142  1 0.144     
#>    I3 Item location                1    -0.163              6.318  1 0.200     
#>    I6 Item location                1     0.167              6.248  1 0.200     
#>    I8 Item location                1    -0.141              4.539  1 0.283     
#>    I5 Item location                1    -0.041              0.422  1 0.746     
#>    I2 Item location                1    -0.037              0.353  1 0.767     
#>  converged p_adj
#>          * 1.000
#>          * 1.000
#>          * 1.000
#>          * 1.000
#>          * 1.000
#>          * 1.000

An item-location departure moves all thresholds of an item together. A threshold-structure departure changes their relative locations and is available only for polytomous items. A departure should be accepted on substantive grounds, not selected solely because it has the smallest probability.

fit <- relax_explanatory(fit, item = "I4", component = "location")

The refit remains a fixed-effects Rasch model: no item distribution or random effect is introduced. Item locations, person measures, residuals, fit statistics and subsequent analyses are all recomputed. Item deletion, DIF splitting and superitem construction retain the explanatory structure and add the required fixed departures for changed items. Response-dependence resolution does the same for its resolved item copies. With keyed multiple-choice data, unchanged and split items retain their raw responses for distractor analysis.

When person factors are supplied to rasch_explanatory(), DIF is tested in the usual way. The optional bootstrap conditions on each person’s score and refits the active explanatory model, including accepted fixed departures. Its familywise probabilities refer to the fitted global invariant null.

dif <- dif_anova(fit)
dif_bootstrap(fit, dif, B = 999, seed = 2026)$summary

The same workflow is available under Explanatory in the Shiny application. Predictor type is selected explicitly in the application; ordinal predictors also require their level order. The project file records the predictor metadata, formula and accepted departures, and the displayed R code reproduces the analysis.

Comparative judgement

btl_explanatory() constrains object rather than item locations. Predictor metadata contain one row per object and an object column. The formulation is

\[ \log\frac{P(a\succ b)}{P(b\succ a)} =\beta_a-\beta_b, \qquad \beta_i=\mathbf z_i^{\mathsf T}\gamma . \]

cj <- btl_explanatory(
  comparisons,
  predictors = object_design,
  formula = ~ domain + format + domain:format,
  object_a = "object_a",
  object_b = "object_b",
  winner = "winner",
  judge = "judge"
)
explanatory_test(cj)
explanatory_diagnostics(cj)

An ordered response may be supplied instead of winner. Its symmetric response thresholds are estimated jointly with the explanatory object effects. Judge-clustered covariance and its existing availability rules continue to apply. relax_btl_explanatory() adds a nominated fixed object-location departure and repeats the complete fit.

Validation

Regression checks compare the LLTM and LPCM calibrations with eRm. In the simulation study, coefficient bias was at most 0.005 logits, empirical standard deviations divided by mean reported standard errors ranged from 0.99 to 1.03, and 95% coverage ranged from 0.938 to 0.955 across LLTM, LPCM, dichotomous comparative judgement, ordered comparative judgement and judge-clustered comparative judgement conditions. Kent-adjusted null rejection ranged from 4.2% to 6.0% (1,000 replicates per condition). Holm familywise error for fixed-departure diagnostics was 4.3% to 4.7%, with 98.3% to 100% power for a planted 0.8-logit departure (300 replicates per condition). A further 1,000-replicate study covered mixed maximum scores and four-category LPCMs at 300 to 2,000 persons. SE ratios were 0.993 to 1.026, coverage was 0.942 to 0.954, Kent-adjusted null rejection was 4.3% to 5.8%, and mean calibration \(R^2\) was 0.872 to 0.994. The unscaled probability rejected 98.6% to 100% and is therefore returned only as p_naive. No fit in these four conditions was refused or failed to converge. Scripts and full result tables are under tools/simval/ in the source repository.