Fits the partial credit model (PCM) or rating scale model (RSM) by pairwise conditional maximum likelihood. Person locations are Warm weighted likelihood estimates. The fitted object contains item and person fit, targeting, reliability, threshold diagnostics, residuals, and a score-to-measure table.
Usage
rasch(
data,
model = c("PCM", "RSM"),
id = NULL,
factors = NULL,
items = NULL,
n_groups = NULL,
anchors = NULL,
na_codes = -1,
key = NULL,
pc_components = NULL,
maxit = 60,
tol = 1e-08
)Arguments
- data
Persons-by-items integer score matrix (categories from 0), or a data frame also containing ID and person-factor columns. Missing values are allowed subject to the identification and ignorability conditions described above.
- model
Either
"PCM"(partial credit) or"RSM"(rating scale).- id
Optional name of an ID column in
data, or a vector of IDs. Repeated values cluster the item-parameter sandwich covariance and define the person unit in repeated-measures DIF. The ordinary item-fit reference distributions are row-based, so their probabilities are withheld when an ID occurs on more than one response row. The support conditions described above then apply to person clusters rather than response rows.- factors
Optional character vector of person-factor column names in
data(for DIF analysis), a data frame of factors, or one grouping vector with one entry per data row.- items
Optional item column names or numeric column indices. By default, columns named in
idorfactorsare excluded. A separate factor data frame may share item names whenitemsis explicit. Without it, matching names exclude columns whose values agree; conflicting values are refused.- n_groups
Number of class intervals for the item-trait chi-square and ANOVA item fit. The default
NULLapplies the rule of Andrich and Marais (2019, ch. 15): as many intervals of at least 50 non-extreme persons as the sample allows, at most 10, at least 2. The resolved value is stored infit$n_groups.- anchors
Optional anchor table for equating: a data frame with columns
item,k, andtau, and optionallyaverage = TRUEfor average item anchoring; seepcml. Column names must be unique. Anchors determine the scale origin. Anchor values are treated as fixed, so their uncertainty is not included in the fitted standard errors.- na_codes
Numeric or character values to read as missing. They are matched before scores are converted to numbers, including numerically equivalent labels (for example,
"09"matches a score of 9). Defaults to-1, the conventional missing-response code; any negative score is also treated as missing, since valid category scores start at zero. For keyed items, codes apply to raw answer options, not to the scores assigned by the key. Structural refits use the prepared scores and do not apply the original raw codes again.- key
Optional multiple-choice key: a named item-to-option vector, an item/key table, or an item/option/score table. Table column names must be unique. See Details.
- pc_components
NULL(the default) estimates all PCM thresholds freely. Values from 1 to 4 use the principal-components form inpcml_pc: location, then spread, skewness, and kurtosis. This can stabilise sparse categories. Component estimates are stored in the estimation details. Available for PCM fits without anchors.- maxit, tol
Newton-Raphson iteration cap and convergence tolerance of the pairwise conditional estimation.
Value
An object of class "rasch". Its principal components are
the item summary, threshold table, person table, score table, residuals,
reliability, targeting, item-trait statistics, threshold diagnostics,
and estimation details. The component summary_stats contains the
distribution summaries, fit-location correlations, and the cell
degrees-of-freedom factor. The item summary carries a disc
column described below. repeated_ids records whether a person
contributes more than one informative calibration row;
repeated_residual_ids records repetition among rows contributing
fitted residuals, which governs the row-based fit references. If estimation
does not converge, locations and
residual patterns are retained for diagnosis, but standard errors,
separation indices and inferential probabilities are NA.
Details
For scores \(x=0,\ldots,m_i\), the PCM is $$P(X_{ni}=x)=\frac{\exp\{x\theta_n-\sum_{k=1}^{x}\delta_{ik}\}} {\sum_{y=0}^{m_i}\exp\{y\theta_n-\sum_{k=1}^{y}\delta_{ik}\}}.$$ The RSM constrains \(\delta_{ik}=\beta_i+\tau_k\), where \(\beta_i\) is the item location and \(\tau_k\) is common across items. Dichotomous items are the one-threshold case of the PCM.
Pairwise conditioning removes \(\theta_n\) from the item likelihood. Missing responses are omitted from pairwise contributions, and person measures are estimated within each observed item pattern. The observed item-pair graph must identify a common scale. This covers planned linked designs and ignorable missingness; informative missingness can still bias the estimates.
Item-parameter uncertainty uses the empirical Godambe sandwich over independent persons, or over person clusters when IDs repeat. It is withheld unless at least 10 contributing units, at least 8 effective units, more effective units than fitted parameters, and a full-rank score covariance support the fitted directions. Effective support reflects the number of informative conditional item pairs contributed by each unit; rows without one do not count. Point estimates and exact anchors remain available when uncertainty is withheld.
The fit residual is the log-of-mean-square statistic described by Andrich
and Marais (2019, ch. 23). Positive values indicate under-discrimination and
negative values indicate over-discrimination. Its standard-normal reading,
the item-trait chi-square and the class-interval F test are asymptotic
approximations. For ordinary Rasch, PCM and RSM fits,
fit_bootstrap supplies calibrated probabilities.
Multiple-choice responses may be scored from a named item-to-key vector,
an item/key table, or an item/option/score table. A slash separates
alternative correct options. The third form assigns integer category
scores to nominated options and fits the resulting item as polytomous;
unlisted options score zero. Raw responses are retained in fit$mc
for distractor analysis.
Estimated item discrimination
The item summary includes a post-estimation slope disc. For item
\(i\), it maximises that item's response likelihood over \(a_i\) while
holding the fitted person locations and thresholds fixed:
$$\hat a_i=\arg\max_{a_i}
\sum_n\log P(X_{ni}=x_{ni}\mid\hat\theta_n,\hat\delta_i,a_i).$$
The same slope multiplies every threshold of a polytomous item. It is a
descriptive index, not a freely estimated parameter of the Rasch model,
and no sampling standard error or hypothesis test is attached to it.
Item-fit probabilities
The item-trait chi-square assesses invariance over class intervals, but its
asymptotic reference treats the estimated person locations used to form
those intervals as known. Its calibration therefore changes with sample
size and test length. The class-interval ANOVA and standardised residual
readings are approximate for the same reason. The item table retains their
raw and Holm-adjusted probabilities as descriptive diagnostics. Each
adjustment retains the full item family when one probability is
unavailable.
fit_bootstrap re-estimates every replicate and should be used
for item-level inference where it is available. With repeated IDs, the
ordinary asymptotic probabilities are withheld and fit_bootstrap()
is unavailable because neither reference models within-person dependence;
the residuals and fit statistics remain descriptive.
References
Rasch, G. (1960). Probabilistic Models for Some Intelligence and Attainment Tests. Copenhagen: Danish Institute for Educational Research. (Expanded edition, 1980, Chicago: University of Chicago Press.)
Rasch, G. (1961). On general laws and the meaning of measurement in psychology. In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability (Vol. 4, pp. 321–333). Berkeley: University of California Press.
Andrich, D. and Luo, G. (2003). Conditional pairwise estimation in the Rasch model for ordered response categories using principal components. Journal of Applied Measurement, 4(3), 205–221.
Andrich, D. and Marais, I. (2019). A Course in Rasch Measurement Theory: Measuring in the Educational, Social and Health Sciences. Springer.
Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. Psychometrika, 54(3), 427–450.
See also
rasch_mfrm, rasch_efrm,
btl, dif_anova,
test_information, and run_app.
Examples
set.seed(1)
d <- seq(-2, 2, length.out = 8)
X <- matrix(rbinom(500 * 8, 1, plogis(outer(rnorm(500), d, "-"))), 500, 8)
colnames(X) <- paste0("I", 1:8)
fit <- rasch(X, model = "PCM")
fit$items
#> item max location se disc fit_resid df_fit natural_resid infit_ms
#> I1 1 -2.160 0.144 1.106 0.167 423.500 0.170 1.027
#> I2 1 -1.352 0.113 1.209 -0.576 423.500 -0.555 1.030
#> I3 1 -0.894 0.106 1.352 -1.427 423.500 -1.331 1.005
#> I4 1 -0.253 0.098 1.327 -0.684 423.500 -0.668 1.061
#> I5 1 0.391 0.098 1.077 0.868 423.500 0.896 1.160
#> I6 1 0.890 0.100 1.243 -0.563 423.500 -0.549 1.074
#> I7 1 1.468 0.113 1.092 0.186 423.500 0.189 1.105
#> I8 1 1.910 0.128 1.174 0.229 423.500 0.233 0.997
#> outfit_ms infit_z outfit_z chisq df p p_adj p_bonf F_anova p_anova
#> 0.988 0.332 -0.007 5.367 6 0.498 1.000 1.000 0.550 0.770
#> 0.920 0.514 -0.694 8.361 6 0.213 1.000 1.000 1.387 0.218
#> 0.876 0.107 -1.473 13.544 6 0.035 0.281 0.281 2.771 0.012
#> 0.975 1.424 -0.373 7.579 6 0.271 1.000 1.000 1.493 0.179
#> 1.094 3.694 1.432 2.837 6 0.829 1.000 1.000 0.499 0.809
#> 0.963 1.566 -0.429 5.526 6 0.478 1.000 1.000 0.941 0.465
#> 1.018 1.767 0.196 5.761 6 0.451 1.000 1.000 0.708 0.643
#> 1.012 -0.018 0.130 6.873 6 0.333 1.000 1.000 0.956 0.455
#> p_anova_adj p_anova_bonf
#> 1.000 1.000
#> 1.000 1.000
#> 0.094 0.094
#> 1.000 1.000
#> 1.000 1.000
#> 1.000 1.000
#> 1.000 1.000
#> 1.000 1.000
fit$psi$PSI
#> [1] 0.4907417