rasch: Pairwise Conditional Rasch Measurement Analysis and Diagnostics

Pairwise conditional maximum likelihood estimation of dichotomous and polytomous Rasch models (partial credit and rating scale) after Andrich and Luo (2003) and Zwinderman (1995) <doi:10.1177/014662169501900406>, with standard errors from a Godambe sandwich estimator. An optional alternative estimator reparameterises each item's thresholds as Andrich's (1978 <doi:10.1007/BF02293814>, 1985) orthogonal-polynomial principal components (location, spread, skewness, and kurtosis; Pedler 1987), exact for items with up to 3 thresholds and a smoothed reduced-rank model for items with more, useful when some categories are sparsely populated. Person measures are Warm's (1989) <doi:10.1007/BF02294627> weighted likelihood estimates, computed per missing-data pattern. The diagnostic suite follows the conventions set out in Andrich and Marais (2019) <doi:10.1007/978-981-13-7496-8>: the log-of-mean-square fit residual with apportioned degrees of freedom (and its natural form), infit and outfit, the item-trait interaction chi-square over automatically sized class intervals with its per-interval detail table, the class-interval ANOVA item-fit F, the person separation index with and without extremes and the item separation index, Cronbach's alpha, summary distribution statistics with skewness and kurtosis, targeting, the score-to-measure table with maximum likelihood and geometric extreme-score extrapolation options, test information, threshold and category diagnostics, residual principal-components dimensionality testing, local dependence by residual correlation, and differential item functioning by two-way residual analysis of variance over any number of person factors, factor-at-a-time (the full two-way table with partial eta-squared effect sizes) or as a full factorial with interaction precedence, Tukey HSD post-hoc comparisons on significant group terms and interaction cells, false-discovery-rate or familywise adjustment, and DIF magnitudes in logits by resolved-item locations with a practical-significance criterion. Violations of independence are quantified, not just flagged: the magnitude of response dependence between two items by the resolution method of Andrich and Kreiner (2010) <doi:10.1177/0146621609360202> (polytomous form Andrich, Humphry and Marais 2012 <doi:10.1177/0146621612441858>), the spread-parameter least-upper-bound screen (Andrich 1985), and the magnitude of multidimensionality (latent subscale correlation and common-variance proportion) from Andrich's (2016) two-calculation reliability comparison. A likelihood-ratio test of the partial credit against the rating parameterisation is reported both raw, as conventionally displayed, and with a first-order composite-likelihood calibration (Kent 1982 <doi:10.1093/biomet/69.1.19>) from the Godambe matrices. Also included: anchored estimation for test equating (individual threshold and average item-location anchors), common-item equating tests and plots, item splitting to resolve invariance violations, tailored analysis for guessing with the four-step anchored comparison (Andrich, Marais and Humphry 2012 <doi:10.3102/1076998611411914>), classical test theory companion statistics, racked and stacked reshaping for repeated measurements, model comparison by composite-likelihood information criteria whose penalty is the Godambe effective parameter count (Varin and Vidoni 2005 <doi:10.1093/biomet/92.3.519>; Gao and Song 2010 <doi:10.1198/jasa.2010.tm09414>), absorbing the pairwise over-counting that a nominal AIC or BIC would ignore, the many-facet Rasch model (Linacre 1989) for rated long-format data with facet severities, fit, and optional item-by-facet interactions, subtest formation for locally dependent items, multiple-choice scoring against a key with double keying and polytomous option scoring of informative distractors (Andrich and Styles 2011, with an evidence-based rescoring proposal), rest-measure distractor analysis and option curves, the Guttman scalogram with the coefficient of reproducibility, the Bradley-Terry-Luce model for paired comparisons (Bradley and Terry 1952 <doi:10.1093/biomet/39.3-4.324>; Luce 1959) as the conditional form of the dichotomous Rasch model (Andrich 1978), estimated by the same conventions with judge-clustered sandwich errors and judge fit diagnostics, and the first software implementation of the extended frame of reference model (Humphry 2005; Humphry and Andrich 2008), in which the unit of the latent scale differs across item-set by person-group frames: group units are estimated by person-free within-frame pairwise conditioning and set units by error-corrected person linking, all reported in a common arbitrary unit; its paired-comparison form estimates judge-panel and object-set units with the linking identified from cross-set comparisons alone. A modern 'shiny' interface and a one-call exporter for every table and plot are included. Implemented from published measurement theory in base R, with no dependence on other estimation engines.

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