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Person fit indices: lz and Snijders-corrected lz*

Source: R/api-person-fit.R
compute_person_fit_indices.Rd

Computes person-level fit statistics for an MFRM bundle, extending the Infit / Outfit / ZSTD columns that diagnose_mfrm()$measures already exposes with the standardized log-likelihood lz and, when justified by the person-estimation method, Snijders' lz*.

Usage

compute_person_fit_indices(diagnostics, fit = NULL)

Arguments

diagnostics

Output from diagnose_mfrm().

fit

Optional mfrm_fit from fit_mfrm(). Required to decide whether the person estimates are JML/fixed-effect estimates for which the Snijders (2001) correction is computed. MML/EAP person scores return NA for lz_star with an explanatory status.

Value

A data frame of class mfrm_person_fit_indices with one row per Person and columns:

Person

Person ID.

N

Number of contributing response opportunities.

LogLik

Sum of log P(X = x | theta) under the fitted model. Computed from the per-observation category probability PrObserved (the model probability of the observed category), not from a Gaussian residual approximation.

lz

Drasgow et al. (1985) standardized log-likelihood, in its proper polytomous form.

lz_star

Snijders-corrected lz* when the source fit used JML/fixed-effect person estimates, conditioning on the fitted non-person calibration, and the diagnostics include the required derivative terms; otherwise NA.

lz_star_status

Status string for lz_star, such as "computed_jml_conditional_calibration", "fit_required", "not_applicable_eap", or "insufficient_information".

lz_star_c

Estimated Snijders projection coefficient c_n for each person, when available.

lz_star_variance

Corrected variance denominator used for lz_star, when available.

lz_flag_5pct, lz_flag_1pct

Logical flags for practical two-sided lz thresholds of |z| > 1.96 and |z| > 2.58.

lz_star_flag_5pct, lz_star_flag_1pct

The same flags for lz_star, returned as FALSE when lz_star is unavailable.

ReportIndex, ReportValue, ReportFlagLevel, ReportFlag, ReviewStatus, ReviewReason, ReportCaveat

Compact reporting columns. ReportIndex prefers lz_star when the Snijders correction was computed; otherwise it falls back to lz with an explicit caveat.

Under the conditional-independence assumption of the MFRM, lz is asymptotically standard normal. Practical reporting thresholds: |lz| > 1.96 flags a person at the 5% level; |lz| > 2.58 at the 1% level. When lz_star_status == "computed_jml_conditional_calibration", lz_star applies Snijders' estimated-ability correction for JML person estimates, conditional on the fitted non-person parameters. This does not propagate non-person calibration uncertainty. For MML/EAP person scores, use lz with its documented caveat rather than treating EAP scores as if they satisfied the Snijders estimating equation.

Note: this implementation reads the model category probabilities directly from the diagnostics bundle. Earlier mfrmr releases used a Gaussian-residual approximation \(\log P(X = x) \approx -\tfrac{1}{2}(R^2/V) - \tfrac{1}{2}\log(2\pi V)\) as a stand-in for \(\log P\), which overstated the per-item variance of \(\log P\) for polytomous items, shrinking the reported lz toward zero. Numerical lz values are therefore not directly comparable across mfrmr releases; treat the values returned here as the polytomous statistic and re-evaluate any historical |lz| > 1.96 flagging that was based on the earlier approximation.

References

  • Drasgow, F., Levine, M. V., & Williams, E. A. (1985). Appropriateness measurement with polychotomous item response models and standardized indices. British Journal of Mathematical and Statistical Psychology, 38(1), 67-86.

  • Snijders, T. A. B. (2001). Asymptotic null distribution of person fit statistics with estimated person parameter. Psychometrika, 66(3), 331-342.

  • Magis, D., Raiche, G., & Beland, S. (2012). A didactic presentation of Snijders's lz* index of person fit with emphasis on response model selection and ability estimation. Journal of Educational and Behavioral Statistics, 37(1), 57-81.

  • Sinharay, S. (2016). Asymptotically correct standardization of person-fit statistics beyond dichotomous items. Psychometrika, 81(4), 992-1013.

See also

diagnose_mfrm()

Examples

if (FALSE) { # interactive()
toy <- load_mfrmr_data("example_core")
fit <- fit_mfrm(toy, "Person", c("Rater", "Criterion"), "Score",
                method = "JML", maxit = 30)
diag <- diagnose_mfrm(fit, residual_pca = "none",
                      diagnostic_mode = "legacy")
pf <- compute_person_fit_indices(diag, fit = fit)
head(pf)
summary(pf)
# Look for: |lz| > 1.96 (5% level) flags a person whose response
#   pattern is statistically inconsistent with the model; > 2.58 is
#   a 1% flag. lz_star is populated for JML/fixed-effect person
#   estimates and left NA for MML/EAP estimates. Use ReportIndex /
#   ReviewStatus for a compact report-ready reading.
}