Likelihood-ratio test of the partial credit against the rating scale model

A likelihood-ratio test in the tradition of Andersen (1973): an unrestricted (partial credit) analysis is compared with the rating re-parameterisation of the same model on the same data. Twice the difference in the pairwise conditional log-likelihoods is referred to a chi-square on the difference in the number of threshold parameters. A non-significant outcome supports adopting the simpler rating parameterisation.

Usage

lr_test(fit, maxit = 60, tol = 1e-08)

Arguments

fit

A "PCM" fit from rasch with equal maximum scores across items (the rating parameterisation requires them).

maxit, tol

Passed to the rating-scale refit.

Value

A list of class "rasch_lr": raw chisq, df, p (the conventional display); adjusted chisq_adj, p_adj, and the eigenvalues lambda; the two log-likelihoods; and the rating-scale refit (fit_rsm).

Details

The likelihood here is the pairwise composite likelihood, not a full likelihood, and twice its difference is not chi-square distributed: each response enters every pair its item forms, so the raw statistic is inflated. Two statistics are therefore reported. chisq is the raw composite value with its naive p, the conventional display. The limiting law of the raw statistic is \(\sum_j \lambda_j \chi^2_1\) (Kent 1982; Varin, Reid and Firth 2011) with \(\lambda_j\) the eigenvalues of \((C'H^{-1}C)^{-1}\,C'H^{-1}JH^{-1}C\) over the \(r\) constrained directions \(C\) (the part of the partial-credit threshold space outside the rating subspace), estimated from the same Godambe \(H\) and \(J\) matrices that supply the sandwich standard errors; matching the mean gives chisq_adj \(= r W / \sum_j \lambda_j\) on \(r\) degrees of freedom. Use p_adj for inference; the naive p is severely anticonservative and kept only for comparability with conventional software displays.

References

Kent, J. T. (1982). Robust properties of likelihood ratio tests. Biometrika, 69, 19-27. Varin, C., Reid, N. and Firth, D. (2011). An overview of composite likelihood methods. Statistica Sinica, 21, 5-42.

Examples

set.seed(1)
tau <- c(-0.7, 0.7)
X <- sapply(seq(-1, 1, length.out = 6), function(d) vapply(rnorm(300),
  function(b) sample(0:2, 1, prob = item_moments(b, tau + d)$P), 0L))
colnames(X) <- paste0("Q", 1:6)
lr_test(rasch(X, model = "PCM"))
#> Likelihood-ratio test: partial credit vs rating parameterisation
#>   Raw composite chi-square 8.741 on 5 df, p = 0.120 (conventional display; anticonservative)
#>   Adjusted chi-square 2.062 on 5 df, p = 0.840 (Kent 1982 first-order calibration)
#>   log-likelihood (pairwise composite): PCM -2814.271, RSM -2818.642