Compares a fitted partial credit model with the rating scale reparameterisation of the same data. Both raw and composite-likelihood adjusted statistics are returned.
Arguments
- fit
An unrestricted, unanchored
"PCM"fit fromraschwith 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), retaining keyed scoring, DIF-split
records and superitem definitions for subsequent analyses.
Details
The pairwise conditional likelihood is a composite likelihood: each response contributes to every item pair in which it appears. Consequently, the raw statistic \(W=2(cl_{PCM}-cl_{RSM})\) does not have an ordinary chi-square reference distribution. Its limiting distribution is \(\sum_j\lambda_j\chi^2_1\) (Kent 1982; Varin, Reid and Firth 2011), where the \(\lambda_j\) are obtained from the sensitivity matrix \(H\), variability matrix \(J\), and the constraints defining the RSM. The mean-matched statistic is $$W_{adj}=rW/\sum_j\lambda_j,$$ with \(r\) degrees of freedom.
Use p_adj for inference. The unadjusted p is retained for
descriptive comparison with conventional displays. The adjustment is a
first-order approximation and can be mildly anti-conservative in small
samples with long polytomous tests. Interpret values near the nominal
level cautiously in such designs.
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