Estimate Rasch thresholds using a principal-component parameterisation

Re-expresses each item's thresholds as orthogonal polynomial components: location, spread, skewness, and kurtosis (Andrich and Luo 2003). Estimation uses the same pairwise conditional likelihood as pcml. With at most four thresholds per item the full parameterisation is exact. Items with five or more thresholds are fitted by a reduced-rank polynomial trend, which can stabilise sparse categories at the cost of restricting the threshold pattern.

Usage

pcml_pc(X, n_components = 4, maxit = 60, tol = 1e-08)

Arguments

X: Persons-by-items integer score matrix. Each item must have at least two observed categories, numbered consecutively from 0. Missing values are handled by pairwise deletion.
n_components: Maximum number of components per item: 1 (location only) up to 4 (location, spread, skewness, kurtosis). Capped per item at its own number of thresholds. Kurtosis requires at least four thresholds (five response categories).
maxit, tol: Newton-Raphson iteration cap and convergence tolerance.

Value

A list with the threshold table thr (columns id, item, k, tau, se), the component table components (one row per item, with location, spread, skewness, kurtosis and their standard errors, NA where an item's rank does not support that component), the threshold covariance matrix cov_tau, the pairwise conditional log-likelihood, the iteration count, a convergence flag, and the max-score vector m. If estimation does not converge, the function warns and all standard errors and covariance entries are NA. The independent-person and effective-support conditions described for pcml also apply.

Details

For scores $x=0,\ldots,m$, the cumulative threshold function is $$C(x)=x\omega_1-x(m-x)\omega_2-x(m-x)(2x-m)\omega_3 -x(m-x)(5x^2-5mx+m^2+1)\omega_4.$$ Threshold $k$ is $C(k)-C(k-1)$. The four coefficients are location, spread, skewness and kurtosis, respectively.

References

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.

Zwinderman, A. H. (1995). Pairwise parameter estimation in Rasch models. Applied Psychological Measurement, 19(4), 369–375.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561–573.

Andrich, D. (1985). An elaboration of Guttman scaling with Rasch models for measurement. In N. B. Tuma (Ed.), Sociological Methodology 1985 (pp. 33–80). Jossey-Bass.

Pedler, P. J. (1987). Accounting for psychometric dependence with a class of latent trait models. PhD thesis, University of Western Australia.

Examples

set.seed(1)
d <- seq(-1.5, 1.5, length.out = 6)
X <- matrix(rbinom(400 * 6, 1, plogis(outer(rnorm(400), d, "-"))), 400, 6)
colnames(X) <- paste0("I", 1:6)
pcml_pc(X)$components
#>   item   location location_se spread spread_se skewness skewness_se kurtosis
#> 1   I1 -1.5174835   0.1297013     NA        NA       NA          NA       NA
#> 2   I2 -0.7730904   0.1121427     NA        NA       NA          NA       NA
#> 3   I3 -0.2093836   0.1023242     NA        NA       NA          NA       NA
#> 4   I4  0.3667512   0.1047318     NA        NA       NA          NA       NA
#> 5   I5  0.7889434   0.1105680     NA        NA       NA          NA       NA
#> 6   I6  1.3442630   0.1221786     NA        NA       NA          NA       NA
#>   kurtosis_se
#> 1          NA
#> 2          NA
#> 3          NA
#> 4          NA
#> 5          NA
#> 6          NA