The residual-PCA analogue for paired comparisons. The fitted model predicts
how often each object should beat each other from their locations; the
object-by-object matrix of departures from that prediction (on the
log-odds scale) is skew-symmetric, so its structure decomposes into
rotational planes – Gower's (1977) bimensions – rather than the ordinary
components of a symmetric residual-correlation matrix. A dominant leading
bimension is a coherent “swirl” in the residuals (A over-beats B, B
over-beats C, C over-beats A): a second attribute steering some contests. A
flat spectrum is noise: the single scale suffices. The leading bimension is
judged against a reference built by simulating unidimensional data from the
fitted model with the observed pair counts (a parametric bootstrap, as in
plot_scree); an observed strength above the reference is
structure the one-dimensional model does not explain. For graded fits the
residual log-odds are taken on the points-proportion scale, whose model
mean is not exactly plogis(beta_i - beta_j); the simulated reference
carries the same construction, so the test stays calibrated (verified
mildly conservative on model-true graded data) rather than anticonservative.
Likewise, when the fit carries within-judge dependence effects
(order), the reference is simulated sequentially through each
judge's comparisons WITH the fitted exposure and carry-over coefficients:
order effects push the marginal pair rates around in a structured way, and
a reference without them would read that structure as a second attribute.
The price is power: carry-over and a judge-camp second attribute are
partially confounded (both appear as consistent within-judge deviation),
so with order modelled the test is conservative about attributing
the ambiguous share to a second dimension.
The reference simulates from the point estimates without refitting each
replicate, so it carries sampling noise in the responses but not
estimation noise in the parameters – adequate for the screening use
here, slightly liberal in tiny designs.
Arguments
- fit
A paired-comparison fit from
btl.- reps
Model-simulated replicates for the noise reference.
Value
A list of class "rasch_btl_dim": bimensions (per
bimension: strength and share of residual size; the reference mean, 95th
percentile, and the clears-the-reference flag are reported for the
leading bimension and NA for the rest);
coords (each object's position in the leading bimension plane, for
the residual map); leading_structured (whether bimension 1 clears
its reference); residual_matrix; and notes.
References
Gower, J. C. (1977). The analysis of asymmetry and orthogonality. In J. R. Barra et al. (Eds.), Recent Developments in Statistics (pp. 109-123). North-Holland.
Examples
set.seed(1); objs <- LETTERS[1:6]; beta <- setNames(seq(-1.5, 1.5, len = 6), objs)
pr <- t(utils::combn(objs, 2))
d <- data.frame(a = rep(pr[, 1], each = 30), b = rep(pr[, 2], each = 30))
d$win <- ifelse(runif(nrow(d)) < plogis(beta[d$a] - beta[d$b]), d$a, d$b)
btl_dimensionality(btl(d, "a", "b", "win"), reps = 20)
#> Paired-comparison residual dimensionality: 3 bimension(s)
#> Leading bimension strength 1.370 (91% of residual; reference 95%: 2.440) -> within noise (one scale suffices)