Tests explanatory item, threshold or object restrictions against the
corresponding free calibration of the same responses. The inferential
result uses the first-order Kent calibration for the fitted likelihood and
sandwich covariance. The calibration coefficient of determination is
$$R^2_{cal}=1-\frac{\sum_j(\hat\eta^{free}_j-
\hat\eta^{expl}_j-\bar d)^2}{\sum_j(\hat\eta^{free}_j-
\bar\eta^{free})^2},$$
where \(\bar d\) removes the arbitrary scale origin. It describes the
proportion of variation in the well-determined free threshold calibration
(Rasch models) or free object calibration (comparative judgement)
reproduced by the explanatory model. It is at most one and may be negative.
It is not adjusted for the number of predictors, so with few calibrated
parameters it reads above zero even for an uninformative design.
r_squared_adj divides the unexplained proportion by its share of
the degrees of freedom, \(1-(1-R^2_{cal})(n-1)/\mathit{df}\), where
\(n\) counts the calibrated parameters compared and \(\mathit{df}\)
is \(n\) minus the rank of the retained explanatory design with its
origin, so exclusions that remove a level's only support reduce it. The
correction is exact for independent homoskedastic estimates fitted by
least squares, which these calibrations are not, so read it as a
descriptive optimism adjustment. Read either beside the test rather than
in place of it.
Value
A one-row data frame containing the raw and Kent-calibrated
statistics, degrees of freedom and parameter counts. The primary
p and the retained p_kent are the Kent-calibrated
probability. p_naive is the unscaled composite-likelihood
probability and is provided for methodological inspection, not
inference. r_squared is the calibration coefficient of
determination, r_squared_adj its degrees-of-freedom-adjusted
counterpart, and r2_basis names the calibrated parameters used.