Equate two test calibrations through their common items

Places two calibrations on a common origin using their shared items, then tests the shared items for drift. The reference may be a fitted model or an item bank.

Usage

equate_tests(fit, reference, shift = c("mean", "none"), independent = NULL)

Arguments

fit: A fitted object from rasch.
reference: A second rasch fit, or a data frame with columns item, location, and optionally se. Item names and column names must be unique. Numeric fields may be numeric columns, numeric text, or factors with numeric labels; other column classes are refused. Locations must be finite. For bank-based drift inference with an estimated mean shift, attach the bank's joint item-location covariance as a square matrix in attr(reference, "cov_location"), ordered like the bank rows (or named by item); marginal SEs alone do not carry the centring covariance. They are sufficient with shift = "none". A bank treated as fixed may instead have zero SEs. A polytomous bank must also include max, the maximum item score. A bank covariance estimated from a finite number of independent sampling units may carry its positive residual degrees of freedom in attr(reference, "df_location").
shift: "mean" (default) allows a scale shift between the two analyses; "none" compares raw locations, appropriate when both analyses are already on a shared (anchored) scale.
independent: Whether the two calibrations use independent sampling units. For two fitted objects this must be stated explicitly: the default NULL withholds drift tests because cross-fit covariance is otherwise unknown. A bank table is treated as independent unless FALSE is supplied. When FALSE, descriptive equating is returned but inferential drift columns are withheld.

Value

A list with the comparison table (locations, standard errors, difference, its standard error, t statistic, reference degrees of freedom, raw and Holm-adjusted p, drift flag), the estimated shift and its shift_method, the location correlation, the root mean square difference after shifting (rmsd), the number of common items n_common, the number with usable standard errors n, and whether drift inference was available (inferential). The note component records exclusions and the reason inference was withheld, where applicable. An individual drift probability is also withheld when its contrast has zero estimated uncertainty. Common items with unavailable drift probabilities remain in the multiplicity family.

Details

Let $d_j$ be the location difference for common item $j$ and $v_j$ its marginal variance. With shift = "mean" the scale shift is the precision-weighted mean $$\hat s=\frac{\sum_j d_j/v_j}{\sum_j 1/v_j},$$ and each item is tested using $d_j-\hat s$ with a variance that accounts for the estimated shift through the items' joint covariance. If fewer than two common items have usable variances but at least two have finite locations, the function returns their unweighted mean difference as a descriptive fallback and records shift_method = "unweighted". An exact common anchor determines the shift even when it is the only common item with usable uncertainty; unavailable SEs do not override it. When the shift is estimated, drift inference requires independent calibrations and at least three common items with usable, positive-semidefinite joint covariance information. With shift = "none", the origin is fixed before the comparison and each item's variance is the sum of its two marginal variances; joint covariance information and a three-item link are then unnecessary. One common item is sufficient for that fixed-origin comparison; estimating a shift still requires at least two. Otherwise the function returns a descriptive link. A fitted calibration's empirical covariance must also pass the informative-person count, effective-support and projected-rank checks used by rasch. Supported independent rows use the limiting normal reference. When either covariance comes from repeated person clusters, drift probabilities use contrast-specific Welch–Satterthwaite degrees of freedom. The corresponding residual degrees of freedom are the number of independent person clusters minus one. A fixed anchor contributes zero variance and does not consume cluster degrees of freedom. Fitted calibrations must have converged.

Examples

set.seed(1); d <- seq(-1.5, 1.5, length.out = 8)
mk <- function() {
  X <- matrix(rbinom(400 * 8, 1, plogis(outer(rnorm(400), d, "-"))), 400, 8)
  colnames(X) <- paste0("I", 1:8); rasch(X)
}
eq <- equate_tests(mk(), mk(), independent = TRUE)
eq$table
#>   item location_1      se_1 location_2      se_2  difference adj_difference
#> 1   I1 -1.5760281 0.1292289 -1.4780195 0.1246379 -0.09800863    -0.10727215
#> 2   I2 -1.1398494 0.1155055 -1.1006867 0.1168127 -0.03916271    -0.04842624
#> 3   I3 -0.5314218 0.1083620 -0.8227470 0.1123970  0.29132523     0.28206171
#> 4   I4 -0.2287656 0.1063021 -0.2085726 0.1093047 -0.02019291    -0.02945643
#> 5   I5  0.3229450 0.1053953  0.1917743 0.1077741  0.13117072     0.12190720
#> 6   I6  0.5366178 0.1097245  0.7088231 0.1124051 -0.17220529    -0.18146881
#> 7   I7  1.1874751 0.1180524  1.1279739 0.1189861  0.05950128     0.05023776
#> 8   I8  1.4290269 0.1263670  1.5814546 0.1284329 -0.15242768    -0.16169120
#>     se_diff          t  df          p     p_adj drift
#> 1 0.1844083 -0.5817100 Inf 0.56076205 1.0000000 FALSE
#> 2 0.1640769 -0.2951436 Inf 0.76788421 1.0000000 FALSE
#> 3 0.1541893  1.8293214 Inf 0.06735147 0.5388118 FALSE
#> 4 0.1489597 -0.1977476 Inf 0.84324253 1.0000000 FALSE
#> 5 0.1466567  0.8312417 Inf 0.40583712 1.0000000 FALSE
#> 6 0.1551459 -1.1696652 Inf 0.24213573 1.0000000 FALSE
#> 7 0.1692070  0.2969011 Inf 0.76654200 1.0000000 FALSE
#> 8 0.1850922 -0.8735713 Inf 0.38235177 1.0000000 FALSE