Mean-mean equating coefficients

Formula: the scale linking theta_R = A theta_F + B carries the IRT item parameters as b_R = A b_F + B and a_R = a_F / A. Averaging each relation over the common items gives A = mean(a_F) / mean(a_R) and B = mean(b_R) - A mean(b_F).

Usage

Eqmm(y, a_R, b_R, a_F, b_F)

Arguments

y

Scores on the Form F metric to place on the Form R metric.

a_R, b_R

Common-item discriminations and difficulties on the Form R (reference) metric.

a_F, b_F

The same items' parameters on the Form F metric.

Value

List with estimate, A, B, equated, n_items, n, method.

Details

Note the orientation: A = mean(a_F)/mean(a_R), not its reciprocal. Discrimination is inversely proportional to the scale factor – stretching the ability metric by A flattens the item characteristic curve by the same factor – so the ratio must be a_F over a_R, and only then is A consistent with B.

References

Loyd & Hoover (1980), Journal of Educational Measurement 17(3):179-193, doi:10.1111/j.1745-3984.1980.tb00825.x.

Examples

Eqmm(y = c(1, 2, 3, 4, 5, 6, 7, 8), a_R = c(1, 2, 3, 4, 5, 6, 7, 8),
  b_R = c(1, 2, 3, 4, 5, 6, 7, 8), a_F = c(1, 2, 3, 4, 5, 6, 7, 8),
  b_F = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] 1
#> 
#> $A
#> [1] 1
#> 
#> $B
#> [1] 0
#> 
#> $equated
#> [1] 1 2 3 4 5 6 7 8
#> 
#> $n_items
#> [1] 8
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "Mean-mean equating coefficients"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"