The arrow that represents a compensatory item in the ability plane. It points in the direction in which the probability rises fastest, at the angle \(\arccos(a_1 / A)\) to the \(\theta_1\) axis; its length is the multidimensional discrimination \(A = \sqrt{a_1^2 + a_2^2}\); and it starts on the line where the probability is halfway between the asymptotes, at the signed distance \((a_1 b_1 + a_2 b_2) / A\), the multidimensional difficulty, from the origin.
Value
A list with start and end, the coordinates of the two
ends of the arrow, discrimination, the multidimensional
discrimination \(A\), and difficulty, the multidimensional
difficulty.
Examples
item_vector(a = c(1, 1), b = c(0, 0))
#> $start
#> [1] 0 0
#>
#> $end
#> [1] 1 1
#>
#> $discrimination
#> [1] 1.414214
#>
#> $difficulty
#> [1] 0
#>
item_vector(a = c(2, 0.5), b = c(1, -1))
#> $start
#> [1] 0.7058824 0.1764706
#>
#> $end
#> [1] 2.7058824 0.6764706
#>
#> $discrimination
#> [1] 2.061553
#>
#> $difficulty
#> [1] 0.7276069
#>