How Interaction Denies Objectivity

Suppose the five Tables below show the average gains observed for the same numbers of Boys and Girls when taught from books X and Y. Each Table shows a different hypothetical situation.

Table 1: Book Y gains more
Gender
BookBoysGirlsAll
X 20 20 20
Y 40 40 40
Both books 30 30 30

Table 1 tells us that performance gain on book Y was better than on book X regardless of gender.

Table 2: Girls gain more
Gender
BookBoysGirlsAll
X 20 40 30
Y 20 40 30
Both books 20 40 30

Table 2 tells us that Girls gain more than Boys regardless of book used.

Table 3: Book Y and Girls gain more
Gender
BookBoysGirlsAll
X 20 40 30
Y 40 80 60
Both books 30 60

In Table 3, even though Boys gain more than Girls and performance gain on book Y is better than on book X, each of these conclusions can still be drawn separately.

In Tables 1, 2 and 3, the gender conclusion does not depend on which book was used and the book conclusion does not depend on which gender. Data which allow the book conclusion to be gender-free and the gender conclusion to be book-free are data which enable objective inference about book and gender.

Now inspect Tables 4 and 5 in which there is an interaction between book and gender.

Table 4: Book Y and Girls gain more?
Gender
BookBoysGirlsAll
X 40 20 30
Y 40 80 60
Both books 40 50

In Table 4, gain on book X is better for Boys while book Y is better for Girls. We might want to conclude from the "Both books" row and the "All" column that the performance gain on book Y was better for everyone and that Girls gained more on both books, but that would be a conclusion which these data do not support.

Table 5: Who gains more?
Gender
BookBoysGirlsAll
X40 20 30
Y 20 40 30
Both books 30 30

In Table 5, book X is better for Boys and book Y is better for Girls, but there is no basis at all for concluding that performance gain on either book is better, or that either gender gains more. The presence of interaction in these data denies objective inferences about book and gender.

If we intend our research to lead to objective inference, i.e., to provide useful contributions to knowledge, then we must model, construct and organize our data so that interactions of the kinds shown in Tables 4 and 5 do not occur.

Objectivity and IRT Models

The "2" and "3" parameter IRT models (2-PL, 3-PL) introduce guessing and discrimination parameters which model interaction and thereby destroy the possibility of objective inference concerning person abilities or item difficulties.

It also follows that item response data which are allowed to manifest substantial variation in guessing and discrimination will not sustain objective inferences concerning persons or items.

Andrew Stephanou contributed to this version.

How interaction denies objectivity. Wright BD. … Rasch Measurement Transactions, 1988, 1:2 p.12

Please help with Standard Dataset 4: Andrich Rating Scale Model

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