To illustrate a one-facet crossed design, we created an example that assesses eight persons' responses on four multiple-choice items, and the data are in Table 19.3.
The person scores on the items are decomposed into four parts:
Xpi = M + - M) + (jU, -H) + (Xp. -flp- fj,. + ¡A),
where Xpj is the score of person p on item i; ju is the grand mean, the expectation over persons and items; (ju - pi) is the person effect, the expectation of the persons' score over items; (jUj - /J.) is the item effect, the expectation of the item difficulty over persons; and (Xpj- /Up- ¡u. + ¡J.) is the residual, which includes both the interaction effect between items and persons and all error components. These two effects cannot be distinguished because we have only one observation for each person-item combination.
Each effect in Equation (14), except the grand mean, has a distribution with a mean of zero and a nonzero variance (Shavelson & Webb, 1991). Standard AN OVA estimates the mean squares as MS
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