where X is the score of person p on item i; ¡1 is the grand mean; - ¡1) is the person effect; and (Xpi -jUp) is the residual. There is no separate term for the item effect. Because all students have answered different items, the item effect cannot be estimated, and the item effect becomes part of the residual.
Each effect in Equation (22), except the grand mean, has a distribution with a mean of zero and a variance. The ANOVA estimates (people are random, items not) are MSperson = 0.348 and MSmsidual = 0.188, and the calculated variance components (see
Equations  and ) are a2p = 0.080and <72 ., ,= 0.188. Thus, 30% of the variance is residual '
associated with the people, the remainder with the items, the interaction, and error.
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