ON CLIFF FORMULA, THE KAISER-GUTTMAN RULE, AND THE NUMBER OF FACTORS

Cliff (1988) has presented a formula for the reliability of a principal component which is different from my long-known formula (Kaiser, 1957, 1991) for coefficient alpha of a principal component. Cliff claims that his approach is "correct" and mine "is the result of a misapplication of the formula for internal consistency reliability." Actually, both developments are correct but are based on different premises: Cliff considers measurement error within-but not between-attributes, while I consider measurement error between-but not within-attributes. The application of my formula to the knotty problem of the "number of factors"-the Kaiser-Guttman Rule-appears often to give the "right" result, when "right" means agreement with the subjective judgment of factor-analytic grandmasters. But when it fails it is approximately equally likely to overfactor as to underfactor. Cliff's formula, on the other hand, when used to establish the number of factors, almost invariably overfactors and, in the limit, as the within-attribute reliabilities all approach one (as with, say, physical attributes), nonsensically will declare all principal components perfectly reliable no matter how small their associated eigenvalues, yielding an absurd answer if used to establish the number of factors.