From a decision theoretic point of view a general
coefficient for tests, d, is derived. The coefficient is
applied to three kinds of decision situations. First,
the situation is considered in which a true score is
estimated by a function of the observed score of a
subject on a test (point estimation). Using the
squared error loss function and Kelley’s formula for
estimating the true score, it is shown that d equals
the reliability coefficient from classical test theory.
Second, the situation is considered in which the observed
scores are split into more than two categories
and different decisions are made for the
categories (multiple decision). The general form of
the coefficient is derived, and two loss functions
suited to multiple decision situations are described.
It is shown that for the loss function specifying constant
losses for the various combinations of categories
on the true and on the observed scores, the
coefficient can be computed under the assumptions
of the beta-binomial model. Third, the situation is
considered in which the observed scores are split
into only two categories and different decisions are
made for each category (dichotomous decisions).
Using a loss function that specifies constant losses
for combinations of categories on the true and observed
score and the assumption of an increasing
regression function of t on x, it is shown that coefficient
d equals Loevinger’s coefficient H between true
and observed scores. The coefficient can be computed
under the assumption of the beta-binomial
model. Finally, it is shown that for a linear loss
function and Kelley’s formula for the regression of
the true score on the observed score, the coefficient
equals the reliability coefficient of classical test
Van der Linden, Wim J & Mellenbergh, Gideon J. (1978). Coefficients for tests from a decision theoretic point of view. Applied Psychological Measurement, 2, 119-134. doi:10.1177/014662167800200112
Van der Linden, Wim J.; Mellenbergh, Gideon J..
Coefficients for tests from a decision theoretic point of view.
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