Shows that optimal linear combinations provide better measures of latent traits than sum scores, suggesting improved evaluation methods.
We show that for a unidimensional behavioral measuring instrument with uncorrelated errors and normal components, their linear combination with maximal reliability is a minimal sufficient statistic for the latent trait evaluated by the instrument. The popular simple scale sum score is in general not a sufficient statistic for the underlying construct, and becomes sufficient for it only under particular restrictive conditions. The findings are illustrated with data showing that this optimal linear combination, as a minimal sufficient statistic, (i) is more strongly correlated with the studied ability than the sum score is with the latter; and (ii) is associated with a lower average squared distance from the latent variable than that distance is for the widely used sum score.
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Zhang et al. (2026) studied this question.
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