A note on residual M-distances for identifying aberrant response patterns

Although a statistical model might fit well to a large proportion of the individuals of a random sample, some individuals might give ‘unusual’ responses that are not well explained by the hypothesized model. If individual responses are given as continuous response vectors, M-distances can be used to produce real valued indicators of how well an individual's response vector corresponds to a covariance structure implied by a psychometric model. In this note, we focus on the so-called one-factor model. Two M-distances, d_si and d_ri, which are sensitive to different aspects of the assumed factor model, have been proposed. While one of the M-distances, d_ri, has been derived based on Bartlett factor scores, in this note we show that the second M-distance, d_si, can be derived in an analogous fashion based on Thomson factor scores.