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Modelling non‐normal data: The relationship between the skew‐normal factor model and the quadratic factor model
Author(s) -
Smits Iris A. M.,
Timmerman Marieke E.,
Stegeman Alwin
Publication year - 2016
Publication title -
british journal of mathematical and statistical psychology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.157
H-Index - 51
eISSN - 2044-8317
pISSN - 0007-1102
DOI - 10.1111/bmsp.12062
Subject(s) - skew , quadratic equation , normality , mathematics , factor (programming language) , factor analysis , skew normal distribution , statistics , skewness , econometrics , computer science , telecommunications , geometry , programming language
Maximum likelihood estimation of the linear factor model for continuous items assumes normally distributed item scores. We consider deviations from normality by means of a skew‐normally distributed factor model or a quadratic factor model. We show that the item distributions under a skew‐normal factor are equivalent to those under a quadratic model up to third‐order moments. The reverse only holds if the quadratic loadings are equal to each other and within certain bounds. We illustrate that observed data which follow any skew‐normal factor model can be so well approximated with the quadratic factor model that the models are empirically indistinguishable, and that the reverse does not hold in general. The choice between the two models to account for deviations of normality is illustrated by an empirical example from clinical psychology.