Abstract
In this paper we present a new implication of the unidimensional factor model. We prove that the partial correlation between two observed variables that load on one factor given any subset of other observed variables that load on this factor lies between zero and the zero-order correlation between these two observed variables. We implement this result in an empirical bootstrap test that rejects the unidimensional factor model when partial correlations are identified that are either stronger than the zero-order correlation or have a different sign than the zero-order correlation. We demonstrate the use of the test in an empirical data example with data consisting of fourteen items that measure extraversion.
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Notes
Let \(x = \rho _{y_iy_j\cdot Z}/\rho _{y_iy_j}\). It follows that \(0< x< 1 \iff 0< (x - 0.5) + 0.5< 1 \iff -0.5< x - 0.5< 0.5 \iff |x - 0.5|< 0.5 \iff 2|x - 0.5|< 1 \iff |2x - 1| < 1\).
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Funding was provided by European Research Council (Career Integration Grant) (Grand No 631145) and European Research Council (Consolidator Grant) (Grand No 647209).
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We would like to thank Mijke Rhemtulla and Denny Borsboom for their help in constructing the theory that later resulted in the proof presented in this paper. We would like to thank Sacha Epskamp for his helpful comments.
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van Bork, R., Grasman, R.P.P.P. & Waldorp, L.J. Unidimensional factor models imply weaker partial correlations than zero-order correlations. Psychometrika 83, 443–452 (2018). https://doi.org/10.1007/s11336-018-9607-z
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DOI: https://doi.org/10.1007/s11336-018-9607-z