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Heteroscedastic Latent Trait Models for Dichotomous Data

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Abstract

Effort has been devoted to account for heteroscedasticity with respect to observed or latent moderator variables in item or test scores. For instance, in the multi-group generalized linear latent trait model, it could be tested whether the observed (polychoric) covariance matrix differs across the levels of an observed moderator variable. In the case that heteroscedasticity arises across the latent trait itself, existing models commonly distinguish between heteroscedastic residuals and a skewed trait distribution. These models have valuable applications in intelligence, personality and psychopathology research. However, existing approaches are only limited to continuous and polytomous data, while dichotomous data are common in intelligence and psychopathology research. Therefore, in present paper, a heteroscedastic latent trait model is presented for dichotomous data. The model is studied in a simulation study, and applied to data pertaining alcohol use and cognitive ability.

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Notes

  1. A sensible designation as ‘skedasis’ is the Greek word for scatter or dispersion.

  2. Truly continuous observed scores are rare in psychological and educational measurement. It has been shown that pragmatically, a scale with 7 or more ordered levels can be treated as continuous (Dolan, 1994). Therefore, in this paper, the term ‘continuous variable’ is used to denote variables with 7 or more ordered levels.

  3. In fact, Millsap and Yun-Tein (2004) use the restriction of \(\hbox {VAR}(\hbox {y}_{pi}^{*}) = 1\) instead of fixing \(\sigma _{\varepsilon i}^{2} = 1\) as is done here. Because, \(\hbox {VAR}(\hbox {y}_{pi}^{*})=\alpha _{i}^{2} \times \sigma _{\theta }^{2}+\sigma _{\varepsilon i}^{2}\) in which \(\sigma _{\theta }^{2}\) is already identified by fixing \(\sigma _{\theta }^{2 }= 1\) (or \(\alpha _{i} = 1\)), fixing \(\hbox {VAR}(\hbox {y}_{pi}) = 1\) will result in \(\sigma _{\varepsilon i}^{2} = 1 - \alpha _{i}^{2}\) (or \(\sigma _{\varepsilon i}^{2} = 1- \sigma _{\theta }^{2})\) which thus implies a fixed \(\sigma _{\varepsilon i}^{2}\). The opposite holds as well, that is, fixing \(\sigma _{\varepsilon i}^{2} = 1\) implies a fixed \(\hbox {VAR}(\hbox {y}_{pi}^{*})\).

  4. Note that Molenaar et al. (2012) used \(\delta _{0i}\) = 1.5 instead of \(\delta _{0i} = 1,\); therefore, in the present case, \(\delta _{1i}\) is rescaled to correspond to the effect size in Molenaar et al (i.e. due to the difference in \(\delta _{0i}\), there is no one-to-one correspondence).

  5. The opinions expressed in this article are those of the author and do not necessarily reflect the views of the ICPSR. 

  6. RMSEA values smaller than 0.05 are generally considered to indicate good model fit. CFI and TLI values larger than 0.95 are considered to indicate good model fit, see Hu and Bentler (1999)

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Acknowledgments

In memory of Roger Millsap whose comments and guidance were of great help in preparing the final version of this paper. I am also indebted to Mijke Rhemtulla, Conor Dolan, and two anonymous reviewers for their valuable comments on a previous draft of the manuscript.

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  1. Psychological Methods, Department of Psychology, University of Amsterdam, Weesperplein 4, 1018 XA , Amsterdam, The Netherlands

    Dylan Molenaar

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Correspondence should be sent to Dylan Molenaar, Psychological Methods, Department of Psychology, University of Amsterdam, Weesperplein 4, 1018 XA Amsterdam, The Netherlands. E-mail: D.Molenaar@uva.nl

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Molenaar, D. Heteroscedastic Latent Trait Models for Dichotomous Data. Psychometrika 80, 625–644 (2015). https://doi.org/10.1007/s11336-014-9406-0

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