Sample Size Requirements for Bifactor Models

Abstract

Despite the widespread application of bifactor models, little research has considered required sample sizes for this type of model. As universal sample size recommendations are often misleading, we illustrate how to determine sample size requirements of bifactor models using Monte Carlo simulations in R. Furthermore, we present results of an extensive simulation study investigating the effects of the number of specific factors and indicators, loading magnitude, the relative general factor strength, and the validity of the proportionality condition on sample size requirements. Although a sample size of 500 was often sufficient to obtain acceptable convergence rates and parameter estimates, the exact sample size requirements depended on various model characteristics.

Keywords:

• Bifactor model
• Monte Carlo simulation
• R
• sample size
• simsem

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Correction

Data availability

The Supplementary materials are uploaded at the Open Science Framework (https://osf.io/uq5yb/).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article was originally published with errors, which have now been corrected in the online version. Please see Correction (http://dx.doi.org/10.1080/10705511.2022.2123641)

Notes

1 Also, several exploratory bifactor analysis methods have been developed (for a review and comparison, see Giordano & Waller, 2020). However, this study focuses exclusively on the confirmatory variant.

2 The analyses in this article were performed using simsem development version 0.5-16.908.

3 Alternatively, simsem allows to specify the population model using a matrix style approach or the OpenMx software. For details, see https://github.com/simsem/simsem/wiki/Vignette.

4 Users may test the stability of their results by running the simulation multiple times specifying different random numbers in the seed argument of the sim function.

5 As the number of proper solutions specified in the nRep argument indicates only the minimum target number, the final number of proper solutions may exceed this number. This is because if there are non-converged solutions, simsem increases the total number of replications based on the estimated convergence rate such that it is expected to yield the target number of proper solutions. However, it is possible that within those replications more proper solutions are obtained.

6 Specifically, ${\lambda }_G$ = 0.70 combined with ECV = 0.50 would translate into ${\lambda }_S$ = 0.70, which yields a very high indicator communality (0.70² + 0.70² =0.98). Also, ${\lambda }_G$ = 0.70 combined with ECV = 0.25 translating into an inadmissibly high ${\lambda }_S$ = 1.21, and ${\lambda }_G$ = 0.50 combined with ECV = 0.25 translating into ${\lambda }_S$ = 0.87 would yield indicator communalities > 1.

7 Note that in 6% of all conditions (55 out of 960), no 500 properly converged solutions were obtained within the maximum number of 5,000 replications (see Table S1). For these conditions, the remaining outcomes should be interpreted with caution.

8 Note that although the approach by Vale and Maurelli (1983) to generate non-normal data is widely used, it is subject to criticism, such as being limited in the range of realizable skewness and kurtosis values or having multivariate properties similar to a multivariate normal distribution (e.g., Foldnes & Grønneberg, 2015; Olvera Astivia & Zumbo, 2015). Thus, users may employ alternative methods for data generation, for instance, using copulas (Mair et al., 2012) or a non-linear structural model (Auerswald & Moshagen, 2015).

9 Fitting higher-order models (assuming perfect proportionality) to the population variance-covariance matrices of the bifactor models deviating from the proportionality condition, revealed only a small degree of misspecification: The average population minimum of the maximum likelihood fit function (F₀) was 0.10 (SD = 0.12), the average root-mean-square error of approximation (RMSEA) was 0.012 (SD = 0.01), and the average standardized-root-mean-square residual (SRMR) was 0.013 (SD = 0.01).