help for swain, swaini

Correct the SEM chi-square overidentification test in small sample sizes or complex models

Description

swain and swaini provide a suite of corrections to the chi-square overidentification
test (i.e., likelihood ratio test of fit) for structural equation models whether with or without
latent variables. The chi-square statistic is asymptotically correct; however, it does not behave as
expected in small samples and/or when the model is complex (cf. Herzog, Boomsma, & Reinecke, 2007).
In situations where the ratio of the sample size to the number of parameters estimated is relatively
small, the chi-square test will tend to overreject correctly specified models. Applied researchers
should thus use a statistic having a more appropriate Type 1 reject error rate to judge model fit
in finite samples (Bastardoz & Antonakis, 2014, 2016).

The module provides three corrections: (a) the Swain (1975) scaling factor (see Herzog & Boomsma,
2009; Herzog, et al., 2007); (b) an empirical correction T(MLEC1) by Yuan, Tian, and Yanagihara (2015),
and (c) the F-test, which can be thought of as the small sample correction to the chi-square test (McNeish,
2020).

Swain can be executed:

1. after -sem- as a postestimation command by simply typing swain

2. or immediately (see immed) as follows: swaini #vars #df #N #chi

Examples

Note, the examples below are from correctly specified models. They should not be rejected
by the chi-square overidentification statistic (yet they are). As shown in the examples, -swain-
and -swaini- appropriately correct the statistic, which, on the basis of all the corrections,
becomes nonsignificant. In practice the true model is unknown; however, there is sufficient
simulation evidence to show that on repeated sampling, Type 1 error will be held to approximately
5% for true models.

To run the commands, simply click on the relevant links.

1. Immediate version: -swaini-

Enter data in the following order: #vars #df #N #chi. Note, the example data for the
immediate version are based on the full structural equation model in the 3rd example below.
We will use the chi-square value to two decimal places only, as reported in the output of
the model; thus the minor difference in the -swaini- as compared to the -swain-
chi-square value is because of rounding error.

. swaini 22 208 50 265.71

2. Postestimation for confirmatory factor analysis: -swain-

. use http://www.hec.unil.ch/jantonakis/swaindata.dta

. sem (X->x*)

. swain

3. Postestimation for full structural equation model: -swain-

. use http://www.hec.unil.ch/jantonakis/swaindata.dta

. sem (X->x*) (X->m) (m->y), cov(e.m*e.y)

. swain

Citation

swain is not an official Stata command. Please cite it as:

Antonakis, J., & Bastardoz, N. (2020) Swain: Stata module to correct the SEM
chi-square overidentification test in small sample sizes or complex models.
http://econpapers.repec.org/software/bocbocode/s457617.htm

Authors

John Antonakis, University of Lausanne, Switzerland
john.antonakis@unil.ch

Nicolas Bastardoz, University of Zurich, Switzerland
nicolas.bastardoz@business.uzh.ch

References and readings

Bastardoz, N., & Antonakis, J. (2014). Sample size requirement for unbiased
estimation of structural equation models. A Monte Carlo study. Academy of Management
Proceedings, 13405.

Bastardoz, N., & Antonakis, J. (2016). How should the fit of structural
equation models be judged? Insights from Monte Carlo simulations. Academy of
Management Proceedings, 12634.

Curran, P. J., Bollen, K. A., Paxton, P., Kirby, J., & Chen, F. N. (2002).
The noncentral chi-square distribution in misspecified structural equation models:
Finite sample results from a Monte Carlo simulation. Multivariate Behavioral Research,
37(1), 1-36.

Herzog, W., & Boomsma, W. (2009). Small-sample robust estimators of
noncentrality-based and incremental model fit. Structural Equation Modeling, 16(1), 1–27.

Herzog, W., Boomsma, W., & Reinecke, S. (2007). The model-size effect on traditional
and modified tests of covariance structures. Structural Equation Modeling, 14(3), 361–90.

Jackson, D. L., Voth, J., & Frey, M. P. (2013). A Note on Sample Size and Solution
Propriety for Confirmatory Factor Analytic Models. Structural Equation Modeling, 20(1), 86-97.

McNeish, D. (2020). Should We Use F-Tests for Model Fit Instead of Chi-Square in
Overidentified Structural Equation Models? Organizational Research Methods, 23(3), 487-510.

Swain, A. J. (1975). Analysis of parametric structures for variance matrices
(doctoral thesis). University of Adelaide, Adelaide.

Yuan, K.-H., Tian, Y., & Yanagihara, H. (2015). Empirical correction to the likelihood ratio
statistic for structural equation modeling with many variables. Psychometrika, 80(2), 379-405.

Saved results

swain and swaini save the following results in r():

r(swain_corr) Swain correction factor
r(swain_chi) Swain-corrected chi-square
r(swain_p) p-value of the Swain chi-square
r(f_test) F-test value
r(f_test_p) p-value of the F-test
r(yuan_chi) Yuan-Tian-Yanagihara corrected chi-square value
r(yuan_p) p-value of Yuan-Tian-Yanagihara empirically corrected chi-square

Also see

Manual: [R] sem

On-line: help for sem