scholarly journals Supplemental Material for Type I Error Rates and Power of Several Versions of Scaled Chi-Square Difference Tests in Investigations of Measurement Invariance

2016 ◽  
Keyword(s):  
Measurement Invariance ◽  
Type I Error ◽  
Error Rates ◽  
Type I ◽  
Chi Square ◽  
Type I Error Rates

scholarly journals Bias, Type I Error Rates, and Statistical Power of a Latent Mediation Model in the Presence of Violations of Invariance

2017 ◽  
Vol 78 (3) ◽  
pp. 460-481 ◽  
Author(s):  
Margarita Olivera-Aguilar ◽  
Samuel H. Rikoon ◽  
Oscar Gonzalez ◽  
Yasemin Kisbu-Sakarya ◽  
David P. MacKinnon
Keyword(s):  
Measurement Invariance ◽  
Statistical Power ◽  
Type I Error ◽  
Error Rates ◽  
Parameter Estimates ◽  
Type I ◽  
Mediation Model ◽  
Type I Error Rates ◽  
Mediated Effects ◽  
The Impact

When testing a statistical mediation model, it is assumed that factorial measurement invariance holds for the mediating construct across levels of the independent variable X. The consequences of failing to address the violations of measurement invariance in mediation models are largely unknown. The purpose of the present study was to systematically examine the impact of mediator noninvariance on the Type I error rates, statistical power, and relative bias in parameter estimates of the mediated effect in the single mediator model. The results of a large simulation study indicated that, in general, the mediated effect was robust to violations of invariance in loadings. In contrast, most conditions with violations of intercept invariance exhibited severely positively biased mediated effects, Type I error rates above acceptable levels, and statistical power larger than in the invariant conditions. The implications of these results are discussed and recommendations are offered.

scholarly journals Robust Chi-Square in Extreme and Boundary Conditions: Comments on Jak et al. (2021)

Psych ◽  
2021 ◽  
Vol 3 (3) ◽  
pp. 542-551
Author(s):  
Tihomir Asparouhov ◽  
Bengt Muthén
Keyword(s):  
Boundary Conditions ◽  
Type I Error ◽  
Error Rates ◽  
Type I ◽  
Chi Square ◽  
Type I Error Rates ◽  
Chi Square Test

In this article we describe a modification of the robust chi-square test of fit that yields more accurate type I error rates when the estimated model is at the boundary of the admissible space.

scholarly journals Evaluating Equivalence Testing Methods for Measurement Invariance

2020 ◽  
Author(s):  
Alyssa Counsell ◽  
Rob Cribbie
Keyword(s):  
Measurement Invariance ◽  
Goodness Of Fit ◽  
Type I Error ◽  
Error Rates ◽  
Type I ◽  
Equivalence Testing ◽  
Fit Indices ◽  
Equivalence Test ◽  
Chi Square ◽  
Difference Test

Measurement Invariance (MI) is often concluded from a nonsignificant chi-square difference test. Researchers have also proposed using change in goodness-of-fit indices (ΔGOFs) instead. Both of these commonly used methods for testing MI have important limitations. To combat these issues, To combat these issues, it was proposed using an equivalence test (EQ) to replace the chi-square difference test commonly used to test MI. Due to concerns with the EQ's power, and adjusted version (EQ-A) was created, but provides little evaluation of either procedure. The current study evaluated the Type I error and power of both the EQ and EQ-A, and compared their performance to that of the traditional chi-square difference test and ΔGOFs. The EQ was the only procedure that maintained empirical error rates below the nominal alpha level. Results also highlight that the EQ requires larger sample sizes than traditional difference-based approaches or using equivalence bounds based on larger than conventional RMSEA values (e.g., > .05) to ensure adequate power rates. We do not recommend the proposed adjustment (EQ-A) over the EQ.

ModL: exploring and restoring regularity when testing for positive selection

2018 ◽  
Vol 35 (15) ◽  
pp. 2545-2554 ◽  
Author(s):  
Joseph Mingrone ◽  
Edward Susko ◽  
Joseph P Bielawski
Keyword(s):  
Positive Selection ◽  
Likelihood Ratio ◽  
Type I Error ◽  
Error Rates ◽  
Ratio Test ◽  
Type I ◽  
Chi Square ◽  
Type I Error Rates ◽  
Modified Likelihood Ratio Test ◽  
Modified Likelihood

Abstract Motivation Likelihood ratio tests are commonly used to test for positive selection acting on proteins. They are usually applied with thresholds for declaring a protein under positive selection determined from a chi-square or mixture of chi-square distributions. Although it is known that such distributions are not strictly justified due to the statistical irregularity of the problem, the hope has been that the resulting tests are conservative and do not lose much power in comparison with the same test using the unknown, correct threshold. We show that commonly used thresholds need not yield conservative tests, but instead give larger than expected Type I error rates. Statistical regularity can be restored by using a modified likelihood ratio test. Results We give theoretical results to prove that, if the number of sites is not too small, the modified likelihood ratio test gives approximately correct Type I error probabilities regardless of the parameter settings of the underlying null hypothesis. Simulations show that modification gives Type I error rates closer to those stated without a loss of power. The simulations also show that parameter estimation for mixture models of codon evolution can be challenging in certain data-generation settings with very different mixing distributions giving nearly identical site pattern distributions unless the number of taxa and tree length are large. Because mixture models are widely used for a variety of problems in molecular evolution, the challenges and general approaches to solving them presented here are applicable in a broader context. Availability and implementation https://github.com/jehops/codeml_modl Supplementary information Supplementary data are available at Bioinformatics online.

scholarly journals Evaluating Equivalence Testing Methods for Measurement Invariance

2020 ◽  
Author(s):  
Alyssa Counsell ◽  
Rob Cribbie ◽  
David B Flora
Keyword(s):  
Measurement Invariance ◽  
Goodness Of Fit ◽  
Type I Error ◽  
Error Rates ◽  
Type I ◽  
Equivalence Testing ◽  
Fit Indices ◽  
Equivalence Test ◽  
Chi Square ◽  
Difference Test

Measurement Invariance (MI) is often concluded from a nonsignificant chi-square difference test. Researchers have also proposed using change in goodness of fit indices (∆GOFs) instead. Both of these commonly used methods for testing MI have important limitations. To combat these issues, Yuan and Chan (2016) proposed using an equivalence test (EQ) to replace the chi-square difference test commonly used to test MI. Due to their concerns with the EQ’s power, Yuan and Chan also created an adjusted version (EQ-A), but provide little evaluation of either procedure. The current study evaluated the Type I error and power of both the EQ and EQ-A, and compared their performance to that of the traditional chi-square difference test and ∆GOFs. The EQ for nested model comparisons was the only procedure that always maintained empirical error rates below the nominal alpha level. Results also highlight that the EQ requires larger sample sizes than traditional difference-based approaches or using equivalence bounds based on larger than conventional RMSEA values (e.g., > .05) to ensure adequate power rates. We do not recommend Yuan and Chan’s proposed adjustment (EQ-A) over the EQ.

scholarly journals Chi-square and F Ratio: Which should be used when?

2018 ◽  
Vol 8 (2) ◽  
pp. 58-71
Author(s):  
Richard L. Gorsuch ◽  
Curtis Lehmann
Keyword(s):  
Count Data ◽  
Type I Error ◽  
Statistical Significance ◽  
Error Rates ◽  
P Value ◽  
Type I ◽  
Chi Square ◽  
P Values ◽  
Type I Error Rates ◽  
Criterion Variables

Approximations for Chi-square and F distributions can both be computed to provide a p-value, or probability of Type I error, to evaluate statistical significance. Although Chi-square has been used traditionally for tests of count data and nominal or categorical criterion variables (such as contingency tables) and F ratios for tests of non-nominal or continuous criterion variables (such as regression and analysis of variance), we demonstrate that either statistic can be applied in both situations. We used data simulation studies to examine when one statistic may be more accurate than the other for estimating Type I error rates across different types of analysis (count data/contingencies, dichotomous, and non-nominal) and across sample sizes (Ns) ranging from 20 to 160 (using 25,000 replications for simulating p-value derived from either Chi-squares or F-ratios). Our results showed that those derived from F ratios were generally closer to nominal Type I error rates than those derived from Chi-squares. The p-values derived from F ratios were more consistent for contingency table count data than those derived from Chi-squares. The smaller than 100 the N was, the more discrepant p-values derived from Chi-squares were from the nominal p-value. Only when the N was greater than 80 did the p-values from Chi-square tests become as accurate as those derived from F ratios in reproducing the nominal p-values. Thus, there was no evidence of any need for special treatment of dichotomous dependent variables. The most accurate and/or consistent p's were derived from F ratios. We conclude that Chi-square should be replaced generally with the F ratio as the statistic of choice and that the Chi-square test should only be taught as history.

The Robustness of the Likelihood Ratio Chi-Square Test for Structural Equation Models: A Meta-Analysis

2001 ◽  
Vol 26 (1) ◽  
pp. 105-132 ◽  
Author(s):  
Douglas A. Powell ◽  
William D. Schafer
Keyword(s):  
Structural Equation ◽  
Structural Equation Models ◽  
Type I Error ◽  
Meta Analysis ◽  
Generalized Least Squares ◽  
Error Rates ◽  
Type I ◽  
Chi Square ◽  
Distribution Free ◽  
Projection Techniques

The robustness literature for the structural equation model was synthesized following the method of Harwell which employs meta-analysis as developed by Hedges and Vevea. The study focused on the explanation of empirical Type I error rates for six principal classes of estimators: two that assume multivariate normality (maximum likelihood and generalized least squares), elliptical estimators, two distribution-free estimators (asymptotic and others), and latent projection. Generally, the chi-square tests for overall model fit were found to be sensitive to non-normality and the size of the model for all estimators (with the possible exception of the elliptical estimators with respect to model size and the latent projection techniques with respect to non-normality). The asymptotic distribution-free (ADF) and latent projection techniques were also found to be sensitive to sample sizes. Distribution-free methods other than ADF showed, in general, much less sensitivity to all factors considered.

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