Smoothed Quantiles for     Type Test Statistics with Applications

Ke Hai Yuan; Brenna Gomer; Katerina M. Marcoulides

doi:10.1080/00273171.2020.1858018

Smoothed Quantiles for Type Test Statistics with Applications

Ke Hai Yuan
, Brenna Gomer
, Katerina M. Marcoulides

Psychology (Twin Cities)

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-square distribution due to violation of conditions encountered with real data. In such instances, the bootstrap or Monte Carlo methodology might be used to approximate the distribution of the statistic. However, the sample quantile may be a poor estimate of the population counterpart when either the sample size is small or the number of different values of the replicated statistic is limited. Using statistical learning, this article develops a method that yields more accurate quantiles for chi-square type test statistics. Formulas for smoothing the quantiles of chi-square type statistics are obtained. Combined with the bootstrap methodology, the smoothed quantiles are further used to conduct equivalence testing in mean and covariance structure analysis. Two real data examples illustrate the applications of the developed formulas in quantifying the size of model misspecification under equivalence testing. The idea developed in the article can also be used to develop formulas for smoothing the quantiles of other types of test statistics or parameter estimates.

Original language	English (US)
Pages (from-to)	223-242
Number of pages	20
Journal	Multivariate Behavioral Research
Volume	57
Issue number	2-3
DOIs	https://doi.org/10.1080/00273171.2020.1858018
State	Published - 2022

Bibliographical note

Funding Information:
This work was supported by Grant 31971029 from the Natural Science Foundation of China (NSFC). The authors would like to thank the Action Editor Dr. Michael Edwards and three anonymous reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institutions or the NSFC is not intended and should not be inferred.

Publisher Copyright:
© 2020 Taylor & Francis Group, LLC.

Keywords

Bootstrap simulation
equivalence testing
smoothed quantile
statistical learning
Sample Size
Models, Statistical
Chi-Square Distribution
Monte Carlo Method
Research Design

PubMed: MeSH publication types

Journal Article

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10.1080/00273171.2020.1858018

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Cite this

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title = "Smoothed Quantiles for Type Test Statistics with Applications",

abstract = "Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-square distribution due to violation of conditions encountered with real data. In such instances, the bootstrap or Monte Carlo methodology might be used to approximate the distribution of the statistic. However, the sample quantile may be a poor estimate of the population counterpart when either the sample size is small or the number of different values of the replicated statistic is limited. Using statistical learning, this article develops a method that yields more accurate quantiles for chi-square type test statistics. Formulas for smoothing the quantiles of chi-square type statistics are obtained. Combined with the bootstrap methodology, the smoothed quantiles are further used to conduct equivalence testing in mean and covariance structure analysis. Two real data examples illustrate the applications of the developed formulas in quantifying the size of model misspecification under equivalence testing. The idea developed in the article can also be used to develop formulas for smoothing the quantiles of other types of test statistics or parameter estimates.",

keywords = "Bootstrap simulation, equivalence testing, smoothed quantile, statistical learning, Sample Size, Models, Statistical, Chi-Square Distribution, Monte Carlo Method, Research Design",

author = "Yuan, \{Ke Hai\} and Brenna Gomer and Marcoulides, \{Katerina M.\}",

note = "Funding Information: This work was supported by Grant 31971029 from the Natural Science Foundation of China (NSFC). The authors would like to thank the Action Editor Dr. Michael Edwards and three anonymous reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institutions or the NSFC is not intended and should not be inferred. Publisher Copyright: {\textcopyright} 2020 Taylor \& Francis Group, LLC.",

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Y1 - 2022

N2 - Chi-square type test statistics are widely used in assessing the goodness-of-fit of a theoretical model. The exact distributions of such statistics can be quite different from the nominal chi-square distribution due to violation of conditions encountered with real data. In such instances, the bootstrap or Monte Carlo methodology might be used to approximate the distribution of the statistic. However, the sample quantile may be a poor estimate of the population counterpart when either the sample size is small or the number of different values of the replicated statistic is limited. Using statistical learning, this article develops a method that yields more accurate quantiles for chi-square type test statistics. Formulas for smoothing the quantiles of chi-square type statistics are obtained. Combined with the bootstrap methodology, the smoothed quantiles are further used to conduct equivalence testing in mean and covariance structure analysis. Two real data examples illustrate the applications of the developed formulas in quantifying the size of model misspecification under equivalence testing. The idea developed in the article can also be used to develop formulas for smoothing the quantiles of other types of test statistics or parameter estimates.

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KW - Monte Carlo Method

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