Mean and Mean-and-Variance Corrections With Big Data

Abstract

Mean and mean-and-variance corrections are the 2 major principles to develop test statistics with violation of conditions. In structural equation modeling (SEM), mean-rescaled and mean-and-variance-adjusted test statistics have been recommended under different contexts. However, recent studies indicated that their Type I error rates vary from 0% to 100% as the number of variables p increases. Can we still trust the 2 principles and what alternative rules can be used to develop test statistics for SEM with “big data”? This article addresses the issues by a large-scale Monte Carlo study. Results indicate that empirical means and standard deviations of each statistic can differ from their expected values many times in standardized units when p is large. Thus, the problems in Type I error control with the 2 statistics are because they do not possess the properties to which they are entitled, not because of the wrongdoing of the mean and mean-and-variance corrections. However, the 2 principles need to be implemented using small sample methodology instead of asymptotics. Results also indicate that distributions other than chi-square might better describe the behavior of test statistics in SEM with big data.

Keywords:

• adjusted statistic
• big data
• chi-square distribution
• relative multivariate kurtosis
• rescaled statistic

FUNDING

The research was supported by the National Science Foundation under Grant No. SES-1461355.

Notes

¹ We use d for degrees of freedom in this section instead of df to simplify the presentation of equations.

² The conditions of the 6,966 different combinations of p, N, m and distributions are documented at http://www3.nd.edu/kyuan/bigdata/Design_condition.txt.

Additional information

Funding

The research was supported by the National Science Foundation under Grant No. SES-1461355.