Robustness to nonnormality of regression F-tests

Gaussian models are robust as long as there are no outliers that occur in X and Y simultaneously.

“leading to the conclusion that Gaussian models are robust as long as there are no outliers that occur in X and Y simultaneously.”

When the predictor variable X contains a single outlier that coincides with an outlier in Y, it creates an extreme observation with high leverage on the regression line.

“Specifically, when the predictor variable X contains a single outlier, then it is possible that the case coincides with an outlier in Y, creating an extreme observation with high leverage on the regression line.”

With large sample sizes ≥ 500, the Central Limit Theorem guarantees that regression coefficients are on average normally distributed.

“With large sample sizes ≥ 500 the Central Limit Theorem guarantees that the regression coefficients are on average normally distributed (Ali & Sharma, 1996; Lumley et al., 2002).”

Linear models are typically said to be robust to violation of the normality assumption for hypothesis testing and parameter estimation as long as outliers are handled properly.

“Throughout the scientific literature, linear models are typically said to be robust to the violation of the normality assumption when it comes to hypothesis testing and parameter estimation as long as outliers are handled properly (Ali & Sharma, 1996; Box & Watson, 1962; Gelman & Hill, 2007; Lumley et al., 2002; Miller, 1986; Puth et al., 2014; Ramsey & Schafer, 2013; Schielzeth et al., 2020; W”

Statistical significance gets seriously misestimated based on the assumption of Gaussian errors in Y only when an outlier in Y coincides with an outlier in X.

“This is the only case where statistical significance gets seriously misestimated based on the assumption of Gaussian errors in Y which is violated by the outlier in Y.”