This is not normal! (Re-) Evaluating the lower $n$ guildelines for regression analysis

The commonly cited rule of thumb for regression analysis, which suggests that a sample size of $n \geq 30$ is sufficient to ensure valid inferences, is frequently referenced but rarely scrutinized. This research note evaluates the lower bound for the number of observations required for regression analysis by exploring how different distributional characteristics, such as skewness and kurtosis, influence the convergence of t-values to the t-distribution in linear regression models. Through an extensive simulation study involving over 22 billion regression models, this paper examines a range of symmetric, platykurtic, and skewed distributions, testing sample sizes from 4 to 10,000. The results reveal that it is sufficient that either the dependent or independent variable follow a symmetric distribution for the t-values to converge to the t-distribution at much smaller sample sizes than $n=30$. This is contrary to previous guidance which suggests that the error term needs to be normally distributed for this convergence to happen at low $n$. On the other hand, if both dependent and independent variables are highly skewed the required sample size is substantially higher. In cases of extreme skewness, even sample sizes of 10,000 do not ensure convergence. These findings suggest that the $n\geq30$ rule is too permissive in certain cases but overly conservative in others, depending on the underlying distributional characteristics. This study offers revised guidelines for determining the minimum sample size necessary for valid regression analysis.