# Normal distribution assumptions in ANOVA vs. Linear Regression

I don't get it. I am planning to conduct a 2x2 between-subjects measurement with a sample size of 30 per cell, hence four groups that I want to test for the difference.

Is there a difference in the normality assumptions of two-way ANOVA vs. Linear Regression? I've read numerous times that they're in fact the same analysis, but can somebody give me an example that can help me better understand it.

From what I know:

ANOVA: Assumes that the residuals are within each of the four groups are normally distributed with residuals being the difference of each data point to the mean. Moreover, within each group, the variance needs to be similar across all groups. (and of course independence of observations). Thus, the assumptions are checked four times (once per cell). Given my sample size (n= 30 per cell) and the Central Limit Theorem, I can assume that this normality assumption will be met as it is concerned with the mean within each of the four groups.

Linear Regression: With respect to normality, the residuals need to be normally distributed with residual being the difference between every single data point and the regression line. Here, it is only checked once whether the assumption is met with respect to the regression line across all data points (n=120). (and of course other assumptions like homoscedasticity, etc.)

My questions:

1. Will the results for checking the assumptions always be the same when checking for ANOVA (once per group) and Linear Regression (once per model)?

2. How does the Central Limit Theorem come into play in the regression? I mean, how could one argue for the normality with the Central Limit Theorem? Can I say that because I have n= 120 it is expected that residuals across all conditions will be normally distributed? Would that mean that the full sample normality assumption for Linear Regression (n=120) is more likely to hold true than for each of the four groups in isolation, as tested for ANOVA (n=30)?

• anova is a special case of linear regression, hence anova assumptions are also linear regression assumptions. May 28 '20 at 6:57
• Thank you Gordon! But can the Central Limit Theorem be used to argue for normality in a regression model? May 28 '20 at 7:14
• The CLT applies to regression the same as it applies in anova. It does not guarantee normality of residuals (nothing can do that) but it does guarantee asymptotic normality of the regression coefficients under any asymptotic sequence for which the coefficient standard errors tend to zero. May 29 '20 at 12:02

Summary: Do a pooled test, but plot by group.

1. Will the results for checking the assumptions always be the same when checking for ANOVA (once per group) and Linear Regression (once per model)?

One cannot guarantee that it will always give the same result, but the usual approach is to test once, that is, pooling the residuals from the different groups. Anyhow, with typical sample sizes there would not be sufficient power to test by group. But, if we assume sufficient sample size in each group, what could be reasons for different results of tests?

• normality by group, but not overall: If the groups have unequal variances, but otherwise normal, the pooled residuals distribution will be a mixture of normals. That could lead to rejection of normality, so do a check for homoskedasticity also, and qqplots per group could be informative.

• if residuals by group have different variance and shape, then everything could happen, so plot!

• You could still have sufficient power for the pooled test, but not for the by-group tests.

• If there is some natural ordering of the groups, that could be utilized in testing for unequal variances.

1. How does the Central Limit Theorem come into play in the regression? I mean, how could one argue for the normality with the Central Limit Theorem? Can I say that because I have $$n= 120$$ it is expected that residuals across all conditions will be normally distributed? Would that mean that the full sample normality assumption for Linear Regression ($$n=120$$) is more likely to hold true than for each of the four groups in isolation, as tested for ANOVA ($$n=30$$)?