What should be normally distributed for a two-way repeated measures ANOVA? R

Question

When running a two-way repeated measures ANOVA, should:

A) the independent varibles be normally distributed?
B) should the scores between the variables be normally distributed?

Overall: should the variables be normally distributed, the difference of scores (as such in a paired) test, anything else ?

• I'm struggling with the dataset I've linked below, I know that we can interpret this Anova as a kind of lmer, so I started to wonder what should be normally distributed. For regression, I know that the variable's distribution doesn't matter (only the residuals), what about for a repeated measures two-way ANOVA, what should be normally distributed?

• data:

CONT_Y ~ CATEGORIES * MY_GROUPS

structure(list(PARTICIPANTS = c(1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 
3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 
8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 
13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 
17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 
21, 21, 21, 21), CONT_Y = c(19.44, 20.07, 19.21, 16.35, 11.37, 
12.82, 19.42, 18.94, 19.59, 20.01, 19.7, 17.92, 18.78, 19.21, 
19.27, 18.46, 19.52, 20.02, 16.19, 19.97, 13.83, 15.93, 14.79, 
21.55, 18.8, 19.42, 19.27, 19.37, 17.14, 14.45, 17.63, 20.01, 
20.28, 17.93, 19.36, 20.15, 16.06, 17.04, 19.16, 20.1, 16.44, 
18.39, 18.01, 19.05, 18.04, 19.69, 19.61, 16.88, 19.02, 20.42, 
18.27, 18.43, 18.08, 17.1, 19.98, 19.43, 19.71, 19.93, 20.11, 
18.41, 20.31, 20.1, 20.38, 20.29, 13.6, 18.92, 19.05, 19.13, 
17.75, 19.15, 20.19, 18.3, 19.43, 19.8, 19.83, 19.53, 16.14, 
21.14, 17.37, 18.73, 16.51, 17.51, 17.06, 19.42), CATEGORIES = c("A", 
"A", "B", "B", "A", "A", "B", "B", "A", "A", "B", "B", "A", "A", 
"B", "B", "A", "A", "B", "B", "A", "A", "B", "B", "A", "A", "B", 
"B", "A", "A", "B", "B", "A", "A", "B", "B", "A", "A", "B", "B", 
"A", "A", "B", "B", "A", "A", "B", "B", "A", "A", "B", "B", "A", 
"A", "B", "B", "A", "A", "B", "B", "A", "A", "B", "B", "A", "A", 
"B", "B", "A", "A", "B", "B", "A", "A", "B", "B", "A", "A", "B", 
"B", "A", "A", "B", "B"), MY_GROUP = c("G1", "G2", "G1", "G2", 
"G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", 
"G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", 
"G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", 
"G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", 
"G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", 
"G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", 
"G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", "G2", "G1", 
"G2", "G1", "G2")), row.names = c(NA, -84L), spec = structure(list(
    cols = list(PARTICIPANTS = structure(list(), class = c("collector_double", 
    "collector")), CONT_Y = structure(list(), class = c("collector_double", 
    "collector")), CATEGORIES = structure(list(), class = c("collector_character", 
    "collector")), MY_GROUP = structure(list(), class = c("collector_character", 
    "collector"))), default = structure(list(), class = c("collector_guess", 
    "collector")), delim = ","), class = "col_spec"), problems = <pointer: 0x00000279757d98f0>, class = c("spec_tbl_df", 
"tbl_df", "tbl", "data.frame"))

Thanks in advance! :)

---

• Bonus: by the way, I've researched a lot about a non-paremetric alternative, I've seen many interesting posts here, as far as I'm concerned, there's no alternative for the two-way repeated measures Anova, right?

maybe this package WRS2 can be an alternative ? (link for documentation

• maybe a robust lmer may be a solution?

Answer 1

The linear regression model gives us a prediction that theoretically can take on any value in a continuous range when the regressor input is changed. In your data with the two categorical regressors, each of which can take on only two values, the two-way ANOVA (reaped measures or not) will give us only four predictions depending on the input combination of values of these two categories. These four predictions won't match the observed values in your sample. The differences between those four predictions and the observed data are basically the residuals.

Yes, for the two-way repeated measures ANOVA (like for the regular two-way ANOVA), those residuals must be normally distributed. In other words, the dependent variable must be normally distributed within the categorical combo "bins" (there are four of them in your dataset A∩G1, B∩G1, A∩G2, B∩G2).

Answer 2

In any kind of linear regression that requires normality, that normality applies to the error term.

A) No, and this is impossible in an ANOVA, since your independent $x$ variables are categorical indicator variables.

B) I do not follow what this means, but the couple of ways I could interpret it would mean that normality is not required or even desired.

What you need is for the error terms, estimates by the residuals, to be independent normal variables with zero mean and constant variance. This is equivalent to the conditional distributions being normal and all having the same variance (but they need not have the same mean).