Generally, standardized betas are not recommended for dichotomies because they represent a change of one standard deviation in the independent variable but the only change possible in a two-category variable is from one category to the other. I believe the same logic would apply to the interaction effect between two dichotomies.

I agree with David L Morgan that this might not be a good idea. A difference in means is a perfectly fine effect size measure in most cases and easy to calculate for the interaction contrast from cell means.

Putting that aside and assuming an independent design:

1) As the difference is an unstandardised slope you can convert it to a beta weight by multiplying by the ratio of the SD of X over the SD of Y. This can be calculated if you know the cell sizes and the SDs of Y in each cell and how X is coded (e.g., 0 or 1, -1 or 1 etc). It would be more usual to scale differences using the SD of Y to get a d like measure (there's various ways to do this depending on which SD you use).

2) Not with further information like the MSe (and assumptions like assuming homogeneity of variances).

3) Not fundamentally. A lot depends on how the model is parameterised.

It's all quite fiddly. I wonder if a neater approach would just be to simulate data from the sufficient statistics (n, means and SD per group) and run the analysis as a regression and pool estimates of the quantities you want over multiple runs.

Qing Tang, Yes, you can estimate the standardized regression coefficient (β) and SE for the interaction in a 2×2 ANOVA using:

bA×B​=M11​−M10​−M01​+M00​

β=SDY​bA×B​⋅SDA×B​​

For binary variables, SDA×B​≈0.5. Use pooled SD (from cell SDs or ANOVA MSE) for SDYSD_YSDY​.

If SDs or MSE are missing, you cannot compute standardized β or SE—you’ll need at least one measure of variability.

For a 2×4 ANOVA, the idea is the same, but interaction has multiple degrees of freedom (df), so you’d compute separate βs for each contrast or use partial η² for the overall interaction.