# Three way ANOVA - very small p-values. Is my data normal?

I have a data set with three groups: {Hi, Low}, {Drug amount 1, Drug amount 2...}, {Time 1, time 2, Time 3...}

The dependent variable is a percentage, which I have log transformed (as the distribution of the error was dependent on the value... i.e. I would be seeing 1% +/- 0.1, but 90% +/- 9%)

I am really interested in whether or not the effect of the first group is significant. I have performed a three-way ANOVA, but the resulting p-values seem too small (type II SS, categorical variables, testing only for linear interactions).

In my research as to whether or not this sort of p-value might be expected, I have come across some similar questions (such as Very small p-value after performing ANOVA TEST (small sample size)), with answers suggesting that this issue might be a result of non-normally distributed data.

What I don't understand, is how I can tell if this sort of data is normally distributed (my sample size is small, n=3).

So I have

Time 1 Time 2 Time 3 Time 4
Amount One n=3 n=3 n=3 n=3
Amount Two n=3 n=3 n=3 n=3
Amount Three n=3 n=3 n=3 n=3
Amount Four n=3 n=3 n=3 n=3
Amount Five n=3 n=3 n=3 n=3

And I have this twice, once for {Hi} and once for {Low}. Is there a way to check for normality here, with n=3, or do I not really have enough samples to be performing a three-way ANOVA?

• Maybe post your full output to get advice on the statistical side of the issue; however I suspect N=3 per cell is just too few observations to get any meaningful results. Jan 27, 2023 at 18:33
• Thanks @sointu - I've ended up grouping my exposure amounts and times into a single categorical variable, treatment condition, and performing a Two-Way ANOVA. I've then checked a qq plot and can see that normality is a reasonable assumption, so I'm relatively happy. I'm worried about being asked to defend not having done a three-way ANOVA - is there a sort of rule of thumb on a number of observations at which a Three-way ANOVA becomes meaningful? I haven't been able to find anything
– MDN
Feb 2, 2023 at 13:06
• Technically, you can run an ANOVA with your sample size or even smaller but it's impossible to check for normality with only 3 observations per cell, and the role of random factors affecting the outcome is likely to be large. If your new way of dealing with your data makes sense substantially and from the perspective of your research questions, then you are probably able to justify that approach. Also, this response stats.stackexchange.com/questions/567015/… may be helpful. Feb 6, 2023 at 10:11