# Does a p-value of 1 for an interaction where the plot has crossing lines signal something is wrong in a 3-way ANOVA?

I have run a 3-way ANOVA and the interaction between two of the factors looks significant when plotted (i.e. the two lines cross). However the p-value is 1.

1. Is a p-value of 1 in a 3-way ANOVA weird?
2. If the lines cross in a plot of a 2-way interaction, does this mean that there is definitely an interaction between the two factors or can the lines cross without there being an interaction?

The data comes from a yoked study on two different datasets and in the other dataset, there was a similar looking plot and the interaction was highly significant. If I were to interpret these results as the data is trending in that direction, but there is no significant interaction, would that be a not-completely false conclusion?

What I'm worried about is someone saying either 'oh well the p-value is 1 and that always means x, y, z' or 'oh well the lines might cross, but crossing lines isn't always a sign of an interaction, it can just be a result of x, y, z'. The p-value of 1 feels weird to me, as does the crossing lines without a significant p-value. Should it, or do p-values of 1 occur often and crossing lines without significant p-values occur often?

This is the graph:

If two lines are not exactly parallel, they will inevitably cross at some point. Without loss of generality let's simplify your question. Say that you were using one sample $t$-test and wanted to test $H_0 : \bar X = 0$. Obviously, you could say that if $\bar X \ne 0$, then the hypothesis is false, but we use hypothesis tests to account for the uncertainty of your results. $p$-values tell you how "likely" would such result be if the $H_0$ was true. So basically what your test tells you is "you could, as well, observe such result if there wouldn't be any interaction".