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Using a two-way ANOVA, I can test three null hypotheses:

  1. The means of the response variable are equal for different values of the first factor.
  2. The means are equal for different values of the second factor.
  3. There is no interaction between the two factors (the effects of one factor do not depend on the value of the second factor).

If the p-value for the last null hypothesis is lower than my significance level, I can conclude that there is an interaction between the two factors. However, if the p-value is higher than my significance level, I cannot conclude that there is no interaction between the two factors. A failure to reject the null hypothesis in a significance test does not mean that the null hypothesis is true.

What test should I use if I want to show that there is no interaction between the two factors?

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You are correct in saying that a failure to reject the null hypothesis does not mean that the null hypothesis is true. However, our default position in hypothesis testing is that that null hypothesis is assumed to be true until proven otherwise.

There is no statistical test to prove that an interaction effect does not exist. You will have to design your experiment or study with enough replicates to ensure that you have a very high level of statistical power. If your interaction term is non-significant, that shows that the interaction effect is either very small or does not exist. That is the best you can do.

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