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I have a question which may seem very simple to someone with more experience in statistics.

I have performed a two-factor ANOVA test, without replication, in order to test if there is a significant difference between the means of 4 groups. What is the purpose of doing an ANOVA test when Post-HOC T-tests can be done between each group? What added implication does an ANOVA test have over individual Post-HOC tests?

I am also trying to find a correlation between these means and an independent variable. Once I have done an ANOVA test which shows that there is a significant difference between these means, is it absolutely mandatory to do additional Post-HOC tests when justifying my conclusion?

I appreciate your help.

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ANOVA is joint test for equality of all means, whereas, post-hoc is performed between only two groups. Pair-wise comparison also make sense, but it reduces the power of the test. That is why, ANOVA is performed first (which has high power as compared to doing test for two groups). If null hypothesis is rejected, then only post-hoc test is performed.

For your second part, if you are interested to know, where the differences exist, then you should go for post-hoc analysis.

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