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Supplemental regarding simple effects

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Simple effects* Often you will see people say that one cannot test for simple effects in the presence of a non- significant interaction, which makes sense in that we see simple effects as a way to understand the specifics of an interaction However… There can be significant simple effects without a significant interaction How can this be?

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Simple effects The simple effects of one factor over the multiple levels of another are not partitions of the interaction only Specifically they test in a one-way ANOVA fashion whether an effect is different from zero However in general terms of variance they are the breakdown of the variance of the SS interact and the SS main effect being tested Furthermore if one goes ‘backward’ from simple effects to try and draw conclusions about the interaction, one will not find a straightforward path The interaction can be seen as testing whether or not the simple effects are different from each other whereas the simple effects are testing whether their own effect is different from zero The problem of ‘accepting the null’ in the face of non-sig simple effects comes into play

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Simple effects SS A at B1 + SS A at B2 = SS A + SS A x B The point of doing the simple effects analysis is to see the specifics of how the levels of an effect are changing over the levels of another For example, without any knowledge of the interaction, if we had a one sig and one nonsig simple effect we might assume a moderating effect of one variable for the other However, this would be sketchy due to the presence of the main effect

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Possible outcomes: All signigificant A x B design SS A sig SS B sig SS A x B sig Significant main effects are ambiguous due to interaction. They are not an effect in isolation Proceed to simple effects to determine the nature of the interaction.

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Factor B significant only SS A non SS B sig SS A x B non Simple effects may show presence of a moderated effect, but only one type of simple effect could allow us to assume an interaction SS B at levels of A would not tell us that there is a moderated effect of B by A We have a significant main effect of B Would the effect be coming from B or the AxB interaction? If we tested A at B, we could claim a moderated effect if one simple effect was significant and the other not I.e. the effect is not coming from the main effect of A as it has been shown to be zero A1A2 B1 B2

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Factor A significant only SS A sig SS B non SS A x B non Here we have the converse situation In this we can test the simple effect of B at the levels of A for possible moderation effects A1A2 B1 B2

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Non-significance throughout SS A non SS B non SS A x B non Can test either simple effect without ambiguity A1A2 B1 B2

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Significant main effects, no significant interaction SS A sig SS B sig SS A x B non One should not test for simple effects in this situation as they would be reflections of main effects only But consider as an example, low n, ‘smallish’ p, noticeable effect size Simple effects testing might show differential effects for one treatment across levels of another i.e. all of this discussion focuses on a p-value and we must be concerned about that Technically though, one should stick to main effects post hocs in such a situation Possible capitalization of chance

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Last example* In this example we have a non-statistically significant interaction (p =.08) Tests of simple effects reveal a sig effect for the addicted group ( p =.02) but not the non-addicted (p =.09) Conclusion? Effect of Treatment for the Addicted group but not for the Non-Addicted group? Incorrect Concluding a zero for the non-addicted group is something we cannot do based on the type of test we employ Remember, we can reject the null, but there are many hypotheses (of varying small effects) that might be viable if we do not. Again, relying on effect sizes can help alleviate confusions arising from NHST

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Summary Note, that in the face of a significant interaction, one should do simple effects Otherwise one is essentially claiming something without showing exactly what’s going on However, even in the presence of a non-significant interaction, there are cases where one might still be interested in simple effects, and situations in which one can proceed to test them Just remember that simple effects are not just a breakdown of the interaction by itself, and this is why we can

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