Question about pairwise comparisons

[Phoenix91]

Hey guys,

I'm looking at the SNARC and SQUARC effects, in which responses to numerical and non-numerical stimuli invoke spatial biases in response mapping. I won't go into great detail as I feel that my question is so basic that it may be unnecessary. I am conducting a repeated measures 2 by 2 ANOVA and have completely forgotten everything about pairwise comparisons.

If I get two main effects (one of each factor), and there are only 2 levels within each factor, do I need to conduct some kind of orthogonal contrasts to assess whether level 1 of the factor is statistically significantly different from level 2? At this stage I feel not, but was just hoping to get some clarification!

[Jake]

No. The significant main effects already tell you that the 2 levels differ. You typically don't need to begin thinking about different contrast coding schemes until you have factors with >2 levels.

[Phoenix91]

Thanks so much for your help. I have actually included two other contrasts for the interaction, given that you said that they shouldn't be used where the factor count = 2, I'm now wondering whether this is suitable.

I want to show that when these two effects operate simultaneously (i.e. when the SNARC and SQUARC effect occur on a single trial), the outcome is greater than when only one occurs (i.e. SNARC occurs but SQUARC doesn't, or SQUARC occurs but SNARC doesn't).

I was going to do this with an orthogonal contrast of SNARC compatible/SQUARC compatible (1) + (-1) SNARC compatible/SQUARC incompatible + (0) SN I/SQ C + (0) SN I/SQ I

[Jake]

Thanks so much for your help. I have actually included two other contrasts for the interaction, given that you said that they shouldn't be used where the factor count = 2, I'm now wondering whether this is suitable.

I don't know what you mean about having two other contrasts for the interaction. For a 2*2 setup, there is one and only one possible contrast that represents the interaction effect. How is it that you think you have 2 contrasts that represent the interaction?

I want to show that when these two effects operate simultaneously (i.e. when the SNARC and SQUARC effect occur on a single trial), the outcome is greater than when only one occurs (i.e. SNARC occurs but SQUARC doesn't, or SQUARC occurs but SNARC doesn't).

What about when neither factor occurs? The pattern of means that you just described could follow from an interaction, or simply from an additive effect of the two factors. We would need to know the prediction for the 4th cell to differentiate between these two cases.

I was going to do this with an orthogonal contrast of SNARC compatible/SQUARC compatible (1) + (-1) SNARC compatible/SQUARC incompatible + (0) SN I/SQ C + (0) SN I/SQ I

If I understand you correctly here, this contrast compares the effect of SQUARC compatible vs. incompatible for only those cases where SNARC=compatible. Maybe this is an interesting contrast for you, but it is not what we would call an interaction. It also does not seem like it tests what you said you wanted to test above (although, like I mentioned, you need to be more specific about the pattern of means you are predicting).

[Phoenix91]

Hi Jake, thanks so much for all your help thus far. I've thrown together two (very non APA) figures to demonstrate the two possible patterns of means I am predicting (one being an interaction, the other being an additive effect). (Hopefully the rationale for these predictions will become evident below)
http://i.imgur.com/cEAOl.png

I don't know what you mean about having two other contrasts for the interaction. For a 2*2 setup, there is one and only one possible contrast that represents the interaction effect. How is it that you think you have 2 contrasts that represent the interaction?

I think I may be going about this contrast approach wrong based on what you are saying. From what I gather, the one possible contrast you are talking about is factor 1 with factor 2. I have (seemingly incorrectly) identified a number of contrasts by including the actual levels of the factors i.e. Factor1Level1, Factor1Level2, Factor2Level1 and Factor2Level2.

What about when neither factor occurs? The pattern of means that you just described could follow from an interaction, or simply from an additive effect of the two factors. We would need to know the prediction for the 4th cell to differentiate between these two cases.

Absolutely, I think I follow you here, and a bit more information on my part may have helped. I am following Sternberg's Additive Factor Method and am attempting to determine whether the SNARC and SQUARC effects are due to shared or distinct processes (indicated by an interaction or additive effect respectively).

If there is evidence of an interaction (as indicated by the ANOVA), then this will be taken as evidence of shared processes. Conversely, if there is not a significant interaction, then this will be taken as evidence of distinct processes. So, if no interaction is present, I feel like I still need to compare means to say that the effects are additive.

In order to do this, I wanted to make two contrasts - (Both effects compatible) with (1st effect compatible whilst 2nd is not) and (Both effects compatible) with (2nd effect compatible whilst 1st is not). If both comparisons showed response times are significantly lower for the (both effects compatible) combination compared to the other two possible combinations, then this would be taken as evidence of additivity.

If I understand you correctly here, this contrast compares the effect of SQUARC compatible vs. incompatible for only those cases where SNARC=compatible. Maybe this is an interesting contrast for you, but it is not what we would call an interaction. It also does not seem like it tests what you said you wanted to test above (although, like I mentioned, you need to be more specific about the pattern of means you are predicting).

Somehow I failed to include the second contrast I was interested in before, I hope this is evident in what I said above.

[Jake]

From what I gather, the one possible contrast you are talking about is factor 1 with factor 2.

Hopefully this will help make sure we are both on the same page about what is and is not an interaction:

                          A  B AB
SnarcComp/SquarcComp      1  1  1
SnarcComp/SquarcIncomp    1 -1 -1
SnarcIncomp/SquarcComp   -1  1 -1
SnarcIncomp/SquarcIncomp -1 -1  1

This matrix gives the traditional contrasts for a 2*2 factorial ANOVA. The "AB" contrast tests the interaction effect. Notice that it equals the product of the two main effect contrasts. This is what an interaction is. There is no other contrast or combination of contrasts that directly tests the interaction effect.

So, if no interaction is present, I feel like I still need to compare means to say that the effects are additive

Well, no. If there is no interaction, then the effects are additive. This is by definition, since additivity of factors simply means that the two factors do not interact.

In order to do this, I wanted to make two contrasts - (Both effects compatible) with (1st effect compatible whilst 2nd is not) and (Both effects compatible) with (2nd effect compatible whilst 1st is not).

I guess you are saying you want a contrast matrix like this one:

                         CCvsCI CCvsIC CCvsII
SnarcComp/SquarcComp          0      0      0
SnarcComp/SquarcIncomp        1      0      0
SnarcIncomp/SquarcComp        0      1      0
SnarcIncomp/SquarcIncomp      0      0      1

These are simply dummy codes with the compatible/compatible group set as the baseline category.

[Phoenix91]

Hi Jake, again, thanks so much, you've been a huge help to me.

In light of everything that you have said, I can't see any need for pairwise comparisons at all in my research.

My whole rationale for the comparisons was to show that the effects were additive, however given that "If there is no interaction, then the effects are additive", this seems to be unnecessary.

In a nutshell, my hypotheses are

1. There will be a main effect of SNARC compatibility (Snarc compatible RTs lower than snarc incompatible)

2. There will be a main effect of SQUARC compatibility (Squarc compatible RTs lower than squarc incompatible)

3. There will be an interaction between snarc and squarc compatibility. (if the two effects involve some shared mechanism, there will be an interaction. if they involve independent mechanisms, there will be an additive effect)

Hypotheses 1 and 2 are answered by the main effects of each factor in a 2x2 repeated measures ANOVA. Hypothesis 3 is tested by the interaction.

So in summary, all I need to test these predictions is the basic ANOVA - no contrasts or anything