Solved – 3-way repeated mesures ANOVA

anovainteractionrrepeated measures

I conducted reaction-time experiment with three factors – set size (4, 16, 64), presense of the target (present, absent) and the third factor with two levels (C, F).

After that I ran a repeated measures ANOVA in R:

m2 <- aov(time ~ pres*fact*size + Error(1/subj), data = data)

fact and pres are factorized. The result is that all main effects and all interactions (including 3-way) are significant.

I have no idea how to interpret 3-way interaction in this case, can I somehow exclude it from the model? And did I even specify an ANOVA correctly?

Best Answer

Although this is five years late, I will post as an FYI for anyone else who happens to stumble upon this thread. The first part of the model is properly specified, i.e.

time ~ pres*fact*size

However, the error term is incorrectly specified. Note that in repeated measures ANOVA, "the error term for any within-subject effect is the interaction of that effect with subjects [or whatever entity that is repeatedly measured] (Keppel, G., & Wickens, T. D., 2004, p. 40)."

For this particular example, if using R's aov() function, the error term would be specified as such:

Error(subj/(pres*fact*size))

So in all, we have:

m2 <- aov(time ~ pres*fact*size + Error(subj/(pres*fact*size)), data = data)

WRT the interpretation of a three-way interaction: The implication is that a two-way interaction varied over the levels of a third factor. In other words, there was a two-way interaction that was only manifest at a specific level or levels of a third factor. To get to the heart of the matter, you would need to follow up your omnibus ANOVA with interaction contrasts. To determine what interaction contrasts you should test, it would be helpful to plot your data using a line graph. As it stands, there is not enough information available in your post to provide further guidance.

Edit Perhaps it goes without saying, but this is in the case that you obtain such a result with the model properly specified.

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Solved – Blocked repeated measures ANOVA in R

Your data there works fine for me as soon as I put 'Block' in the model instead of 'Year'

Check the data reads in okay:

> summary(TWP)
       Block   Severity Plot     Richness    
 2003-2004:6   High:9   A:6   Min.   : 8.00  
 2005-2006:6   Low :9   B:6   1st Qu.:14.00  
 2007-2009:6            C:6   Median :18.50  
                              Mean   :18.67  
                              3rd Qu.:23.25  
                              Max.   :29.00

Fit the model, then fix it so it has the right names:

> TWP.aov <- aov(Richness ~ Severity * Year + Error(Plot/Severity), data = TWP)
Error in eval(expr, envir, enclos) : object 'Year' not found
> TWP.aov <- aov(Richness ~ Severity * Block + Error(Plot/Severity), data = TWP)

After that, it seems to work:

> summary(TWP.aov)

Error: Plot
          Df Sum Sq Mean Sq F value Pr(>F)
Residuals  2  49.33   24.67               

Error: Plot:Severity
          Df Sum Sq Mean Sq F value Pr(>F)  
Severity   1 304.22  304.22   85.56 0.0115 *
Residuals  2   7.11    3.56                 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Error: Within
               Df Sum Sq Mean Sq F value Pr(>F)
Block           2  43.00  21.500   0.745  0.505
Severity:Block  2   1.44   0.722   0.025  0.975
Residuals       8 230.89  28.861

What message did you get?

---

Warning message: In aov(Richness ~ Severity * Block + Error(Plot/Severity), data = TWP) : Error() model is singular

Please note that "model is singular" has nothing to do with there being extra "=" in the formula. (How did you get that interpretation of the message?).

Looking at the code for aov, it looks like it relates to the rank of the design matrix being lower than the number of coefficients on completing the QR deomposition. This would suggest you have multicollinearity.

Solved – 2x2x5 repeated measures ANOVA: significant 3-way interaction

I'm not sure what you plan to test but looking at your first graph it seems pretty clear. There's generally an effect of colour but on the left side and at larger offsets it disappears.

I'm guessing you wanted to test all of the effects of colour to see where they were significant and where they weren't. If you found they were all significant or all not it wouldn't give your more information than your interaction (and the interaction is independent of such findings). If you found some were and some weren't it doesn't test your interaction because the difference between significant and not significant is not itself significant.

I suppose you could try to explore something else about the difference in patterns but given that offset seems like a continuous variable that should have some clear function if it is doing anything I'm not sure establishing anything else different about the waviness of the lines on the two sides would be where you'd want to go.

UPDATE after viewing comments

The explanation for your interaction would be that the effect of colour and side are consistent until offsets 4 and 5 where the effect of colour only exists for one of the sides. That's just a recasting of what I said before in line with your hypotheses.

Keep in mind what you'd have to test post hoc or in planned contrasts to really demonstrate this. Finding an effect just at large offsets is useless because colour itself is interacting with side; therefore you have to show it interacts with offsets to show they're having an effect as well. That's what your ANOVA is telling you. It's already the planned contrast you want. Is there anything else there that could be making that interaction occur? Do you need to explain anything else?

If you do the ANOVA at 4 and 5 you likely won't get an interaction with offset, just one between colour and side, which will be substantially less evidence for what you want to say, not more.

Keep in mind, interactions mean something. Look at your data and figure out what they mean before considering further statistical tests. If they're relatively clear, as in this case, then you're done.

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Solved – Blocked repeated measures ANOVA in R

Solved – 2x2x5 repeated measures ANOVA: significant 3-way interaction

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