# Following up a three-way interaction found with (mixed) model comparison

I hope this isn't a silly question, I'd like some advice on following up a threeway interaction in a mixed effects model. I've been building my models incrementally, like this:

lmer0<- lmer(predval ~ 1 + (1 + Quantile|Subj), data = d)
lmer1<- update(lmer0, .~. + epoch)
lmer2<- update(lmer1, .~. + Quantile)
lmer3<- update(lmer2., .~. + Congruence)
lmer4<- update(lmer3, .~. + FixDur)
lmer5<- update(lmer4, .~. + Quantile:Congruence)
lmer6<- update(lmer5, .~. + Quantile:FixDur)
lmer7<- update(lmer6, .~. + Congruence:FixDur)
lmer8<- update(lmer7, .~. + Quantile:Congruence:FixDur)


Here's what I get when I test these models against each other. All interaction terms improve the model (indicated by decreasing AIC/BIC values). This printout indicates the fully interactive model (lmer8) is the best model so far.

        Df     AIC     BIC   logLik     Chisq Chi Df Pr(>Chisq)
lmer0   5 2062522 2062573 -1031256
lmer1    6 2053156 2053217 -1026572 9368.2240      1  < 2.2e-16 ***
lmer2    7 2053144 2053215 -1026565   14.2730      1  0.0001581 ***
lmer3    8 2051857 2051938 -1025921 1288.7862      1  < 2.2e-16 ***
lmer4   11 2051604 2051716 -1025791  259.3176      3  < 2.2e-16 ***
lmer5   12 2051606 2051728 -1025791    0.0284      1  0.8662848
lmer6   15 2051168 2051320 -1025569  443.8425      3  < 2.2e-16 ***
lmer7   18 2051152 2051335 -1025558   21.9526      3  6.673e-05 ***
lmer8   21 2050697 2050911 -1025327  460.9543      3  < 2.2e-16 ***


It's the last term in the model that really interests me - the three way interaction. Unfortunately I can't embed the image here (newbie), but you can find a plot of the data shows Congruence (shape) by Quantile (Colour gradient/x-axis) by Fixation Duration (facet_grid) here:

What I think I should do now is as follows:

• Test the Congruence x Quantile 2-way at each level of FixDur (i.e. Begin new model comparison at each level of FixDur, that's each column in the plot above).
• For those two-way interactions that are significant (they all are), follow with t-tests at each Quantile (i.e. 20 paired t-tests per FixDur). I'm not too sure how to conduct these using lmer, I'm basically just using the t.test function.

The obvious issue with this will be multiple comparisons. Essentially I want to do 20 t-tests at each level of Fixation Duration (so I can say something about the timecourse of my effects). That's a lot of t-tests, so using a Bonferroni or Scheffe correction is going to kill me!

Does anyone have a suggestion on what method or approach would be suitable for dealing with this problem? It's a hunch, but I feel as though the co-occurence of significant p-vals should count for something. That is, if the quantiles that yield a t-vals significant at .05 occur all in a row (i.e. show the effect emerging, as in the first column of the graph), that should give us confidence that they're not type 1 errors. Am I going down a dangerous road?

This is a terribly long post i'm sorry. I'd welcome all thoughts though on any part of it! My thanks to you in advance.

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 +1 It's a long post, but at least it's clear that you're really thinking carefully about the problem. I agree wholeheartedly with @John's answer, although this can be a hard sell with some traditionally trained bosses / supervisors / reviewers. If you do decide on multiple comparisons, you can use p.adjust with (say type="BH" to control false discovery rate -- very popular nowadays because it's not as conservative as methods that control experimentwise error ...) – Ben Bolker Jun 27 '12 at 17:07

I was just reading a paper that used a pretty similar setup the other day, looking at congruence effects over a time-course.

You can find it here: "The Flexibility of Nonconsciously Deployed Cognitive Processes: Evidence from Masked Congruence Priming" (Finkbeiner and Friedman, 2011)

To control for the multiple comparisons over the time-course, they:

used a permutation procedure described by Blair and Karniski [47] to produce a reference distribution of maximum t-values (N=216). We set our critical t-value equal to the value in the reference distribution that corresponded to the 99th percentile. The advantage of this procedure for continuous data is that it allows the researcher to maintain experimentwise error at a prescribed level (we used 0.01) whereas alternative methods (e.g. Bonferroni correction) simply ensure that a desired level is not exceeded.

I haven't had time to look into the details of the permutation procedure, but given that it seems to be a pretty similar experiment, I thought I'd suggest it as an option.

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 Thanks very much Marius, I'm going to take a look at implementing perm tests now. It does seem like a convincing reviewer-friendly option. – quekles Jun 27 '12 at 23:40

It sounds like what you want to know is if each little detail in the pattern of data is significant or not. This is a bad path. The worst reason is probably because the difference between significant and not significant is not a direct comparison of two things and may not itself be significant. Therefore, what you really need to be doing, if you want to really do that, is comparing each and every effect with every other effect directly, not to whether the effect itself is significant. It's a combinatorial explosion!

So stop.

You already have an interaction. Avoiding multiple testing was a major reason for the analysis you already ran. If you model to avoid multiple comparisons, you have to draw something useful out of the model. Going back and then doing all the multiple comparisons you'd do without doing the model in the first place makes the modelling rather pointless.

You have modelled your data with a linear regression. Your 3 way interaction is that your congruency effect depends on lift off latency and fixation duration. You start with an increasing congruency effect across lift off latency at your lowest fixation duration and that pattern shifts to a reducing effect across lift off latency and back to approximately parallel across fixation durations.

That's your interaction. It has to be true, you've got a substantial interaction and that's the shape of the data. Is there any particular bit of extra information you need that testing all of your data points is going to give you?

The interaction itself is telling you something. The pattern in the data you see has to mean something. Go about figuring out what's in an interaction from tons of tests is not a good idea and usually a terrible one. You only should do the multiple testing if the interaction is somehow unclear or there's something particular you need to highlight in a particular area of it.

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 Thank you John! That is a very sensible and calming answer to a convoluted question. I'm new to model comparison, but it does seem odd to build up incrementally and then go about deconstructing what you just built. Thank you for your kind advice. – quekles Jun 27 '12 at 23:49 I have encountered another post that suggests a similar approach - take the interaction to be significant and use your noodle (rather than extensive further testing) to interpret it. stats.stackexchange.com/questions/18873/… – quekles Jul 10 '12 at 7:18