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I've fit a Linear Mixed Effects model to some "accuracy" scores for a study with rats. The fixed effects are TrialNumber and Age, and the random effect is the individual rat (Rat). Age is included because the trials span a number of months (and Age is measured in days; not all rats were the same age at the start of the study, for a number of reasons).

The resulting model in R was:

library(lme)
library(car) #for the logit() function, since scores are percentages
lmer(logit(Score) ~ Age*TrialNumber + (1|Rat))

I get negative values for Trial and Age (the study is designed such that we predicted scores to actually go down with more experience, which indeed turned out to be the case). However, the interaction term is also significant, but with a positive estimate.

Results:

Fixed effects:
                 Estimate   Std. Error  t value
(Intercept)      1.182e+00  1.689e-02   70.00
Age             -2.788e-03  2.849e-04   -9.78
TrialNumber     -1.872e-06  9.344e-08  -20.04
Age:TrialNumber  2.123e-08  1.741e-09   12.20 

How do I interpret this interaction? Does it mean that for repeated trials on the same day, Score went up, or does it mean that older rats did better on the same TrialNumbers?

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It means that for a given rat of Age=0, logit(Score) should be expected to be lower when TrialNumber is higher; -1.872e-06 units lower for each unit of TrialNumber. But this effect changes when we look at higher ages; the effect of TrialNumber is 2.123e0-08 units higher for additional unit of Age. (You can make up similar interpretations for the Age effects at specific values of TrialNumber) –  guest Apr 6 '12 at 23:40
    
@guest Thanks -- does that mean that the negative effect of TrialNumber increases with increasing Age? I.e., is Score predicted to be even lower for older rats as TrialNumber increases? –  Julie Apr 6 '12 at 23:57
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The effect of TrialNumber is negative when Age=0, but then increases as Age gets bigger. It'll go positive eventually, but whether it happens at plausible values of Age will depend on what those plausible values are. –  guest Apr 7 '12 at 6:14
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@guest Feel free to put this explanation in an answer rather than comments. It's more visible there and the questioner can accept it. –  conjugateprior Apr 7 '12 at 13:54
    
I'd definitely accept @guest 's answer. –  Julie Apr 7 '12 at 16:58
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up vote 3 down vote accepted

It means that for a given rat of Age=0, logit(Score) should be expected to be lower when TrialNumber is higher; -1.872e-06 units lower for each unit of TrialNumber. But this effect changes when we look at higher ages; the effect of TrialNumber is 2.123e0-08 units higher for additional unit of Age. As Age gets bigger, the effect will go positive, eventually, but whether that happens at plausible values of Age will depend on what those plausible values are.

Similar interpretations hold for the Age effects at specific values of TrialNumber; the fitted model says that at TrialNumber=0, the expected logit(Score) is -2.788e-03 units lower, for each additional unit of Age. This effect differs by 2.123e0-08 for each additional unit of TrialNumber.

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