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I've constructed a mixed effects regression model to investigate the interaction between three, categorical predictors (S_condition, C_condition and E_condition) – with three levels each (S1, S2, S3; C1, C2, C3; E1, E2, E3) – in predicting a continuous DV (trust). There are random effects by subject (which also has a random slope) and claim.

model3 <- lmer(trust ~ S_condition*C_condition*E_condition + (1+stance|subject) + (1|claim), data = dataC, REML=FALSE)

The fixed effects of output from this model is as follows.

                                            Estimate Std. Error         df t value Pr(>|t|)  
(Intercept)                                 -0.33582    0.38341  138.93163  -0.876   0.3826  
S_conditionS2                               -0.28344    0.47676 2683.40160  -0.595   0.5522  
S_conditionS3                               -0.46068    0.47957 2679.28922  -0.961   0.3368  
C_conditionC2                                0.25793    0.47493 2649.02712   0.543   0.5871  
C_conditionC3                                0.05433    0.47507 2649.41999   0.114   0.9090  
E_conditionE2                               -0.02748    0.47476 2648.58893  -0.058   0.9539  
E_conditionE3                                0.14434    0.47552 2650.55022   0.304   0.7615  
S_conditionS2:C_conditionC2                 -0.02042    0.66883 2649.93697  -0.031   0.9756  
S_conditionS3:C_conditionC2                  0.69522    0.67363 2649.56439   1.032   0.3021  
S_conditionS2:C_conditionC3                  0.85942    0.66985 2651.65264   1.283   0.1996  
S_conditionS3:C_conditionC3                  0.88873    0.67362 2649.55228   1.319   0.1872  
S_conditionS2:E_conditionE2                  0.08978    0.66830 2648.93336   0.134   0.8931  
S_conditionS3:E_conditionE2                  0.63116    0.67342 2649.17937   0.937   0.3487  
S_conditionS2:E_conditionE3                  0.72908    0.66942 2650.95145   1.089   0.2762  
S_conditionS3:E_conditionE3                  0.26589    0.67389 2650.04088   0.395   0.6932  
C_conditionC2:E_conditionE2                  0.47762    0.67135 2648.46205   0.711   0.4769  
C_conditionC3:E_conditionE2                  0.67541    0.67135 2648.44933   1.006   0.3145  
C_conditionC2:E_conditionE3                  0.02980    0.67182 2649.36016   0.044   0.9646  
C_conditionC3:E_conditionE3                  0.59804    0.67206 2649.80941   0.890   0.3736  
S_conditionS2:C_conditionC2:E_conditionE2   -0.05959    0.94493 2648.67938  -0.063   0.9497  
S_conditionS3:C_conditionC2:E_conditionE2   -1.61455    0.95237 2649.19981  -1.695   0.0901 .
S_conditionS2:C_conditionC3:E_conditionE2   -1.24787    0.94555 2649.51572  -1.320   0.1870  
S_conditionS3:C_conditionC3:E_conditionE2   -1.39477    0.95265 2649.55567  -1.464   0.1433  
S_conditionS2:C_conditionC2:E_conditionE3   -0.99598    0.94629 2650.45541  -1.053   0.2927  
S_conditionS3:C_conditionC2:E_conditionE3   -1.28928    0.95209 2648.81876  -1.354   0.1758  
S_conditionS2:C_conditionC3:E_conditionE3   -2.01203    0.94586 2649.91207  -2.127   0.0335 *
S_conditionS3:C_conditionC3:E_conditionE3   -1.70194    0.95235 2649.16702  -1.787   0.0740 .

What I cannot figure out is what the intercept is in this model.

Is it "S_conditionS1" or "S_conditionS1:C_conditionC1:E_conditionE1" or something else entirely?

And either way, why do the first levels of each predictor not appear anywhere else in the output? (e.g. If the intercept is indeed "S_conditionS1:C_conditionC1:E_conditionE1", then why is there no row in the output for the coefficient of, say, "S_conditionS1:C_conditionC2:E_conditionE2", etc.?

S_Brown
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  • Are you new to statistical modeling? When you have categorical variables you typically use one-hot-encoding or a reference level encoding in order to make your model identifiable. The intercept is the value when all categorical variables are at their reference level (the first level of the factor). – MrFlick Aug 01 '18 at 17:38
  • @MrFlick Did you read to the end of the question? I proposed two potential answers for what the intercept is, and both involved the first level of either one factor or all of the factors. I thought this made it self-evident that I was aware of an intercept being the reference level, but my apologies if not. If your answer is that the intercept is the interaction of all the factors' first levels (i.e. S_conditionS1:C_conditionC1:E_conditionE1, as I suggested), then my final question is still unanswered; namely, why are there no other coefficients involving any of the first levels? – S_Brown Aug 01 '18 at 21:12
  • @MrFlick ...Or to be more precise, why are there no other coefficients involving a first level interacting with other levels? e.g. S_conditionS1:C_conditionC2:E_conditionE2 – S_Brown Aug 01 '18 at 21:21
  • I guess you could consider it an interaction term in the case where all your variables are categorical, but that's not typically how one usually describes the intercept term. All other coefficients are differences from this "uber" reference level. It wouldn't make mathematical sense to include the same level again when talking about difference. You run into the problem of identifiability. If you have an intro regression textbook handy, this is most likely covered there. But these aren't programming questions, these are stats questions and would probably be better addressed at [stats.se]. – MrFlick Aug 01 '18 at 21:22

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