Hi Tobias,

I'll take a stab at your questions. Others should feel free to jump in and correct me if necessary.

First some general comments. I don't think I understood your motivation for wanting to use the contrasts that you specified rather than dummy codes. In what way would dummies "mess up" your interaction whereas the codes you propose would not?

Also, note that the particular coding scheme you have chosen is commonly called "effect coding." Like all non-orthogonal coding schemes, these do NOT test the mean differences that they naively appear to test based on the specified contrast weights! In your case, contrasts 1 and 2 compute mean differences between, respectively, condition E1 vs. mean of group means, and condition E2 mean vs. mean of group means. The E groups are *not* directly compared to the C group.

Now for your questions:
1. Yes. (Taking into account the usual additional complexities that come with mixed model analyses.)
2. Personally I do not find usually inspecting the covariance/correlation matrix of the fixed effect parameters to be particularly interesting or informative. I always thought it was a little strange that lmer printed this as part of the default output. Maybe others feel more strongly about it than I do.
3. Sure, I guess so, but interpret the estimated correlations among the random effects with caution, for at least two reasons. First, the correlations among the random effects depend to some extent on the scaling of the predictors associated with random slopes (i.e., you can rescale your predictors and substantially "push around" the estimated correlations even though the model has not changed in a deep sense). Second, there could be considerable uncertainty around these estimates. If you really want to make firm statements about these, you should probably try testing them more systematically using something like the tools in the RLRsim package (or maybe others have even better suggestions).
4. Yes.

Hope this helps,
Jake

From: tobias.heed@uni-hamburg.de
Date: Mon, 20 Jan 2014 15:23:53 +0100
To: r-sig-mixed-models@r-project.org
Subject: [R-sig-ME] non-orthogonal contrasts

Dear List,
 
I am wondering about the consequences of using non-orthogonal contrasts in an LMM.
I come from psychological experiments in which we often code factorial effects in contrasts as -1 and 1 (rather than use treatment coding, 0 and 1). This coding affects interactions differently from treatment coding which seems to be what most other disciplines use. 
Now, I have an experiment with one control condition (C), and two experimental conditions (E1, E2).
(I am measuring tactile localization performance under 3 different head postures; C is head straight; E1 and E2 are 2 different head postures. I have an additional 2-level factor, hand posture, and I have acquired all 3 conditions with both levels of this hand posture factor).
 
I am interested in the difference between the control condition and each experimental condition. I would like to specify the contrasts coding for conditions C, E1, and E2 like this:
contrast 1 (C, E1, E2): -1, 1, 0
contrast 2 (C, E1, E2): -1, 0, 1
These contrasts are not independent.
I am aware that I could use treatment coding (contrasts 0, 1, 0 and 0, 0, 1), but this will “mess up” the interactions with my second factor: what I really want to know is how the difference between C and each E is changed in interaction with the hand posture factor. 
 
When I use the contrasts as specified, I see in the fixed effects correlations that the two contrasts are correlated (-0.5 for contrast 1 with contrast 2), and similarly, the interaction of contrast 1 with the hand posture factor correlates negatively with the interaction of contrast 2 and the hand posture factor. Other correlations are small.
 
Questions:
1, can I still interpret the fixed effects as I would in a “classical" ANOVA?
2, is it the correct approach to check the correlation structure of the fixed effects, and if so, what should I infer from it?
3, can I interpret the random effects in the model? Specifically, I get negative correlations also for the random effects of my contrasts (i.e., the random effect correlations more or less mirror the fixed effect correlations)?
4, I have read (http://www.utdallas.edu/~herve/abdi-contrasts2010-pretty.pdf) that the regression model deals with non-orthogonal contrasts by partialing out the other variables from each contrast. Would the interpretation in an LMM be similar?
 
Thanks for any help.
Best,
Tobias
 
Model output:
 
“crossingStatus.c” is the hand posture factor (2 levels, coded -1 and 1 in the contrast)
“condition.c” is the factor with conditions C, E1, and E2. It is coded with 2 contrasts that are named “hemifield” (coded -1, 1, 0) and “hemispace” (coded -1, 0, 1).
> summary(model.body.head.3)
Generalized linear mixed model fit by the Laplace approximation 
Formula: accuracyClean ~ crossingStatus.c * condition.c + (1 + crossingStatus.c *      condition.c | subNo) 
   Data: dataSet.1 
   AIC   BIC logLik deviance
 42374 42611 -21160    42320
Random effects:
 Groups Name                                                Variance  Std.Dev. Corr                               
 subNo  (Intercept)                                         0.2423548 0.492295                                    
        crossingStatus.ccrossingStatus                      0.0826755 0.287534  0.184                             
        condition.chemifield                                0.0050947 0.071377  0.133  0.074                      
        condition.chemispace                                0.0030593 0.055311 -0.044  0.360 -0.696               
        crossingStatus.ccrossingStatus:condition.chemifield 0.0090415 0.095087  0.015 -0.474  0.470 -0.284        
        crossingStatus.ccrossingStatus:condition.chemispace 0.0169650 0.130250  0.002  0.508  0.132  0.191 -0.624 
Number of obs: 46928, groups: subNo, 23
 
Fixed effects:
                                                     Estimate Std. Error z value Pr(>|z|)    
(Intercept)                                          1.601920   0.103541  15.471  < 2e-16 ***
crossingStatus.ccrossingStatus                      -0.480008   0.061458  -7.810  5.7e-15 ***
condition.chemifield                                 0.061866   0.023619   2.619  0.00881 ** 
condition.chemispace                                 0.004818   0.021711   0.222  0.82439    
crossingStatus.ccrossingStatus:condition.chemifield  0.080060   0.027042   2.961  0.00307 ** 
crossingStatus.ccrossingStatus:condition.chemispace -0.038528   0.032899  -1.171  0.24156    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
 
Correlation of Fixed Effects:
                                  (Intr) crsS.S cndtn.chmf cndtn.chms crssngStts.ccrssngStts:cndtn.chmf
crssngStt.S                        0.170                                                               
cndtn.chmfl                        0.084  0.050                                                        
cndtn.chmsp                       -0.021  0.183 -0.576                                                 
crssngStts.ccrssngStts:cndtn.chmf  0.013 -0.339  0.098     -0.036                                      
crssngStts.ccrssngStts:cndtn.chms  0.000  0.411  0.125     -0.045     -0.577         
 
 
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