[R-sig-ME] Random structure more complex than fixed structure

[Tibor Kiss]

Dear list members,

I have a model where I assume a more complex structure for the random effects than for the fixed effects. Schematically, the model looks like the following, it is a cumulative link mixed model (ordinal::clmm), because the response consists of a five point Likert scale (the same would apply to a binomial GLMM). You can find the output of the model for the fixed and random structures below.

model <- 
  clmm(ANSWER  ~ ADVERBIAL  + POSITION + 
         (0 + ADVERBIAL * POSITION | subjects) +
         (0 + POSITION | items),  # ADVERBIAL constant for items
       data)

The reason for assuming random slopes with interaction, but no such interaction in the fixed effects are as follows:

1. I am interested in the by-subject variability for each condition. This is a 3 x 2 design, with three ADVERBIAL types, and two POSITIONs, and I assume that there is less by-subject variance in the second level of POSITION. Hence, I need an interaction to get simple effects for each combination of the two predictors.

2. If I compare this model with a simpler one, where I assume a random slope for POSITION only, then the more complex structure is indicated as significant in comparison to the simpler model (p < 0.01). 

3. Also, an interaction of the fixed effects does not prove to be significant.

In addition, the output of the model makes perfect sense to me: for each pair of POSITION and ADVERBIAL the by-subject variance for POSITION 2 is lower than the one for POSITION 1, which I interpret as a stronger confidence in judgments for POSITION 2 across the participants of the experiment. 

My question is: is it ok/justified to assume a random structure that is more complex than the fixed structure, if I want to find out about the by-subject variability for each condition in an experiment. Also: would there be a simpler way to do the same? 

Thanks for your help.

With kind regards


Tibor

## Coefficients:
##                     			 Estimate Std. Error z value Pr(>|z|)    
## ADVERBIAL COM(O)  	-0.9571     0.3479  -2.751  0.00594 ** 
## ADVERBIAL ILOC    		-1.0353     0.3396  -3.049  0.00230 ** 
## POSITION  2                       1.1916     0.1738   6.856 7.08e-12 ***
## ---

## Threshold coefficients:
##     Estimate Std. Error z value
## 1|2  -4.5321     0.3557 -12.740
## 2|3  -2.0566     0.3301  -6.231
## 3|4  -1.3708     0.3278  -4.182
## 4|5   1.7521     0.3286   5.333


## Random effects:
##  Groups   Name                                		Variance Std.Dev. Corr                               
##  subjects ADVERBIAL INSTR                 	2.3987   1.5488                                      
##           ADVERBIAL COM(O)                	2.3873   1.5451    0.748                             
##           ADVERBIAL ILOC                  		2.1802   1.4766    0.898  0.561                      
##           POSITION 2                      		0.8843   0.9404   -0.078 -0.109 -0.088               
##           ADVERBIAL COM(O):POSITION 2 0.7158   0.8460    0.240 -0.334  0.487 -0.178        
##           ADVERBIAL ILOC:POSITION 2   	0.6667   0.8165   -0.430  0.083 -0.595 -0.386 -0.790 
##  items    POSITION 1                      		0.5416   0.7359                                      
##              POSITION 2                      		0.6100   0.7810   0.781                              
## Number of groups:  subjects 51,  items 36 



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Prof. Dr. Tibor Kiss
Linguistic Data Science Lab
Ruhr-Universität Bochum