# Mixed model with crossed effects and repeated measures - what is correct ‘maximal model’ for use in lmer/lme4? [R]

Objective

I have a crossed and implicitly nested design and am trying to validate the correct ‘maximal’ model (including all linear and pairwise interactions of the variables) for use in lmer(). I intend to use this as the starting point for some kind of backward stepwise regression, possibly making use of the function mixed() in the {afex} package.

Experimental design

This a linguistics study. We have 20 Subjects, each speaking 180 utterances, amounting to 3600 observations in total. Each utterance is initiated via prompting, and an associated Response Time is measured. Log Response Time is the dependent variable.

Conditions & Blocks

The Response Time for the utterances is affected by 3 Conditions (coded 1 to 3). Each Condition is implemented by prompting the Subject to recite 1 of 4 Blocks of utterances (coded 1 to 12).

Words & Tones

Each Block brings about its associated Condition via 15-utterance repetition of 3 carefully chosen Words. There are a total of 12 Words used in the experiment (coded 1 to 12). The Words within each Block can also be categorized by Tone (coded 1 to 2). There are 6 Words per Tone.

Summary

Each of the 20 Subjects utter all 12 Blocks of 15 utterances each. In doing so, they repeatedly utter all 12 Words (15 utterances per Word), and thereby use both Tones (90 utterances per Tone).

I would like to consider Block, Word, and Subject as random effects, and Condition and Tone as fixed.

Proposed Model

I think the model can be written in the following way…

RT_log ~ Condition*Tone + (Condition*Tone|Subject) + (Condition|Word) + (Tone|Block)

Questions

1. Is this the 'maximal' model (with linear plus pairwise interactions) appropriate for my experimental design?_

2. There is correlation between Block and Condition (there are only 4 possible blocks - out of the total 12 - for each Condition). There is, similarly, correlation between Word and Tone. Is it 'okay' to leave this correlation in the model? I don't see a good way of removing it.

3. How will lme4 handle implicit nesting: I.e., the blocks, which are implicitly nested in the 3 conditions (i.e., only 4 blocks are applicable to each of the 3 conditions, even though the blocks are coded from 1 to 12), and the words, which are implicitly nested within the 2 tones (only 6 words are applicable to each tone, even though words are coded from 1 to 12)?

4. Some Blocks utilize Words of only a single Tone, whereas other Blocks utilize words of both Tones. Will that cause problems for the (Tone|Block) term in the model? It will only make sense for certain values of Block.

5. It has been suggested by some that we might need a "Subject:Word" grouping (random effect). Why might we need this grouping?

• I am very skeptical about the keeping-it-maximal rationale. It really seems like one over-parameterizes a model. From a practical standpoint there are a lot of parameters to estimate. From a theoretical standpoint one tests against really spurious phenomena. Such models are often singular and having problematic convergence (something that it is a essential for this "maximal" rationale to work). Check Bates at al. 2015 for more. – usεr11852 Jan 29 '16 at 2:16
• Thanks. I am only looking for maximal as a starting point...guidance for which terms to investigate. – clarpaul Jan 29 '16 at 4:01
• @user11852, do you have any thoughts regarding the specific terms I have listed or not listed? I am totally new to lmer modeling and to mixed-effects modeling, so any thoughts along these lines would be much appreciated. – clarpaul Jan 29 '16 at 15:05
• Given you have recognised three terms as random I would start with those. At first instance I would use a fully crossed design between Subject, Word and Block (so something like) ...+ (1|Subject) + (1|Word) + (1|Block) and see were this takes me, what are the estimated variance amplitude, etc. I would avoid using Tone in any of the r.e. structures. – usεr11852 Jan 29 '16 at 17:47
• Additional comment: Surely some effect for a grouping Subject:Word can be present. By definition none of the variables are orthogonal to each other. I am pretty sure that you might be able to argue a Subject:Block:Word grouping too, but what good is that for your research question? – usεr11852 Jan 29 '16 at 17:50