model specification: crossed random effect for item and subject with fixed effect nested (??) within items

Question

In a previous thread I got the advice to run a crossed random effects linear mixed model with my data. Whilst working on the model specification, I came across a new question.

In short, all participants have judged 14 written statements (the same statements for all participants) in terms of various overall impressions. I want to examine which overall impressions (e.g. fluency in reading, perceived friendliness) predict the credibility rating of statements.

Whereas the statements and participants are crossed (i.e. all participants rated all statements), some of the statements are truthful whereas others are deceptive (veracity of statement). Now I am trying to determine how a model would look like if veracity is included in the model.

I reckon a model without veracity would be specified with the lmer (R, lme4 package) function as follows:

credibility ~ predictors + (1|ID) + (1|statement) 

Question: I am wondering how the model would change if veracity of statement would be added as a fixed factor into that model.

Bayeen and colleagues (2008) work through an interesting example in their publication. In their example, participants were presented with three items (which would be statements in my case). Each item was presented under two different conditions, i.e. the same item once under condition A (long SOA) and once under condition B (short SOA). This is where the difference to my case lies as in my study some statements were truthful and some deceptive, i.e. the statements in the two veracity conditions are different.

Based on the article mentioned above, I would specify my model as follows:

credibility ~ predictors + veracity + (1|statement) + (1 + veracity|ID)

I would appreciate any help with respect to the question whether and if so how the fact that the statements in the two veracity conditions are different impacts the model specification.

Answer 1

The model formula

credibility ~ predictors + veracity + (1|statement) + (1|ID)

says that in addition to predictors, veracity is also a fixed effect. The observations are grouped within statement and also grouped within ID, however neither statement nor ID are nested within each other. Such a model will estimate random intercepts for ID and statement. Since veracity appears as a fixed effect only, this model assumes that the "effect" of veracity on credibility is the same across all groups.

In the model formula

credibility ~ predictors + veracity + (1|statement) + (1 + veracity|ID)

the difference is that veracity now also appears as a random slope / random coefficient within ID, that is for every level of ID, veracity is also allowed to vary. This means that each participant has it's own slope for veracity (the fixed part plus for random part).

You would use the second model to assess whether the association of veracity with credibility differs between participants.