# Linear Mixed Model for partially nested AND crossed factors

We are struggling with random effects for a linear mixed effects model that has partial nesting AND crossing.

In our data, all individuals are coupled (male and female only, one couple per case). 2/3 of the couples have mediators (treatment) and 1/3 do not (Non-treatment or TAU). Regardless of the mediator assignment, all couples are assigned to a judge. We have about 18 mediators and 15 judges.

There are three conditions, two of which are for cases that have mediators, and one for the cases that have no mediators. Thus, we have partial nesting due to the mediator groupings, and crossing by the mediators and judges (all cases have judges and 2/3 have mediators). We gave all cases without mediators dummy mediators that are unique to each case to prevent grouping of the non-mediation cases.

We are trying to figure out what model makes most sense. Below is a working draft. The outcome is a subjective continuous measure of Satisfaction. Fixed effects are the condition and sex.

We want to control for lack of independence due to couples, due to the mediator, and due to the judge. Since couples are nested and not crossed with mediators, we have an interacting random effect of couples nested within mediators. A new variable "d" indicates treatment vs. non-treatment where 1 is treatment and 0 is nontreatment. Since mediators are nested within treatment and non-treatment, we included an interacting random term for that too (1|d:Mediators). Furthermore, since couples are nested within treatment and non-treatment, we included a random term for that as well (1|d:Couples). We are also wondering whether we need to nest couples within mediators (1|Mediators:Couples). Since mediators are crossed with Judges, we did not add an interacting term there.

LMM4 <- lmerTest::lmer(Satisfaction ~ condition + sex + ( 1 | Mediator ) + (1 | Couples) + (1 | Judge) + (1 | Mediator:Couples ) + (1 | d:Mediator) + ( 1 | d:Couples), data=df)

Otherwise, we are also considering the following since the treatment/no-treatment are captured in the fixed effect condition:

LMM4 <- lmerTest::lmer(Satisfaction ~ condition + sex + ( 1 | Mediator ) + (1 | Couples) + (1 | Judge) + (1 | Mediator:Couples ) + ( 1 | Judge:Couples), data=df)

Is this on the right track? Is there anything we are missing, or do any of the interactions not make sense? We would greatly appreciate any help or direction given.