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I am attempting to fit a linear mixed-effect model in R using lme4 that is quite a bit more complex than any example I've seen in forums or in textbooks. I am having trouble finding the correct code for the random effects in particular. I have two fixed factors (parental environment and germination treatment) and random factor (genetic line); I want to test all 2 and 3-way interactions between these factors. I have two additional fixed factors that I want to test without interactions (provisioning and block). I want to specify random intercepts and slopes for genetic line, but I am not sure how to do this. The dependent variable is biomass.

Is this model specification correct?:

fullmod <- lmer(biomass ~ parental.environment* germination.treatment*
   (1+parental.environment*germination.treatment|genetic.line)
                + provisioning + block, biomass.data)

I am also interested in testing the significance of the 3-way interaction parental environment x germination treatment x genetic line by comparing the fit of the reduced model to the full model. What would be the correct model specification for the full model minus this 3-way interaction?

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You're nearly right.

The main effects of the fixed terms plus their interaction is/are represented as

1+parental.environment* germination.treatment

Interactions between random and fixed factors are coded as variation of the fixed effects across levels of the random effects grouping variables, like so:

(1+parental.environment* germination.treatment|genetic.line)

As with all R model formulae, you can choose to include or not include the intercept term explicitly.

The only confusion on your part is that these terms are combined with + rather than *,

fullmod <- lmer(biomass ~ parental.environment* germination.treatment +
   (1+parental.environment*germination.treatment|genetic.line) +
                provisioning + block, biomass.data)

I am also interested in testing the significance of the 3-way interaction parental environment x germination treatment x genetic line by comparing the fit of the reduced model to the full model.

The reduced model would be

biomass ~ parental.environment* germination.treatment +
       (1+parental.environment+germination.treatment|genetic.line) +
                    provisioning + block

Two notes (1) if there are $m$ levels of parental environment and $n$ levels of germination treatment, you will be fitting an $mn \times mn$ variance-covariance matrix (with $mn(mn+1)/2$ parameters), so you'd better have enough genetic lines to make that practical; (2) significance tests of variance components are a little tricky (the $p$-values from a naive likelihood ratio test are typically conservative by a factor of about 2; see Pinheiro and Bates 2000).

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  • $\begingroup$ Thanks for your comment. I would appreciate some clarification. When you say that my only confusion is "these terms are combined with + rather than *", I took you to mean that I should use + to combine parental environment and germination treatment in the random effects specification of the full model. If so, I do not see a difference between the full model and the reduced model that you suggested. Am I correct in my interpretation, and if so, what is the correct reduced model? $\endgroup$
    – Jacob Herman
    Commented Jul 23, 2015 at 21:55
  • $\begingroup$ the character at the end of the first line of your formula, connecting the fixed- and random-effects parts of the formula, is a *; it should be + instead. I would expect that if you entered the formula exactly as shown in your question it would generate some interesting error messages. $\endgroup$
    – Ben Bolker
    Commented Jul 23, 2015 at 22:37

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