The APA style manual does not provide specific guidelines for linear mixed models. Additionally, a review of studies using linear mixed models reported that the psychological papers surveyed differed 'substantially' in how they reported on these models (Barr, Levy, Scheepers and Tily, 2013). It depends greatly on your study, in other words. Normatively speaking, I find this checklist (Ferron et al., 2008) extremely helpful in such cases.
Besides the more general checklist, Barr, Levy, Scheepers and Tily (2013) also provide some guidelines for reporting on LMMs. The whole paper is well worth a read, but excerpts of their "Reporting results" section are provided below.
One needs to provide sufficient information for the reader to be able to recreate the analyses. One way of satisfying this requirement is to report the variance–covariance matrix, which includes all the information about the random effects, including their estimates. ... A simpler option is to mention that one attempted to use a maximal LMEM and, as an added check, also state which factors had random slopes associated with them.
If it is seen as necessary or desirable in a confirmatory analysis to determine the random effects structure using a data-driven approach, certain minimal guidelines should be followed. First, it is critical to report the criteria that have been used, including the a-level for exclusion/inclusion of
random slopes and the order in which random slopes were tested. Furthermore, authors should explicitly report the changed assumptions about the generative process underlying the data that result from excluding the random slope (rather than just stating that the slopes did not ‘‘improve model fit’’), and should do so in non-technical language that non-experts can understand. Readers with only background in ANOVA will not understand that removing the random slope corresponds to pooling error across strata in a mixed-model ANOVA analysis. It is therefore preferable to clearly state the underlying assumption of a constant effect, e.g., ‘‘by excluding the random slope for the priming manipulation, we assume that the priming effect is invariant across subjects (or items) in the population.’’
Searching the Barr article, I also found a highly relevant CrossVal.SE question which asks for examples rather than guidelines. Since no guidelines appear to exist, I would suggest that.
- Barr, D. J., Levy, R., Scheepers, C., & Tily, H. J. (2013). Random effects structure for confirmatory hypothesis testing: Keep it maximal. Journal of Memory and Language, 68(3), 255–278. doi:10.1016/j.jml.2012.11.001
- Ferron, J. M., Hogarty, K. Y., Dedrick, R. F., Hess, M. R., Niles, J. D., & Kromrey, J. D. (2008). Reporting results from multilevel analyses. Multilevel modeling of educational data, 391-426. (Statistical checklist only.)