Internals of Mixed Models

Linear mixed models are widely used in Agriculture and Plant Breeding, as of recent. With access to genotype data high resolution phenotype data, it has become more of a requirement to use this family of model.

Mixed models allow for experimental (design or outcome) variables’ parameter estimates to have probabilistic distributions – most commonly normal – with opportunity to specify different variance-covariance components among the levels of those variables. In this post, I wish to discuss on some of the popular mixed modeling tools and techniques in the R community with links and discussion of the concepts surrounding variations of modeling techniques.

1. For a discussion of when should nesting of certain factor inside other is important what are obvious implications of implicit vs. explicit nesting, refer to:

2. Paul Schmidt has a nice blog discussing different types of variance structure in mixed models and how they are specified using popular statistical packages: https://schmidtpaul.github.io/MMFAIR/variance_structures.html# . His github repo is equally rich in codes on mixed modeling options:

3. Partial pooling in mixed effects models (Links were learnt from https://stats.stackexchange.com/a/396755/126964)

A visual tutorial (https://www.tjmahr.com/plotting-partial-pooling-in-mixed-effects-models/) by Tristan Mahr is excellent in showing how partial pooling works in a mixed model. The blog shows beautiful plots comparing different model families (with no pooling – linear model for each group, with partial pooling – group intercept specified random effects model, complete pooling – average model disregarding group; notice that group is a categorical variable) with respect to coefficient estimates. Also, an interesting graph is that of scatterplot showing intercept and slope estimates in x and y respectively, with arrow depiction of how they are shrunk/pulled towards common center (estimates from complete pooling model). I link graphic below, but for the code generating it refer to the blog. The author (in bonus) also provides a way to visualize the distribution of posterior samples of all coefficient estimates for individual grouping factor from a bayesian model fitting using rstanarm.

For model representation of various pooled, unpooled and partially pooled models, refer to the lecture slides at http://www.utstat.toronto.edu/~guerzhoy/303/lec/lec9/multi.pdf

Here is another resource that shows effects of partial pooling for variance components estimates. https://win-vector.com/2017/09/28/partial-pooling-for-lower-variance-variable-encoding/#more-5245

These slides talk about the degree of pooling and how to quantify it: https://web.as.uky.edu/statistics/users/pbreheny/701/S13/notes/4-18.pdf.

knitr::include_url("https://www.tjmahr.com/figs//2017-06-22-plotting-partial-pooling-in-mixed-effects-models/topgraphic-map-1-1.png")