Mixed-effects models in Julia

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This package defines linear mixed models (LinearMixedModel) and generalized linear mixed models (GeneralizedLinearMixedModel). Users can use the abstraction for statistical model API to build, fit (fit/fit!), and query the fitted models.

A mixed-effects model is a statistical model for a response variable as a function of one or more covariates. For a categorical covariate the coefficients associated with the levels of the covariate are sometimes called effects, as in "the effect of using Treatment 1 versus the placebo". If the potential levels of the covariate are fixed and reproducible, e.g. the levels for Sex could be "F" and "M", they are modeled with fixed-effects parameters. If the levels constitute a sample from a population, e.g. the Subject or the Item at a particular observation, they are modeled as random effects.

A mixed-effects model contains both fixed-effects and random-effects terms.

With fixed-effects it is the coefficients themselves or combinations of coefficients that are of interest. For random effects it is the variability of the effects over the population that is of interest.

In this package random effects are modeled as independent samples from a multivariate Gaussian distribution of the form 𝓑 ~ 𝓝(0, 𝚺). For the response vector, 𝐲, only the mean of conditional distribution, 𝓨|𝓑 = 𝐛 depends on 𝐛 and it does so through a linear predictor expression, 𝛈 = 𝐗𝛃 + 𝐙𝐛, where 𝛃 is the fixed-effects coefficient vector and 𝐗 and 𝐙 are model matrices of the appropriate sizes,

In a LinearMixedModel the conditional mean, 𝛍 = 𝔼[𝓨|𝓑 = 𝐛], is the linear predictor, 𝛈, and the conditional distribution is multivariate Gaussian, (𝓨|𝓑 = 𝐛) ~ 𝓝(𝛍, σ²𝐈).

In a GeneralizedLinearMixedModel, the conditional mean, 𝔼[𝓨|𝓑 = 𝐛], is related to the linear predictor via a link function. Typical distribution forms are Bernoulli for binary data or Poisson for count data.

Currently Supported Platforms

Upon release of the next Julia LTS, Tier 1 will become Tier 2 and Julia LTS will become Tier 1.

Version 3.0.0

Version 3.0.0 contains some user-visible changes and many changes in the underlying code.

Please see NEWS for a complete overview. The most dramatic user-facing change is a re-working of parametricbootstrap for speed and convenience. Additionally, several formula features have been added and the handling of rank deficiency has changed.

Quick Start

julia> using MixedModels

julia> m1 = fit(MixedModel, @formula(yield ~ 1 + (1|batch)), MixedModels.dataset(:dyestuff))
Linear mixed model fit by maximum likelihood
 yield ~ 1 + (1 | batch)
   logLik   -2 logLik     AIC        BIC
  -163.6635   327.3271   333.3271   337.5307

Variance components:
            Column   VarianceStd.Dev.
batch    (Intercept)  1388.33 37.2603
Residual              2451.25 49.5101
 Number of obs: 30; levels of grouping factors: 6

  Fixed-effects parameters:
────────────────────────────────────────────────
              Coef.  Std. Error      z  Pr(>|z|)
────────────────────────────────────────────────
(Intercept)  1527.5     17.6946  86.33    <1e-99
────────────────────────────────────────────────


julia> using Random

julia> bs = parametricbootstrap(MersenneTwister(42), 1000, m1);
Progress: 100%%|████████████████████████████████████████████████| Time: 0:00:00

julia> propertynames(bs)
13-element Vector{Symbol}:
 :allpars
 :objective
 :σ
 :β
 :se
 :coefpvalues
 :θ
 :σs
 :λ
 :inds
 :lowerbd
 :fits
 :fcnames

julia> bs.coefpvalues # returns a row table
1000-element Vector{NamedTuple{(:iter, :coefname, :β, :se, :z, :p), Tuple{Int64, Symbol, Float64, Float64, Float64, Float64}}}:
 (iter = 1, coefname = Symbol("(Intercept)"), β = 1517.0670832927115, se = 20.76271142094811, z = 73.0669059804057, p = 0.0)
 (iter = 2, coefname = Symbol("(Intercept)"), β = 1503.5781855888436, se = 8.1387737362628, z = 184.7425956676446, p = 0.0)
 (iter = 3, coefname = Symbol("(Intercept)"), β = 1529.2236379016574, se = 16.523824785737837, z = 92.54659001356465, p = 0.0)
 ⋮
 (iter = 998, coefname = Symbol("(Intercept)"), β = 1498.3795009457242, se = 25.649682012258104, z = 58.417079019913054, p = 0.0)
 (iter = 999, coefname = Symbol("(Intercept)"), β = 1526.1076747922416, se = 16.22412120273579, z = 94.06411945042063, p = 0.0)
 (iter = 1000, coefname = Symbol("(Intercept)"), β = 1557.7546433870125, se = 12.557577103806015, z = 124.04898098653763, p = 0.0)

julia> using DataFrames

julia> DataFrame(bs.coefpvalues) # puts it into a DataFrame
1000×6 DataFrame
│ Row  │ iter  │ coefname    │ β       │ se      │ z       │ p       │
│      │ Int64 │ Symbol      │ Float64 │ Float64 │ Float64 │ Float64 │
├──────┼───────┼─────────────┼─────────┼─────────┼─────────┼─────────┤
│ 1    │ 1     │ (Intercept) │ 1517.07 │ 20.7627 │ 73.0669 │ 0.0     │
│ 2    │ 2     │ (Intercept) │ 1503.58 │ 8.13877 │ 184.743 │ 0.0     │
│ 3    │ 3     │ (Intercept) │ 1529.22 │ 16.5238 │ 92.5466 │ 0.0     │
⋮
│ 998  │ 998   │ (Intercept) │ 1498.38 │ 25.6497 │ 58.4171 │ 0.0     │
│ 999  │ 999   │ (Intercept) │ 1526.11 │ 16.2241 │ 94.0641 │ 0.0     │
│ 1000 │ 1000  │ (Intercept) │ 1557.75 │ 12.5576 │ 124.049 │ 0.0     │

julia> DataFrame(bs.β)
1000×3 DataFrame
│ Row  │ iter  │ coefname    │ β       │
│      │ Int64 │ Symbol      │ Float64 │
├──────┼───────┼─────────────┼─────────┤
│ 1    │ 1     │ (Intercept) │ 1517.07 │
│ 2    │ 2     │ (Intercept) │ 1503.58 │
│ 3    │ 3     │ (Intercept) │ 1529.22 │
⋮
│ 998  │ 998   │ (Intercept) │ 1498.38 │
│ 999  │ 999   │ (Intercept) │ 1526.11 │
│ 1000 │ 1000  │ (Intercept) │ 1557.75 │

Funding Acknowledgement

The development of this package was supported by the Center for Interdisciplinary Research, Bielefeld (ZiF)/Cooperation Group "Statistical models for psychological and linguistic data".