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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 ๐ ~ ๐(0, ๐บ). For the response vector, ๐ฒ, only the mean of conditional distribution, ๐จ|๐ = ๐ depends on ๐ and it does so through a linear predictor expression, ๐ = ๐๐ + ๐๐, where ๐ is the fixed-effects coefficient vector and ๐ and ๐ are model matrices of the appropriate sizes,
In a LinearMixedModel the conditional mean, ๐ = ๐ผ[๐จ|๐ = ๐], is the linear predictor, ๐, and the conditional distribution is multivariate Gaussian, (๐จ|๐ = ๐) ~ ๐(๐, ฯยฒ๐).
In a GeneralizedLinearMixedModel, the conditional mean, ๐ผ[๐จ|๐ = ๐], is related to the linear predictor via a link function.
Typical distribution forms are Bernoulli for binary data or Poisson for count data.
| OS | OS Version | Arch | Julia |
|---|---|---|---|
| Linux | Ubuntu 20.04 | x64 | v1.8 |
| Linux | Ubuntu 20.04 | x64 | current release |
| Linux | Ubuntu 20.04 | x64 | nightly |
| macOS | Monterey 12 | x64 | v1.8 |
| Windows | Server 2019 | x64 | v1.8 |
Note that previous releases still support older Julia versions.
Version 4.0.0 contains some user-visible changes and many changes in the underlying code.
Please see NEWS for a complete overview, but a few key points are:
- The internal storage of the model matrices in
LinearMixedModelhas changed and been optimized. This change should be transparent to users who are not manipulating the fields of the modelstructdirectly. - The handling of rank deficiency continues to evolve.
- Additional
predictandsimulatemethods have been added for generalizing to new data. saveoptsumandrestoreoptsum!provide for saving and restoring theoptsumand thus offer a way to serialize a model fit.- There is improved support for the runtime construction of model formula, especially
RandomEffectsTerms and nested terms (methods forBase.|(::AbstractTerm, ::AbstractTerm)andBase./(::AbstractTerm, ::AbstractTerm)). - A progress display is shown by default for models taking more than a few hundred milliseconds to fit. This can be disabled with the keyword argument
progress=false.
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 AICc BIC
-163.6635 327.3271 333.3271 334.2501 337.5307
Variance components:
Column Variance Std.Dev.
batch (Intercept) 1388.3332 37.2603
Residual 2451.2501 49.5101
Number of obs: 30; levels of grouping factors: 6
Fixed-effects parameters:
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
Coef. Std. Error z Pr(>|z|)
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
(Intercept) 1527.5 17.6946 86.33 <1e-99
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
julia> using Random
julia> bs = parametricbootstrap(MersenneTwister(42), 1000, m1);
Progress: 100%%|โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ| Time: 0:00:00
julia> propertynames(bs)
13-element Vector{Symbol}:
:allpars
:objective
:ฯ
:ฮฒ
:se
:coefpvalues
:ฮธ
:ฯs
:ฮป
:inds
:lowerbd
:fits
:fcnames
julia> bs.coefpvalues # returns a row table
1000-element Vector{NamedTuple{(:iter, :coefname, :ฮฒ, :se, :z, :p), Tuple{Int64, Symbol, Float64, Float64, Float64, Float64}}}:
(iter = 1, coefname = Symbol("(Intercept)"), ฮฒ = 1517.0670832927115, se = 20.76271142094811, z = 73.0669059804057, p = 0.0)
(iter = 2, coefname = Symbol("(Intercept)"), ฮฒ = 1503.5781855888436, se = 8.1387737362628, z = 184.7425956676446, p = 0.0)
(iter = 3, coefname = Symbol("(Intercept)"), ฮฒ = 1529.2236379016574, se = 16.523824785737837, z = 92.54659001356465, p = 0.0)
โฎ
(iter = 998, coefname = Symbol("(Intercept)"), ฮฒ = 1498.3795009457242, se = 25.649682012258104, z = 58.417079019913054, p = 0.0)
(iter = 999, coefname = Symbol("(Intercept)"), ฮฒ = 1526.1076747922416, se = 16.22412120273579, z = 94.06411945042063, p = 0.0)
(iter = 1000, coefname = Symbol("(Intercept)"), ฮฒ = 1557.7546433870125, se = 12.557577103806015, z = 124.04898098653763, p = 0.0)
julia> using DataFrames
julia> DataFrame(bs.coefpvalues) # puts it into a DataFrame
1000ร6 DataFrame
โ Row โ iter โ coefname โ ฮฒ โ se โ z โ p โ
โ โ Int64 โ Symbol โ Float64 โ Float64 โ Float64 โ Float64 โ
โโโโโโโโผโโโโโโโโผโโโโโโโโโโโโโโผโโโโโโโโโโผโโโโโโโโโโผโโโโโโโโโโผโโโโโโโโโโค
โ 1 โ 1 โ (Intercept) โ 1517.07 โ 20.7627 โ 73.0669 โ 0.0 โ
โ 2 โ 2 โ (Intercept) โ 1503.58 โ 8.13877 โ 184.743 โ 0.0 โ
โ 3 โ 3 โ (Intercept) โ 1529.22 โ 16.5238 โ 92.5466 โ 0.0 โ
โฎ
โ 998 โ 998 โ (Intercept) โ 1498.38 โ 25.6497 โ 58.4171 โ 0.0 โ
โ 999 โ 999 โ (Intercept) โ 1526.11 โ 16.2241 โ 94.0641 โ 0.0 โ
โ 1000 โ 1000 โ (Intercept) โ 1557.75 โ 12.5576 โ 124.049 โ 0.0 โ
julia> DataFrame(bs.ฮฒ)
1000ร3 DataFrame
โ Row โ iter โ coefname โ ฮฒ โ
โ โ Int64 โ Symbol โ Float64 โ
โโโโโโโโผโโโโโโโโผโโโโโโโโโโโโโโผโโโโโโโโโโค
โ 1 โ 1 โ (Intercept) โ 1517.07 โ
โ 2 โ 2 โ (Intercept) โ 1503.58 โ
โ 3 โ 3 โ (Intercept) โ 1529.22 โ
โฎ
โ 998 โ 998 โ (Intercept) โ 1498.38 โ
โ 999 โ 999 โ (Intercept) โ 1526.11 โ
โ 1000 โ 1000 โ (Intercept) โ 1557.75 โThe development of this package was supported by the Center for Interdisciplinary Research, Bielefeld (ZiF)/Cooperation Group "Statistical models for psychological and linguistic data".