# API

In addition to its own functionality, MixedModels.jl also implements extensive support for the StatsAPI.StatisticalModel and StatsAPI.RegressionModel API.

## Types

MixedModels.BlockDescriptionType
BlockDescription

Description of blocks of A and L in a LinearMixedModel

Fields

• blknms: Vector{String} of block names
• blkrows: Vector{Int} of the number of rows in each block
• ALtypes: Matrix{String} of datatypes for blocks in A and L.

When a block in L is the same type as the corresponding block in A, it is described with a single name, such as Dense. When the types differ the entry in ALtypes is of the form Diag/Dense, as determined by a shorttype method.

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MixedModels.BlockedSparseType
BlockedSparse{Tv,S,P}

A SparseMatrixCSC whose nonzeros form blocks of rows or columns or both.

Members

• cscmat: SparseMatrixCSC{Tv, Int32} representation for general calculations
• nzasmat: nonzeros of cscmat as a dense matrix
• colblkptr: pattern of blocks of columns

The only time these are created are as products of ReMats.

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MixedModels.FeMatType
FeMat{T,S}

A matrix and a (possibly) weighted copy of itself.

Typically, an FeMat represents the fixed-effects model matrix with the response (y) concatenated as a final column.

Note

FeMat is not the same as FeTerm.

Fields

• xy: original matrix, called xy b/c in practice this is hcat(fullrank(X), y)
• wtxy: (possibly) weighted copy of xy (shares storage with xy until weights are applied)

Upon construction the xy and wtxy fields refer to the same matrix

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MixedModels.FeTermType
FeTerm{T,S}

Term with an explicit, constant matrix representation

Typically, an FeTerm represents the model matrix for the fixed effects.

Note

FeTerm is not the same as FeMat!

Fields

• x: full model matrix
• piv: pivot Vector{Int} for moving linearly dependent columns to the right
• rank: computational rank of x
• cnames: vector of column names
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MixedModels.GaussHermiteNormalizedType
GaussHermiteNormalized{K}

A struct with 2 SVector{K,Float64} members

• z: abscissae for the K-point Gauss-Hermite quadrature rule on the Z scale
• wt: Gauss-Hermite weights normalized to sum to unity
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MixedModels.GeneralizedLinearMixedModelType
GeneralizedLinearMixedModel

Generalized linear mixed-effects model representation

Fields

• LMM: a LinearMixedModel - the local approximation to the GLMM.
• β: the pivoted and possibly truncated fixed-effects vector
• β₀: similar to β. Used in the PIRLS algorithm if step-halving is needed.
• θ: covariance parameter vector
• b: similar to u, equivalent to broadcast!(*, b, LMM.Λ, u)
• u: a vector of matrices of random effects
• u₀: similar to u. Used in the PIRLS algorithm if step-halving is needed.
• resp: a GlmResp object
• η: the linear predictor
• wt: vector of prior case weights, a value of T[] indicates equal weights.

The following fields are used in adaptive Gauss-Hermite quadrature, which applies only to models with a single random-effects term, in which case their lengths are the number of levels in the grouping factor for that term. Otherwise they are zero-length vectors.

• devc: vector of deviance components
• devc0: vector of deviance components at offset of zero
• sd: approximate standard deviation of the conditional density
• mult: multiplier

Properties

In addition to the fieldnames, the following names are also accessible through the . extractor

• theta: synonym for θ
• beta: synonym for β
• σ or sigma: common scale parameter (value is NaN for distributions without a scale parameter)
• lowerbd: vector of lower bounds on the combined elements of β and θ
• formula, trms, A, L, and optsum: fields of the LMM field
• X: fixed-effects model matrix
• y: response vector
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MixedModels.GroupingType
struct Grouping <: StatsModels.AbstractContrasts end

A placeholder type to indicate that a categorical variable is only used for grouping and not for contrasts. When creating a CategoricalTerm, this skips constructing the contrasts matrix which makes it robust to large numbers of levels, while still holding onto the vector of levels and constructing the level-to-index mapping (invindex field of the ContrastsMatrix.).

Note that calling modelcols on a CategoricalTerm{Grouping} is an error.

Examples

julia> schema((; grp = string.(1:100_000)))
# out-of-memory error

julia> schema((; grp = string.(1:100_000)), Dict(:grp => Grouping()))
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MixedModels.LikelihoodRatioTestType
LikelihoodRatioTest

Results of MixedModels.likelihoodratiotest

Fields

• formulas: Vector of model formulae
• models: NamedTuple of the dof and deviance of the models
• tests: NamedTuple of the sequential dofdiff, deviancediff, and resulting pvalues

Properties

• deviance : note that this is actually -2 log likelihood for linear models (i.e. without subtracting the constant for a saturated model)
• pvalues
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MixedModels.LinearMixedModelType
LinearMixedModel

Linear mixed-effects model representation

Fields

• formula: the formula for the model
• reterms: a Vector{AbstractReMat{T}} of random-effects terms.
• Xymat: horizontal concatenation of a full-rank fixed-effects model matrix X and response y as an FeMat{T}
• feterm: the fixed-effects model matrix as an FeTerm{T}
• sqrtwts: vector of square roots of the case weights. Can be empty.
• parmap : Vector{NTuple{3,Int}} of (block, row, column) mapping of θ to λ
• dims : NamedTuple{(:n, :p, :nretrms),NTuple{3,Int}} of dimensions. p is the rank of X, which may be smaller than size(X, 2).
• A: a Vector{AbstractMatrix} containing the row-major packed lower triangle of hcat(Z,X,y)'hcat(Z,X,y)
• L: the blocked lower Cholesky factor of Λ'AΛ+I in the same Vector representation as A
• optsum: an OptSummary object

Properties

• θ or theta: the covariance parameter vector used to form λ
• β or beta: the fixed-effects coefficient vector
• λ or lambda: a vector of lower triangular matrices repeated on the diagonal blocks of Λ
• σ or sigma: current value of the standard deviation of the per-observation noise
• b: random effects on the original scale, as a vector of matrices
• u: random effects on the orthogonal scale, as a vector of matrices
• lowerbd: lower bounds on the elements of θ
• X: the fixed-effects model matrix
• y: the response vector
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MixedModels.LinearMixedModelType
LinearMixedModel(y, Xs, form, wts=[], σ=nothing)

Private constructor for a LinearMixedModel.

To construct a model, you only need the response (y), already assembled model matrices (Xs), schematized formula (form) and weights (wts). Everything else in the structure can be derived from these quantities.

Note

This method is internal and experimental and so may change or disappear in a future release without being considered a breaking change.

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MixedModels.LinearMixedModelMethod
LinearMixedModel(y, feterm, reterms, form, wts=[], σ=nothing)

Private constructor for a LinearMixedModel given already assembled fixed and random effects.

To construct a model, you only need a vector of FeMats (the fixed-effects model matrix and response), a vector of AbstractReMat (the random-effects model matrices), the formula and the weights. Everything else in the structure can be derived from these quantities.

Note

This method is internal and experimental and so may change or disappear in a future release without being considered a breaking change.

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MixedModels.MixedModelType
MixedModel

Abstract type for mixed models. MixedModels.jl implements two subtypes: LinearMixedModel and GeneralizedLinearMixedModel. See the documentation for each for more details.

This type is primarily used for dispatch in fit. Without a distribution and link function specified, a LinearMixedModel will be fit. When a distribution/link function is provided, a GeneralizedLinearModel is fit, unless that distribution is Normal and the link is IdentityLink, in which case the resulting GLMM would be equivalent to a LinearMixedModel anyway and so the simpler, equivalent LinearMixedModel will be fit instead.

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MixedModels.MixedModelBootstrapType
MixedModelBootstrap{T<:AbstractFloat} <: MixedModelFitCollection{T}

Object returned by parametericbootstrap with fields

• fits: the parameter estimates from the bootstrap replicates as a vector of named tuples.
• λ: Vector{LowerTriangular{T,Matrix{T}}} containing copies of the λ field from ReMat model terms
• inds: Vector{Vector{Int}} containing copies of the inds field from ReMat model terms
• lowerbd: Vector{T} containing the vector of lower bounds (corresponds to the identically named field of OptSummary)
• fcnames: NamedTuple whose keys are the grouping factor names and whose values are the column names

The schema of fits is, by default,

Tables.Schema:
:objective  T
:σ          T
:β          NamedTuple{β_names}{NTuple{p,T}}
:se         StaticArrays.SArray{Tuple{p},T,1,p}
:θ          StaticArrays.SArray{Tuple{k},T,1,k}

where the sizes, p and k, of the β and θ elements are determined by the model.

Characteristics of the bootstrap replicates can be extracted as properties. The σs and σρs properties unravel the σ and θ estimates into estimates of the standard deviations and correlations of the random-effects terms.

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MixedModels.OptSummaryType
OptSummary

Summary of an NLopt optimization

Fields

• initial: a copy of the initial parameter values in the optimization
• finitial: the initial value of the objective
• lowerbd: lower bounds on the parameter values
• ftol_rel: as in NLopt
• ftol_abs: as in NLopt
• xtol_rel: as in NLopt
• xtol_abs: as in NLopt
• initial_step: as in NLopt
• maxfeval: as in NLopt (maxeval)
• maxtime: as in NLopt
• final: a copy of the final parameter values from the optimization
• fmin: the final value of the objective
• feval: the number of function evaluations
• optimizer: the name of the optimizer used, as a Symbol
• returnvalue: the return value, as a Symbol
• nAGQ: number of adaptive Gauss-Hermite quadrature points in deviance evaluation for GLMMs
• REML: use the REML criterion for LMM fits
• sigma: a priori value for the residual standard deviation for LMM
• fitlog: A vector of tuples of parameter and objectives values from steps in the optimization

The latter four fields are MixedModels functionality and not related directly to the NLopt package or algorithms.

Note

The internal storage of the parameter values within fitlog may change in the future to use a different subtype of AbstractVector (e.g., StaticArrays.SVector) for each snapshot without being considered a breaking change.

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MixedModels.PCAType
PCA{T<:AbstractFloat}

Principal Components Analysis

Fields

• covcorr covariance or correlation matrix
• sv singular value decomposition
• rnames rownames of the original matrix
• corr is this a correlation matrix?
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MixedModels.RaggedArrayType
RaggedArray{T,I}

A "ragged" array structure consisting of values and indices

Fields

• vals: a Vector{T} containing the values
• inds: a Vector{I} containing the indices

For this application a RaggedArray is used only in its sum! method.

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MixedModels.ReMatType
ReMat{T,S} <: AbstractMatrix{T}

A section of a model matrix generated by a random-effects term.

Fields

• trm: the grouping factor as a StatsModels.CategoricalTerm
• refs: indices into the levels of the grouping factor as a Vector{Int32}
• levels: the levels of the grouping factor
• cnames: the names of the columns of the model matrix generated by the left-hand side of the term
• z: transpose of the model matrix generated by the left-hand side of the term
• wtz: a weighted copy of z (z and wtz are the same object for unweighted cases)
• λ: a LowerTriangular matrix of size S×S
• inds: a Vector{Int} of linear indices of the potential nonzeros in λ
• adjA: the adjoint of the matrix as a SparseMatrixCSC{T}
• scratch: a Matrix{T}
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MixedModels.VarCorrType
VarCorr

Information from the fitted random-effects variance-covariance matrices.

Members

• σρ: a NamedTuple of NamedTuples as returned from σρs
• s: the estimate of the per-observation dispersion parameter

The main purpose of defining this type is to isolate the logic in the show method.

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## Exported Functions

LinearAlgebra.logdetMethod
logdet(m::LinearMixedModel)

Return the value of log(det(Λ'Z'ZΛ + I)) + m.optsum.REML * log(det(LX*LX')) evaluated in place.

Here LX is the diagonal term corresponding to the fixed-effects in the blocked lower Cholesky factor.

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MixedModels.GHnormMethod
GHnorm(k::Int)

Return the (unique) GaussHermiteNormalized{k} object.

The function values are stored (memoized) when first evaluated. Subsequent evaluations for the same k have very low overhead.

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MixedModels.condVarMethod
condVar(m::LinearMixedModel)

Return the conditional variances matrices of the random effects.

The random effects are returned by ranef as a vector of length k, where k is the number of random effects terms. The ith element is a matrix of size vᵢ × ℓᵢ where vᵢ is the size of the vector-valued random effects for each of the ℓᵢ levels of the grouping factor. Technically those values are the modes of the conditional distribution of the random effects given the observed data.

This function returns an array of k three dimensional arrays, where the ith array is of size vᵢ × vᵢ × ℓᵢ. These are the diagonal blocks from the conditional variance-covariance matrix,

s² Λ(Λ'Z'ZΛ + I)⁻¹Λ'
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MixedModels.condVartablesMethod
condVartables(m::LinearMixedModel)

Return the conditional covariance matrices of the random effects as a NamedTuple of columntables

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MixedModels.fitted!Method
fitted!(v::AbstractArray{T}, m::LinearMixedModel{T})

Overwrite v with the fitted values from m.

See also fitted.

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MixedModels.fixefMethod
fixef(m::MixedModel)

Return the fixed-effects parameter vector estimate of m.

In the rank-deficient case the truncated parameter vector, of length rank(m) is returned. This is unlike coef which always returns a vector whose length matches the number of columns in X.

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MixedModels.fulldummyMethod
fulldummy(term::CategoricalTerm)

Assign "contrasts" that include all indicator columns (dummy variables) and an intercept column.

This will result in an under-determined set of contrasts, which is not a problem in the random effects because of the regularization, or "shrinkage", of the conditional modes.

The interaction of fulldummy with complex random effects is subtle and complex with numerous potential edge cases. As we discover these edge cases, we will document and determine their behavior. Until such time, please check the model summary to verify that the expansion is working as you expected. If it is not, please report a use case on GitHub.

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MixedModels.issingularFunction
issingular(m::MixedModel, θ=m.θ)

Test whether the model m is singular if the parameter vector is θ.

Equality comparisons are used b/c small non-negative θ values are replaced by 0 in fit!.

Note

For GeneralizedLinearMixedModel, the entire parameter vector (including β in the case fast=false) must be specified if the default is not used.

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MixedModels.lowerbdMethod
lowerbd{T}(A::ReMat{T})

Return the vector of lower bounds on the parameters, θ associated with A

These are the elements in the lower triangle of A.λ in column-major ordering. Diagonals have a lower bound of 0. Off-diagonals have a lower-bound of -Inf.

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MixedModels.parametricbootstrapMethod
parametricbootstrap([rng::AbstractRNG], nsamp::Integer, m::MixedModel{T}, ftype=T;
β = coef(m), σ = m.σ, θ = m.θ, use_threads=false, hide_progress=false)

Perform nsamp parametric bootstrap replication fits of m, returning a MixedModelBootstrap.

The default random number generator is Random.GLOBAL_RNG.

ftype can be used to store the computed bootstrap values in a lower precision. ftype is not a named argument because named arguments are not used in method dispatch and thus specialization. In other words, having ftype as a positional argument has some potential performance benefits.

Named Arguments

• β, σ, and θ are the values of m's parameters for simulating the responses.
• σ is only valid for LinearMixedModel and GeneralizedLinearMixedModel for

families with a dispersion parameter.

• use_threads determines whether or not to use thread-based parallelism.
• hide_progress can be used to disable the progress bar. Note that the progress

bar is automatically disabled for non-interactive (i.e. logging) contexts.

Note

Note that use_threads=true may not offer a performance boost and may even decrease peformance if multithreaded linear algebra (BLAS) routines are available. In this case, threads at the level of the linear algebra may already occupy all processors/processor cores. There are plans to provide better support in coordinating Julia- and BLAS-level threads in the future.

Warning

The PRNG shared between threads is locked using Threads.SpinLock, which should not be used recursively. Do not wrap parametricbootstrap in an outer SpinLock.

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MixedModels.pirls!Method
pirls!(m::GeneralizedLinearMixedModel)

Use Penalized Iteratively Reweighted Least Squares (PIRLS) to determine the conditional modes of the random effects.

When varyβ is true both u and β are optimized with PIRLS. Otherwise only u is optimized and β is held fixed.

Passing verbose = true provides verbose output of the iterations.

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MixedModels.ranefMethod
ranef(m::LinearMixedModel; uscale=false)

Return, as a Vector{Matrix{T}}, the conditional modes of the random effects in model m.

If uscale is true the random effects are on the spherical (i.e. u) scale, otherwise on the original scale.

For a named variant, see raneftables.

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MixedModels.raneftablesMethod
raneftables(m::MixedModel; uscale = false)

Return the conditional means of the random effects as a NamedTuple of columntables

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MixedModels.refit!Method
refit!(m::GeneralizedLinearMixedModel[, y::Vector];
fast::Bool = (length(m.θ) == length(m.optsum.final)),
nAGQ::Integer = m.optsum.nAGQ,
kwargs...)

Refit the model m after installing response y.

If y is omitted the current response vector is used.

If not specified, the fast and nAGQ options from the previous fit are used. kwargs are the same as fit!

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MixedModels.refit!Method
refit!(m::LinearMixedModel[, y::Vector]; REML=m.optsum.REML, kwargs...)

Refit the model m after installing response y.

If y is omitted the current response vector is used. kwargs are the same as fit!.

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MixedModels.replicateMethod
replicate(f::Function, n::Integer; use_threads=false)

Return a vector of the values of n calls to f() - used in simulations where the value of f is stochastic.

hide_progress can be used to disable the progress bar. Note that the progress bar is automatically disabled for non-interactive (i.e. logging) contexts.

Warning

If f() is not thread-safe or depends on a non thread-safe RNG, then you must set use_threads=false. Also note that ordering of replications is not guaranteed when use_threads=true, although the replications are not otherwise affected for thread-safe f().

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MixedModels.restoreoptsum!Method
restoreoptsum!(m::LinearMixedModel, io::IO)
restoreoptsum!(m::LinearMixedModel, filename)

Read, check, and restore the optsum field from a JSON stream or filename.

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MixedModels.saveoptsumMethod
saveoptsum(io::IO, m::LinearMixedModel)
saveoptsum(filename, m::LinearMixedModel)

Save m.optsum (w/o the lowerbd field) in JSON format to an IO stream or a file

The reason for omitting the lowerbd field is because it often contains -Inf values that are not allowed in JSON.

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MixedModels.sdestMethod
sdest(m::GeneralizedLinearMixedModel)

Return the estimate of the dispersion, i.e. the standard deviation of the per-observation noise.

For models with a dispersion parameter ϕ, this is simply ϕ. For models without a dispersion parameter, this value is missing. This differs from disperion, which returns 1 for models without a dispersion parameter.

For Gaussian models, this parameter is often called σ.

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MixedModels.setθ!Method
setθ!(bsamp::MixedModelFitCollection, i::Integer)

Install the values of the i'th θ value of bsamp.fits in bsamp.λ

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MixedModels.shortestcovintMethod
shortestcovint(bsamp::MixedModelFitCollection, level = 0.95)

Return the shortest interval containing level proportion for each parameter from bsamp.allpars.

Warning

Currently, correlations that are systematically zero are included in the the result. This may change in a future release without being considered a breaking change.

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MixedModels.simulate!Method
simulate!([rng::AbstractRNG,] y::AbstractVector, m::MixedModel{T}[, newdata];
β = coef(m), σ = m.σ, θ = T[], wts=m.wts)
simulate([rng::AbstractRNG,] m::MixedModel{T}[, newdata];
β = coef(m), σ = m.σ, θ = T[], wts=m.wts)

Simulate a new response vector, optionally overwriting a pre-allocated vector.

New data can be optionally provided in tabular format.

This simulation includes sampling new values for the random effects. Thus in contrast to predict, there is no distinction in between "new" and "old" / previously observed random-effects levels.

Unlike predict, there is no type parameter for GeneralizedLinearMixedModel because the noise term in the model and simulation is always on the response scale.

The wts argument is currently ignored except for GeneralizedLinearMixedModel models with a Binomial distribution.

Warning

Models are assumed to be full rank.

Note

Note that simulate! methods with a y::AbstractVector as the first argument (besides the RNG) and simulate methods return the simulated response. This is in contrast to simulate! methods with a m::MixedModel as the first argument, which modify the model's response and return the entire modified model.

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MixedModels.simulate!Method
simulate!(rng::AbstractRNG, m::MixedModel{T}; β=m.β, σ=m.σ, θ=T[])
simulate!(m::MixedModel; β=m.β, σ=m.σ, θ=m.θ)

Overwrite the response (i.e. m.trms[end]) with a simulated response vector from model m.

This simulation includes sampling new values for the random effects.

Note

Note that simulate! methods with a y::AbstractVector as the first argument (besides the RNG) and simulate methods return the simulated response. This is in contrast to simulate! methods with a m::MixedModel as the first argument, which modify the model's response and return the entire modified model.

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MixedModels.sparseLMethod
sparseL(m::LinearMixedModel; fname::Symbol=first(fnames(m)), full::Bool=false)

Return the lower Cholesky factor L as a SparseMatrix{T,Int32}.

full indicates whether the parts of L associated with the fixed-effects and response are to be included.

fname specifies the first grouping factor to include. Blocks to the left of the block corresponding to fname are dropped. The default is the first, i.e., leftmost block and hence all blocks.

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MixedModels.stderror!Method
stderror!(v::AbstractVector, m::LinearMixedModel)

Overwrite v with the standard errors of the fixed-effects coefficients in m

The length of v should be the total number of coefficients (i.e. length(coef(m))). When the model matrix is rank-deficient the coefficients forced to -0.0 have an undefined (i.e. NaN) standard error.

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MixedModels.updateL!Method
updateL!(m::LinearMixedModel)

Update the blocked lower Cholesky factor, m.L, from m.A and m.reterms (used for λ only)

This is the crucial step in evaluating the objective, given a new parameter value.

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MixedModels.varestMethod
varest(m::LinearMixedModel)

Returns the estimate of σ², the variance of the conditional distribution of Y given B.

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MixedModels.varestMethod
varest(m::GeneralizedLinearMixedModel)

Returns the estimate of ϕ², the variance of the conditional distribution of Y given B.

For models with a dispersion parameter ϕ, this is simply ϕ². For models without a dispersion parameter, this value is missing. This differs from disperion, which returns 1 for models without a dispersion parameter.

For Gaussian models, this parameter is often called σ².

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Statistics.stdMethod
std(m::MixedModel)

Return the estimated standard deviations of the random effects as a Vector{Vector{T}}.

FIXME: This uses an old convention of isfinite(sdest(m)). Probably drop in favor of m.σs

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StatsAPI.devianceMethod
deviance(m::GeneralizedLinearMixedModel{T}, nAGQ=1)::T where {T}

Return the deviance of m evaluated by the Laplace approximation (nAGQ=1) or nAGQ-point adaptive Gauss-Hermite quadrature.

If the distribution D does not have a scale parameter the Laplace approximation is the squared length of the conditional modes, $u$, plus the determinant of $Λ'Z'WZΛ + I$, plus the sum of the squared deviance residuals.

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StatsAPI.dof_residualMethod
dof_residual(m::MixedModel)

Return the residual degrees of freedom of the model.

Note

The residual degrees of freedom for mixed-effects models is not clearly defined due to partial pooling. The classical nobs(m) - dof(m) fails to capture the extra freedom granted by the random effects, but nobs(m) - nranef(m) would overestimate the freedom granted by the random effects. nobs(m) - sum(leverage(m)) provides a nice balance based on the relative influence of each observation, but is computationally expensive for large models. This problem is also fundamentally related to long-standing debates about the appropriate treatment of the denominator degrees of freedom for $F$-tests. In the future, MixedModels.jl may provide additional methods allowing the user to choose the computation to use.

Warning

Currently, the residual degrees of freedom is computed as nobs(m) - dof(m), but this may change in the future without being considered a breaking change because there is no canonical definition of the residual degrees of freedom in a mixed-effects model.

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StatsAPI.fit!Method
fit!(m::GeneralizedLinearMixedModel; fast=false, nAGQ=1,
verbose=false, progress=true,
thin::Int=1,
init_from_lmm=Set())

Optimize the objective function for m.

When fast is true a potentially much faster but slightly less accurate algorithm, in which pirls! optimizes both the random effects and the fixed-effects parameters, is used.

If progress is true, the default, a ProgressMeter.ProgressUnknown counter is displayed. during the iterations to minimize the deviance. There is a delay before this display is initialized and it may not be shown at all for models that are optimized quickly.

If verbose is true, then both the intermediate results of both the nonlinear optimization and PIRLS are also displayed on standard output.

At every thinth iteration is recorded in fitlog, optimization progress is saved in m.optsum.fitlog.

By default, the starting values for model fitting are taken from a (non mixed, i.e. marginal ) GLM fit. Experience with larger datasets (many thousands of observations and/or hundreds of levels of the grouping variables) has suggested that fitting a (Gaussian) linear mixed model on the untransformed data may provide better starting values and thus overall faster fits even though an entire LMM must be fit before the GLMM can be fit. init_from_lmm can be used to specify which starting values from an LMM to use. Valid options are any collection (array, set, etc.) containing one or more of :β and :θ, the default is the empty set.

Note

Initializing from an LMM requires fitting the entire LMM first, so when progress=true, there will be two progress bars: first for the LMM, then for the GLMM.

Warning

The init_from_lmm functionality is experimental and may change or be removed entirely without being considered a breaking change.

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StatsAPI.fit!Method
fit!(m::LinearMixedModel; progress::Bool=true, REML::Bool=false,
σ::Union{Real, Nothing}=nothing,
thin::Int=typemax(Int))

Optimize the objective of a LinearMixedModel. When progress is true a ProgressMeter.ProgressUnknown display is shown during the optimization of the objective, if the optimization takes more than one second or so.

At every thinth iteration is recorded in fitlog, optimization progress is saved in m.optsum.fitlog.

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StatsAPI.leverageMethod
leverage(::LinearMixedModel)

Return the diagonal of the hat matrix of the model.

For a linear model, the sum of the leverage values is the degrees of freedom for the model in the sense that this sum is the dimension of the span of columns of the model matrix. With a bit of hand waving a similar argument could be made for linear mixed-effects models. The hat matrix is of the form $[ZΛ X][L L']⁻¹[ZΛ X]'$.

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StatsAPI.modelmatrixMethod
modelmatrix(m::MixedModel)

Returns the model matrix X for the fixed-effects parameters, as returned by coef.

This is always the full model matrix in the original column order and from a field in the model struct. It should be copied if it is to be modified.

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StatsAPI.predictMethod
StatsAPI.predict(m::LinearMixedModel, newdata;
new_re_levels=:missing)
StatsAPI.predict(m::GeneralizedLinearMixedModel, newdata;
new_re_levels=:missing, type=:response)

Predict response for new data.

Note

Currently, no in-place methods are provided because these methods internally construct a new model and therefore allocate not just a response vector but also many other matrices.

Warning

newdata should contain a column for the response (dependent variable) initialized to some numerical value (not missing), because this is used to construct the new model used in computing the predictions. missing is not valid because missing data are dropped before constructing the model matrices.

Warning

Models are assumed to be full rank.

Warning

These methods construct an entire MixedModel behind the scenes and as such may use a large amount of memory when newdata is large.

The keyword argument new_re_levels specifies how previously unobserved values of the grouping variable are handled. Possible values are:

• :population: return population values for the relevant grouping variable. In other words, treat the associated random effect as 0. If all grouping variables have new levels, then this is equivalent to just the fixed effects.
• :missing: return missing.
• :error: error on this condition. The error type is an implementation detail: you should not rely on a particular type of error being thrown.

If you want simulated values for unobserved levels of the grouping variable, consider the simulate! and simulate methods.

Predictions based purely on the fixed effects can be obtained by specifying previously unobserved levels of the random effects and setting new_re_levels=:population. Similarly, the contribution of any grouping variable can be excluded by specifying previously unobserved levels, while including previously observed levels of the other grouping variables. In the future, it may be possible to specify a subset of the grouping variables or overall random-effects structure to use, but not at this time.

Note

new_re_levels impacts only the behavior for previously unobserved random effects levels, i.e. new RE levels. For previously observed random effects levels, predictions take both the fixed and random effects into account.

For GeneralizedLinearMixedModel, the type parameter specifies whether the predictions should be returned on the scale of linear predictor (:linpred) or on the response scale (:response). If you don't know the difference between these terms, then you probably want type=:response.

Regression weights are not yet supported in prediction. Similarly, offsets are also not supported for GeneralizedLinearMixedModel.

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StatsAPI.responseMethod
response(m::MixedModel)

Return the response vector for the model.

For a linear mixed model this is a view of the last column of the XyMat field. For a generalized linear mixed model this is the m.resp.y field. In either case it should be copied if it is to be modified.

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StatsAPI.vcovMethod
vcov(m::MixedModel; corr=false)

Returns the variance-covariance matrix of the fixed effects. If corr is true, the correlation of the fixed effects is returned instead.

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Tables.columntableMethod
columntable(s::OptSummary, [stack::Bool=false])

Return s.fitlog as a Tables.columntable.

When stack is false (the default), there will be 3 columns in the result:

• iter: the sample number
• objective: the value of the objective at that sample
• θ: the parameter vector at that sample

(The term sample here refers to the fact that when the thin argument to the fit or refit! call is greater than 1 only a subset of the iterations have results recorded.)

When stack is true, there will be 4 columns: iter, objective, par, and value where value is the stacked contents of the θ vectors (the equivalent of vcat(θ...)) and par is a vector of parameter numbers.

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## Methods from StatsAPI.jl, StatsBase.jl, StatsModels.jl and GLM.jl

aic
aicc
bic
coef
coefnames
coeftable
deviance
dispersion
dispersion_parameter
dof
dof_residual
fit
fit!
fitted
formula
isfitted
islinear
leverage
loglikelihood
meanresponse
modelmatrix
model_response
nobs
predict
residuals
response
responsename
StatsModels.lrtest # not exported
std
stderror
vcov
weights

### MixedModels.jl "alternatives" and extensions to StatsAPI and GLM functions

The following are MixedModels.jl-specific functions and not simply methods for functions defined in StatsAPI and GLM.jl.

coefpvalues
condVar
condVarTables
fitted!
fixef
fixefnames
likelihoodratiotest # not exported
ranef
raneftables
refit!
shortestcovint
sdest
simulate
simulate!
stderrror!
varest

## Non-Exported Functions

Note that unless discussed elsewhere in the online documentation, non-exported functions should be considered implementation details.

Base.sizeMethod
size(m::MixedModel)

Returns the size of a mixed model as a tuple of length four: the number of observations, the number of (non-singular) fixed-effects parameters, the number of conditional modes (random effects), the number of grouping variables

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GLM.wrkresp!Method
GLM.wrkresp!(v::AbstractVector{T}, resp::GLM.GlmResp{AbstractVector{T}})

A copy of a method from GLM that generalizes the types in the signature

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MixedModels.LDMethod
LD(A::Diagonal)
LD(A::HBlikDiag)
LD(A::DenseMatrix)

Return log(det(tril(A))) evaluated in place.

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MixedModels.adjAMethod
adjA(refs::AbstractVector, z::AbstractMatrix{T})

Returns the adjoint of an ReMat as a SparseMatrixCSC{T,Int32}

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MixedModels.allparsMethod
allpars(bsamp::MixedModelFitCollection)

Return a tidy (column)table with the parameter estimates spread into columns of iter, type, group, name and value.

Warning

Currently, correlations that are systematically zero are included in the the result. This may change in a future release without being considered a breaking change.

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MixedModels.blockMethod
block(i, j)

Return the linear index of the [i,j] position ("block") in the row-major packed lower triangle.

Use the row-major ordering in this case because the result depends only on i and j, not on the overall size of the array.

When i == j the value is the same as kp1choose2(i).

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MixedModels.cholUnblocked!Function
cholUnblocked!(A, Val{:L})

Overwrite the lower triangle of A with its lower Cholesky factor.

The name is borrowed from [https://github.com/andreasnoack/LinearAlgebra.jl] because these are part of the inner calculations in a blocked Cholesky factorization.

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MixedModels.corrmatMethod
corrmat(A::ReMat)

Return the estimated correlation matrix for A. The diagonal elements are 1 and the off-diagonal elements are the correlations between those random effect terms

Example

Note that trailing digits may vary slightly depending on the local platform.

julia> using MixedModels

julia> mod = fit(MixedModel,
@formula(rt_trunc ~ 1 + spkr + prec + load + (1 + spkr + prec | subj)),
MixedModels.dataset(:kb07));

julia> VarCorr(mod)
Variance components:
Column      Variance  Std.Dev.  Corr.
subj     (Intercept)     136591.782 369.583
spkr: old        22922.871 151.403 +0.21
prec: maintain   32348.269 179.856 -0.98 -0.03
Residual                 642324.531 801.452

julia> MixedModels.corrmat(mod.reterms[1])
3×3 LinearAlgebra.Symmetric{Float64,Array{Float64,2}}:
1.0        0.214816   -0.982948
0.214816   1.0        -0.0315607
-0.982948  -0.0315607   1.0
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MixedModels.cpadMethod
cpad(s::AbstractString, n::Integer)

Return a string of length n containing s in the center (more-or-less).

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MixedModels.datasetMethod
dataset(nm)

Return, as an Arrow.Table, the test data set named nm, which can be a String or Symbol

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MixedModels.densifyFunction

densify(S::SparseMatrix, threshold=0.1)

Convert sparse S to Diagonal if S is diagonal or to Array(S) if the proportion of nonzeros exceeds threshold.

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MixedModels.deviance!Function
deviance!(m::GeneralizedLinearMixedModel, nAGQ=1)

Update m.η, m.μ, etc., install the working response and working weights in m.LMM, update m.LMM.A and m.LMM.R, then evaluate the deviance.

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MixedModels.feLMethod
feL(m::LinearMixedModel)

Return the lower Cholesky factor for the fixed-effects parameters, as an LowerTriangular p × p matrix.

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MixedModels.fixef!Method
fixef!(v::Vector{T}, m::MixedModel{T})

Overwrite v with the pivoted fixed-effects coefficients of model m

For full-rank models the length of v must be the rank of X. For rank-deficient models the length of v can be the rank of X or the number of columns of X. In the latter case the calculated coefficients are padded with -0.0 out to the number of columns.

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MixedModels.getθ!Method
getθ!(v::AbstractVector{T}, A::ReMat{T}) where {T}

Overwrite v with the elements of the blocks in the lower triangle of A.Λ (column-major ordering)

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MixedModels.isnestedMethod
isnested(A::ReMat, B::ReMat)

Is the grouping factor for A nested in the grouping factor for B?

That is, does each value of A occur with just one value of B?

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MixedModels.kchoose2Method
kchoose2(k)

The binomial coefficient k choose 2 which is the number of elements in the packed form of the strict lower triangle of a matrix.

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MixedModels.kp1choose2Method
kp1choose2(k)

The binomial coefficient k+1 choose 2 which is the number of elements in the packed form of the lower triangle of a matrix.

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MixedModels.likelihoodratiotestMethod
likelihoodratiotest(m::MixedModel...)
likelihoodratiotest(m0::LinearModel, m::MixedModel...)
likelihoodratiotest(m0::GeneralizedLinearModel, m::MixedModel...)
likelihoodratiotest(m0::TableRegressionModel{LinearModel}, m::MixedModel...)
likelihoodratiotest(m0::TableRegressionModel{GeneralizedLinearModel}, m::MixedModel...)

Likeihood ratio test applied to a set of nested models.

Note

The nesting of the models is not checked. It is incumbent on the user to check this. This differs from StatsModels.lrtest as nesting in mixed models, especially in the random effects specification, may be non obvious.

Note

For comparisons between mixed and non-mixed models, the deviance for the non-mixed model is taken to be -2 log likelihood, i.e. omitting the additive constant for the fully saturated model. This is in line with the computation of the deviance for mixed models.

This functionality may be deprecated in the future in favor of StatsModels.lrtest.

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MixedModels.nranefMethod
nranef(A::ReMat)

Return the number of random effects represented by A. Zero unless A is an ReMat.

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MixedModels.ranef!Method
ranef!(v::Vector{Matrix{T}}, m::MixedModel{T}, β, uscale::Bool) where {T}

Overwrite v with the conditional modes of the random effects for m.

If uscale is true the random effects are on the spherical (i.e. u) scale, otherwise on the original scale

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MixedModels.rankUpdate!Function
rankUpdate!(C, A)
rankUpdate!(C, A, α)
rankUpdate!(C, A, α, β)

A rank-k update, C := αA'A + βC, of a Hermitian (Symmetric) matrix.

α and β both default to 1.0. When α is -1.0 this is a downdate operation. The name rankUpdate! is borrowed from [https://github.com/andreasnoack/LinearAlgebra.jl]

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MixedModels.rePCAMethod
rePCA(m::LinearMixedModel; corr::Bool=true)

Return a named tuple of the normalized cumulative variance of a principal components analysis of the random effects covariance matrices or correlation matrices when corr is true.

The normalized cumulative variance is the proportion of the variance for the first principal component, the first two principal components, etc. The last element is always 1.0 representing the complete proportion of the variance.

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MixedModels.reevaluateAend!Method
reevaluateAend!(m::LinearMixedModel)

Reevaluate the last column of m.A from m.Xymat. This function should be called after updating the response.

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MixedModels.sdcorrMethod
sdcorr(A::AbstractMatrix{T}) where {T}

Transform a square matrix A with positive diagonals into an NTuple{size(A,1), T} of standard deviations and a tuple of correlations.

A is assumed to be symmetric and only the lower triangle is used. The order of the correlations is row-major ordering of the lower triangle (or, equivalently, column-major in the upper triangle).

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MixedModels.setβθ!Method
setβθ!(m::GeneralizedLinearMixedModel, v)

Set the parameter vector, :βθ, of m to v.

βθ is the concatenation of the fixed-effects, β, and the covariance parameter, θ.

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MixedModels.ssqdenomMethod
ssqdenom(m::LinearMixedModel)

Return the denominator for penalized sums-of-squares.

For MLE, this value is the number of observations. For REML, this value is the number of observations minus the rank of the fixed-effects matrix. The difference is analagous to the use of n or n-1 in the denominator when calculating the variance.

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MixedModels.statsrankMethod
statsrank(x::Matrix{T}, ranktol::Real=1e-8) where {T<:AbstractFloat}

Return the numerical column rank and a pivot vector.

The rank is determined from the absolute values of the diagonal of R from a pivoted QR decomposition, relative to the first (and, hence, largest) element of this vector.

In the full-rank case the pivot vector is collect(axes(x, 2)).

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MixedModels.tidyβMethod
tidyβ(bsamp::MixedModelFitCollection)

Return a tidy (row)table with the parameter estimates spread into columns of iter, coefname and β

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MixedModels.tidyσsMethod
tidyσs(bsamp::MixedModelFitCollection)

Return a tidy (row)table with the estimates of the variance components (on the standard deviation scale) spread into columns of iter, group, column and σ.

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MixedModels.unscaledre!Function
unscaledre!(y::AbstractVector{T}, M::ReMat{T}) where {T}
unscaledre!(rng::AbstractRNG, y::AbstractVector{T}, M::ReMat{T}) where {T}

Add unscaled random effects simulated from M to y.

These are unscaled random effects (i.e. they incorporate λ but not σ) because the scaling is done after the per-observation noise is added as a standard normal.

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MixedModels.updateA!Method
updateA!(m::LinearMixedModel)

Update the cross-product array, m.A, from m.reterms and m.Xymat

This is usually done after a reweight! operation.

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MixedModels.updateη!Method
updateη!(m::GeneralizedLinearMixedModel)

Update the linear predictor, m.η, from the offset and the B-scale random effects.

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StatsModels.isnestedMethod
isnested(m1::MixedModel, m2::MixedModel; atol::Real=0.0)

Indicate whether model m1 is nested in model m2, i.e. whether m1 can be obtained by constraining some parameters in m2. Both models must have been fitted on the same data. This check is conservative for MixedModels and may reject nested models with different parameterizations as being non nested.

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