API

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.

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.

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.

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

FeTerm{T,S}

Term with an explicit, constant matrix representation

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

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

FeTerm(X::SparseMatrixCSC, cnms)

Convenience constructor for a sparse FeTerm assuming full rank, identity pivot and unit weights.

Note: automatic rank deficiency handling may be added to this method in the future, as discussed in the vignette "Rank deficiency in mixed-effects models" for general FeTerm.

FeTerm(X::AbstractMatrix, cnms)

Convenience constructor for FeTerm that computes the rank and pivot with unit weights.

See the vignette "Rank deficiency in mixed-effects models" for more information on the computation of the rank and pivot.

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

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

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()))

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

LinearMixedModel(y, Xs, form, wts=[], σ=nothing, amalgamate=true)

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.

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

LinearMixedModel(y, feterm, reterms, form, wts=[], σ=nothing; amalgamate=true)

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.

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.

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.

MixedModelFitCollection{T<:AbstractFloat}

Abstract supertype for MixedModelBootstrap and related functionality in other packages.

 MixedModelProfile{T<:AbstractFloat}

Type representing a likelihood profile of a LinearMixedModel, including associated interpolation splines.

The function profile is used for computing profiles, while confint provides a useful method for constructing confidence intervals from a MixedModelProfile.

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
• xtol_zero_abs: the tolerance for a near zero parameter to be considered practically zero
• ftol_zero_abs: the tolerance for change in the objective for setting a near zero parameter to zero
• fitlog: A vector of tuples of parameter and objectives values from steps in the optimization
• 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

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

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?

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.

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 or Diagonal 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}

TableColumns

A structure containing the column names for the numeric part of the profile table.

The struct also contains a Dict giving the column ranges for Symbols like :σ and :β. Finally it contains a scratch vector used to accumulate to values in a row of the profile table.

UniformBlockDiagonal{T}

Homogeneous block diagonal matrices. k diagonal blocks each of size m×m

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.

cond(m::MixedModel)

Return a vector of condition numbers of the λ matrices for the random-effects terms

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.

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.

coefpvalues(bsamp::MixedModelFitCollection)

Return a rowtable with columns (:iter, :coefname, :β, :se, :z, :p)

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'

condVartables(m::LinearMixedModel)

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

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

Overwrite v with the fitted values from m.

See also fitted.

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.

fixefnames(m::MixedModel)

Return a (permuted and truncated in the rank-deficient case) vector of coefficient names.

fnames(m::MixedModel)

Return the names of the grouping factors for the random-effects terms.

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.

issingular(m::MixedModel, θ=m.θ; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps)

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!.

issingular(bsamp::MixedModelFitCollection;
           atol::Real=0, rtol::Real=atol>0 ? 0 : √eps))

Test each bootstrap sample for singularity of the corresponding fit.

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

See also issingular(::MixedModel).

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.

objective!(m::LinearMixedModel, θ)
objective!(m::LinearMixedModel)

Equivalent to objective(updateL!(setθ!(m, θ))).

When m has a single, scalar random-effects term, θ can be a scalar.

The one-argument method curries and returns a single-argument function of θ.

Note that these methods modify m. The calling function is responsible for restoring the optimal θ.

objective(m::LinearMixedModel)

Return negative twice the log-likelihood of model m

parametricbootstrap([rng::AbstractRNG], nsamp::Integer, m::MixedModel{T}, ftype=T;
    β = fixef(m), σ = m.σ, θ = m.θ, progress=true, optsum_overrides=(;))

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.

Keyword 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.

• progress controls whether the progress bar is shown. Note that the progress

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

• optsum_overrides is used to override values of OptSummary in the models

fit during the bootstrapping process. For example, optsum_overrides=(;ftol_rel=1e-08) reduces the convergence criterion, which can greatly speed up the bootstrap fits. Taking advantage of this speed up to increase n can often lead to better estimates of coverage intervals.

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.

profile(m::LinearMixedModel; threshold = 4)

Return a MixedModelProfile for the objective of m with respect to the fixed-effects coefficients.

m is refit! if !isfitted(m).

Profiling starts at the parameter estimate and continues until reaching a parameter bound or the absolute value of ζ exceeds threshold.

 profilevc(m::LinearMixedModel{T}, val::T, rowj::AbstractVector{T}) where {T}

Profile an element of the variance components.

profileσ(m::LinearMixedModel, tc::TableColumns; threshold=4)

Return a Table of the profile of σ for model m. The profile extends to where the magnitude of ζ exceeds threshold.

pwrss(m::LinearMixedModel)

The penalized, weighted residual sum-of-squares.

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.

raneftables(m::MixedModel; uscale = false)

Return the conditional means of the random effects as a NamedTuple of Tables.jl-compliant tables.

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!

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!.

replicate(f::Function, n::Integer; progress=true)

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

progress controls whether the progress bar is shown. Note that the progress bar is automatically disabled for non-interactive (i.e. logging) contexts.

restoreoptsum!(m::MixedModel, io::IO; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps)
restoreoptsum!(m::MixedModel, filename; atol::Real=0, rtol::Real=atol>0 ? 0 : √eps)

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

restorereplicates(f, m::MixedModel{T})
restorereplicates(f, m::MixedModel{T}, ftype::Type{<:AbstractFloat})
restorereplicates(f, m::MixedModel{T}, ctype::Type{<:MixedModelFitCollection{S}})

Restore replicates from f, using m to create the desired subtype of MixedModelFitCollection.

f can be any entity supported by Arrow.Table. m does not have to be fitted, but it must have been constructed with the same structure as the source of the saved replicates.

The two-argument method constructs a MixedModelBootstrap with the same eltype as m. If an eltype is specified as the third argument, then a MixedModelBootstrap is returned. If a subtype of MixedModelFitCollection is specified as the third argument, then that is the return type.

See also savereplicates, restoreoptsum!.

saveoptsum(io::IO, m::MixedModel)
saveoptsum(filename, m::MixedModel)

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.

savereplicates(f, b::MixedModelFitCollection)

Save the replicates associated with a MixedModelFitCollection, e.g. MixedModelBootstrap as an Arrow file.

See also restorereplicates, saveoptsum

sdest(m::LinearMixedModel)

Return the estimate of σ, the standard deviation of the per-observation noise.

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.

setθ!(m::LinearMixedModel, v)

Install v as the θ parameters in m.

setθ!(bsamp::MixedModelFitCollection, θ::AbstractVector)
setθ!(bsamp::MixedModelFitCollection, i::Integer)

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

shortestcovint(v, level = 0.95)

Return the shortest interval containing level proportion of the values of v

shortestcovint(bsamp::MixedModelFitCollection, level = 0.95)

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

See simulate!

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.

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

β can be specified either as a pivoted, full rank coefficient vector (cf. fixef) or as an unpivoted full dimension coefficient vector (cf. coef), where the entries corresponding to redundant columns will be ignored.

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.

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.

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.

varest(m::LinearMixedModel)

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

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.

zerocorr(term::RandomEffectsTerm)

Remove correlations between random effects in term.

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

confint(pr::MixedModelProfile; level::Real=0.95)

Compute profile confidence intervals for coefficients and variance components, with confidence level level (by default 95%).

confint(pr::MixedModelBootstrap; level::Real=0.95, method=:shortest)

Compute bootstrap confidence intervals for coefficients and variance components, with confidence level level (by default 95%).

The keyword argument method determines whether the :shortest, i.e. highest density, interval is used or the :equaltail, i.e. quantile-based, interval is used. For historical reasons, the default is :shortest, but :equaltail gives the interval that is most comparable to the profile and Wald confidence intervals.

See also shortestcovint.

confint(pr::MixedModelProfile; level::Real=0.95)

Compute profile confidence intervals for (fixed effects) coefficients, with confidence level level (by default 95%).

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.

dof_residual(m::MixedModel)

Return the residual degrees of freedom of the model.

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.

fit!(m::LinearMixedModel; progress::Bool=true, REML::Bool=m.optsum.REML,
                          σ::Union{Real, Nothing}=m.optsum.sigma,
                          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.

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 $[ZΛ X][L L']⁻¹[ZΛ X]'$.

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.

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.

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.

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.

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.

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.

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.

Base.copy(ReMat{T,S})

Return a shallow copy of ReMat.

A shallow copy shares as much internal storage as possible with the original ReMat. Only the vector λ and the scratch matrix are copied.

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

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

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

LD(A::Diagonal)
LD(A::HBlikDiag)
LD(A::DenseMatrix)

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

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

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

allpars(bsamp::MixedModelFitCollection)

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

amalgamate(reterms::Vector{AbstractReMat})

Combine multiple ReMat with the same grouping variable into a single object.

average(a::T, b::T) where {T<:AbstractFloat}

Return the average of a and b

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).

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.

copyscaleinflate!(L::AbstractMatrix, A::AbstractMatrix, Λ::ReMat)

Overwrite L with Λ'AΛ + I

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

cpad(s::AbstractString, n::Integer)

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

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.

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.

feL(m::LinearMixedModel)

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

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.

fname(A::ReMat)

Return the name of the grouping factor as a Symbol

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)

getθ(m::LinearMixedModel)

Return the current covariance parameter vector.

indmat(A::ReMat)

Return a Bool indicator matrix of the potential non-zeros in A.λ

isconstant(x::Array)
isconstant(x::Tuple)

Are all elements of the iterator the same? That is, is it constant?

isfullrank(A::FeTerm)

Does A have full column rank?

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?

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.

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.

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.

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

nranef(A::ReMat)

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

nθ(A::ReMat)

Return the number of free parameters in the relative covariance matrix λ

optsumj(os::OptSummary, j::Integer)

Return an OptSummary with the j'th component of the parameter omitted.

os.final with its j'th component omitted is used as the initial parameter.

parsej(sym::Symbol)

Return the index from symbol names like :θ1, :θ01, etc.

pivot(m::MixedModel)
pivot(A::FeTerm)

Return the pivot associated with the FeTerm.

 profileσs!(val::NamedTuple, tc::TableColumns{T}; nzlb=1.0e-8) where {T}

Profile the variance components.

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

β is the truncated, pivoted coefficient vector.

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]

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.

reevaluateAend!(m::LinearMixedModel)

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

refitσ!(m::LinearMixedModel{T}, σ::T, tc::TableColumns{T}, obj::T, neg::Bool)

Refit the model m with the given value of σ and return a NamedTuple of information about the fit.

obj and neg allow for conversion of the objective to the ζ scale and tc is used to return a NamedTuple

schematize(f, tbl, contrasts::Dict{Symbol}, Mod=LinearMixedModel)

Find and apply the schema for f in a way that automatically uses Grouping() contrasts when appropriate.

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).

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.

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 analogous to the use of n or n-1 in the denominator when calculating the variance.

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)).

tidyβ(bsamp::MixedModelFitCollection)

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

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.

unfit!(model::MixedModel)

Mark a model as unfitted.

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.

updateA!(m::LinearMixedModel)

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

This is usually done after a reweight! operation.

updateη!(m::GeneralizedLinearMixedModel)

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

σvals!(v::AbstractVector, A::ReMat, sc::Number)

Overwrite v with the standard deviations of the random effects associated with A

σρ!(v, t, σ)

push! σ times the row lengths (σs) and the inner products of normalized rows (ρs) of t onto v

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.