API

Basic

@covstr

Metida.@covstrMacro
@covstr(ex)

Macros for random/repeated effect model.

Example

@covstr(factor|subject)
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@lmmformula

Metida.@lmmformulaMacro
@lmmformula(formula, args...)

Macro for made formula with variance-covariance structure representation. @lmmformula could be used for shorter LMM construction.

Example:

lmm = Metida.LMM(@lmmformula(var~sequence+period+formulation,
random = formulation|subject:CSH,
repeated = formulation|subject:DIAG),
df0)

equal to:

lmm = LMM(@formula(var~sequence+period+formulation), df0;
random = VarEffect(@covstr(formulation|subject), CSH),
repeated = VarEffect(@covstr(formulation|subject), DIAG),
)

@lmmformula have 3 components - 1'st is a formula for fixed effect, it's defined like in StstsModels (1st argument just provided to @formula macro). Other arguments should be defined like keywords. repeated keyword define repeated effect part, random - define random effect part. You can use several random factors as in example bellow:

lmm = LMM(@lmmformula(var~sequence+period+formulation,
random = formulation|subject:CSH,
random = 1|subject:DIAG,
repeated = formulation|subject:DIAG),
df0)

random or repeated structure made by template:

effect formula | blocking factor [/ nested factor] [: covariance structure]

| - devide effect formula form blocking factor definition (necessarily), / and : modificator are optional.

/ work like in MixedModels or in RegressionFormulae - expand factor f|a/b to f|a + f|a&b. It can't be used in repeated effect definition.

: - covariance structure defined right after : (SI, DIAG, CS, CSH, ets...), if : not used then SI used for this effect.

Terms like a+b or a*b shuould not be used as a blocking factors.

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Metida.CovarianceType

LMM

Metida.LMMType
LMM(model, data; contrasts=Dict{Symbol,Any}(),  random::Union{Nothing, VarEffect, Vector{VarEffect}} = nothing, repeated::Union{Nothing, VarEffect} = nothing)

Make Linear-Mixed Model object.

model: is a fixed-effect model (@formula)

data: tabular data

contrasts: contrasts for fixed factors

random: vector of random effects or single random effect

repeated: is a repeated effect (only one)

See also: @lmmformula

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VarEffect

Metida.VarEffectType
VarEffect(formula, covtype::T, coding) where T <: AbstractCovarianceType

VarEffect(formula, covtype::T; coding = nothing) where T <: AbstractCovarianceType

VarEffect(formula; coding = nothing)

Random/repeated effect.

  • formula from @covstr(ex) macros.

  • covtype - covariance type (SI, DIAG, CS, CSH, AR, ARH, ARMA, TOEP, TOEPH, TOEPP, TOEPHP)

Note

Categorical factors are coded with FullDummyCoding() by default, use coding for other contrast codeing.

Example

VarEffect(@covstr(1+factor|subject), CSH)

VarEffect(@covstr(1 + formulation|subject), CSH; coding = Dict(:formulation => StatsModels.DummyCoding()))
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Covariance structures

Metida.Autoregressive

Metida.AutoregressiveFunction
Autoregressive()

Autoregressive covariance type.

AR = Autoregressive()

\[\begin{bmatrix} 1 & \rho & \rho^2 & \rho^3 \\ \rho & 1 & \rho & \rho^2 \\ \rho^2 & \rho & 1 & \rho \\ \rho^3 & \rho^2 & \rho & 1 \end{bmatrix}\sigma^2\]

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Metida.AutoregressiveMovingAverage

Metida.AutoregressiveMovingAverageFunction
AutoregressiveMovingAverage()

Autoregressive moving average covariance type.

ARMA = AutoregressiveMovingAverage()

\[\begin{bmatrix} 1 & \gamma & \gamma\rho & \gamma\rho^2 \\ \gamma & 1 & \gamma & \gamma\rho \\ \gamma\rho & \gamma & 1 & \gamma \\ \gamma\rho^2 & \gamma\rho & \gamma & 1 \end{bmatrix}\sigma^2\]

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Metida.CompoundSymmetry

Metida.CompoundSymmetryFunction
CompoundSymmetry()

Compound symmetry covariance type.

CS = CompoundSymmetry()

\[\begin{bmatrix} 1 & \rho & \rho & \rho \\ \rho & 1 & \rho & \rho \\ \rho & \rho & 1 & \rho \\ \rho & \rho & \rho & 1 \end{bmatrix}\sigma^2\]

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Metida.Diag

Metida.DiagFunction
Diag()

Diagonal covariance type.

DIAG = Diag()

\[\begin{bmatrix} \sigma_a^2 & 0 & 0 \\ 0 & \sigma_b^2 & 0 \\ 0 & 0 & \sigma_c^2 \end{bmatrix}\]

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Metida.HeterogeneousAutoregressive

Metida.HeterogeneousAutoregressiveFunction
HeterogeneousAutoregressive()

Heterogeneous autoregressive covariance type.

ARH = HeterogeneousAutoregressive()

\[\begin{bmatrix} \sigma_a^2 & \rho\sigma_a\sigma_b & \rho^2\sigma_a\sigma_c & \rho^3\sigma_a\sigma_d \\ \rho\sigma_b\sigma_a & \sigma_b^2 & \rho\sigma_b\sigma_c & \rho^2\sigma_b\sigma_d \\ \rho^2\sigma_c\sigma_a & \rho\sigma_c\sigma_b & \sigma_c^2 & \rho\sigma_c\sigma_d \\ \rho^3\sigma_d\sigma_a & \rho^2\sigma_d\sigma_b & \rho\sigma_d\sigma_c & \sigma_d^2 \end{bmatrix}\]

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Metida.HeterogeneousCompoundSymmetry

Metida.HeterogeneousCompoundSymmetryFunction
HeterogeneousCompoundSymmetry()

Heterogeneous compound symmetry covariance type.

CSH = HeterogeneousCompoundSymmetry()

\[\begin{bmatrix} \sigma_a^2 & \rho\sigma_a\sigma_b & \rho\sigma_a\sigma_c & \rho\sigma_a\sigma_d \\ \rho\sigma_b\sigma_a & \sigma_b^2 & \rho\sigma_b\sigma_c & \rho\sigma_b\sigma_d \\ \rho\sigma_c\sigma_a & \rho\sigma_c\sigma_b & \sigma_c^2 & \rho\sigma_c\sigma_d \\ \rho\sigma_d\sigma_a & \rho\sigma_d\sigma_b & \rho\sigma_d\sigma_c & \sigma_d^2 \end{bmatrix}\]

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Metida.HeterogeneousToeplitz

Metida.HeterogeneousToeplitzFunction
HeterogeneousToeplitz()

Heterogeneous toeplitz covariance type. Only for G matrix.

TOEPH = HeterogeneousToeplitz()

\[\begin{bmatrix} \sigma_a^2 & \rho_1 \sigma_a \sigma_b & \rho_2 \sigma_a \sigma_c & \rho_3 \sigma_a \sigma_d \\ \rho_1 \sigma_b \sigma_a & \sigma_b^2 & \rho_1 \sigma_b \sigma_c & \rho_2 \sigma_b \sigma_d \\ \rho_2 \sigma_c \sigma_a & \rho_1 \sigma_c \sigma_b & \sigma_c^2 & \rho_1 \sigma_c \sigma_d \\ \rho_3 \sigma_d \sigma_a & \rho_2 \sigma_d \sigma_b & \rho_1 \sigma_d \sigma_c & \sigma_d^2 \end{bmatrix}\]

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Metida.HeterogeneousToeplitzParameterized

Metida.HeterogeneousToeplitzParameterizedFunction
HeterogeneousToeplitzParameterized(p::Int)

Heterogeneous toeplitz covariance type with parameter p, (number of bands = p - 1, if p = 1 it's equal DIAG structure).

TOEPHP(p) = HeterogeneousToeplitzParameterized(p)

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Metida.ScaledIdentity

Metida.ScaledIdentityFunction
ScaledIdentity()

Scaled identity covariance type.

SI = ScaledIdentity()

\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\sigma^{2}\]

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Metida.SpatialExponential

Metida.SpatialExponentialFunction
SpatialExponential()

Spatian Exponential covariance structure. Used only for repeated effect.

\[R_{i,j} = \sigma^{2} * exp(-dist(i,j)/\theta)\]

where dist - Euclidean distance between row-vectors of repeated effect matrix for subject i and j, θ > 0.

SPEXP = SpatialExponential()

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Metida.SpatialGaussian

Metida.SpatialGaussianFunction
SpatialGaussian()

Spatian Gaussian covariance structure. Used only for repeated effect.

\[R_{i,j} = \sigma^{2} * exp(- dist(i,j)^2 / \theta^2)\]

where dist - Euclidean distance between row-vectors of repeated effect matrix for subject i and j, θ ≠ 0.

SPGAU = SpatialGaussian()

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Metida.SpatialPower

Metida.SpatialPowerFunction
SpatialPower()

Spatian Power covariance structure. Used only for repeated effect.

\[R_{i,j} = \sigma^{2} * \rho^{dist(i,j)}\]

where dist - Euclidean distance between row-vectors of repeated effect matrix for subject i and j, 1 > ρ > -1.

SPPOW = SpatialPower()

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Metida.Toeplitz

Metida.ToeplitzFunction
Toeplitz()

Toeplitz covariance type. Only for G matrix.

TOEP = Toeplitz()

\[\begin{bmatrix} 1 & \rho_1 & \rho_2 & \rho_3 \\ \rho_1 & 1 & \rho_1 & \rho_2 \\ \rho_2 & \rho_1 & 1 & \rho_1 \\ \rho_3 & \rho_2 & \rho_1 & 1 \end{bmatrix}\sigma^2\]

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Metida.ToeplitzParameterized

Metida.ToeplitzParameterizedFunction
ToeplitzParameterized(p::Int)

Toeplitz covariance type with parameter p, (number of bands = p - 1, if p = 1 it's equal SI structure).

TOEPP(p) = ToeplitzParameterized(p)

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Metida.Unstructured

Metida.UnstructuredFunction
Unstructured()

Unstructured covariance structure with t*(t+1)/2-t paremeters where t - number of factor levels, t*(t+1)/2-2t of them is covariance (ρ) patemeters.

UN = Unstructured()

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Methods

Metida.caic

Metida.caicFunction
caic(lmm::LMM)

Conditional Akaike Information Criterion.

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Metida.dof_satter

Metida.dof_satterFunction
dof_satter(lmm::LMM{T}, l) where T

Return Satterthwaite approximation for the denominator degrees of freedom, where l is a contrast vector (estimable linear combination of fixed effect coefficients vector (β).

\[df = \frac{2(LCL')^{2}}{g'Ag}\]

Where: $A = 2H^{-1}$, $g = \triangledown_{\theta}(LC^{-1}_{\theta}L')$

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dof_satter(lmm::LMM{T}, n::Int) where T

Return Satterthwaite approximation for the denominator degrees of freedom, where n - coefficient number.

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dof_satter(lmm::LMM{T}) where T

Return Satterthwaite approximation for the denominator degrees of freedom for all coefficients.

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dof_satter(lmm::LMM{T}, l::Matrix) where T

Return Satterthwaite approximation for the denominator degrees of freedom for conrast matrix l.

For size(l, 1) > 1:

\[df = \frac{2E}{E - rank(LCL')}\]

where:

  • let $LCL' = QΛQ^{-1}$, where $QΛQ^{-1}$ - spectral decomposition of $LCL'$
  • $Lq_i$ is the i-th row of $Q^{-1}L$
  • $A = 2H^{-1}$, $g = \triangledown_{\theta}(Lq_i C^{-1}_{\theta} Lq_i')$
  • $v_i = \frac{2*Λ_{i,i}^2}{g' * A * g}$
  • $E = \sum_{i=1}^n {\frac{v_i}(v_i - 2)}$ for $v_i > 2$
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Metida.estimate

Metida.estimateFunction
estimate(lmm, l::AbstractVector; level = 0.95, name = "Estimate")

Estimate table for l vector. Satter DF used.

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estimate(lmm; level = 0.95)

Estimates table. Satter DF used.

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Metida.getlog

Metida.gmatrix

Metida.hessian

Metida.lcontrast

Metida.nblocks

Metida.rand

Base.randFunction
rand(rng::AbstractRNG, lmm::LMM{T}) where T

Generate random responce vector for fitted 'lmm' model.

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rand(rng::AbstractRNG, lmm::LMM{T}; theta) where T

Generate random responce vector 'lmm' model, theta covariance vector, and zero means.

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rand(rng::AbstractRNG, lmm::LMM{T}; theta, beta) where T

Generate random responce vector 'lmm' model, theta covariance vector and mean's vector.

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Metida.rand!

Missing docstring.

Missing docstring for Metida.rand!. Check Documenter's build log for details.

Metida.raneff

Metida.raneffFunction
raneff(lmm::LMM{T}, i)

Vector of random effect coefficients for block i.

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raneff(lmm::LMM{T})

Vector of random effect coefficients for all subjects by each random effect.

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Metida.rankx

Metida.rmatrix

Metida.theta

Metida.thetalength

Metida.vmatrix!

Metida.vmatrix!Function
vmatrix!(V, θ, lmm, i)

Update variance-covariance matrix V for i bolock. Upper triangular updated.

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StatsAPI

Metida.aic

Metida.aicc

StatsAPI.aiccFunction
StatsBase.aicc(lmm::LMM)

Corrected Akaike Information Criterion.

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Metida.bic

Metida.coef

StatsAPI.coefFunction
StatsBase.coef(lmm::LMM) = copy(lmm.result.beta)

Model coefficients (β).

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Metida.coefnames

Metida.coeftable

Metida.confint

StatsAPI.confintFunction
StatsBase.confint(lmm::LMM{T}; level::Real=0.95, ddf::Symbol = :satter) where T

Confidece interval for coefficients.

ddf = :satter/:residual

\[CI_{U/L} = β ± SE * t_{ddf, 1-α/2}\]

See also: dof_satter, dof_residual

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StatsBase.confint(lmm::LMM{T}, i::Int; level::Real=0.95, ddf::Symbol = :satter) where T

Confidece interval for coefficient i.

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StatsBase.confint(br::BootstrapResult, n::Int; level::Float64=0.95, method=:bp, metric = :coef, delrml = false)

Confidence interval for bootstrap result.

*method:

  • :bp - bootstrap percentile;
  • :rbp - reverse bootstrap percentile;
  • :norm - Normal distribution;
  • :bcnorm - Bias corrected Normal distribution;
  • :jn - bias corrected (jackknife resampling).
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Metida.crossmodelmatrix

Metida.dof

Metida.dof_residual

StatsAPI.dof_residualFunction
StatsBase.dof_residual(lmm::LMM)

DOF residuals: N - rank(X), where N - total number of observations.

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Metida.fit

StatsAPI.fitFunction
fit(::Type{T}, f::FormulaTerm, data;
contrasts=Dict{Symbol,Any}(),  
random::Union{Nothing, VarEffect, Vector{VarEffect}} = nothing, 
repeated::Union{Nothing, VarEffect} = nothing,
kwargs...)

Fit LMM model with @formula.

Keywords see fit!

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fit(::Type{T}, f::LMMformula, data;
contrasts=Dict{Symbol,Any}(),  
kwargs...) where T <: LMM

Fit LMM model with @lmmformula.

Keywords see fit!

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Metida.fit!

StatsAPI.fit!Function
fit!(lmm::LMM{T}; kwargs...
) where T

Fit LMM model.

Keywords:

  • solver - :default / :nlopt for using with MetidaNLopt.jl/ :cuda for using with MetidaCu.jl
  • verbose - :auto / 1 / 2 / 3 - - 1 - only log, 2 - log and print, 3 - print only errors, other log, 0 (or any other value) - no logging
  • varlinkf - :exp / :sq / :identity ref
  • rholinkf - :sigm / :atan / :sqsigm / :psigm
  • aifirst - first iteration with AI-like method - :default / :ai / :score
  • g_tol - absolute tolerance in the gradient
  • x_tol - absolute tolerance of theta vector
  • f_tol - absolute tolerance in changes of the REML
  • hes - calculate REML Hessian
  • init - initial theta values
  • io - output IO
  • time_limit - time limit = 120 sec
  • iterations - maximum iterations = 300
  • refitinit - true/false - if true - use last values for initial condition (false by default)
  • optmethod - Optimization method. Look at Optim.jl documentation. (Newton by default)
  • singtol - singular tolerance = 1e-8
  • maxthreads - maximum threads = num_cores()
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isfitted

Metida.loglikelihood

Metida.modelmatrix

Metida.nobs

Metida.response

Metida.responsename

Metida.stderror

Metida.vcov

Experimental

Metida.SpatialExponentialD

Metida.SpatialExponentialDFunction
SpatialExponentialD()
Warning

Experimental

Same as SpatialExponential, but add D to all diagonal elements.

SPEXPD = SpatialExponentialD()

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Metida.SpatialGaussianD

Metida.SpatialGaussianDFunction
SpatialGaussianD()
Warning

Experimental

Same as SpatialGaussianD, but add D to all diagonal elements.

SPGAUD = SpatialGaussianD()

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Metida.SpatialPowerD

Metida.SpatialPowerDFunction
SpatialPowerD()
Warning

Experimental

Same as SpatialPower, but add D to all diagonal elements.

SPPOWD = SpatialPowerD()

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Metida.dof_contain

Metida.dof_containFunction
dof_contain(lmm, i)
Warning

Experimental! Compute rank(XZi) for each random effect that syntactically contain factor assigned for β[i] element (Where Zi - Z matrix for random effect i). Minimum returned. If no random effect found N - rank(XZ) returned.

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Metida.typeiii

Metida.typeiiiFunction
typeiii(lmm::LMM{T}; ddf::Symbol = :satter) where T
Warning

Experimental

Type III table.

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Not API functions

Metida.contrast

Metida.contrastFunction
contrast(lmm, l::AbstractMatrix; name::String = "Contrast", ddf = :satter)

User contrast table. ddf = :satter or :residual or any number for direct ddf setting.

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Metida.fvalue

Metida.fvalueFunction
fvalue(lmm::LMM, l::Matrix)

F value for contrast matrix l.

\[F = \frac{\beta'L'(LCL')^{-1}L\beta}{rank(LCL')}\]

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Metida.mulαβαtinc!

Metida.mulαβαtinc!Function
mulαβαtinc!(θ::AbstractMatrix, A::AbstractMatrix, B::AbstractMatrix)

θ + A * B * A'

Change θ (only upper triangle). B is symmetric.

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mulαβαtinc!(θ::AbstractMatrix, A::AbstractMatrix, B::AbstractMatrix, alpha)

θ + A * B * A' * alpha

Change θ (only upper triangle). B is symmetric.

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mulαβαtinc!(θ::AbstractVector{T}, A::AbstractMatrix, B::AbstractMatrix, a::AbstractVector, b::AbstractVector, alpha) where T

θ + A * B * (a - b) * alpha

Change θ (only upper triangle). B is symmetric.

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Metida.mulθ₃

Metida.mulθ₃Function
mulθ₃(y, X, β, V::AbstractMatrix{T})::T where T

(y - X * β)' * (-V) * (y - X * β)

use only upper triangle of V

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Metida.mulαtβinc!

Metida.mulαtβinc!Function
mulαtβinc!(θ::AbstractVector{T}, A::AbstractMatrix, b::AbstractVector) where T

θ + A' * b

Change θ.

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