API

DensePredChol{T}

A LinPred type with a dense Cholesky factorization of X'X

Members

• X: model matrix of size n × p with n ≥ p. Should be full column rank.
• beta0: base coefficient vector of length p
• delbeta: increment to coefficient vector, also of length p
• scratchbeta: scratch vector of length p, used in linpred! method
• chol: a Cholesky object created from X'X, possibly using row weights.
• scratchm1: scratch Matrix{T} of the same size as X
• scratchm2: scratch Matrix{T} os the same size as X'X

DensePredQR

A LinPred type with a dense, unpivoted QR decomposition of X

Members

• X: Model matrix of size n × p with n ≥ p. Should be full column rank.
• beta0: base coefficient vector of length p
• delbeta: increment to coefficient vector, also of length p
• scratchbeta: scratch vector of length p, used in linpred! method
• qr: a QRCompactWY object created from X, with optional row weights.

GlmResp

The response vector and various derived vectors in a generalized linear model.

LinearModel

A combination of a LmResp and a LinPred

Members

• rr: a LmResp object
• pp: a LinPred object

LmResp

Encapsulates the response for a linear model

Members

• mu: current value of the mean response vector or fitted value
• offset: optional offset added to the linear predictor to form mu
• wts: optional vector of prior weights
• y: observed response vector

Either or both offset and wts may be of length 0

LinPred

Abstract type representing a linear predictor

ModResp

Abstract type representing a model response vector

cancancel(r::GlmResp{V,D,L})

Returns true if dμ/dη for link L is the variance function for distribution D

When L is the canonical link for D the derivative of the inverse link is a multiple of the variance function for D. If they are the same a numerator and denominator term in the expression for the working weights will cancel.

delbeta!(p::LinPred, r::Vector)

Evaluate and return p.delbeta the increment to the coefficient vector from residual r

deviance(obj::LinearModel)

For linear models, the deviance is equal to the residual sum of squares (RSS).

dispersion(m::AbstractGLM, sqr::Bool=false)

Return the estimated dispersion (or scale) parameter for a model's distribution, generally written σ for linear models and ϕ for generalized linear models. It is, by definition, equal to 1 for the Bernoulli, Binomial, and Poisson families.

If sqr is true, the squared dispersion parameter is returned.

installbeta!(p::LinPred, f::Real=1.0)

Install pbeta0 .+= f * p.delbeta and zero out p.delbeta. Return the updated p.beta0.

A helper function to determine if mod1 is nested in mod2

linpred!(out, p::LinPred, f::Real=1.0)

Overwrite out with the linear predictor from p with factor f

The effective coefficient vector, p.scratchbeta, is evaluated as p.beta0 .+ f * p.delbeta, and out is updated to p.X * p.scratchbeta

linpred(p::LinPred, f::Read=1.0)

Return the linear predictor p.X * (p.beta0 .+ f * p.delbeta)

lm(X, y, allowrankdeficient::Bool=false)

An alias for fit(LinearModel, X, y, allowrankdeficient)

The arguments X and y can be a Matrix and a Vector or a Formula and a DataFrame.

nobs(obj::LinearModel)
nobs(obj::GLM)

For linear and generalized linear models, returns the number of rows, or, when prior weights are specified, the sum of weights.

nulldeviance(obj::LinearModel)

For linear models, the deviance of the null model is equal to the total sum of squares (TSS).

predict(mm::LinearModel, newx::AbstractMatrix;
        interval::Union{Symbol,Nothing} = nothing, level::Real = 0.95)

If interval is nothing (the default), return a vector with the predicted values for model mm and new data newx. Otherwise, return a 3-column matrix with the prediction and the lower and upper confidence bounds for a given level (0.95 equates alpha = 0.05). Valid values of interval are :confidence delimiting the uncertainty of the predicted relationship, and :prediction delimiting estimated bounds for new data points.

predict(mm::AbstractGLM, newX::AbstractMatrix; offset::FPVector=Vector{eltype(newX)}(0))

Form the predicted response of model mm from covariate values newX and, optionally, an offset.

updateμ!{T<:FPVector}(r::GlmResp{T}, linPr::T)

Update the mean, working weights and working residuals, in r given a value of the linear predictor, linPr.

wrkresp(r::GlmResp)

The working response, r.eta + r.wrkresid - r.offset.

wrkresp!{T<:FPVector}(v::T, r::GlmResp{T})

Overwrite v with the working response of r

Link

An abstract type whose subtypes determine methods for linkfun, linkinv, mueta, and inverselink.

Link01

An abstract subtype of Link which are links defined on (0, 1)

CauchitLink

A Link01 corresponding to the standard Cauchy distribution, Distributions.Cauchy.

CloglogLink

A Link01 corresponding to the extreme value (or log-Wiebull) distribution. The link is the complementary log-log transformation, log(1 - log(-μ)).

IdentityLink

The canonical Link for the Normal distribution, defined as η = μ.

InverseLink

The canonical Link for Distributions.Gamma distribution, defined as η = inv(μ).

InverseSquareLink

The canonical Link for Distributions.InverseGaussian distribution, defined as η = inv(abs2(μ)).

LogitLink

The canonical Link01 for Distributions.Bernoulli and Distributions.Binomial. The inverse link, linkinv, is the c.d.f. of the standard logistic distribution, Distributions.Logistic.

LogLink

The canonical Link for Distributions.Poisson, defined as η = log(μ).

NegativeBinomialLink

The canonical Link for Distributions.NegativeBinomial distribution, defined as η = log(μ/(μ+θ)). The shape parameter θ has to be fixed for the distribution to belong to the exponential family.

ProbitLink

A Link01 whose linkinv is the c.d.f. of the standard normal distribution, Distributions.Normal().

SqrtLink

A Link defined as η = √μ

linkfun(L::Link, μ)

Return η, the value of the linear predictor for link L at mean μ.

Examples

julia> μ = inv(10):inv(5):1
0.1:0.2:0.9

julia> show(linkfun.(LogitLink(), μ))
[-2.19722, -0.847298, 0.0, 0.847298, 2.19722]

linkinv(L::Link, η)

Return μ, the mean value, for link L at linear predictor value η.

Examples

julia> μ = 0.1:0.2:1
0.1:0.2:0.9

julia> η = logit.(μ);

julia> linkinv.(LogitLink(), η) ≈ μ
true

mueta(L::Link, η)

Return the derivative of linkinv, dμ/dη, for link L at linear predictor value η.

Examples

julia> mueta(LogitLink(), 0.0)
0.25

julia> mueta(CloglogLink(), 0.0) ≈ 0.36787944117144233
true

julia> mueta(LogLink(), 2.0) ≈ 7.38905609893065
true

inverselink(L::Link, η)

Return a 3-tuple of the inverse link, the derivative of the inverse link, and when appropriate, the variance function μ*(1 - μ).

The variance function is returned as NaN unless the range of μ is (0, 1)

Examples

julia> inverselink(LogitLink(), 0.0)
(0.5, 0.25, 0.25)

julia> μ, oneminusμ, variance = inverselink(CloglogLink(), 0.0);

julia> μ + oneminusμ ≈ 1
true

julia> μ*(1 - μ) ≈ variance
true

julia> isnan(last(inverselink(LogLink(), 2.0)))
true

canonicallink(D::Distribution)

Return the canonical link for distribution D, which must be in the exponential family.

Examples

julia> canonicallink(Bernoulli())
LogitLink()

glmvar(D::Distribution, μ)

Return the value of the variance function for D at μ

The variance of D at μ is the product of the dispersion parameter, ϕ, which does not depend on μ and the value of glmvar. In other words glmvar returns the factor of the variance that depends on μ.

Examples

julia> μ = 1/6:1/3:1;

julia> glmvar.(Normal(), μ)    # constant for Normal()
3-element Array{Float64,1}:
 1.0
 1.0
 1.0

julia> glmvar.(Bernoulli(), μ) ≈ μ .* (1 .- μ)
true

julia> glmvar.(Poisson(), μ) == μ
true

mustart(D::Distribution, y, wt)

Return a starting value for μ.

For some distributions it is appropriate to set μ = y to initialize the IRLS algorithm but for others, notably the Bernoulli, the values of y are not allowed as values of μ and must be modified.

Examples

julia> mustart(Bernoulli(), 0.0, 1) ≈ 1/4
true

julia> mustart(Bernoulli(), 1.0, 1) ≈ 3/4
true

julia> mustart(Binomial(), 0.0, 10) ≈ 1/22
true

julia> mustart(Normal(), 0.0, 1) ≈ 0
true

devresid(D, y, μ)

Return the squared deviance residual of μ from y for distribution D

The deviance of a GLM can be evaluated as the sum of the squared deviance residuals. This is the principal use for these values. The actual deviance residual, say for plotting, is the signed square root of this value

sign(y - μ) * sqrt(devresid(D, y, μ))

Examples

julia> devresid(Normal(), 0, 0.25) ≈ abs2(0.25)
true

julia> devresid(Bernoulli(), 1, 0.75) ≈ -2*log(0.75)
true

julia> devresid(Bernoulli(), 0, 0.25) ≈ -2*log1p(-0.25)
true

dispersion_parameter(D)  # not exported

Does distribution D have a separate dispersion parameter, ϕ?

Returns false for the Bernoulli, Binomial and Poisson distributions, true otherwise.

Examples

julia> show(GLM.dispersion_parameter(Normal()))
true
julia> show(GLM.dispersion_parameter(Bernoulli()))
false

loglik_obs(D, y, μ, wt, ϕ)  # not exported

Returns wt * logpdf(D(μ, ϕ), y) where the parameters of D are derived from μ and ϕ.

The wt argument is a multiplier of the result except in the case of the Binomial where wt is the number of trials and μ is the proportion of successes.

The loglikelihood of a fitted model is the sum of these values over all the observations.