# GLM.jl Manual

Linear and generalized linear models in Julia

## Installation

Pkg.add("GLM")

will install this package and its dependencies, which includes the Distributions package.

The RDatasets package is useful for fitting models on standard R datasets to compare the results with those from R.

## Fitting GLM models

Two methods can be used to fit a Generalized Linear Model (GLM): glm(formula, data, family, link) and glm(X, y, family, link). Their arguments must be:

• formula: a StatsModels.jl Formula object referring to columns in data; for example, if column names are :Y, :X1, and :X2, then a valid formula is @formula(Y ~ X1 + X2)
• data: a table in the Tables.jl definition, e.g. a data frame; rows with missing values are ignored
• X a matrix holding values of the independent variable(s) in columns
• y a vector holding values of the dependent variable (including if appropriate the intercept)
• family: chosen from Bernoulli(), Binomial(), Gamma(), Geometric(), Normal(), Poisson(), or NegativeBinomial(θ)
• link: chosen from the list below, for example, LogitLink() is a valid link for the Binomial() family

Typical distributions for use with glm and their canonical link functions are

       Bernoulli (LogitLink)
Poisson (LogLink)

Currently the available Link types are

CauchitLink
SqrtLink

Note that the canonical link for negative binomial regression is NegativeBinomialLink, but in practice one typically uses LogLink. The NegativeBinomial distribution belongs to the exponential family only if θ (the shape parameter) is fixed, thus θ has to be provided if we use glm with NegativeBinomial family. If one would like to also estimate θ, then negbin(formula, data, link) should be used instead.

An intercept is included in any GLM by default.

## Categorical variables

Categorical variables will be dummy coded by default if they are non-numeric or if they are CategoricalVectors within a Tables.jl table (DataFrame, JuliaDB table, named tuple of vectors, etc). Alternatively, you can pass an explicit contrasts argument if you would like a different contrast coding system or if you are not using DataFrames.

The response (dependent) variable may not be categorical.

Using a CategoricalVector constructed with categorical or categorical!:

julia> using CategoricalArrays, DataFrames, GLM, StableRNGs

julia> rng = StableRNG(1); # Ensure example can be reproduced

julia> data = DataFrame(y = rand(rng, 100), x = categorical(repeat([1, 2, 3, 4], 25)));

julia> lm(@formula(y ~ x), data)
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

y ~ 1 + x

Coefficients:
───────────────────────────────────────────────────────────────────────────
Coef.  Std. Error      t  Pr(>|t|)   Lower 95%  Upper 95%
───────────────────────────────────────────────────────────────────────────
(Intercept)   0.490985    0.0564176   8.70    <1e-13   0.378997    0.602973
x: 2          0.0527655   0.0797865   0.66    0.5100  -0.105609    0.21114
x: 3          0.0955446   0.0797865   1.20    0.2341  -0.0628303   0.25392
x: 4         -0.032673    0.0797865  -0.41    0.6831  -0.191048    0.125702
───────────────────────────────────────────────────────────────────────────

Using contrasts:

julia> using StableRNGs

julia> data = DataFrame(y = rand(StableRNG(1), 100), x = repeat([1, 2, 3, 4], 25));

julia> lm(@formula(y ~ x), data, contrasts = Dict(:x => DummyCoding()))
StatsModels.TableRegressionModel{LinearModel{GLM.LmResp{Vector{Float64}}, GLM.DensePredChol{Float64, LinearAlgebra.CholeskyPivoted{Float64, Matrix{Float64}, Vector{Int64}}}}, Matrix{Float64}}

y ~ 1 + x

Coefficients:
───────────────────────────────────────────────────────────────────────────
Coef.  Std. Error      t  Pr(>|t|)   Lower 95%  Upper 95%
───────────────────────────────────────────────────────────────────────────
(Intercept)   0.490985    0.0564176   8.70    <1e-13   0.378997    0.602973
x: 2          0.0527655   0.0797865   0.66    0.5100  -0.105609    0.21114
x: 3          0.0955446   0.0797865   1.20    0.2341  -0.0628303   0.25392
x: 4         -0.032673    0.0797865  -0.41    0.6831  -0.191048    0.125702
───────────────────────────────────────────────────────────────────────────

## Comparing models with F-test

Comparisons between two or more linear models can be performed using the ftest function, which computes an F-test between each pair of subsequent models and reports fit statistics:

julia> using DataFrames, GLM, StableRNGs

julia> data = DataFrame(y = (1:50).^2 .+ randn(StableRNG(1), 50), x = 1:50);

julia> ols_lin = lm(@formula(y ~ x), data);

julia> ols_sq = lm(@formula(y ~ x + x^2), data);

julia> ftest(ols_lin.model, ols_sq.model)
F-test: 2 models fitted on 50 observations
─────────────────────────────────────────────────────────────────────────────────
DOF  ΔDOF           SSR           ΔSSR      R²     ΔR²            F*   p(>F)
─────────────────────────────────────────────────────────────────────────────────
[1]    3        1731979.2266                 0.9399
[2]    4     1       40.7581  -1731938.4685  1.0000  0.0601  1997177.0357  <1e-99
─────────────────────────────────────────────────────────────────────────────────

## Methods applied to fitted models

Many of the methods provided by this package have names similar to those in R.

• adjr2: adjusted R² for a linear model (an alias for adjr²)
• aic: Akaike's Information Criterion
• aicc: corrected Akaike's Information Criterion for small sample sizes
• bic: Bayesian Information Criterion
• coef: estimates of the coefficients in the model
• confint: confidence intervals for coefficients
• cooksdistance: Cook's distance for each observation
• deviance: measure of the model fit, weighted residual sum of squares for lm's
• dispersion: dispersion (or scale) parameter for a model's distribution
• dof: number of degrees of freedom consumed in the model
• dof_residual: degrees of freedom for residuals, when meaningful
• fitted: fitted values of the model
• glm: fit a generalized linear model (an alias for fit(GeneralizedLinearModel, ...))
• lm: fit a linear model (an alias for fit(LinearModel, ...))
• loglikelihood: log-likelihood of the model
• modelmatrix: design matrix
• nobs: number of rows, or sum of the weights when prior weights are specified
• nulldeviance: deviance of the model with all predictors removed
• nullloglikelihood: log-likelihood of the model with all predictors removed
• predict: predicted values of the dependent variable from the fitted model
• r2: R² of a linear model (an alias for r²)
• residuals: vector of residuals from the fitted model
• response: model response (a.k.a the dependent variable)
• stderror: standard errors of the coefficients
• vcov: variance-covariance matrix of the coefficient estimates

Note that the canonical link for negative binomial regression is NegativeBinomialLink, but in practice one typically uses LogLink.

julia> using GLM, DataFrames, StatsBase

julia> data = DataFrame(X=[1,2,3], y=[2,4,7]);

julia> mdl = lm(@formula(y ~ X), data);

julia> round.(coef(mdl); digits=8)
2-element Vector{Float64}:
-0.66666667
2.5

julia> round(r2(mdl); digits=8)
0.98684211

julia> round(aic(mdl); digits=8)
5.84251593

The predict method returns predicted values of response variable from covariate values in an input newX. If newX is omitted then the fitted response values from the model are returned.

julia> test_data = DataFrame(X=[4]);

julia> round.(predict(mdl, test_data); digits=8)
1-element Vector{Float64}:
9.33333333

The cooksdistance method computes Cook's distance for each observation used to fit a linear model, giving an estimate of the influence of each data point. Note that it's currently only implemented for linear models without weights.

julia> round.(cooksdistance(mdl); digits=8)
3-element Vector{Float64}:
2.5
0.25
2.5

## Separation of response object and predictor object

The general approach in this code is to separate functionality related to the response from that related to the linear predictor. This allows for greater generality by mixing and matching different subtypes of the abstract type LinPred and the abstract type ModResp.

A LinPred type incorporates the parameter vector and the model matrix. The parameter vector is a dense numeric vector but the model matrix can be dense or sparse. A LinPred type must incorporate some form of a decomposition of the weighted model matrix that allows for the solution of a system X'W * X * delta=X'wres where W is a diagonal matrix of "X weights", provided as a vector of the square roots of the diagonal elements, and wres is a weighted residual vector.

Currently there are two dense predictor types, DensePredQR and DensePredChol, and the usual caveats apply. The Cholesky version is faster but somewhat less accurate than that QR version. The skeleton of a distributed predictor type is in the code but not yet fully fleshed out. Because Julia by default uses OpenBLAS, which is already multi-threaded on multicore machines, there may not be much advantage in using distributed predictor types.

A ModResp type must provide methods for the wtres and sqrtxwts generics. Their values are the arguments to the updatebeta methods of the LinPred types. The Float64 value returned by updatedelta is the value of the convergence criterion.

Similarly, LinPred types must provide a method for the linpred generic. In general linpred takes an instance of a LinPred type and a step factor. Methods that take only an instance of a LinPred type use a default step factor of 1. The value of linpred is the argument to the updatemu method for ModResp types. The updatemu method returns the updated deviance.