Estimating Equations Regression in Julia

This package fits regression models to data using estimating equations. Estimating equations are useful for carrying out regression analysis when the data are not independent, or when there are certain forms of heteroscedasticity. This package currently support three methods:

  • Generalized Estimating Equations (GEE)

  • Quadratic Inference Functions (QIF)

  • Generalized Expectile Estimating Equations (GEEE)

using EstimatingEquationsRegression, Random, RDatasets, StatsModels, Plots

# The example below fits linear GEE models to test score data that are clustered
# by classroom, using two different working correlation structures.
da = dataset("SASmixed", "SIMS")
da = sort(da, :Class)
f = @formula(Gain ~ Pretot)

# m1 uses an independence working correlation (by default)
m1 = fit(GeneralizedEstimatingEquationsModel, f, da, da[:, :Class])

# m2 uses an exchangeable working correlation
m2 = fit(GeneralizedEstimatingEquationsModel, f, da, da[:, :Class],
         IdentityLink(), ConstantVar(), ExchangeableCor())

StatsModels.TableRegressionModel{EstimatingEquationsRegression.GeneralizedEstimatingEquationsModel{EstimatingEquationsRegression.GEEResp{Float64}, EstimatingEquationsRegression.DensePred{Float64}}, Matrix{Float64}}

Gain ~ 1 + Pretot

Coefficients:
──────────────────────────────────────────────────────────────────────────
                 Coef.  Std. Error       z  Pr(>|z|)  Lower 95%  Upper 95%
──────────────────────────────────────────────────────────────────────────
(Intercept)   6.93691    0.36197     19.16    <1e-81   6.22746    7.64636
Pretot       -0.185577   0.0160356  -11.57    <1e-30  -0.217006  -0.154148
──────────────────────────────────────────────────────────────────────────

The within-classroom correlation:

corparams(m2)

0.2569238150456968

The standard deviation of the unexplained variation:

sqrt(dispersion(m2.model))

5.584595437738074

Plot the fitted values with a 95% pointwise confidence band:

x = range(extrema(da[:, :Pretot])..., 20)
xm = [ones(20) x]
se = sum((xm * vcov(m2)) .* xm, dims=2).^0.5 # standard errors
yy = xm * coef(m2) # fitted values
plt = plot(x, yy; ribbon=2*se, color=:grey, xlabel="Pretot", ylabel="Gain",
           label=nothing, size=(400,300))
plt = plot!(plt, x, yy, label=nothing)
Plots.savefig(plt, "assets/readme1.svg")

"/home/kshedden/Projects/julia/EstimatingEquationsRegression.jl/assets/readme1.svg"

Example plot 1

For more examples, see the examples folder and the unit tests in the test folder.

References

Longitudinal Data Analysis Using Generalized Linear Models. KY Liang, S Zeger (1986). https://www.biostat.jhsph.edu/~fdominic/teaching/bio655/references/extra/liang.bka.1986.pdf

Efficient estimation for longitudinal data by combining large-dimensional moment condition. H Cho, A Qu (2015). https://projecteuclid.org/journals/electronic-journal-of-statistics/volume-9/issue-1/Efficient-estimation-for-longitudinal-data-by-combining-large-dimensional-moment/10.1214/15-EJS1036.full

A new GEE method to account for heteroscedasticity, using assymetric least-square regressions. A Barry, K Oualkacha, A Charpentier (2018). https://arxiv.org/abs/1810.09214

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