In easystats/parameters: Processing of Model Parameters

library(knitr)
options(knitr.kable.NA = "")
options(digits = 2)
knitr::opts_chunk$set(comment = "#>")

if (!requireNamespace("poorman", quietly = TRUE) ||
  !requireNamespace("clubSandwich", quietly = TRUE) ||
  !requireNamespace("sandwich", quietly = TRUE) ||
  !requireNamespace("lme4", quietly = TRUE)) {
  knitr::opts_chunk$set(eval = FALSE)
} else {
  library(parameters)
  library(poorman)
  library(clubSandwich)
  library(lme4)
}

set.seed(333)

The model_parameters() function also allows the computation of standard errors, confidence intervals and p-values based on robust covariance matrix estimation from model parameters. Robust estimation is based on the packages sandwich and clubSandwich, so all models supported by either of these packages work with model_parameters().

There are two arguments responsible to define the heteroscedasticity-consistent covariance matrix, that should be used for robust estimation: vcov and vcov_args.

The vcov argument defines the variance-covariance matrix used to compute uncertainty estimates. This argument accepts following inputs:

A covariance matrix
A function which returns a covariance matrix (e.g., stats::vcov())
A string which indicates the kind of uncertainty estimates to return.
Heteroskedasticity-consistent: "HC", "HC0", "HC1", "HC2", "HC3", "HC4", "HC4m", "HC5". See ?sandwich::vcovHC
Cluster-robust: "CR", "CR0", "CR1", "CR1p", "CR1S", "CR2", "CR3". See ?clubSandwich::vcovCR
Bootstrap: "BS", "xy", "residual", "wild", "mammen", "fractional", "jackknife", "norm", "webb". See ?sandwich::vcovBS
Other sandwich package functions: "HAC", "PC", "vCL", "PL".

Classical Regression Models

Robust Covariance Matrix Estimation from Model Parameters

The simplest option, model_parameters(vcov = "HC), internally calls sandwich::vcovHC(type = "HC3"). However, vcov and vcov_args can be used for different options to calculate the heteroscedasticity-consistent covariance matrix (see ?standard_error_robust for further details).

Let us start with a simple example, which uses a heteroskedasticity-consistent covariance matrix estimation with estimation-type "HC3" (i.e. sandwich::vcovHC(type = "HC3") is called):

data(iris)
model <- lm(Petal.Length ~ Sepal.Length * Species + Sepal.Width, data = iris)

# model parameters, where SE, CI and p-values are based on robust estimation
mp <- model_parameters(model, vcov = "HC")
mp

# compare standard errors to result from sandwich-package
mp$SE
unname(sqrt(diag(sandwich::vcovHC(model))))

Cluster-Robust Covariance Matrix Estimation (sandwich)

The specific estimation type can be either passed as string to vcov, or changed with the type option that is passed to the vcov_args argument:

vcov = "HC1"
vcov = "HC", vcov_args = list(type = "HC1").

If the type is acceptes by different functions, e.g., both sandwich::vcovHC() and sandwich::vcovCL() accepts estimation types HC0 to HC3, we need the "long" form to specify all relevant options. In the next example, we use a clustered covariance matrix estimation with HC1-estimation type.

# change estimation-type
mp <- model_parameters(model, vcov = "CL", vcov_args = list(type = "HC2"))
mp

# compare standard errors to result from sandwich-package
mp$SE
unname(sqrt(diag(sandwich::vcovCL(model, type = "HC2"))))

Usually, clustered covariance matrix estimation is used when there is a cluster-structure in the data. The variable indicating the cluster-structure can be defined in sandwich::vcovCL() with the cluster-argument. As mentioned above, in model_parameters(), additional arguments that should be passed down to functions from the sandwich package can be specified in vcov_args:

iris$cluster <- factor(rep(LETTERS[1:8], length.out = nrow(iris)))
# change estimation-type, defining additional arguments
mp <- model_parameters(
  model,
  vcov = "CL",
  vcov_args = list(type = "HC2", cluster = iris$cluster)
)
mp

# compare standard errors to result from sandwich-package
mp$SE
unname(sqrt(diag(sandwich::vcovCL(model, type = "HC2", cluster = iris$cluster))))

Cluster-Robust Covariance Matrix Estimation (clubSandwich)

Cluster-robust estimation of the variance-covariance matrix can also be achieved using clubSandwich::vcovCR(). Thus, when vcov = "CR", the related function from the clubSandwich package is called. Note that this function requires the specification of the cluster-argument.

# create fake-cluster-variable, to demonstrate cluster robust standard errors
iris$cluster <- factor(rep(LETTERS[1:8], length.out = nrow(iris)))

# cluster-robust estimation
mp <- model_parameters(
  model,
  vcov = "CR",
  vcov_args = list(type = "CR1", cluster = iris$cluster)
)
mp

# compare standard errors to result from clubSsandwich-package
mp$SE
unname(sqrt(diag(clubSandwich::vcovCR(model, type = "CR1", cluster = iris$cluster))))

Robust Covariance Matrix Estimation on Standardized Model Parameters

Finally, robust estimation can be combined with standardization. However, robust covariance matrix estimation only works for standardize = "refit".

# model parameters, robust estimation on standardized model
model_parameters(model, standardize = "refit", vcov = "HC")

Mixed Models

Robust Covariance Matrix Estimation for Mixed Models

For linear mixed models, that by definition have a clustered ("hierarchical" or multilevel) structure in the data, it is also possible to estimate a cluster-robust covariance matrix. This is possible due to the clubSandwich package, thus we need to define the same arguments as in the above example.

library(lme4)
data(iris)
set.seed(1234)
iris$grp <- as.factor(sample(1:3, nrow(iris), replace = TRUE))

# fit example model
model <- lme4::lmer(
  Sepal.Length ~ Species * Sepal.Width + Petal.Length + (1 | grp),
  data = iris
)

# normal model parameters, like from 'summary()'
model_parameters(model)

# model parameters, cluster robust estimation for mixed models
model_parameters(
  model,
  vcov = "CR",
  vcov_args = list(type = "CR1", cluster = iris$grp)
)

Robust Covariance Matrix Estimation on Standardized Mixed Model Parameters

Again, robust estimation can be combined with standardization for linear mixed models as well, which in such cases also only works for standardize = "refit".

# model parameters, cluster robust estimation on standardized mixed model
model_parameters(
  model,
  standardize = "refit",
  vcov = "CR",
  vcov_args = list(type = "CR1", cluster = iris$grp)
)