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Showing content from https://cran.r-project.org/web/packages/rms/../cluster/../sjPlot/vignettes/tab_model_robust.html below:

Robust Estimation of Standard Errors, Confidence Intervals and p-values

The tab_model() 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 tab_model().

Classical Regression Models Robust Covariance Matrix Estimation from Model Parameters

There are two arguments that allow for choosing different methods and options of robust estimation: vcov.fun and vcov.args. 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
tab_model(model, vcov.fun = "HC3", show.se = TRUE)
  Petal Length Predictors Estimates std. Error CI p (Intercept) 0.87 0.45 -0.03 – 1.76 0.059 Sepal Length 0.04 0.12 -0.19 – 0.28 0.711 Species [versicolor] -0.78 0.69 -2.15 – 0.59 0.265 Species [virginica] -0.41 0.63 -1.66 – 0.83 0.513 Sepal Width 0.11 0.08 -0.05 – 0.27 0.190 Sepal Length × Species
[versicolor] 0.61 0.13 0.35 – 0.87 <0.001 Sepal Length × Species
[virginica] 0.68 0.12 0.45 – 0.91 <0.001 Observations 150 R2 / R2 adjusted 0.979 / 0.978 

# compare standard errors to result from sandwich-package
unname(sqrt(diag(sandwich::vcovHC(model))))
#> [1] 0.45382603 0.11884474 0.69296611 0.63031982 0.08318559 0.13045539 0.11841325
Cluster-Robust Covariance Matrix Estimation (sandwich)

If another covariance matrix estimation is required, use the vcov.fun-argument. This argument needs the suffix for the related vcov*()-functions as value, i.e. vcov.fun = "CL" would call sandwich::vcovCL(), or vcov.fun = "HAC" would call sandwich::vcovHAC().

The specific estimation type can be changed with vcov.args. E.g., sandwich::vcovCL() accepts estimation types HC0 to HC3. In the next example, we use a clustered covariance matrix estimation with HC1-estimation type.

# change estimation-type
tab_model(model, vcov.fun = "CL", vcov.args = list(type = "HC1"), show.se = TRUE)
  Petal Length Predictors Estimates std. Error CI p (Intercept) 0.87 0.42 0.03 – 1.70 0.042 Sepal Length 0.04 0.11 -0.18 – 0.26 0.692 Species [versicolor] -0.78 0.65 -2.07 – 0.51 0.237 Species [virginica] -0.41 0.59 -1.57 – 0.75 0.483 Sepal Width 0.11 0.08 -0.05 – 0.27 0.170 Sepal Length × Species
[versicolor] 0.61 0.12 0.37 – 0.85 <0.001 Sepal Length × Species
[virginica] 0.68 0.11 0.46 – 0.90 <0.001 Observations 150 R2 / R2 adjusted 0.979 / 0.978 

# compare standard errors to result from sandwich-package
unname(sqrt(diag(sandwich::vcovCL(model))))
#> [1] 0.42197635 0.11148130 0.65274212 0.58720711 0.07934029 0.12251570 0.11058144

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. In tab_model(), additional arguments that should be passed down to functions from the sandwich package can be specified in vcov.args:

iris$cluster <- factor(rep(LETTERS[1:8], length.out = nrow(iris)))
# change estimation-type, defining additional arguments
tab_model(
  model, 
  vcov.fun = "CL", 
  vcov.args = list(type = "HC1", cluster = iris$cluster),
  show.se = TRUE
)
  Petal Length Predictors Estimates std. Error CI p (Intercept) 0.87 0.34 0.20 – 1.53 0.011 Sepal Length 0.04 0.07 -0.10 – 0.19 0.540 Species [versicolor] -0.78 0.52 -1.80 – 0.25 0.137 Species [virginica] -0.41 0.26 -0.94 – 0.11 0.120 Sepal Width 0.11 0.07 -0.03 – 0.25 0.131 Sepal Length × Species
[versicolor] 0.61 0.10 0.42 – 0.80 <0.001 Sepal Length × Species
[virginica] 0.68 0.05 0.58 – 0.78 <0.001 Observations 150 R2 / R2 adjusted 0.979 / 0.978 

# compare standard errors to result from sandwich-package
unname(sqrt(diag(sandwich::vcovCL(model, cluster = iris$cluster))))
#> [1] 0.33714287 0.07192334 0.51893777 0.26415406 0.07201145 0.09661348 0.05123446
Cluster-Robust Covariance Matrix Estimation (clubSandwich)

Cluster-robust estimation of the variance-covariance matrix can also be achieved using clubSandwich::vcovCR(). Thus, when vcov.fun = "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
tab_model(
  model, 
  vcov.fun = "CR1", 
  vcov.args = list(cluster = iris$cluster),
  show.se = TRUE
)
  Petal Length Predictors Estimates std. Error CI p (Intercept) 0.87 0.33 0.21 – 1.52 0.010 Sepal Length 0.04 0.07 -0.10 – 0.18 0.531 Species [versicolor] -0.78 0.51 -1.78 – 0.23 0.129 Species [virginica] -0.41 0.26 -0.92 – 0.10 0.112 Sepal Width 0.11 0.07 -0.03 – 0.25 0.123 Sepal Length × Species
[versicolor] 0.61 0.09 0.42 – 0.79 <0.001 Sepal Length × Species
[virginica] 0.68 0.05 0.58 – 0.78 <0.001 Observations 150 R2 / R2 adjusted 0.979 / 0.978 

# compare standard errors to result from clubSsandwich-package
unname(sqrt(diag(clubSandwich::vcovCR(model, type = "CR1", cluster = iris$cluster))))
#> [1] 0.33028501 0.07046034 0.50838200 0.25878087 0.07054666 0.09464825 0.05019229
Robust Covariance Matrix Estimation on Standardized Model Parameters

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

# model parameters, robust estimation on standardized model
tab_model(
  model, 
  show.std = "std",
  vcov.fun = "HC"
)
  Petal Length Predictors Estimates std. Beta CI standardized CI p std. p (Intercept) 0.87 -1.30 -0.03 – 1.76 -1.44 – -1.16 0.059 <0.001 Sepal Length 0.04 0.02 -0.19 – 0.28 -0.09 – 0.13 0.711 0.711 Species [versicolor] -0.78 1.57 -2.15 – 0.59 1.40 – 1.74 0.265 <0.001 Species [virginica] -0.41 2.02 -1.66 – 0.83 1.84 – 2.20 0.513 <0.001 Sepal Width 0.11 0.03 -0.05 – 0.27 -0.01 – 0.07 0.190 0.190 Sepal Length × Species
[versicolor] 0.61 0.28 0.35 – 0.87 0.16 – 0.41 <0.001 <0.001 Sepal Length × Species
[virginica] 0.68 0.32 0.45 – 0.91 0.21 – 0.43 <0.001 <0.001 Observations 150 R2 / R2 adjusted 0.979 / 0.978 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()'
tab_model(model)
  Sepal Length Predictors Estimates CI p (Intercept) 1.55 0.76 – 2.35 <0.001 Species [versicolor] 0.41 -0.67 – 1.50 0.454 Species [virginica] -0.41 -1.56 – 0.74 0.483 Sepal Width 0.66 0.44 – 0.89 <0.001 Petal Length 0.82 0.69 – 0.95 <0.001 Species [versicolor] ×
Sepal Width -0.48 -0.85 – -0.12 0.010 Species [virginica] ×
Sepal Width -0.36 -0.71 – -0.00 0.048 Random Effects σ2 0.09 τ00 grp 0.01 ICC 0.07 N grp 3 Observations 150 Marginal R2 / Conditional R2 0.860 / 0.870 

# model parameters, cluster robust estimation for mixed models
tab_model(
  model, 
  vcov.fun = "CR1", 
  vcov.args = list(cluster = iris$grp)
)
  Sepal Length Predictors Estimates CI p (Intercept) 1.55 0.76 – 2.35 <0.001 Species [versicolor] 0.41 -1.17 – 1.99 0.608 Species [virginica] -0.41 -0.78 – -0.03 0.033 Sepal Width 0.66 0.46 – 0.86 <0.001 Petal Length 0.82 0.72 – 0.91 <0.001 Species [versicolor] ×
Sepal Width -0.48 -1.18 – 0.21 0.172 Species [virginica] ×
Sepal Width -0.36 -0.57 – -0.15 0.001 Random Effects σ2 0.09 τ00 grp 0.01 ICC 0.07 N grp 3 Observations 150 Marginal R2 / Conditional R2 0.860 / 0.870 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 show.std = "std".

# model parameters, cluster robust estimation on standardized mixed model
tab_model(
  model, 
  show.std = "std",
  vcov.fun = "CR1", 
  vcov.args = list(cluster = iris$grp)
)
  Sepal Length Predictors Estimates std. Beta CI standardized CI p std. p (Intercept) 1.55 0.97 0.76 – 2.35 0.82 – 1.12 <0.001 <0.001 Species [versicolor] 0.41 -1.29 -1.17 – 1.99 -1.95 – -0.63 0.608 <0.001 Species [virginica] -0.41 -1.81 -0.78 – -0.03 -2.26 – -1.37 0.033 <0.001 Sepal Width 0.66 0.35 0.46 – 0.86 0.24 – 0.45 <0.001 <0.001 Petal Length 0.82 1.74 0.72 – 0.91 1.54 – 1.94 <0.001 <0.001 Species [versicolor] ×
Sepal Width -0.48 -0.25 -1.18 – 0.21 -0.62 – 0.11 0.172 0.172 Species [virginica] ×
Sepal Width -0.36 -0.19 -0.57 – -0.15 -0.30 – -0.08 0.001 0.001 Random Effects σ2 0.09 τ00 grp 0.01 ICC 0.07 N grp 3 Observations 150 Marginal R2 / Conditional R2 0.860 / 0.870

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