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() when robust = TRUE.

Classical Regression Models

Robust Covariance Matrix Estimation from Model Parameters

By default, when model_parameters(robust = TRUE), it internally calls sandwich::vcovHC(type = "HC3"). However, there are three arguments that allow for choosing different methods and options of robust estimation: vcov_estimation, vcov_type and vcov_args (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, robust = TRUE)
mp
#> Parameter                           | Coefficient |   SE |        95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                         |        0.87 | 0.45 | [-0.03, 1.76] |  1.91 | 143 | 0.059 
#> Sepal.Length                        |        0.04 | 0.12 | [-0.19, 0.28] |  0.37 | 143 | 0.711 
#> Species [versicolor]                |       -0.78 | 0.69 | [-2.15, 0.59] | -1.12 | 143 | 0.265 
#> Species [virginica]                 |       -0.41 | 0.63 | [-1.66, 0.83] | -0.66 | 143 | 0.513 
#> Sepal.Width                         |        0.11 | 0.08 | [-0.05, 0.27] |  1.32 | 143 | 0.190 
#> Sepal.Length * Species [versicolor] |        0.61 | 0.13 | [ 0.35, 0.87] |  4.65 | 143 | < .001
#> Sepal.Length * Species [virginica]  |        0.68 | 0.12 | [ 0.45, 0.91] |  5.75 | 143 | < .001
# compare standard errors to result from sandwich-package
mp$SE
#> [1] 0.454 0.119 0.693 0.630 0.083 0.130 0.118
unname(sqrt(diag(sandwich::vcovHC(model))))
#> [1] 0.454 0.119 0.693 0.630 0.083 0.130 0.118

Cluster-Robust Covariance Matrix Estimation (sandwich)

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

The specific estimation type can be changed with vcov_type. 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
mp <- model_parameters(model, robust = TRUE, vcov_estimation = "CL", vcov_type = "HC1")
mp
#> Parameter                           | Coefficient |   SE |        95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                         |        0.87 | 0.42 | [ 0.03, 1.70] |  2.05 | 143 | 0.042 
#> Sepal.Length                        |        0.04 | 0.11 | [-0.18, 0.26] |  0.40 | 143 | 0.692 
#> Species [versicolor]                |       -0.78 | 0.65 | [-2.07, 0.51] | -1.19 | 143 | 0.237 
#> Species [virginica]                 |       -0.41 | 0.59 | [-1.57, 0.75] | -0.70 | 143 | 0.483 
#> Sepal.Width                         |        0.11 | 0.08 | [-0.05, 0.27] |  1.38 | 143 | 0.170 
#> Sepal.Length * Species [versicolor] |        0.61 | 0.12 | [ 0.37, 0.85] |  4.96 | 143 | < .001
#> Sepal.Length * Species [virginica]  |        0.68 | 0.11 | [ 0.46, 0.90] |  6.15 | 143 | < .001
# compare standard errors to result from sandwich-package
mp$SE
#> [1] 0.422 0.111 0.653 0.587 0.079 0.123 0.111
unname(sqrt(diag(sandwich::vcovCL(model))))
#> [1] 0.422 0.111 0.653 0.587 0.079 0.123 0.111

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 model_parameters(), additional arguments that should be passed down to functions from the sandwich package can be specified in vcov_args:

iris$cluster <- factor(rep(LETTERS[1:8], length.out = nrow(iris)))
# change estimation-type, defining additional arguments
mp <- model_parameters(
  model,
  robust = TRUE,
  vcov_estimation = "CL",
  vcov_type = "HC1",
  vcov_args = list(cluster = iris$cluster)
)
mp
#> Parameter                           | Coefficient |   SE |        95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                         |        0.87 | 0.34 | [ 0.20, 1.53] |  2.57 | 143 | 0.011 
#> Sepal.Length                        |        0.04 | 0.07 | [-0.10, 0.19] |  0.61 | 143 | 0.540 
#> Species [versicolor]                |       -0.78 | 0.52 | [-1.80, 0.25] | -1.49 | 143 | 0.137 
#> Species [virginica]                 |       -0.41 | 0.26 | [-0.94, 0.11] | -1.56 | 143 | 0.120 
#> Sepal.Width                         |        0.11 | 0.07 | [-0.03, 0.25] |  1.52 | 143 | 0.131 
#> Sepal.Length * Species [versicolor] |        0.61 | 0.10 | [ 0.42, 0.80] |  6.29 | 143 | < .001
#> Sepal.Length * Species [virginica]  |        0.68 | 0.05 | [ 0.58, 0.78] | 13.28 | 143 | < .001
# compare standard errors to result from sandwich-package
mp$SE
#> [1] 0.337 0.072 0.519 0.264 0.072 0.097 0.051
unname(sqrt(diag(sandwich::vcovCL(model, cluster = iris$cluster))))
#> [1] 0.337 0.072 0.519 0.264 0.072 0.097 0.051

Cluster-Robust Covariance Matrix Estimation (clubSandwich)

Cluster-robust estimation of the variance-covariance matrix can also be achieved using clubSandwich::vcovCR(). Thus, when vcov_estimation = "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,
  robust = TRUE,
  vcov_estimation = "CR",
  vcov_type = "CR1",
  vcov_args = list(cluster = iris$cluster)
)
mp
#> Parameter                           | Coefficient |   SE |        95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                         |        0.87 | 0.33 | [ 0.21, 1.52] |  2.62 | 143 | 0.010 
#> Sepal.Length                        |        0.04 | 0.07 | [-0.10, 0.18] |  0.63 | 143 | 0.531 
#> Species [versicolor]                |       -0.78 | 0.51 | [-1.78, 0.23] | -1.53 | 143 | 0.129 
#> Species [virginica]                 |       -0.41 | 0.26 | [-0.92, 0.10] | -1.60 | 143 | 0.112 
#> Sepal.Width                         |        0.11 | 0.07 | [-0.03, 0.25] |  1.55 | 143 | 0.123 
#> Sepal.Length * Species [versicolor] |        0.61 | 0.09 | [ 0.42, 0.79] |  6.42 | 143 | < .001
#> Sepal.Length * Species [virginica]  |        0.68 | 0.05 | [ 0.58, 0.78] | 13.56 | 143 | < .001
# compare standard errors to result from clubSsandwich-package
mp$SE
#> [1] 0.330 0.070 0.508 0.259 0.071 0.095 0.050
unname(sqrt(diag(clubSandwich::vcovCR(model, type = "CR1", cluster = iris$cluster))))
#> [1] 0.330 0.070 0.508 0.259 0.071 0.095 0.050

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", robust = TRUE)
#> Parameter                           | Coefficient |   SE |         95% CI |      t |  df |      p
#> -------------------------------------------------------------------------------------------------
#> (Intercept)                         |       -1.30 | 0.07 | [-1.44, -1.16] | -18.70 | 143 | < .001
#> Sepal.Length                        |        0.02 | 0.06 | [-0.09,  0.13] |   0.37 | 143 | 0.711 
#> Species [versicolor]                |        1.57 | 0.09 | [ 1.40,  1.74] |  17.84 | 143 | < .001
#> Species [virginica]                 |        2.02 | 0.09 | [ 1.84,  2.20] |  22.49 | 143 | < .001
#> Sepal.Width                         |        0.03 | 0.02 | [-0.01,  0.07] |   1.32 | 143 | 0.190 
#> Sepal.Length * Species [versicolor] |        0.28 | 0.06 | [ 0.16,  0.41] |   4.65 | 143 | < .001
#> Sepal.Length * Species [virginica]  |        0.32 | 0.06 | [ 0.21,  0.43] |   5.75 | 143 | < .001

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)
#> Loading required package: Matrix
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)
#> Parameter                          | Coefficient |   SE |         95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                        |        1.55 | 0.40 | [ 0.77,  2.34] |  3.87 | 141 | < .001
#> Species [versicolor]               |        0.41 | 0.55 | [-0.66,  1.49] |  0.75 | 141 | 0.453 
#> Species [virginica]                |       -0.41 | 0.58 | [-1.55,  0.73] | -0.70 | 141 | 0.482 
#> Sepal.Width                        |        0.66 | 0.11 | [ 0.44,  0.88] |  5.83 | 141 | < .001
#> Petal.Length                       |        0.82 | 0.07 | [ 0.69,  0.95] | 12.52 | 141 | < .001
#> Species [versicolor] * Sepal.Width |       -0.48 | 0.19 | [-0.85, -0.12] | -2.60 | 141 | 0.009 
#> Species [virginica] * Sepal.Width  |       -0.36 | 0.18 | [-0.71, -0.01] | -1.99 | 141 | 0.046
# model parameters, cluster robust estimation for mixed models
model_parameters(
  model,
  robust = TRUE,
  vcov_estimation = "CR",
  vcov_type = "CR1",
  vcov_args = list(cluster = iris$grp)
)
#> Parameter                          | Coefficient |   SE |         95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                        |        1.55 | 0.40 | [ 0.76,  2.35] |  3.87 | 141 | < .001
#> Species [versicolor]               |        0.41 | 0.80 | [-1.17,  1.99] |  0.75 | 141 | 0.608 
#> Species [virginica]                |       -0.41 | 0.19 | [-0.78, -0.03] | -0.70 | 141 | 0.033 
#> Sepal.Width                        |        0.66 | 0.10 | [ 0.46,  0.86] |  5.83 | 141 | < .001
#> Petal.Length                       |        0.82 | 0.05 | [ 0.72,  0.91] | 12.52 | 141 | < .001
#> Species [versicolor] * Sepal.Width |       -0.48 | 0.35 | [-1.18,  0.21] | -2.60 | 141 | 0.172 
#> Species [virginica] * Sepal.Width  |       -0.36 | 0.11 | [-0.57, -0.15] | -1.99 | 141 | < .001

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",
  robust = TRUE,
  vcov_estimation = "CR",
  vcov_type = "CR1",
  vcov_args = list(cluster = iris$grp)
)
#> Parameter                          | Coefficient |   SE |         95% CI |     t |  df |      p
#> -----------------------------------------------------------------------------------------------
#> (Intercept)                        |        0.97 | 0.08 | [ 0.82,  1.12] |  4.74 | 141 | < .001
#> Species [versicolor]               |       -1.29 | 0.33 | [-1.95, -0.63] | -4.91 | 141 | < .001
#> Species [virginica]                |       -1.81 | 0.23 | [-2.26, -1.37] | -5.33 | 141 | < .001
#> Sepal.Width                        |        0.35 | 0.05 | [ 0.24,  0.45] |  5.83 | 141 | < .001
#> Petal.Length                       |        1.74 | 0.10 | [ 1.54,  1.94] | 12.52 | 141 | < .001
#> Species [versicolor] * Sepal.Width |       -0.25 | 0.19 | [-0.62,  0.11] | -2.60 | 141 | 0.172 
#> Species [virginica] * Sepal.Width  |       -0.19 | 0.06 | [-0.30, -0.08] | -1.99 | 141 | < .001