# Parameters from (General) Linear Models

Source:`R/1_model_parameters.R`

, `R/methods_censReg.R`

, `R/methods_mass.R`

`model_parameters.default.Rd`

Extract and compute indices and measures to describe parameters of (general) linear models (GLMs).

## Usage

```
# Default S3 method
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
exponentiate = FALSE,
p_adjust = NULL,
summary = getOption("parameters_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
vcov = NULL,
vcov_args = NULL,
...
)
# S3 method for class 'glm'
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
exponentiate = FALSE,
p_adjust = NULL,
summary = getOption("parameters_summary", FALSE),
keep = NULL,
drop = NULL,
vcov = NULL,
vcov_args = NULL,
verbose = TRUE,
...
)
# S3 method for class 'censReg'
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
exponentiate = FALSE,
p_adjust = NULL,
summary = getOption("parameters_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
vcov = NULL,
vcov_args = NULL,
...
)
# S3 method for class 'ridgelm'
model_parameters(model, verbose = TRUE, ...)
```

## Arguments

- model
Model object.

- ci
Confidence Interval (CI) level. Default to

`0.95`

(`95%`

).- ci_method
Method for computing degrees of freedom for confidence intervals (CI) and the related p-values. Allowed are following options (which vary depending on the model class):

`"residual"`

,`"normal"`

,`"likelihood"`

,`"satterthwaite"`

,`"kenward"`

,`"wald"`

,`"profile"`

,`"boot"`

,`"uniroot"`

,`"ml1"`

,`"betwithin"`

,`"hdi"`

,`"quantile"`

,`"ci"`

,`"eti"`

,`"si"`

,`"bci"`

, or`"bcai"`

. See section*Confidence intervals and approximation of degrees of freedom*in`model_parameters()`

for further details. When`ci_method=NULL`

, in most cases`"wald"`

is used then.- bootstrap
Should estimates be based on bootstrapped model? If

`TRUE`

, then arguments of Bayesian regressions apply (see also`bootstrap_parameters()`

).- iterations
The number of bootstrap replicates. This only apply in the case of bootstrapped frequentist models.

- standardize
The method used for standardizing the parameters. Can be

`NULL`

(default; no standardization),`"refit"`

(for re-fitting the model on standardized data) or one of`"basic"`

,`"posthoc"`

,`"smart"`

,`"pseudo"`

. See 'Details' in`standardize_parameters()`

.**Importantly**:The

`"refit"`

method does*not*standardize categorical predictors (i.e. factors), which may be a different behaviour compared to other R packages (such as**lm.beta**) or other software packages (like SPSS). to mimic such behaviours, either use`standardize="basic"`

or standardize the data with`datawizard::standardize(force=TRUE)`

*before*fitting the model.For mixed models, when using methods other than

`"refit"`

, only the fixed effects will be standardized.Robust estimation (i.e.,

`vcov`

set to a value other than`NULL`

) of standardized parameters only works when`standardize="refit"`

.

- exponentiate
Logical, indicating whether or not to exponentiate the coefficients (and related confidence intervals). This is typical for logistic regression, or more generally speaking, for models with log or logit links. It is also recommended to use

`exponentiate = TRUE`

for models with log-transformed response values.**Note:**Delta-method standard errors are also computed (by multiplying the standard errors by the transformed coefficients). This is to mimic behaviour of other software packages, such as Stata, but these standard errors poorly estimate uncertainty for the transformed coefficient. The transformed confidence interval more clearly captures this uncertainty. For`compare_parameters()`

,`exponentiate = "nongaussian"`

will only exponentiate coefficients from non-Gaussian families.- p_adjust
Character vector, if not

`NULL`

, indicates the method to adjust p-values. See`stats::p.adjust()`

for details. Further possible adjustment methods are`"tukey"`

,`"scheffe"`

,`"sidak"`

and`"none"`

to explicitly disable adjustment for`emmGrid`

objects (from**emmeans**).- summary
Logical, if

`TRUE`

, prints summary information about the model (model formula, number of observations, residual standard deviation and more).- keep
Character containing a regular expression pattern that describes the parameters that should be included (for

`keep`

) or excluded (for`drop`

) in the returned data frame.`keep`

may also be a named list of regular expressions. All non-matching parameters will be removed from the output. If`keep`

is a character vector, every parameter name in the*"Parameter"*column that matches the regular expression in`keep`

will be selected from the returned data frame (and vice versa, all parameter names matching`drop`

will be excluded). Furthermore, if`keep`

has more than one element, these will be merged with an`OR`

operator into a regular expression pattern like this:`"(one|two|three)"`

. If`keep`

is a named list of regular expression patterns, the names of the list-element should equal the column name where selection should be applied. This is useful for model objects where`model_parameters()`

returns multiple columns with parameter components, like in`model_parameters.lavaan()`

. Note that the regular expression pattern should match the parameter names as they are stored in the returned data frame, which can be different from how they are printed. Inspect the`$Parameter`

column of the parameters table to get the exact parameter names.- drop
See

`keep`

.- verbose
Toggle warnings and messages.

- vcov
Variance-covariance matrix used to compute uncertainty estimates (e.g., for robust standard errors). This argument accepts a covariance matrix, a function which returns a covariance matrix, or a string which identifies the function to be used to compute the covariance matrix.

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:

`"vcovHC"`

,`"HC"`

,`"HC0"`

,`"HC1"`

,`"HC2"`

,`"HC3"`

,`"HC4"`

,`"HC4m"`

,`"HC5"`

. See`?sandwich::vcovHC`

.Cluster-robust:

`"vcovCR"`

,`"CR0"`

,`"CR1"`

,`"CR1p"`

,`"CR1S"`

,`"CR2"`

,`"CR3"`

. See`?clubSandwich::vcovCR`

.Bootstrap:

`"vcovBS"`

,`"xy"`

,`"residual"`

,`"wild"`

,`"mammen"`

,`"webb"`

. See`?sandwich::vcovBS`

.Other

`sandwich`

package functions:`"vcovHAC"`

,`"vcovPC"`

,`"vcovCL"`

,`"vcovPL"`

.

- vcov_args
List of arguments to be passed to the function identified by the

`vcov`

argument. This function is typically supplied by the**sandwich**or**clubSandwich**packages. Please refer to their documentation (e.g.,`?sandwich::vcovHAC`

) to see the list of available arguments.- ...
Arguments passed to or from other methods. For instance, when

`bootstrap = TRUE`

, arguments like`type`

or`parallel`

are passed down to`bootstrap_model()`

.

## Confidence intervals and approximation of degrees of freedom

There are different ways of approximating the degrees of freedom depending
on different assumptions about the nature of the model and its sampling
distribution. The `ci_method`

argument modulates the method for computing degrees
of freedom (df) that are used to calculate confidence intervals (CI) and the
related p-values. Following options are allowed, depending on the model
class:

**Classical methods:**

Classical inference is generally based on the **Wald method**.
The Wald approach to inference computes a test statistic by dividing the
parameter estimate by its standard error (Coefficient / SE),
then comparing this statistic against a t- or normal distribution.
This approach can be used to compute CIs and p-values.

`"wald"`

:

Applies to

*non-Bayesian models*. For*linear models*, CIs computed using the Wald method (SE and a*t-distribution with residual df*); p-values computed using the Wald method with a*t-distribution with residual df*. For other models, CIs computed using the Wald method (SE and a*normal distribution*); p-values computed using the Wald method with a*normal distribution*.

`"normal"`

Applies to

*non-Bayesian models*. Compute Wald CIs and p-values, but always use a normal distribution.

`"residual"`

Applies to

*non-Bayesian models*. Compute Wald CIs and p-values, but always use a*t-distribution with residual df*when possible. If the residual df for a model cannot be determined, a normal distribution is used instead.

**Methods for mixed models:**

Compared to fixed effects (or single-level) models, determining appropriate df for Wald-based inference in mixed models is more difficult. See the R GLMM FAQ for a discussion.

Several approximate methods for computing df are available, but you should
also consider instead using profile likelihood (`"profile"`

) or bootstrap ("`boot"`

)
CIs and p-values instead.

`"satterthwaite"`

Applies to

*linear mixed models*. CIs computed using the Wald method (SE and a*t-distribution with Satterthwaite df*); p-values computed using the Wald method with a*t-distribution with Satterthwaite df*.

`"kenward"`

Applies to

*linear mixed models*. CIs computed using the Wald method (*Kenward-Roger SE*and a*t-distribution with Kenward-Roger df*); p-values computed using the Wald method with*Kenward-Roger SE and t-distribution with Kenward-Roger df*.

`"ml1"`

Applies to

*linear mixed models*. CIs computed using the Wald method (SE and a*t-distribution with m-l-1 approximated df*); p-values computed using the Wald method with a*t-distribution with m-l-1 approximated df*. See`ci_ml1()`

.

`"betwithin"`

Applies to

*linear mixed models*and*generalized linear mixed models*. CIs computed using the Wald method (SE and a*t-distribution with between-within df*); p-values computed using the Wald method with a*t-distribution with between-within df*. See`ci_betwithin()`

.

**Likelihood-based methods:**

Likelihood-based inference is based on comparing the likelihood for the maximum-likelihood estimate to the the likelihood for models with one or more parameter values changed (e.g., set to zero or a range of alternative values). Likelihood ratios for the maximum-likelihood and alternative models are compared to a \(\chi\)-squared distribution to compute CIs and p-values.

`"profile"`

Applies to

*non-Bayesian models*of class`glm`

,`polr`

,`merMod`

or`glmmTMB`

. CIs computed by*profiling the likelihood curve for a parameter*, using linear interpolation to find where likelihood ratio equals a critical value; p-values computed using the Wald method with a*normal-distribution*(note: this might change in a future update!)

`"uniroot"`

Applies to

*non-Bayesian models*of class`glmmTMB`

. CIs computed by*profiling the likelihood curve for a parameter*, using root finding to find where likelihood ratio equals a critical value; p-values computed using the Wald method with a*normal-distribution*(note: this might change in a future update!)

**Methods for bootstrapped or Bayesian models:**

Bootstrap-based inference is based on **resampling** and refitting the model
to the resampled datasets. The distribution of parameter estimates across
resampled datasets is used to approximate the parameter's sampling
distribution. Depending on the type of model, several different methods for
bootstrapping and constructing CIs and p-values from the bootstrap
distribution are available.

For Bayesian models, inference is based on drawing samples from the model posterior distribution.

`"quantile"`

(or `"eti"`

)

Applies to

*all models (including Bayesian models)*. For non-Bayesian models, only applies if`bootstrap = TRUE`

. CIs computed as*equal tailed intervals*using the quantiles of the bootstrap or posterior samples; p-values are based on the*probability of direction*. See`bayestestR::eti()`

.

`"hdi"`

Applies to

*all models (including Bayesian models)*. For non-Bayesian models, only applies if`bootstrap = TRUE`

. CIs computed as*highest density intervals*for the bootstrap or posterior samples; p-values are based on the*probability of direction*. See`bayestestR::hdi()`

.

`"bci"`

(or `"bcai"`

)

Applies to

*all models (including Bayesian models)*. For non-Bayesian models, only applies if`bootstrap = TRUE`

. CIs computed as*bias corrected and accelerated intervals*for the bootstrap or posterior samples; p-values are based on the*probability of direction*. See`bayestestR::bci()`

.

`"si"`

Applies to

*Bayesian models*with proper priors. CIs computed as*support intervals*comparing the posterior samples against the prior samples; p-values are based on the*probability of direction*. See`bayestestR::si()`

.

`"boot"`

Applies to

*non-Bayesian models*of class`merMod`

. CIs computed using*parametric bootstrapping*(simulating data from the fitted model); p-values computed using the Wald method with a*normal-distribution)*(note: this might change in a future update!).

For all iteration-based methods other than `"boot"`

(`"hdi"`

, `"quantile"`

, `"ci"`

, `"eti"`

, `"si"`

, `"bci"`

, `"bcai"`

),
p-values are based on the probability of direction (`bayestestR::p_direction()`

),
which is converted into a p-value using `bayestestR::pd_to_p()`

.

## See also

`insight::standardize_names()`

to
rename columns into a consistent, standardized naming scheme.

## Examples

```
library(parameters)
model <- lm(mpg ~ wt + cyl, data = mtcars)
model_parameters(model)
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> ------------------------------------------------------------------
#> (Intercept) | 39.69 | 1.71 | [36.18, 43.19] | 23.14 | < .001
#> wt | -3.19 | 0.76 | [-4.74, -1.64] | -4.22 | < .001
#> cyl | -1.51 | 0.41 | [-2.36, -0.66] | -3.64 | 0.001
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
# bootstrapped parameters
model_parameters(model, bootstrap = TRUE)
#> Parameter | Coefficient | 95% CI | p
#> ---------------------------------------------------
#> (Intercept) | 39.64 | [35.53, 43.88] | < .001
#> wt | -3.24 | [-4.82, -1.98] | < .001
#> cyl | -1.47 | [-2.15, -0.73] | < .001
#>
#> Uncertainty intervals (equal-tailed) are naıve bootstrap intervals.
# standardized parameters
model_parameters(model, standardize = "refit")
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> ---------------------------------------------------------------------
#> (Intercept) | 5.37e-17 | 0.08 | [-0.15, 0.15] | 7.13e-16 | > .999
#> wt | -0.52 | 0.12 | [-0.77, -0.27] | -4.22 | < .001
#> cyl | -0.45 | 0.12 | [-0.70, -0.20] | -3.64 | 0.001
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
# robust, heteroskedasticity-consistent standard errors
model_parameters(model, vcov = "HC3")
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> ------------------------------------------------------------------
#> (Intercept) | 39.69 | 2.30 | [34.97, 44.40] | 17.22 | < .001
#> wt | -3.19 | 0.78 | [-4.78, -1.60] | -4.10 | < .001
#> cyl | -1.51 | 0.39 | [-2.30, -0.72] | -3.90 | < .001
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
model_parameters(model,
vcov = "vcovCL",
vcov_args = list(cluster = mtcars$cyl)
)
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> ------------------------------------------------------------------
#> (Intercept) | 39.69 | 1.50 | [36.61, 42.76] | 26.43 | < .001
#> wt | -3.19 | 1.20 | [-5.65, -0.73] | -2.65 | 0.013
#> cyl | -1.51 | 0.40 | [-2.32, -0.70] | -3.82 | < .001
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
# different p-value style in output
model_parameters(model, p_digits = 5)
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> -----------------------------------------------------------------------
#> (Intercept) | 39.69 | 1.71 | [36.18, 43.19] | 23.14 | 3.04318e-20
#> wt | -3.19 | 0.76 | [-4.74, -1.64] | -4.22 | 0.00022
#> cyl | -1.51 | 0.41 | [-2.36, -0.66] | -3.64 | 0.00106
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
model_parameters(model, digits = 3, ci_digits = 4, p_digits = "scientific")
#> Parameter | Coefficient | SE | 95% CI | t(29) | p
#> -----------------------------------------------------------------------------
#> (Intercept) | 39.686 | 1.715 | [36.1787, 43.1938] | 23.141 | 3.04318e-20
#> wt | -3.191 | 0.757 | [-4.7390, -1.6429] | -4.216 | 2.22020e-04
#> cyl | -1.508 | 0.415 | [-2.3559, -0.6597] | -3.636 | 1.06428e-03
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation.
# \donttest{
# logistic regression model
model <- glm(vs ~ wt + cyl, data = mtcars, family = "binomial")
model_parameters(model)
#> Parameter | Log-Odds | SE | 95% CI | z | p
#> --------------------------------------------------------------
#> (Intercept) | 10.62 | 4.17 | [ 4.79, 22.66] | 2.55 | 0.011
#> wt | 2.10 | 1.55 | [-0.53, 6.24] | 1.36 | 0.174
#> cyl | -2.93 | 1.38 | [-6.92, -1.07] | -2.12 | 0.034
#>
#> Uncertainty intervals (profile-likelihood) and p-values (two-tailed)
#> computed using a Wald z-distribution approximation.
# show odds ratio / exponentiated coefficients
model_parameters(model, exponentiate = TRUE)
#> Parameter | Odds Ratio | SE | 95% CI | z | p
#> ------------------------------------------------------------------------
#> (Intercept) | 40911.34 | 1.71e+05 | [120.16, 6.95e+09] | 2.55 | 0.011
#> wt | 8.17 | 12.63 | [ 0.59, 514.10] | 1.36 | 0.174
#> cyl | 0.05 | 0.07 | [ 0.00, 0.34] | -2.12 | 0.034
#>
#> Uncertainty intervals (profile-likelihood) and p-values (two-tailed)
#> computed using a Wald z-distribution approximation.
# bias-corrected logistic regression with penalized maximum likelihood
model <- glm(
vs ~ wt + cyl,
data = mtcars,
family = "binomial",
method = "brglmFit"
)
model_parameters(model)
#> Parameter | Log-Odds | SE | 95% CI | z | p
#> --------------------------------------------------------------
#> (Intercept) | 7.71 | 2.66 | [ 2.49, 12.93] | 2.89 | 0.004
#> wt | 1.46 | 1.08 | [-0.65, 3.57] | 1.35 | 0.176
#> cyl | -2.09 | 0.85 | [-3.76, -0.41] | -2.44 | 0.015
#>
#> Uncertainty intervals (profile-likelihood) and p-values (two-tailed)
#> computed using a Wald z-distribution approximation.
# }
```