# Parameters from Mixed Models

Source:`R/methods_cplm.R`

, `R/methods_glmmTMB.R`

, `R/methods_lme4.R`

, and 4 more
`model_parameters.merMod.Rd`

Parameters from (linear) mixed models.

## Usage

```
# S3 method for cpglmm
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
include_sigma = FALSE,
keep = NULL,
drop = NULL,
verbose = TRUE,
...
)
# S3 method for glmmTMB
model_parameters(
model,
ci = 0.95,
ci_method = "wald",
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
component = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
wb_component = TRUE,
summary = getOption("parameters_mixed_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
include_sigma = FALSE,
...
)
# S3 method for merMod
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
wb_component = TRUE,
summary = getOption("parameters_mixed_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
include_sigma = FALSE,
vcov = NULL,
vcov_args = NULL,
...
)
# S3 method for mixed
model_parameters(
model,
ci = 0.95,
ci_method = "wald",
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
component = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
wb_component = TRUE,
summary = getOption("parameters_mixed_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
include_sigma = FALSE,
...
)
# S3 method for MixMod
model_parameters(
model,
ci = 0.95,
ci_method = "wald",
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
component = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
wb_component = TRUE,
summary = getOption("parameters_mixed_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
include_sigma = FALSE,
...
)
# S3 method for lme
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
wb_component = TRUE,
summary = getOption("parameters_mixed_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
include_sigma = FALSE,
vcov = NULL,
vcov_args = NULL,
...
)
# S3 method for clmm2
model_parameters(
model,
ci = 0.95,
bootstrap = FALSE,
iterations = 1000,
component = c("all", "conditional", "scale"),
standardize = NULL,
exponentiate = FALSE,
p_adjust = NULL,
summary = getOption("parameters_summary", FALSE),
keep = NULL,
drop = NULL,
verbose = TRUE,
...
)
# S3 method for clmm
model_parameters(
model,
ci = 0.95,
ci_method = NULL,
ci_random = NULL,
bootstrap = FALSE,
iterations = 1000,
standardize = NULL,
effects = "all",
group_level = FALSE,
exponentiate = FALSE,
p_adjust = NULL,
include_sigma = FALSE,
keep = NULL,
drop = NULL,
verbose = TRUE,
...
)
```

## Arguments

- model
A mixed model.

- 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.- ci_random
Logical, if

`TRUE`

, includes the confidence intervals for random effects parameters. Only applies if`effects`

is not`"fixed"`

and if`ci`

is not`NULL`

. Set`ci_random = FALSE`

if computation of the model summary is too much time consuming. By default,`ci_random = NULL`

, which uses a heuristic to guess if computation of confidence intervals for random effects is fast enough or not. For models with larger sample size and/or more complex random effects structures, confidence intervals will not be computed by default, for simpler models or fewer observations, confidence intervals will be included. Set explicitly to`TRUE`

or`FALSE`

to enforce or omit calculation of confidence intervals.- bootstrap
Should estimates be based on bootstrapped model? If

`TRUE`

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

).- iterations
The number of draws to simulate/bootstrap.

- 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"`

.

- effects
Should parameters for fixed effects (

`"fixed"`

), random effects (`"random"`

), or both (`"all"`

) be returned? Only applies to mixed models. May be abbreviated. If the calculation of random effects parameters takes too long, you may use`effects = "fixed"`

.- group_level
Logical, for multilevel models (i.e. models with random effects) and when

`effects = "all"`

or`effects = "random"`

, include the parameters for each group level from random effects. If`group_level = FALSE`

(the default), only information on SD and COR are shown.- 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**).- include_sigma
Logical, if

`TRUE`

, includes the residual standard deviation. For mixed models, this is defined as the sum of the distribution-specific variance and the variance for the additive overdispersion term (see`insight::get_variance()`

for details). Defaults to`FALSE`

for mixed models due to the longer computation time.- 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.

- ...
Arguments passed to or from other methods. For instance, when

`bootstrap = TRUE`

, arguments like`type`

or`parallel`

are passed down to`bootstrap_model()`

.- component
Should all parameters, parameters for the conditional model, for the zero-inflation part of the model, or the dispersion model be returned? Applies to models with zero-inflation and/or dispersion component.

`component`

may be one of`"conditional"`

,`"zi"`

,`"zero-inflated"`

,`"dispersion"`

or`"all"`

(default). May be abbreviated.- wb_component
Logical, if

`TRUE`

and models contains within- and between-effects (see`datawizard::demean()`

), the`Component`

column will indicate which variables belong to the within-effects, between-effects, and cross-level interactions. By default, the`Component`

column indicates, which parameters belong to the conditional or zero-inflation component of the model.- summary
Logical, if

`TRUE`

, prints summary information about the model (model formula, number of observations, residual standard deviation and more).- 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.

## Note

If the calculation of random effects parameters takes too long, you may
use `effects = "fixed"`

. There is also a `plot()`

-method
implemented in the **see**-package.

## Confidence intervals for random effects variances

For models of class `merMod`

and `glmmTMB`

, confidence intervals for random
effect variances can be calculated.

For models of from package

**lme4**, when`ci_method`

is either`"profile"`

or`"boot"`

, and`effects`

is either`"random"`

or`"all"`

, profiled resp. bootstrapped confidence intervals are computed for the random effects.For all other options of

`ci_method`

, and only when the**merDeriv**package is installed, confidence intervals for random effects are based on normal-distribution approximation, using the delta-method to transform standard errors for constructing the intervals around the log-transformed SD parameters. These are than back-transformed, so that random effect variances, standard errors and confidence intervals are shown on the original scale. Due to the transformation, the intervals are asymmetrical, however, they are within the correct bounds (i.e. no negative interval for the SD, and the interval for the correlations is within the range from -1 to +1).For models of class

`glmmTMB`

, confidence intervals for random effect variances always use a Wald t-distribution approximation.

## Singular fits (random effects variances near zero)

If a model is "singular", this means that some dimensions of the variance-covariance matrix have been estimated as exactly zero. This often occurs for mixed models with complex random effects structures.

There is no gold-standard about how to deal with singularity and which
random-effects specification to choose. One way is to fully go Bayesian
(with informative priors). Other proposals are listed in the documentation
of `performance::check_singularity()`

. However, since version 1.1.9, the
**glmmTMB** package allows to use priors in a frequentist framework, too. One
recommendation is to use a Gamma prior (*Chung et al. 2013*). The mean may
vary from 1 to very large values (like `1e8`

), and the shape parameter should
be set to a value of 2.5. You can then `update()`

your model with the specified
prior. In **glmmTMB**, the code would look like this:

```
# "model" is an object of class gmmmTMB
prior <- data.frame(
prior = "gamma(1, 2.5)", # mean can be 1, but even 1e8
class = "ranef" # for random effects
)
model_with_priors <- update(model, priors = prior)
```

Large values for the mean parameter of the Gamma prior have no large impact
on the random effects variances in terms of a "bias". Thus, if `1`

doesn't
fix the singular fit, you can safely try larger values.

## Dispersion parameters in *glmmTMB*

For some models from package **glmmTMB**, both the dispersion parameter and
the residual variance from the random effects parameters are shown. Usually,
these are the same but presented on different scales, e.g.

```
model <- glmmTMB(Sepal.Width ~ Petal.Length + (1|Species), data = iris)
exp(fixef(model)$disp) # 0.09902987
sigma(model)^2 # 0.09902987
```

For models where the dispersion parameter and the residual variance are the same, only the residual variance is shown in the output.

## 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()`

.

## References

Chung Y, Rabe-Hesketh S, Dorie V, Gelman A, and Liu J. 2013. "A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models." Psychometrika 78 (4): 685–709. doi:10.1007/s11336-013-9328-2

## See also

`insight::standardize_names()`

to
rename columns into a consistent, standardized naming scheme.

## Examples

```
library(parameters)
data(mtcars)
model <- lme4::lmer(mpg ~ wt + (1 | gear), data = mtcars)
model_parameters(model)
#> # Fixed Effects
#>
#> Parameter | Coefficient | SE | 95% CI | t(28) | p
#> ------------------------------------------------------------------
#> (Intercept) | 36.19 | 2.19 | [31.70, 40.68] | 16.52 | < .001
#> wt | -5.05 | 0.64 | [-6.36, -3.73] | -7.89 | < .001
#>
#> # Random Effects
#>
#> Parameter | Coefficient | SE | 95% CI
#> --------------------------------------------------------
#> SD (Intercept: gear) | 1.26 | 1.12 | [0.22, 7.17]
#> SD (Residual) | 2.91 | 0.39 | [2.24, 3.78]
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald t-distribution approximation. Uncertainty intervals for
#> random effect variances computed using a Wald z-distribution
#> approximation.
# \donttest{
data(Salamanders, package = "glmmTMB")
model <- glmmTMB::glmmTMB(
count ~ spp + mined + (1 | site),
ziformula = ~mined,
family = poisson(),
data = Salamanders
)
model_parameters(model, effects = "all")
#> # Fixed Effects (Count Model)
#>
#> Parameter | Log-Mean | SE | 95% CI | z | p
#> ---------------------------------------------------------------
#> (Intercept) | -0.36 | 0.28 | [-0.90, 0.18] | -1.30 | 0.194
#> spp [PR] | -1.27 | 0.24 | [-1.74, -0.80] | -5.27 | < .001
#> spp [DM] | 0.27 | 0.14 | [ 0.00, 0.54] | 1.95 | 0.051
#> spp [EC-A] | -0.57 | 0.21 | [-0.97, -0.16] | -2.75 | 0.006
#> spp [EC-L] | 0.67 | 0.13 | [ 0.41, 0.92] | 5.20 | < .001
#> spp [DES-L] | 0.63 | 0.13 | [ 0.38, 0.87] | 4.96 | < .001
#> spp [DF] | 0.12 | 0.15 | [-0.17, 0.40] | 0.78 | 0.435
#> mined [no] | 1.27 | 0.27 | [ 0.74, 1.80] | 4.72 | < .001
#>
#> # Fixed Effects (Zero-Inflation Component)
#>
#> Parameter | Log-Odds | SE | 95% CI | z | p
#> ---------------------------------------------------------------
#> (Intercept) | 0.79 | 0.27 | [ 0.26, 1.32] | 2.90 | 0.004
#> mined [no] | -1.84 | 0.31 | [-2.46, -1.23] | -5.87 | < .001
#>
#> # Random Effects Variances
#>
#> Parameter | Coefficient | 95% CI
#> -------------------------------------------------
#> SD (Intercept: site) | 0.33 | [0.18, 0.63]
#>
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#> using a Wald z-distribution approximation.
model <- lme4::lmer(mpg ~ wt + (1 | gear), data = mtcars)
model_parameters(model, bootstrap = TRUE, iterations = 50, verbose = FALSE)
#> # Fixed Effects
#>
#> Parameter | Coefficient | 95% CI | p
#> ---------------------------------------------------
#> (Intercept) | 35.83 | [31.97, 40.24] | < .001
#> wt | -4.95 | [-6.04, -3.99] | < .001
# }
```