This function extracts the different variance components of a mixed model and
returns the result as list. Functions like `get_variance_residual(x)`

or
`get_variance_fixed(x)`

are shortcuts for `get_variance(x, component = "residual")`

etc.

## Usage

```
get_variance(x, ...)
# S3 method for class 'merMod'
get_variance(
x,
component = c("all", "fixed", "random", "residual", "distribution", "dispersion",
"intercept", "slope", "rho01", "rho00"),
tolerance = 1e-08,
null_model = NULL,
approximation = "lognormal",
verbose = TRUE,
...
)
# S3 method for class 'glmmTMB'
get_variance(
x,
component = c("all", "fixed", "random", "residual", "distribution", "dispersion",
"intercept", "slope", "rho01", "rho00"),
model_component = NULL,
tolerance = 1e-08,
null_model = NULL,
approximation = "lognormal",
verbose = TRUE,
...
)
get_variance_residual(x, verbose = TRUE, ...)
get_variance_fixed(x, verbose = TRUE, ...)
get_variance_random(x, verbose = TRUE, tolerance = 1e-08, ...)
get_variance_distribution(x, verbose = TRUE, ...)
get_variance_dispersion(x, verbose = TRUE, ...)
get_variance_intercept(x, verbose = TRUE, ...)
get_variance_slope(x, verbose = TRUE, ...)
get_correlation_slope_intercept(x, verbose = TRUE, ...)
get_correlation_slopes(x, verbose = TRUE, ...)
```

## Arguments

- x
A mixed effects model.

- ...
Currently not used.

- component
Character value, indicating the variance component that should be returned. By default, all variance components are returned. The distribution-specific (

`"distribution"`

) and residual (`"residual"`

) variance are the most computational intensive components, and hence may take a few seconds to calculate.- tolerance
Tolerance for singularity check of random effects, to decide whether to compute random effect variances or not. Indicates up to which value the convergence result is accepted. The larger tolerance is, the stricter the test will be. See

`performance::check_singularity()`

.- null_model
Optional, a null-model to be used for the calculation of random effect variances. If

`NULL`

, the null-model is computed internally.- approximation
Character string, indicating the approximation method for the distribution-specific (observation level, or residual) variance. Only applies to non-Gaussian models. Can be

`"lognormal"`

(default),`"delta"`

or`"trigamma"`

. For binomial models, the default is the*theoretical*distribution specific variance, however, it can also be`"observation_level"`

. See*Nakagawa et al. 2017*, in particular supplement 2, for details.- verbose
Toggle off warnings.

- model_component
For models that can have a zero-inflation component, specify for which component variances should be returned. If

`NULL`

or`"full"`

(the default), both the conditional and the zero-inflation component are taken into account. If`"conditional"`

, only the conditional component is considered.

## Value

A list with following elements:

`var.fixed`

, variance attributable to the fixed effects`var.random`

, (mean) variance of random effects`var.residual`

, residual variance (sum of dispersion and distribution-specific/observation level variance)`var.distribution`

, distribution-specific (or observation level) variance`var.dispersion`

, variance due to additive dispersion`var.intercept`

, the random-intercept-variance, or between-subject-variance (τ_{00})`var.slope`

, the random-slope-variance (τ_{11})`cor.slope_intercept`

, the random-slope-intercept-correlation (ρ_{01})`cor.slopes`

, the correlation between random slopes (ρ_{00})

## Details

This function returns different variance components from mixed models, which are needed, for instance, to calculate r-squared measures or the intraclass-correlation coefficient (ICC).

## Fixed effects variance

The fixed effects variance, σ^{2}_{f},
is the variance of the matrix-multiplication β∗X
(parameter vector by model matrix).

## Random effects variance

The random effect variance, σ^{2}_{i},
represents the *mean* random effect variance of the model. Since this
variance reflects the "average" random effects variance for mixed models, it
is also appropriate for models with more complex random effects structures,
like random slopes or nested random effects. Details can be found in
*Johnson 2014*, in particular equation 10. For simple random-intercept models,
the random effects variance equals the random-intercept variance.

## Distribution-specific (observation level) variance

The distribution-specific variance,
σ^{2}_{d},
is the conditional variance of the response given the predictors , `Var[y|x]`

,
which depends on the model family.

**Gaussian:**For Gaussian models, it is σ^{2}(i.e.`sigma(model)^2`

).**Bernoulli:**For models with binary outcome, it is π^{2}/3 for logit-link,`1`

for probit-link, and π^{2}/6 for cloglog-links.**Binomial:**For other binomial models, the distribution-specific variance for Bernoulli models is used, divided by a weighting factor based on the number of trials and successes.**Gamma:**Models from Gamma-families use μ^{2}(as obtained from`family$variance()`

).For all other models, the distribution-specific variance is by default based on lognormal approximation, log(1 + var(x) / μ

^{2}) (see*Nakagawa et al. 2017*). Other approximation methods can be specified with the`approximation`

argument.**Zero-inflation models:**The expected variance of a zero-inflated model is computed according to*Zuur et al. 2012, p277*.

## Variance for the additive overdispersion term

The variance for the additive overdispersion term,
σ^{2}_{e},
represents "the excess variation relative to what is expected from a certain
distribution" (*Nakagawa et al. 2017*). In (most? many?) cases, this will be
`0`

.

## Random intercept variance

The random intercept variance, or *between-subject* variance
(τ_{00}), is obtained from
`VarCorr()`

. It indicates how much groups or subjects differ from each other,
while the residual variance σ^{2}_{ε}
indicates the *within-subject variance*.

## Random slope variance

The random slope variance (τ_{11})
is obtained from `VarCorr()`

. This measure is only available for mixed models
with random slopes.

## Random slope-intercept correlation

The random slope-intercept correlation
(ρ_{01}) is obtained from
`VarCorr()`

. This measure is only available for mixed models with random
intercepts and slopes.

## Supported models and model families

This function supports models of class `merMod`

(including models from
**blme**), `clmm`

, `cpglmm`

, `glmmadmb`

, `glmmTMB`

, `MixMod`

, `lme`

, `mixed`

,
`rlmerMod`

, `stanreg`

, `brmsfit`

or `wbm`

. Support for objects of class
`MixMod`

(**GLMMadaptive**), `lme`

(**nlme**) or `brmsfit`

(**brms**) is
not fully implemented or tested, and therefore may not work for all models
of the aforementioned classes.

The results are validated against the solutions provided by *Nakagawa et al. (2017)*,
in particular examples shown in the Supplement 2 of the paper. Other model
families are validated against results from the **MuMIn** package. This means
that the returned variance components should be accurate and reliable for
following mixed models or model families:

Bernoulli (logistic) regression

Binomial regression (with other than binary outcomes)

Poisson and Quasi-Poisson regression

Negative binomial regression (including nbinom1 and nbinom2 families)

Gaussian regression (linear models)

Gamma regression

Tweedie regression

Beta regression

Ordered beta regression

Following model families are not yet validated, but should work:

Zero-inflated and hurdle models

Beta-binomial regression

Compound Poisson regression

Generalized Poisson regression

Log-normal regression

Extracting variance components for models with zero-inflation part is not straightforward, because it is not definitely clear how the distribution-specific variance should be calculated. Therefore, it is recommended to carefully inspect the results, and probably validate against other models, e.g. Bayesian models (although results may be only roughly comparable).

Log-normal regressions (e.g. `lognormal()`

family in **glmmTMB** or
`gaussian("log")`

) often have a very low fixed effects variance (if they were
calculated as suggested by *Nakagawa et al. 2017*). This results in very low
ICC or r-squared values, which may not be meaningful (see
`performance::icc()`

or `performance::r2_nakagawa()`

).

## References

Johnson, P. C. D. (2014). Extension of Nakagawa & Schielzeth’s R2 GLMM to random slopes models. Methods in Ecology and Evolution, 5(9), 944–946. doi:10.1111/2041-210X.12225

Nakagawa, S., Johnson, P. C. D., & Schielzeth, H. (2017). The coefficient of determination R2 and intra-class correlation coefficient from generalized linear mixed-effects models revisited and expanded. Journal of The Royal Society Interface, 14(134), 20170213. doi:10.1098/rsif.2017.0213

Zuur, A. F., Savel'ev, A. A., & Ieno, E. N. (2012). Zero inflated models and generalized linear mixed models with R. Newburgh, United Kingdom: Highland Statistics.

## Examples

```
# \donttest{
library(lme4)
data(sleepstudy)
m <- lmer(Reaction ~ Days + (1 + Days | Subject), data = sleepstudy)
get_variance(m)
#> $var.fixed
#> [1] 908.9534
#>
#> $var.random
#> [1] 1698.084
#>
#> $var.residual
#> [1] 654.94
#>
#> $var.distribution
#> [1] 654.94
#>
#> $var.dispersion
#> [1] 0
#>
#> $var.intercept
#> Subject
#> 612.1002
#>
#> $var.slope
#> Subject.Days
#> 35.07171
#>
#> $cor.slope_intercept
#> Subject
#> 0.06555124
#>
get_variance_fixed(m)
#> var.fixed
#> 908.9534
get_variance_residual(m)
#> var.residual
#> 654.94
# }
```