Compute the *marginal* and *conditional* r-squared value for
mixed effects models with complex random effects structures.

## Usage

```
r2_nakagawa(
model,
by_group = FALSE,
tolerance = 1e-08,
ci = NULL,
iterations = 100,
ci_method = NULL,
null_model = NULL,
approximation = "lognormal",
model_component = NULL,
verbose = TRUE,
...
)
```

## Arguments

- model
A mixed effects model.

- by_group
Logical, if

`TRUE`

, returns the explained variance at different levels (if there are multiple levels). This is essentially similar to the variance reduction approach by*Hox (2010), pp. 69-78*.- tolerance
Tolerance for singularity check of random effects, to decide whether to compute random effect variances for the conditional r-squared or not. Indicates up to which value the convergence result is accepted. When

`r2_nakagawa()`

returns a warning, stating that random effect variances can't be computed (and thus, the conditional r-squared is`NA`

), decrease the tolerance-level. See also`check_singularity()`

.- ci
Confidence resp. credible interval level. For

`icc()`

,`r2()`

, and`rmse()`

, confidence intervals are based on bootstrapped samples from the ICC, R2 or RMSE value. See`iterations`

.- iterations
Number of bootstrap-replicates when computing confidence intervals for the ICC, R2, RMSE etc.

- ci_method
Character string, indicating the bootstrap-method. Should be

`NULL`

(default), in which case`lme4::bootMer()`

is used for bootstrapped confidence intervals. However, if bootstrapped intervals cannot be calculated this way, try`ci_method = "boot"`

, which falls back to`boot::boot()`

. This may successfully return bootstrapped confidence intervals, but bootstrapped samples may not be appropriate for the multilevel structure of the model. There is also an option`ci_method = "analytical"`

, which tries to calculate analytical confidence assuming a chi-squared distribution. However, these intervals are rather inaccurate and often too narrow. It is recommended to calculate bootstrapped confidence intervals for mixed models.- null_model
Optional, a null model to compute the random effect variances, which is passed to

`insight::get_variance()`

. Usually only required if calculation of r-squared or ICC fails when`null_model`

is not specified. If calculating the null model takes longer and you already have fit the null model, you can pass it here, too, to speed up the process.- 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.- 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.- verbose
Toggle warnings and messages.

- ...
Arguments passed down to

`lme4::bootMer()`

or`boot::boot()`

for bootstrapped ICC, R2, RMSE etc.; for`variance_decomposition()`

, arguments are passed down to`brms::posterior_predict()`

.

## Details

Marginal and conditional r-squared values for mixed models are calculated
based on *Nakagawa et al. (2017)*. For more details on the computation of
the variances, see `insight::get_variance()`

. The random effect variances are
actually the mean random effect variances, thus the r-squared value is also
appropriate for mixed models with random slopes or nested random effects
(see *Johnson, 2014*).

**Conditional R2**: takes both the fixed and random effects into account.**Marginal R2**: considers only the variance of the fixed effects.

The contribution of random effects can be deduced by subtracting the
marginal R2 from the conditional R2 or by computing the `icc()`

.

## Supported models and model families

The single variance components that are required to calculate the marginal
and conditional r-squared values are calculated using the `insight::get_variance()`

function. 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 r-squared values returned by `r2_nakagawa()`

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, nbinom2 and nbinom12 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

Skew-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.

## References

Hox, J. J. (2010). Multilevel analysis: techniques and applications (2nd ed). New York: Routledge.

Johnson, P. C. D. (2014). Extension of Nakagawa and 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., and Schielzeth, H. (2013). A general and simple method for obtaining R2 from generalized linear mixed-effects models. Methods in Ecology and Evolution, 4(2), 133–142. doi:10.1111/j.2041-210x.2012.00261.x

Nakagawa, S., Johnson, P. C. D., and 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.

## Examples

```
model <- lme4::lmer(Sepal.Length ~ Petal.Length + (1 | Species), data = iris)
r2_nakagawa(model)
#> # R2 for Mixed Models
#>
#> Conditional R2: 0.969
#> Marginal R2: 0.658
r2_nakagawa(model, by_group = TRUE)
#> # Explained Variance by Level
#>
#> Level | R2
#> ----------------
#> Level 1 | 0.569
#> Species | -0.853
#>
```