The **coefficient of determination**, denoted \(R^2\) and pronounced “R squared”, typically corresponds the proportion of the variance in the dependent variable (the response) that is *explained* (i.e., predicted) by the independent variables (the predictors).

It is an “absolute” index of *goodness-of-fit*, ranging from 0 to 1 (often expressed in percentage), and can be used for model performance assessment or models comparison.

As models become more complex, the computation of an \(R^2\) becomes increasingly less straightforward.

Currently, depending on the context of the regression model object, one can choose from the following measures supported in `performance`

:

- Bayesian \(R^2\)
- Cox & Snell’s \(R^2\)
- Efron’s \(R^2\)
- Kullback-Leibler \(R^2\)
- LOO-adjusted \(R^2\)
- McFadden’s \(R^2\)
- McKelvey & Zavoinas \(R^2\)
- Nagelkerke’s \(R^2\)
- Nakagawa’s \(R^2\) for mixed models
- Somers’ \(D_{xy}\) rank correlation for binary outcomes
- Tjur’s \(R^2\) - coefficient of determination (D)
- Xu’ \(R^2\) (Omega-squared)
- \(R^2\) for models with zero-inflation

*TO BE COMPLETED.*

In the context of a generalized linear model (e.g., a logistic model which outcome is binary), \(R^2\) doesn’t measure the percentage of *“explained variance”*, as this concept doesn’t apply. However, the \(R^2\)s that have been adapted for GLMs have retained the name of “R2”, mostly because of the similar properties (the range, the sensitivity, and the interpretation as the amount of explanatory power).

For mixed models, `performance`

will return two different \(R^2\)s:

- The
**conditional**\(R^2\) - The
**marginal**\(R^2\)

The marginal \(R^2\) considers only the variance of the **fixed effects** (without the random effects), while the conditional \(R^2\) takes *both* the **fixed and random effects** into account (i.e., the total model).

```
library(lme4)
library(performance)
# defining a linear mixed-effects model
model <- lmer(Petal.Length ~ Petal.Width + (1 | Species), data = iris)
r2(model)
```

```
> # R2 for Mixed Models
>
> Conditional R2: 0.933
> Marginal R2: 0.303
```

Note that `r2`

functions only return the \(R^2\) values. We would encourage users to instead always use the `compare_performance`

function to get a more comprehensive set of indices of model fit.

`model_performance(model)`

```
> # Indices of model performance
>
> AIC | BIC | R2 (cond.) | R2 (marg.) | ICC | RMSE | Sigma
> -------------------------------------------------------------------
> 159.036 | 171.079 | 0.933 | 0.303 | 0.904 | 0.373 | 0.378
```

But, in the current vignette, we would like to exclusively focus on this family of functions and will only talk about this measure.

```
> # Bayesian R2 with Compatibility Interval
>
> Conditional R2: 0.816 (95% CI [0.704, 0.895])
```

As discussed above, for mixed-effects models, there will be two components associated with \(R^2\).

```
# defining a Bayesian mixed-effects model
model <- stan_lmer(Petal.Length ~ Petal.Width + (1 | Species), data = iris, refresh = 0)
r2(model)
```

```
> # Bayesian R2 with Compatibility Interval
>
> Conditional R2: 0.953 (95% CI [0.941, 0.963])
> Marginal R2: 0.826 (95% CI [0.719, 0.897])
```

If you want to know more about these indices, you can check out details and references in the functions that compute them **here**.

If you want to know about how to *interpret* these \(R^2\) values, see these **interpretation guidelines**.