Alternative R2?

Problem

R2 is affected by the absence of an intercept:

data <- mtcars |> 
  within({
    M = model.matrix(~1 + hp)
  })

m1 <- lm(mpg ~ 1 + hp, data = data)
m2 <- lm(mpg ~ 0 + M, data = data) # same model?

Same parameters…

parameters::compare_parameters(m1, m2)
#> Warning in mapply(function(.d, .l) {: longer argument not a multiple of length
#> of shorter
#> Warning: Some model terms could not be found in model data.
#>   You probably need to load the data into the environment.
#> Parameter    |                   m1 |                   m2
#> ----------------------------------------------------------
#> (Intercept)  | 30.10 (26.76, 33.44) |                     
#> hp           | -0.07 (-0.09, -0.05) |                     
#> M(Intercept) |                      | 30.10 (26.76, 33.44)
#> Mhp          |                      | -0.07 (-0.09, -0.05)
#> ----------------------------------------------------------
#> Observations |                   32 |                   32

Same dfs…

df.residual(m1)
#> [1] 30
df.residual(m2)
#> [1] 30

Same predictions…

all.equal(predict(m1), predict(m2))
#> [1] TRUE

But…

performance::r2(m1)
#> # R2 for Linear Regression
#>        R2: 0.602
#>   adj. R2: 0.589
performance::r2(m2)
#> # R2 for Linear Regression
#>        R2: 0.968
#>   adj. R2: 0.966

This isn’t an issue in performace, but R seems to give different values based on whether or not there is an intercept in the model

summary(m1)$r.squared
#> [1] 0.6024373
summary(m2)$r.squared
#> [1] 0.9681196

^{Created on 2022-10-14 by the reprex package (v2.0.1)}

Solution

Perhaps we can add an r2_prediction which computes the R2 values based on the prediction?

Either as: $R^2 = r_{\hat{y},y}^2$

Or (my preferred method) as $R^2 = 1 - \frac{Var(y - \hat{y})}{Var(y)}$

This will allow for correct R2 for transformed outcomes (using smearing), for non-linear models, etc...

Start by tracing the existing performance::r2 entry point and how it handles intercept-free models. Compare the proposed prediction-based formulas with current R2 behavior for the supplied m1 and m2 examples, then define the expected result and tests for transformed outcomes and non-linear models.