Correlation ratio (eta) for nonlinear association

[isaactpetersen]

Would it be possible to incorporate estimation of the correlation ratio (eta), as an estimate of the strength of nonlinear association between variables? Thanks for considering!

[mattansb]

This is (kind of) availble with effectsize::eta_squared():

mod <- aov(mpg ~ factor(gear), data = mtcars)

es <- effectsize::eta_squared(mod)


# Take the sqrt() and re-name the effect size
cols_to_sqrt <- intersect(colnames(es), c("Eta2", "CI_low", "CI_high"))
es[cols_to_sqrt] <- sqrt(es[cols_to_sqrt])
datawizard::data_rename(es, select = c("Eta" = "Eta2"))
#> # Effect Size for ANOVA
#> 
#> Parameter    |  Eta |       95% CI
#> ----------------------------------
#> factor(gear) | 0.66 | [0.43, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].

[isaactpetersen]

Thanks very much for showing how to compute this. My summary interpretation of your code example is that, to compute the correlation ratio (eta) for nonlinear associations, one would:

1. create a categorical variable (i.e., factor) of a (semi)continuous variable,
2. use it as the independent variable in an ANOVA predicting the dependent variable,
3. compute eta-squared (proportion of variance explained) from the ANOVA, and
4. take the square root of eta squared.

Please let me know if my understanding is incorrect. I will leave it up to you all to determine if it would be helpful to include the calculation of the correlation ratio in the package. A pro of doing so is that it would facilitate more easily estimating nonlinear associations (e.g., the correlation ratio) by automating these steps.

Here is potentially relevant documentation, in case it's helpful:

• https://en.wikipedia.org/wiki/Correlation_ratio
• https://dictionary.apa.org/correlation-ratio
• https://www.statisticssolutions.com/free-resources/directory-of-statistical-analyses/correlation-ratio/

Thanks again for the helpful example.

[mattansb]

No need to create categorical variables. This is also just multiple R (not squared).

So as long as you only have one predictor, R is equal to the "eta correlation".

Here is an example:

mod0 <- mgcv::gam(mpg ~ 1, data = mtcars)
mod1 <- mgcv::gam(mpg ~ s(hp), data = mtcars)

anova(mod0, mod1)
es <- effectsize::F_to_eta2(27.129, 3.2625, 27.737)

cols_to_sqrt <- intersect(colnames(es), c("Eta2_partial", "CI_low", "CI_high"))
es[cols_to_sqrt] <- sqrt(es[cols_to_sqrt])
datawizard::data_rename(es, select = c("Eta" = "Eta2_partial"))
#> Eta  |       95% CI
#> -------------------
#> 0.87 | [0.78, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].


# Or....
cor(mtcars$mpg, predict(mod1))
#> [1] 0.8706039

Here's another:

mod0 <- lm(mpg ~ 1, data = mtcars)
mod1 <- lm(mpg ~ hp + I(hp > median(hp)), data = mtcars)

anova(mod0, mod1)
es <- effectsize::F_to_eta2(24.713, 2, 29)

cols_to_sqrt <- intersect(colnames(es), c("Eta2_partial", "CI_low", "CI_high"))
es[cols_to_sqrt] <- sqrt(es[cols_to_sqrt])
datawizard::data_rename(es, select = c("Eta" = "Eta2_partial"))
#> Eta  |       95% CI
#> -------------------
#> 0.79 | [0.65, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].


# Or....
cor(mtcars$mpg, predict(mod1))
#> [1] 0.7938686

# Or...
sqrt(performance::r2(mod1)[[1]])
#> 0.7938686