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From what I understood, standardized coefficients can be used as indices of effect size (with the possibility of using rules of thumb such as Cohen's 1988). I also understood that standardized coefs are expressed in terms of standard deviation, which makes them relatively close to a Cohen's d.

I also understood that one way of obtaining standardized coefs is to standardize the data beforehand. Another is to use the std.coef function from the MuMIn package.

These two methods are equivalent when using a linear predictor:

library(tidyverse)
library(MuMIn) # For stds coefs


df <- iris %>% 
  select(Sepal.Length, Sepal.Width) %>% 
  scale() %>% 
  as.data.frame() %>% 
  mutate(Species = iris$Species)


fit <- lm(Sepal.Length ~ Sepal.Width, data=df)
round(coef(fit), 2)
round(MuMIn::std.coef(fit, partial.sd = TRUE), 2)

In both cases, the coefficient is -0.12. I interpret it as follows: for each increase of 1 standard deviation of Sepal.Width, Sepal.Length diminishes of 0.12 of its SD.

And yet, these two methods give different results with a categorical predictor:

fit <- lm(Sepal.Length ~ Species, data=df)
round(coef(fit), 2)
round(MuMIn::std.coef(fit, partial.sd = TRUE), 2)

Which gives, for the effect of versicolor as compared to setosa (the intercept), 1.12 and 0.46.

Which should I believe to be able to say "the difference between versicolor and setosa is ... of Sepal.Length's SD"? Thanks a lot

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  • $\begingroup$ The standard deviation relies on the mean. What do you think the mean species means? The mode I can understand but the mean? If the mean is meaningless then so is the standard deviation. $\endgroup$
    – mdewey
    Jul 11 '18 at 17:14
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First, recall that categorical variables are silently recoded into dummies in lm function.

MuMIn package standardizes these dummy variables (which is straightforward since they contain only 0's and 1's).

On the other hand, you did not standardize them when creating your df object.

This is why you get two different results.

To find difference between setosa and versicolor in units of Sepal Length SD, you need to standardize only Sepal Length, which is exactly what you did in your df object.

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  • $\begingroup$ You're right, I found that standardizing the dummies gives the same coefs as those obtained with MuMin dummies <- scale(contrasts(df$Species)[df$Species,]) fit <- lm(Sepal.Length ~ dummies, data = df). But then, for using rules of thumb for effect size interpretation (for example roughly follow the Cohen's d grid), should I use the fully standardized, or just the "dependent" variable standardized results? Thanks a lot! $\endgroup$ Jul 12 '18 at 7:07
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    $\begingroup$ I believe you should use just the "dependent" variable standardized results. Because Cohen's d quantifies difference between groups with respect to pooled standard deviation. So you sholud keep gruoping variable (dummies in this case) on it's original scale. $\endgroup$ Jul 12 '18 at 7:29

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