# Difference between Cohen's d and beta coefficient in a standardized regression model with dummy coded predictor variable

I have been wondering about this for a while now and I just could not find an answer to it, so I'd be glad if someone here could help me out!

If I have a simple regression model with a standardized criterion and one predictor, which is a dummy coded variable (two levels), it's the same as calculating a t-test. The t-value from the t-test is the same as the one from the regression model, and also the p-value is exactly the same. But why is the beta-coefficient in my regression model not the same as Cohen's d?

The effect size Cohen's d is the difference between two groups in Standard Deviations. The beta-coefficient in a regression model with dummy coded variables is the deviation of the respective group from a reference group (which for two groups equals the difference between those both groups). Hence Cohen's d and beta should be the same number, or where is my error in reasoning? But the values are completely different (-0.495 vs. -1.137), so I don't think it has to do with pooled vs. unpooled SDs or something like this...

Cohen's $$d$$ can be thought of as semi-standardized, that is, standardized on the dependent variable but not on the independent variable. Recall that the relationship between $$\beta$$ (a standardized regression coefficient) and $$b$$, an standardized regression coefficient is: $$\beta = \frac{b(s_x)}{s_y} .$$ However, Cohen's $$d$$ is approximately: $$\beta = \frac{b}{s_y}$$ with the difference being that the denominator is the pooled within-groups $$s$$ and not the overall $$s$$. Thus, Cohen's $$d$$ will be bigger than the above, proportional to the size of the effect. In your example, the effect is big enough to make a rather noticeable difference.
If the two groups have an equal sample size, then $$s_x$$ is 0.5. Thus, we would expect the two estimates to be off by about 1/2 plus the effect of the difference between $$s$$ and $$s_{pooled}$$.