Minimum Detectable Effect Size for multiple linear regression I am trying to calculate the minimum detectable effect size (MDE) of an explanatory variable in a multiple linear regression.
The regression looks like the following:
$$
y_i = \beta_0 + \beta_1X_1 + \beta_2*X_2 + \beta_3*X_3 + \beta_4*X_4 + \epsilon_i
$$
whereby I am only interested in the MDE of $X_1$ in this multiple linear regression.
I have found code for linear regression in R: pwr.f2.test().
However, I am not sure if this can eliminate the minimum detectable effect size for one variable. Does anyone know how to get the minimum detectable effect size of one explanatory variable in a multiple linear regression? How can I implement this in R?

Along the lines of Demetri's answer, the approach greatly simplifies finding MDE in a multivariate model - a point of confusion for many early analysts. The $\sigma_{y|x}$  that is expressed is conditional on all the other x. In other words, there is practically no difference between finding power in a multivariate model and in a bivariate model: inference for the model coefficient boils down to knowing the conditional variance of the Y, and subtracting off all the degrees of freedom for adjusting the other $X$; it is just a t-test.
So proceed by considering MDE for a t-test, supply the conditional variance of $Y$ adjusting all the other covariates which is required as a free-parameter in the power/sample size calculations, then subtract 1 from the overall $n$ for each adjusted variable.
