Regression – Intercept from Standardized Coefficients in Logistic Regression: Explanation

data transformationinterceptlogisticregressionstandardization

I have fit a logistic regression model with original y and standardized x variables. Slope coefficients can be easily converted back to their original scale by $\beta^*_j/\sigma_{x_j}$ where $\beta^*_j$ is the output from the fitted model. However, I am not sure how to get the original intercept estimate from the output. Any help is appreciated.

Thanks

Best Answer

Start with a simple logistic regression:

$\,\,\text{logit}(\mu) \,= \beta_0 + \beta_1 x\quad\quad$ (original)

$\quad\quad\quad\quad= \beta_0^* + \beta_1^* (x-\bar{x})/s_x\quad\quad$ (standardized x)

$\quad\quad\quad\quad= (\beta_0^* -\beta_1^*\bar{x}/s_x)+ (\beta_1^*/s_x) x$

So $\beta_1=\beta_1^*/s_x$ and $\beta_0=\beta_0^* -\beta_1^*\bar{x}/s_x$

More generally, the same manipulation yields

$\beta_0=\beta_0^* -\sum_{i=1}^p\beta_i^*\bar{x_i}/s_{x_i}$ and $\beta_i=\beta_i^*/s_{x_i}$

Related Solutions

Regression – Methods for Back-Transforming Regression Coefficients

One problem is that you've written

$$Y=α+β⋅X$$

That is a simple deterministic (i.e. non-random) model. In that case, you could back transform the coefficients on the original scale, since it's just a matter of some simple algebra. But, in usual regression you only have $E(Y|X)=α+β⋅X $ ; you've left the error term out of your model. If transformation from $Y$ back to $Y_{orig}$ is non-linear, you may have a problem since $E\big(f(X)\big)≠f\big(E(X)\big)$, in general. I think that may have to do with the discrepancy you're seeing.

Edit: Note that if the transformation is linear, you can back transform to get estimates of the coefficients on the original scale, since expectation is linear.

Solved – How to get the standardized beta coefficients from glm.nb regression in R

A quick way to get at the standardized beta coefficients directly from any lm (or glm) model in R, try using lm.beta(model). In the example provided, this would be:

library("MASS")
nb = glm.nb(responseCountVar ~ predictor1 + predictor2 + 
    predictor3 + predictor4 + predictor5 + predictor6 + 
    predictor7 + predictor8 + predictor9 + predictor10 + 
    predictor11 + predictor12 + predictor13 + predictor14 + 
    predictor15 + predictor16 + predictor17 + predictor18 + 
    predictor19 + predictor20 + predictor21,
    data=myData, control=glm.control(maxit=125))
summary(nb)

library(QuantPsyc)
lm.beta(nb)

Best Answer

Related Solutions

Regression – Methods for Back-Transforming Regression Coefficients

Solved – How to get the standardized beta coefficients from glm.nb regression in R

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