# Adding log odds for combined probability from logistic regression coefficients

I have coefficients of a logistic regression model. I know from the statistical package used that the estimates are log odds. I am interested in combining two variables, beta 5 (B5) and beta 6 (B6) and then converting this combined log odds into a probability (as B5 and B6 are both measuring the same variable; B6 is an interaction term with B5 in the regression).

1. Can you add two log odds together, and then convert them to make a combined conditional probability of, in this case, B5+B6?

2. Also, I am following a paper published in a journal that does this and then converts B5+B6 to probability by p = (exp -(B5+B6)) - 1.
This is not the way I have read how to convert log odds to probabilities. Hence I am wondering if the conversion process (from log odds to probability) changes when using combined coefficients?

I'm a little confused by your notion of coefficients interacting (I think you mean covariates, $x$'s). But I'll add some clarity around what it means to add coefficients.

Can you add two log odds together, and then convert them to make a combined conditional probability of, in this case, B5+B6?

We don't really say the "probability of $\beta_5 + \beta_6$", since these are coefficients. We can however ask about the change in log odds associated with a 1 unit increase in covariates $x_5$ and $x_6$, for example.

Let's say you have the following logistic regression model:

$$\text{logit} (P(Y=1|X=x)) = \alpha + \beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2$$

You want to know if you can add $\beta_2 + \beta_3$? It's important to understand what the coefficient represents:

Let's say $X_2$ increased by 1 unit, the new log odds $P(Y=1|X)$ are:

$$\text{logit} (P(Y=1|X_1=x_1, X_2=x_2+1)) = \alpha + \beta_1x_1 + \beta_2(x_2+1) + \beta_3x_1(x_2 +1)$$

So the change in log odds associated with a 1 unit increase in $X_2$ is:

$$\frac{\text{logit} (P(Y=1|X_1=x_1, X_2=x_2+1))}{\text{logit} (P(Y=1|X_1=x_1, X_2=x_2))} = \frac{\alpha + \beta_1x_1 + \beta_2(x_2+1) + \beta_3x_1(x_2+1)}{\alpha + \beta_1x_1 + \beta_2x_2 + \beta_3x_1x_2}$$

$$= \beta_2 + \beta_3x_1$$

So you can add the coefficients if you know your $x_1$ doesn't change. Of course, if it never changed then you should be modeling it as part of the intercept, $\alpha$.

Also, I am following a paper published in a journal that does this and then converts B5+B6 to probability by p = (exp -(B5+B6)) - 1. This is not the way I have read how to convert log odds to probabilities. Hence I am wondering if the conversion process (from log odds to probability) changes when using combined coefficients?

I'm also not familiar with this notation. In fact if

• $\beta_5 + \beta_6 > 0$ then $e^{-(\beta_5+\beta_6)}-1$ is negative
• $\beta_5 + \beta_6 < -\ln(2)$ then $e^{-(\beta_5+\beta_6)}-1$ is greater than 1

both of which are quite odd for a probability.