# Equation for solving for a variable value of a GLMM probability equation

I am trying to solve for a value of x given all the values of the other x's and at a set value for πij.

For example

.5 = e^(20 + 10(1) + 15(1) + 20(x3) + 3 + 2)/ 1+ e^(20 + 10(1) + 15(1) + 20(x3) + 3 + 2)

How would I rewrite the equation to solve for the value of x3?

• Hey @user158565 can you please describe is the beta just the beta for the variable x3? It doesn't take into account information from all the other variables? Commented Dec 11, 2018 at 22:01
• so in this case when I have a probability of .5 the log of .5/.5 is 0. Commented Dec 11, 2018 at 22:02

Assume $$\pi \ne 0$$ and $$\pi \ne 1$$ $$\pi = \frac {\exp(X\beta)}{1+\exp(X\beta)}$$ $$\pi ({1+\exp(X\beta)})= {\exp(X\beta)}$$ $$\pi+\pi \exp(X\beta)= {\exp(X\beta)}$$ $$\pi= {\exp(X\beta)} - \pi \exp(X\beta)$$ $$\pi= (1-\pi){\exp(X\beta)}$$ $$\frac {\pi}{1-\pi}= {\exp(X\beta)}$$ $$\log\frac {\pi}{1-\pi}= X\beta$$
In your case, $$log(0.5/0.5) = 0 =20 + 10(1) + 15(1) + 20(x3) + 3 + 2$$ Then you can get the answer.