Binary Logistic Regression

We use the logistic function to restrict the domain of $x$ to [0,1]$$$p(x) = \frac{e^x}{e^x + 1}$$ If we express p(x)$intermsof$x$ we produce $\text{logit}(p)=\text{log}(\frac{p}{1-p})$

Refer to Odds

We have that the function $p$ is the logistic function and so it is restricted to produce values on $[0,1]$.

However, since we’ve taken the inverse of the logistic function, the logit function can take on any value.

With this, we set $\text{logit}(p(x))={\beta }_0+{\beta }_{1}x$. This makes sense because a line has the range $[-\infty ,\infty ]$.

Once again, we say that the $\text{logit}$ is an example of a Link Function.

This $\text{logit}$ function links the $p$ to a linear function.

We can fit the $\text{logit}$ function to some data

p13.glm <- glm(y ~ x, data = p13.1, family = binomial)

This produces values for ${\beta }_0$ and ${\beta }_1$.

We have $\text{logit}(p)={\beta }_0+{\beta }_{1}x$ We rewrite in terms of $p$ and have $p=\frac{e^{{\beta }_0+{\beta }_{1}x}}{1+e^{{\beta }_0+{\beta }_{1}x}}$ We now simulate from the model since we know ${\beta }_0$ and ${\beta }_1$.

p <- exp(6.0709-0.0177*p13.1$x)/(1+exp(6.0709-0.0177*p13.1$x)
n <- nrow(p13.1)
simy <- rbinom(n, 1, prob = p)