## Link Functions and Links

**The link function links an unbounded continuous variable with a response bounded on (0, 1). The (0, 1) response can be thought as either binary (it happened or it did not) or as a probability of happening (continuous).**

#### Example: Logistic Link

**Logistic function definition:** \(z = log(p/(1-p))\)

\[z = f(size = a) = \beta_0 + \beta_1 a\]

\[z = g(POD) = log(p/(1-p))\]

\[exp(z) = \frac{p}{1-p}\]

**Solve for p:**

\[exp(z) (1-p) = p\]

\[exp(z) – exp(z) p = p\]

\[exp(z) = p + exp(z) p = p(1 + exp(z))\]

\[p = \frac{exp(z)}{1+exp(z)}\]

**logistic link:**

\[POD(a) = \frac{\beta_0 + \beta_1 a}{1 + \beta_0 + \beta_1 a} \tag{1}\]

#### Finding values for \(\beta_0\) and \(\beta_1 \)

While the logistic transformation results in a linear relationship between *a* and *z* the resulting error^{1} structure is binomial, NOT Gaussian. That means the familiar methods of Least Squares regression are *not* appropriate. That doesn’t mean you *can’t* coerce a computer program to give you an answer using OLS with a binary or proportion response. It does mean, however, that the answer will be ** wrong**.

Parameter values are determined by the method of maximum likelihood, *i.e.* finding values that maximize the Likelihood that the experiment turned out the way it did.

This is accomplished by defining a likelihood function and maximizing it numerically. The likelihood of an individual observation is given by \(POD(a_i)\), eqn 1, left, for “hits” and \(1 – POD(a_i)\), for “misses.” The total likelihood is the product of the individual likelihoods.

Logistic (and Probit) regression are discussed in detail in my POD Workshop and Short Course.

#### Notes:

*error*here is the difference between the observed response (either 0 or 1) and the*expected*response provided by the model, \(POD(a) = \frac{\beta_0 + \beta_1 a}{1 + \beta_0 + \beta_1 a}\)