Logistic

The logit of a number $p \in [0, 1]$ is

The logit function is the inverse of the sigmoid, or logistic function. If $p$ is a probability then $p / (1 - p)$ is the corresponding odds, and the logit of the probability is the logarithm of the odds; similarly the difference between the logits of two probabilities is the logarithm of the odds-ratio, thus providing an additive mechanism for combining odds-ratios.

Logits are used for various purposes by statisticians. In particular there is the "logit model" of which the simplest sort is

where $x_i$ is some quantity on which success or failure in the $i$-th in a sequence of Bernoulli trials may depend, and $p_i$ is the probability of success in the $i$-th case. For example, $x$ may be the age of a patient admitted to a hospital with a heart attack, and "success" may be the event that the patient dies before leaving the hospital (another instance of the reason why the words "success" and "failure" in speaking of Bernoulli trials should be taken with large doses of salt). Having observed the values of $x$ in a sequence of cases and whether there was a "success" or a "failure" in each such case, a statistician will often estimate the values of the coefficients $a$ and $b$ by the method of maximum likelihood. The result can then be used to assess the probability of "success" in a subsequent case in which the value of $x$ is known. Estimation and prediction by this method are called <em> logistic regression </em>.

As you may have noticed there is a link between the logistic and the linear regression methods, through the $\mathrm{logit}$ function. In other words,

Applying the exponential in both sides of the equality and doing further calculations, we arrive at the following relation

where $a$ and $b$ are the parameters of the associated linear regression, intercept and slope.

Below you will find the regression graph for a set of arbitrary points, coloured in blue. The regression curve, displayed in red, has been calculated using this class.

#include <codecogs/maths/approximation/regression/logistic.h>
#include <iostream>
#include <iomanip>
using namespace std;

int main() {
    double x[6] = {   28,    29,    30,    31,    32,    33};
    double y[6] = {.3333, .4000, .7778, .7778, .8000, .9333};

    Maths::Approximation::Regression::Logistic A(6, x, y);

    cout << "Logistic regression values" << endl << endl;
    for (int i = 10; i <= 50; i += 5) {
        cout << "x = " << setw(3) << i << "  y = " << A.getValue(i);
        cout << endl;
	}
    return 0;
}

Logistic regression values

x =  10  y = 0.143118
x =  15  y = 0.233325
x =  20  y = 0.356719
x =  25  y = 0.502592
x =  30  y = 0.648024
x =  35  y = 0.770364
x =  40  y = 0.859406
x =  45  y = 0.917614
x =  50  y = 0.95304

References

Wikipedia, http://en.wikipedia.org/wiki/Logistic_regression