Statistics - Logistic Regression

Logistic regression is a statistical method for analyzing a dataset in which there are one or more independent variables that determine an outcome. The outcome is measured with a dichotomous variable (in which there are only two possible outcomes).

Formula

${\pi(x) = \frac{e^{\alpha + \beta x}}{1 + e^{\alpha + \beta x}}}$

Where −

Response - Presence/Absence of characteristic.
Predictor - Numeric variable observed for each case
${\beta = 0 \Rightarrow }$ P (Presence) is the same at each level of x.
${\beta \gt 0 \Rightarrow }$ P (Presence) increases as x increases
${\beta = 0 \Rightarrow }$ P (Presence) decreases as x increases.

Example

Problem Statement:

Solve the logistic regression of the following problem Rizatriptan for Migraine

Response - Complete Pain Relief at 2 hours (Yes/No).

Predictor - Dose (mg): Placebo (0), 2.5,5,10

Dose	#Patients	#Relieved	%Relieved
0	67	2	3.0
2.5	75	7	9.3
5	130	29	22.3
10	145	40	27.6

Solution:

Having ${\alpha = -2.490} and ${\beta = .165}, we've following data:

$ {\pi(0) = \frac{e^{\alpha + \beta \times 0}}{1 + e^{\alpha + \beta \times 0}} \\[7pt] \, = \frac{e^{-2.490 + 0}}{1 + e^{-2.490}} \\[7pt] \\[7pt] \, = 0.03 \\[7pt] \pi(2.5) = \frac{e^{\alpha + \beta \times 2.5}}{1 + e^{\alpha + \beta \times 2.5}} \\[7pt] \, = \frac{e^{-2.490 + .165 \times 2.5}}{1 + e^{-2.490 + .165 \times 2.5}} \\[7pt] \, = 0.09 \\[7pt] \\[7pt] \pi(5) = \frac{e^{\alpha + \beta \times 5}}{1 + e^{\alpha + \beta \times 5}} \\[7pt] \, = \frac{e^{-2.490 + .165 \times 5}}{1 + e^{-2.490 + .165 \times 5}} \\[7pt] \, = 0.23 \\[7pt] \\[7pt] \pi(10) = \frac{e^{\alpha + \beta \times 10}}{1 + e^{\alpha + \beta \times 10}} \\[7pt] \, = \frac{e^{-2.490 + .165 \times 10}}{1 + e^{-2.490 + .165 \times 10}} \\[7pt] \, = 0.29 }$

Dose(${x}$)	${\pi(x)}$
0	0.03
2.5	0.09
5	0.23
10	0.29