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 |
