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