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

π(x)=eα+βx1+eα+βxπ(x)=eα+βx1+eα+βx

Where −

·        Response – Presence/Absence of characteristic.

·        Predictor – Numeric variable observed for each case

·        β=0⇒β=0⇒ P (Presence) is the same at each level of x.

·        β>0⇒β>0⇒ P (Presence) increases as x increases

·        β=0⇒β=0⇒ 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 α=−2.490andα=−2.490and{\beta = .165}, we’ve following data:

π(0)=eα+β×01+eα+β×0=e−2.490+01+e−2.490=0.03π(2.5)=eα+β×2.51+eα+β×2.5=e−2.490+.165×2.51+e−2.490+.165×2.5=0.09π(5)=eα+β×51+eα+β×5=e−2.490+.165×51+e−2.490+.165×5=0.23π(10)=eα+β×101+eα+β×10=e−2.490+.165×101+e−2.490+.165×10=0.29π(0)=eα+β×01+eα+β×0=e−2.490+01+e−2.490=0.03π(2.5)=eα+β×2.51+eα+β×2.5=e−2.490+.165×2.51+e−2.490+.165×2.5=0.09π(5)=eα+β×51+eα+β×5=e−2.490+.165×51+e−2.490+.165×5=0.23π(10)=eα+β×101+eα+β×10=e−2.490+.165×101+e−2.490+.165×10=0.29

Dose(xx)	π(x)π(x)
0	0.03
2.5	0.09
5	0.23
10	0.29