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Statistics - Logistic Regression


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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

π(x)=eα+βx1+eα+βx

Where −

  • Response - Presence/Absence of characteristic.

  • Predictor - Numeric variable observed for each case

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

  • β>0 P (Presence) increases as x increases

  • 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
06723.0
2.57579.3
51302922.3
101454027.6

Solution:

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

π(0)=eα+β×01+eα+β×0=e2.490+01+e2.490=0.03π(2.5)=eα+β×2.51+eα+β×2.5=e2.490+.165×2.51+e2.490+.165×2.5=0.09π(5)=eα+β×51+eα+β×5=e2.490+.165×51+e2.490+.165×5=0.23π(10)=eα+β×101+eα+β×10=e2.490+.165×101+e2.490+.165×10=0.29
Dose(x)π(x)
00.03
2.50.09
50.23
100.29
Logistic Regression