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

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

Where −

Response - Presence/Absence of characteristic.

Predictor - Numeric variable observed for each case

${eta = 0 Rightarrow }$ P (Presence) is the same at each level of x.

${eta gt 0 Rightarrow }$ P (Presence) increases as x increases

${eta = 0 Rightarrow }$ P (Presence) decreases as x increases.

Example

Problem Statement:

Solve the logistic regression of the following problem Rizatriptan for Migraine

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

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

Dose	#Patients	#Repeved	%Repeved
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 ${eta = .165}, we ve following data:

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

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

Statistics - Logistic Regression

Formula

Example

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