Statistics Logistic Regression – lesscss

By | April 20, 2019

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

Logistic Regression

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