Business Intelligence and Decision Modeling

Week 11 Predictive Modeling (2)

Logistic Regression

Regressions Simple Regression

Y = a + bX + e

Multiple Regression Y = a + b1X1 + b2X2 + … + bnXn + e

Logistic Regression p = 1 / 1+e-z

Z = a + b1X1 + b2X2 + …+ bnXn + e

Logistic Regressionand Functionp = 1 / 1+e-z

Where Logit or z = b0+b1x1+b2x2…+bpxp

P=1

P=0

Z -6 -4 -2 0 2 4 6

Outline Logistic Regression

Purpose Odds and Logit Interpretation

Logistic Regression Models Logistic regression (binary target)

Understand risk factors• Assumptions same as linear regression

Forecasting• Split Samples

Logistic Regression:Odds and Probabilities Dichotomised Response (0/1)

Response Probability p Non Response Probability (1-p)

Odds = p /(1-p) P = odds/(1+odds)

Probabilities and Odss Probability

.10 .20 .30 .40 .50 .60 .70 .80 .90

Odds .11 .25 .43 .67 1.00 1.50 2.33 4.00 9.00

Odds = (p / 1-p)

P = odds/(1+odds)

Logistic Regression:Logit Logit Calculation

Logit ln(odds) or ln(p/1-p)

Inverse Processe(Logit) odds or p/1-p If P = odds/(1+odds)

Then p = e(Logit)/1+e(Logit)

Or p = 1 / 1+e-Logit

Logistic Regressionand Functionp = 1 / 1+e-z

Where Logit or z = b0+b1x1+b2x2…+bpxp

P=1

P=0

Z -6 -4 -2 0 2 4 6

Example (1)Z=-10.83 + (.28 x age) +(2.30 x gender)

Where gender=0 Maleet gender =1 Female

If Male 40 years old Z = -10.83+(.28 x 40)+(2.30 x 0)Z = .37 Logite.37 = 1.448 Odds

Thus

p = 1 / 1+e-z p = 1 / 1+e-.37 = .59

orP = odds/(1+odds) p = 1.448/(1+1.448) = .59

Example (2)

Z=-10.83 + .28 x age +2.30 x genderwhere gender=0 MaleAnd gender =1 Female

If Female 40 years old Z = -10.83+(.28 x 40)+(2.30 x 1)Z = 2.67e2.67 = 14.44 Odds

Logitp = 1 / 1+e-z p = 1 / 1+e-2.67 = .94or

P = odds/(1+odds) p = 14.44/(1+14.44) = .93

To Summarize Simple Regression

Y = a + bX + e

Multiple Regression Y = a + b1X1 + b2X2 + … + bnXn + e

Logistic Regression p = 1 / 1+e-z

Z = a + b1X1 + b2X2 + …+ bnXn + e