# l82 - Logistic Regression Part II—Page 1 Logistic...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Logistic Regression, Part II—Page 1 Logistic Regression, Part II: The Logistic Regression Model (LRM) – Interpreting Parameters [This handout steals heavily from Linear probability, logit, and probit models, by John Aldrich and Forrest Nelson, paper # 45 in the Sage series on Quantitative Applications in the Social Sciences.] P ROBABILITIES , O DDS AND L OG O DDS . The linear probability model (LPM) is P(Yi = 1) = Pi = + X ik Among other things, a problem with this model is that the left hand side can only range from 0 to 1, but the right hand side can vary from negative infinity to positive infinity. One way of approaching this problem is to transform Pi to eliminate the 0 to 1 constraint. We can eliminate the upper bound (Pi = 1) by looking at the ratio Pi/(1 - Pi). Pi/(1 - Pi) is referred to as the odds of an event occurring. That is, i i i P P Odds 1 For example, if Pi = .90, the odds are 9 to 1 (or 9) that the event will happen. If Pi = .30, the odds are 3 to 7 (or .429 to 1, or simply .429) against the event occurring. If Pi = .50, the odds are 1 to 1 (even). The following table helps to illustrate how odds and probabilities are related to each other. It starts with odds of 10,000 to 1 against. Then, the odds are multiplied by 10 each row, until the odds become 10,000 to 1 in favor. Odds Probability Change in Probability 0.0001 0.0100% 0.001 0.0999% 0.0899% 0.01 0.9901% 0.8902% 0.1 9.0909% 8.1008% 1 50.0000% 40.9091% 10 90.9091% 40.9091% 100 99.0099% 8.1008% 1000 99.9001% 0.8902% 10000 99.9900% 0.0899% Note the nonlinear relationship between odds and probability. At the low end, you’d prefer to have 100 to 1 odds against you rather than 10,000 to 1 odds against; but either way, you’ve still Logistic Regression, Part II—Page 2 got less than a 1% chance of success. Conversely, it is better to have 10,000 to 1 odds in your favor rather than 100 to 1, but either way you’ve got better than a 99% chance of success. Hence, at the extremes, changes in the odds have little effect on the probability of success. In the middle ranges, however, it is a very different story. As you go from 10 to 1 odds against to even odds of 1 to 1, the probability of success jumps from 9.9% to 50%, almost a 41% increase. And, as you go from even odds to 10 to 1 odds in your favor, the probability of success jumps from 50% to almost 91%. Think how this compares to our previous example about grades. A student with a D average may have 100 to 1 odds against getting an A, compared to 1,000 to 1 odds for an F student. The odds are 10 times better for the D student, but for both the probability of an A is pretty small, less than 1%. Conversely, a C+ student might have 10 to 1 odds against getting an A, while a B+ student might have 1 to 1 odds. In other words, going from an F to a D may have little effect on the probability of success, but going from C+ to B+ may have a huge effect....
View Full Document

## This note was uploaded on 02/29/2012 for the course SOC 63993 taught by Professor Richardwilliams during the Spring '11 term at Notre Dame.

### Page1 / 15

l82 - Logistic Regression Part II—Page 1 Logistic...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online