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Unformatted text preview: Logistic Regression, Part IIPage 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, youd prefer to have 100 to 1 odds against you rather than 10,000 to 1 odds against; but either way, youve still Logistic Regression, Part IIPage 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 youve 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....
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 Spring '11
 RichardWilliams

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