statisticalrethinkin..

278 11. COUNTING AND CLASSIFICATIONlog-odds-4 -2 0 2 4odds0 20 40 60 80 1000.00 0.25 0.50 0.75 1.00probability0.00 0.25 0.50 0.75 1.00probabilityFIGURE 11.1. Log-odds versus ordinary odds. On the le, the log-odds ofan event are plotted against the raw probability. e horizontal dashed lineshows the log-odds value at which the event has probability one-half. Noticethat the log-odds curve is perfectly symmetrical around this line. On theright, ordinary odds are plotted against probability. Again, the horizontaldashed line shows the value of the odds at which the probability is one-half.Now however, the curve is far from symmetrical around this line. Odds arenot invariant to the choice of which event to focus on, while log-odds areinvariant.highly asymmetrical. Again, the horizontal dashed line shows the value on the vertical axis atwhich the event has probability one-half. is is where the odds are equal to one. But now thecurve is far from symmetrical on both sides of this line. Instead it accelerates towards infinityas the probability increases above one-half. Why is the symmetry of the log-odds superior tothe asymmetry of the odds? We want a scale of measurement that isn’t influenced by arbitrarymeasurement choices. e horizontal probability axis in both plots could just as easily gothe other direction, measuring the probability of the event not happening. e log-oddsplot would be unaffected by this decision, while the odds plot would change dramatically.Since the choice of which event to label on the horizontal axis is largely arbitrary—a matterof our convenience—it is important to have a scale of measurement that is invariant to sucharbitrary decisions. Log-odds provide such a scale.Finally, a reasonable model of plain probabilities and odds must be fundamentally multiplicative,because otherwise the model would allow these values to dip below zero, whichshould be impossible. In contrast, multiplicative relations, once logged, become additive.You’ve seen this already, when you learned that joint likelihood is the product of each individuallikelihood, but the joint log-likelihood is the sum of each log-likelihood. And so likelog-likelihood is additive, log-odds are fundamentally additive, and therefore lend themselvesmore easily to the generalized linear modeling framework.11.1.2. e logit link. So we’ll adopt the strategy of modeling the log-odds, instead of theprobability p directly. More precisely, what this means is that instead of defining the binomial