Logistic regression

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What is the logistic curve? What is the base of the natural logarithm? Why do statisticians prefer logistic regression to ordinary linear regression when the DV is binary? How are probabilities, odds and logits related? What is an odds ratio? How can logistic regression be considered a linear regression? What is a loss function? What is a maximum likelihood estimate? How is the b weight in logistic regression for a categorical variable related to the odds ratio of its constituent categories? For this chapter only, we are going to deal with a dependent variable that is binary (a categorical variable that has two values such as "yes" and "no") rather than continuous.

[Technical note: Logistic regression can also be applied to ordered categories (ordinal data), that is, variables with more than two ordered categories, such as what you find in many surveys. However, we won't be dealing with that in this course and you probably will never be taught it. If our dependent variable has several unordered categories (e.g., suppose our DV was state of origin in the U.S.), then we can use something called discriminant analysis, which will be taught to you in a course on multivariate statistics.]

It is customary to code a binary DV either 0 or 1. For example, we might code a successfully kicked field goal as 1 and a missed field goal as 0 or we might code yes as 1 and no as 0 or admitted as 1 and rejected as 0 or Cherry flavor ice cream as 1 and all other flavors as zero. If we code like this, then the mean of the distribution is equal to the proportion of 1s in the distribution. For example if there are 100 people in the distribution and 30 of them are coded 1, then the mean of the distribution is .30, which is the proportion of 1s. The mean of the distribution is also the probability of drawing a person labeled as 1 at random from the distribution. That is, if we grab a person at random from our sample of 100 that I just described, the probability that the person will be a 1 is .30. Therefore, proportion and probability of 1 are the same in such cases. The mean of a binary distribution so coded is denoted as P, the proportion of 1s. The proportion of zeros is (1-P), which is sometimes denoted as Q. The variance of such a distribution is PQ, and the standard deviation is Sqrt(PQ).

The Logistic Curve

The logistic curve relates the independent variable, X, to the rolling mean of the DV, P (v). The formula to do so may be written either.

Suppose we only know a person's height and we want to predict whether that person is male or female. We can talk about the probability of being male or female, or we can talk about the odds of being male or female. Let's say that the probability of being male at a given height is .90. Then the odds of being male would be {odds = [P/(1-P)]}. (Odds can also be found by counting the number of people in each group and dividing one number by the other. Clearly, the probability is not the same as the odds.) In our example, the odds would be .90/.10 or 9 to one. Now the odds of being female would be .10/.90 or 1/9 or .11. This asymmetry is unappealing, because the odds of being a male should be the opposite of the odds of being a female. We can take care of this asymmetry though the natural logarithm, ln. The natural log of 9 is 2.217 (ln(.9/.1)=2.217). The natural log of 1/9 is -2.217 (ln(.1/.9)=-2.217), so the log odds of being male is exactly opposite to the log odds of being female.

Notes & References

[1] http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html

Credits & Notices

Authors-contributors to this page (listed alphabetically, last name, first & middle initial only, no institutional affiliations, no scientific titles):

Stawicki SP

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