Logit(log odds)

The logit function is the inverse of the sigmoidal “logistic” function or logistic transform used in mathematics, especially in statistics. When the function’s variable represents a probability p, the logit function gives the log-odds, or the logarithm of the odds p/(1 − p).

The logit of a number p between 0 and 1 is given by the formula:

The base of the logarithm function used is of little importance in the present article, as long as it is greater than 1, but the natural logarithm with base e is the one most often used. The choice of base corresponds to the choice of logarithmic unit for the value: base 2 corresponds to a shannon, base e to a nat, and base 10 to a hartley; these units are particularly used in information-theoretic interpretations. For each choice of base, the logit function takes values between negative and positive infinity.
The “logistic” function of any number alpha（α） is given by the inverse-logit:

If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds. Similarly, the difference between the logits of two probabilities is the logarithm of the odds ratio (R), thus providing a shorthand for writing the correct combination of odds ratios only by adding and subtracting:

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