Logit

In statistics, the logit (/ˈloʊdʒɪt/ LOH-jit) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations.

Mathematically, the logit is the inverse of the standard logistic function ${\displaystyle \sigma (x)=1/(1+e^{-x})}$, so the logit is defined as

${\displaystyle \operatorname {logit} p=\sigma ^{-1}(p)=\ln {\frac {p}{1-p}}\quad {\text{for}}\quad p\in (0,1).}$

Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds ${\displaystyle {\frac {p}{1-p}}}$ where p is a probability. Thus, the logit is a type of function that maps probability values from ${\displaystyle (0,1)}$ to real numbers in ${\displaystyle (-\infty ,+\infty )}$,^[1] akin to the probit function.

Definition

If p is a probability, then p/(1 − p) is the corresponding odds; the logit of the probability is the logarithm of the odds, i.e.:

${\displaystyle \operatorname {logit} (p)=\ln \left({\frac {p}{1-p}}\right)=\ln(p)-\ln(1-p)=-\ln \left({\frac {1}{p}}-1\right)=2\operatorname {atanh} (2p-1).}$

The base of the

The “logistic” function of any number ${\displaystyle \alpha }$ is given by the inverse-logit:

${\displaystyle \operatorname {logit} ^{-1}(\alpha )=\operatorname {logistic} (\alpha )={\frac {1}{1+\exp(-\alpha )}}={\frac {\exp(\alpha )}{\exp(\alpha )+1}}={\frac {\tanh({\frac {\alpha }{2}})+1}{2}}}$

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:

${\displaystyle \ln(R)=\ln \left({\frac {p_{1}/(1-p_{1})}{p_{2}/(1-p_{2})}}\right)=\ln \left({\frac {p_{1}}{1-p_{1}}}\right)-\ln \left({\frac {p_{2}}{1-p_{2}}}\right)=\operatorname {logit} (p_{1})-\operatorname {logit} (p_{2})\,.}$

History

Several approaches have been explored to adapt linear regression methods to a domain where the output is a probability value ${\displaystyle (0,1)}$, instead of any real number ${\displaystyle (-\infty ,+\infty )}$. In many cases, such efforts have focused on modeling this problem by mapping the range ${\displaystyle (0,1)}$ to ${\displaystyle (-\infty ,+\infty )}$ and then running the linear regression on these transformed values.^[2]

In 1934, Chester Ittner Bliss used the cumulative normal distribution function to perform this mapping and called his model probit, an abbreviation for "probability unit". This is, however, computationally more expensive.^[2]

In 1944, Joseph Berkson used log of odds and called this function logit, an abbreviation for "logistic unit", following the analogy for probit:

"I use this term [logit] for ${\displaystyle \ln p/q}$ following Bliss, who called the analogous function which is linear on ${\displaystyle x}$ for the normal curve 'probit'."

— Joseph Berkson (1944)^[3]

Log odds was used extensively by

• The logit in
  link function for the Bernoulli distribution
  .
• The logit function is the negative of the derivative of the binary entropy function.
• The logit is also central to the probabilistic Rasch model for measurement, which has applications in psychological and educational assessment, among other areas.
• The inverse-logit function (i.e., the logistic function) is also sometimes referred to as the expit function.^[10]
• In plant disease epidemiology, the logistic, Gompertz, and monomolecular models are collectively known as the Richards family models.
• The log-odds function of probabilities is often used in state estimation algorithms^[11] because of its numerical advantages in the case of small probabilities. Instead of multiplying very small floating point numbers, log-odds probabilities can just be summed up to calculate the (log-odds) joint probability.^[12][13]

Comparison with probit

logit function with a scaled probit (i.e. the inverse CDF of the normal distribution

), comparing ${\displaystyle \operatorname {logit} (x)}$ vs. ${\displaystyle {\tfrac {\Phi ^{-1}(x)}{\,{\sqrt {\pi /8\,}}\,}}}$, which makes the slopes the same at the y-origin.

. The probit function is denoted ${\displaystyle \Phi ^{-1}(x)}$, where ${\displaystyle \Phi (x)}$ is the

${\displaystyle \Phi (x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-y^{2}/2}dy.}$

As shown in the graph on the right, the logit and probit functions are extremely similar when the probit function is scaled, so that its slope at y = 0 matches the slope of the logit. As a result,

• Sigmoid function, inverse of the logit function
• Discrete choice on binary logit, multinomial logit, conditional logit, nested logit, mixed logit, exploded logit, and ordered logit
• Limited dependent variable
• Logit analysis in marketing
• Multinomial logit
• Ogee, curve with similar shape
• Perceptron
• Probit, another function with the same domain and range as the logit
• Ridit scoring
• Data transformation (statistics)
• Arcsin
  (transformation)
• Rasch model

References

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