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This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the model, which in turn makes it easy to consider various sorts of extensions.

See the example below. The choice of the type-1 extreme value distribution seems fairly arbitrary, but it makes the mathematics work out, and it may be possible to justify its use through rational choice theory.

It turns out that this model is equivalent to the previous model, although this seems non-obvious, since there are now two sets of regression coefficients and error variables, and the error variables have a different distribution.

In fact, this model reduces directly to the previous one with the following substitutions:. An intuition for this comes from the fact that, since we choose based on the maximum of two values, only their difference matters, not the exact values — and this effectively removes one degree of freedom.

Another critical fact is that the difference of two type-1 extreme-value-distributed variables is a logistic distribution, i. As an example, consider a province-level election where the choice is between a right-of-center party, a left-of-center party, and a secessionist party e.

We would then use three latent variables, one for each choice. Then, in accordance with utility theory , we can then interpret the latent variables as expressing the utility that results from making each of the choices.

We can also interpret the regression coefficients as indicating the strength that the associated factor i. A voter might expect that the right-of-center party would lower taxes, especially on rich people.

This would give low-income people no benefit, i. On the other hand, the left-of-center party might be expected to raise taxes and offset it with increased welfare and other assistance for the lower and middle classes.

This would cause significant positive benefit to low-income people, perhaps a weak benefit to middle-income people, and significant negative benefit to high-income people.

Finally, the secessionist party would take no direct actions on the economy, but simply secede. Yet another formulation combines the two-way latent variable formulation above with the original formulation higher up without latent variables, and in the process provides a link to one of the standard formulations of the multinomial logit.

Here, instead of writing the logit of the probabilities p i as a linear predictor, we separate the linear predictor into two, one for each of the two outcomes:.

This term, as it turns out, serves as the normalizing factor ensuring that the result is a distribution. This can be seen by exponentiating both sides:.

In this form it is clear that the purpose of Z is to ensure that the resulting distribution over Y i is in fact a probability distribution , i. This means that Z is simply the sum of all un-normalized probabilities, and by dividing each probability by Z , the probabilities become " normalized ".

That is:. This shows clearly how to generalize this formulation to more than two outcomes, as in multinomial logit. Note that this general formulation is exactly the softmax function as in.

In fact, it can be seen that adding any constant vector to both of them will produce the same probabilities:. As a result, we can simplify matters, and restore identifiability, by picking an arbitrary value for one of the two vectors.

Note that most treatments of the multinomial logit model start out either by extending the "log-linear" formulation presented here or the two-way latent variable formulation presented above, since both clearly show the way that the model could be extended to multi-way outcomes.

In general, the presentation with latent variables is more common in econometrics and political science , where discrete choice models and utility theory reign, while the "log-linear" formulation here is more common in computer science , e.

This functional form is commonly called a single-layer perceptron or single-layer artificial neural network. A single-layer neural network computes a continuous output instead of a step function.

With this choice, the single-layer neural network is identical to the logistic regression model. This function has a continuous derivative, which allows it to be used in backpropagation.

This function is also preferred because its derivative is easily calculated:. A closely related model assumes that each i is associated not with a single Bernoulli trial but with n i independent identically distributed trials, where the observation Y i is the number of successes observed the sum of the individual Bernoulli-distributed random variables , and hence follows a binomial distribution :.

An example of this distribution is the fraction of seeds p i that germinate after n i are planted. In terms of expected values , this model is expressed as follows:.

In a Bayesian statistics context, prior distributions are normally placed on the regression coefficients, usually in the form of Gaussian distributions.

There is no conjugate prior of the likelihood function in logistic regression. When Bayesian inference was performed analytically, this made the posterior distribution difficult to calculate except in very low dimensions.

However, when the sample size or the number of parameters is large, full Bayesian simulation can be slow, and people often use approximate methods such as variational Bayesian methods and expectation propagation.

A detailed history of the logistic regression is given in Cramer The logistic function was independently developed in chemistry as a model of autocatalysis Wilhelm Ostwald , This naturally gives rise to the logistic equation for the same reason as population growth: the reaction is self-reinforcing but constrained.

They were initially unaware of Verhulst's work and presumably learned about it from L. Gustave du Pasquier , but they gave him little credit and did not adopt his terminology.

The probit model influenced the subsequent development of the logit model and these models competed with each other. By , the logit model achieved parity with the probit model in use in statistics journals and thereafter surpassed it.

This relative popularity was due to the adoption of the logit outside of bioassay, rather than displacing the probit within bioassay, and its informal use in practice; the logit's popularity is credited to the logit model's computational simplicity, mathematical properties, and generality, allowing its use in varied fields.

Various refinements occurred during that time, notably by David Cox , as in Cox The multinomial logit model was introduced independently in Cox and Thiel , which greatly increased the scope of application and the popularity of the logit model.

Most statistical software can do binary logistic regression. Notably, Microsoft Excel 's statistics extension package does not include it.

From Wikipedia, the free encyclopedia. Statistical model for a binary dependent variable. It is not to be confused with Logit function.

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Archived from the original PDF on New York: Cambridge University Press. Cox, David R. J R Stat Soc B. David ed. London: Wiley. The origins of logistic regression PDF Technical report.

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In regression analysis , logistic regression [1] or logit regression is estimating the parameters of a logistic model a form of binary regression.

In the logistic model, the log-odds the logarithm of the odds for the value labeled "1" is a linear combination of one or more independent variables "predictors" ; the independent variables can each be a binary variable two classes, coded by an indicator variable or a continuous variable any real value.

The corresponding probability of the value labeled "1" can vary between 0 certainly the value "0" and 1 certainly the value "1" , hence the labeling; the function that converts log-odds to probability is the logistic function, hence the name.

The unit of measurement for the log-odds scale is called a logit , from log istic un it , hence the alternative names. Analogous models with a different sigmoid function instead of the logistic function can also be used, such as the probit model ; the defining characteristic of the logistic model is that increasing one of the independent variables multiplicatively scales the odds of the given outcome at a constant rate, with each independent variable having its own parameter; for a binary dependent variable this generalizes the odds ratio.

In a binary logistic regression model, the dependent variable has two levels categorical. Outputs with more than two values are modeled by multinomial logistic regression and, if the multiple categories are ordered , by ordinal logistic regression for example the proportional odds ordinal logistic model [2].

The logistic regression model itself simply models probability of output in terms of input and does not perform statistical classification it is not a classifier , though it can be used to make a classifier, for instance by choosing a cutoff value and classifying inputs with probability greater than the cutoff as one class, below the cutoff as the other; this is a common way to make a binary classifier.

Logistic regression is used in various fields, including machine learning, most medical fields, and social sciences. Conditional random fields , an extension of logistic regression to sequential data, are used in natural language processing.

Let us try to understand logistic regression by considering a logistic model with given parameters, then seeing how the coefficients can be estimated from data.

However in some cases it can be easier to communicate results by working in base 2, or base To be concrete, the model is.

A group of 20 students spends between 0 and 6 hours studying for an exam. How does the number of hours spent studying affect the probability of the student passing the exam?

The reason for using logistic regression for this problem is that the values of the dependent variable, pass and fail, while represented by "1" and "0", are not cardinal numbers.

The table shows the number of hours each student spent studying, and whether they passed 1 or failed 0. The graph shows the probability of passing the exam versus the number of hours studying, with the logistic regression curve fitted to the data.

These coefficients are entered in the logistic regression equation to estimate the odds probability of passing the exam:.

One additional hour of study is estimated to increase log-odds of passing by 1. Similarly, for a student who studies 4 hours, the estimated probability of passing the exam is 0.

Logistic regression can be binomial, ordinal or multinomial. Binomial or binary logistic regression deals with situations in which the observed outcome for a dependent variable can have only two possible types, "0" and "1" which may represent, for example, "dead" vs.

Multinomial logistic regression deals with situations where the outcome can have three or more possible types e. Ordinal logistic regression deals with dependent variables that are ordered.

In binary logistic regression, the outcome is usually coded as "0" or "1", as this leads to the most straightforward interpretation. Binary logistic regression is used to predict the odds of being a case based on the values of the independent variables predictors.

The odds are defined as the probability that a particular outcome is a case divided by the probability that it is a noninstance.

Like other forms of regression analysis , logistic regression makes use of one or more predictor variables that may be either continuous or categorical.

Unlike ordinary linear regression, however, logistic regression is used for predicting dependent variables that take membership in one of a limited number of categories treating the dependent variable in the binomial case as the outcome of a Bernoulli trial rather than a continuous outcome.

Given this difference, the assumptions of linear regression are violated. In particular, the residuals cannot be normally distributed.

In addition, linear regression may make nonsensical predictions for a binary dependent variable. What is needed is a way to convert a binary variable into a continuous one that can take on any real value negative or positive.

To do that, binomial logistic regression first calculates the odds of the event happening for different levels of each independent variable, and then takes its logarithm to create a continuous criterion as a transformed version of the dependent variable.

The logarithm of the odds is the logit of the probability, the logit is defined as follows:. Although the dependent variable in logistic regression is Bernoulli, the logit is on an unrestricted scale.

The logit of the probability of success is then fitted to the predictors. The predicted value of the logit is converted back into predicted odds, via the inverse of the natural logarithm — the exponential function.

In some applications, the odds are all that is needed. The assumption of linear predictor effects can easily be relaxed using techniques such as spline functions.

Logistic regression measures the relationship between the categorical dependent variable and one or more independent variables by estimating probabilities using a logistic function , which is the cumulative distribution function of logistic distribution.

Thus, it treats the same set of problems as probit regression using similar techniques, with the latter using a cumulative normal distribution curve instead.

Equivalently, in the latent variable interpretations of these two methods, the first assumes a standard logistic distribution of errors and the second a standard normal distribution of errors.

Logistic regression can be seen as a special case of the generalized linear model and thus analogous to linear regression. The model of logistic regression, however, is based on quite different assumptions about the relationship between the dependent and independent variables from those of linear regression.

In particular, the key differences between these two models can be seen in the following two features of logistic regression.

Second, the predicted values are probabilities and are therefore restricted to 0,1 through the logistic distribution function because logistic regression predicts the probability of particular outcomes rather than the outcomes themselves.

Logistic regression is an alternative to Fisher's method, linear discriminant analysis. The converse is not true, however, because logistic regression does not require the multivariate normal assumption of discriminant analysis.

If the standard normal distribution is used instead, it is a probit model. The estimation approach is explained below. An explanation of logistic regression can begin with an explanation of the standard logistic function.

It is easy to see that it satisfies:. This illustrates how the logit serves as a link function between the probability and the linear regression expression.

Given that the logit ranges between negative and positive infinity, it provides an adequate criterion upon which to conduct linear regression and the logit is easily converted back into the odds.

Logistic regression is an important machine learning algorithm. This leads to the intuition that by maximizing the log-likelihood of a model, you are minimizing the KL divergence of your model from the maximal entropy distribution.

Intuitively searching for the model that makes the fewest assumptions in its parameters. A widely used rule of thumb, the " one in ten rule ", states that logistic regression models give stable values for the explanatory variables if based on a minimum of about 10 events per explanatory variable EPV ; where event denotes the cases belonging to the less frequent category in the dependent variable.

However, there is considerable debate about the reliability of this rule, which is based on simulation studies and lacks a secure theoretical underpinning.

Others have found results that are not consistent with the above, using different criteria. A useful criterion is whether the fitted model will be expected to achieve the same predictive discrimination in a new sample as it appeared to achieve in the model development sample.

For that criterion, 20 events per candidate variable may be required. The regression coefficients are usually estimated using maximum likelihood estimation.

This process begins with a tentative solution, revises it slightly to see if it can be improved, and repeats this revision until no more improvement is made, at which point the process is said to have converged.

In some instances, the model may not reach convergence. Non-convergence of a model indicates that the coefficients are not meaningful because the iterative process was unable to find appropriate solutions.

A failure to converge may occur for a number of reasons: having a large ratio of predictors to cases, multicollinearity , sparseness , or complete separation.

In machine learning applications where logistic regression is used for binary classification, the MLE minimises the Cross entropy loss function.

More details can be found in the literature. Goodness of fit in linear regression models is generally measured using R 2.

Since this has no direct analog in logistic regression, various methods [31] : ch. In linear regression analysis, one is concerned with partitioning variance via the sum of squares calculations — variance in the criterion is essentially divided into variance accounted for by the predictors and residual variance.

In logistic regression analysis, deviance is used in lieu of a sum of squares calculations. In the above equation, D represents the deviance and ln represents the natural logarithm.

The log of this likelihood ratio the ratio of the fitted model to the saturated model will produce a negative value, hence the need for a negative sign.

D can be shown to follow an approximate chi-squared distribution. When assessed upon a chi-square distribution, nonsignificant chi-square values indicate very little unexplained variance and thus, good model fit.

Conversely, a significant chi-square value indicates that a significant amount of the variance is unexplained. Two measures of deviance are particularly important in logistic regression: null deviance and model deviance.

The null deviance represents the difference between a model with only the intercept which means "no predictors" and the saturated model.

The model deviance represents the difference between a model with at least one predictor and the saturated model.

Given that deviance is a measure of the difference between a given model and the saturated model, smaller values indicate better fit. If the model deviance is significantly smaller than the null deviance then one can conclude that the predictor or set of predictors significantly improved model fit.

This is analogous to the F -test used in linear regression analysis to assess the significance of prediction. This is the most analogous index to the squared multiple correlations in linear regression.

The highest this upper bound can be is 0. Logistic regression will always be heteroscedastic — the error variances differ for each value of the predicted score.

For each value of the predicted score there would be a different value of the proportionate reduction in error. This test is considered to be obsolete by some statisticians because of its dependence on arbitrary binning of predicted probabilities and relative low power.

After fitting the model, it is likely that researchers will want to examine the contribution of individual predictors. To do so, they will want to examine the regression coefficients.

In linear regression, the regression coefficients represent the change in the criterion for each unit change in the predictor.

Given that the logit is not intuitive, researchers are likely to focus on a predictor's effect on the exponential function of the regression coefficient — the odds ratio see definition.

In linear regression, the significance of a regression coefficient is assessed by computing a t test. In logistic regression, there are several different tests designed to assess the significance of an individual predictor, most notably the likelihood ratio test and the Wald statistic.

The likelihood-ratio test discussed above to assess model fit is also the recommended procedure to assess the contribution of individual "predictors" to a given model.

If the predictor model has significantly smaller deviance c. Although some common statistical packages e. SPSS do provide likelihood ratio test statistics, without this computationally intensive test it would be more difficult to assess the contribution of individual predictors in the multiple logistic regression case.

Alternatively, when assessing the contribution of individual predictors in a given model, one may examine the significance of the Wald statistic.

The Wald statistic, analogous to the t -test in linear regression, is used to assess the significance of coefficients.

The Wald statistic is the ratio of the square of the regression coefficient to the square of the standard error of the coefficient and is asymptotically distributed as a chi-square distribution.

Although several statistical packages e. When the regression coefficient is large, the standard error of the regression coefficient also tends to be larger increasing the probability of Type-II error.

The Wald statistic also tends to be biased when data are sparse. Suppose cases are rare. Then we might wish to sample them more frequently than their prevalence in the population.

For example, suppose there is a disease that affects 1 person in 10, and to collect our data we need to do a complete physical.

It may be too expensive to do thousands of physicals of healthy people in order to obtain data for only a few diseased individuals.

Thus, we may evaluate more diseased individuals, perhaps all of the rare outcomes. This is also retrospective sampling, or equivalently it is called unbalanced data.

As a rule of thumb, sampling controls at a rate of five times the number of cases will produce sufficient control data.

Logistic regression is unique in that it may be estimated on unbalanced data, rather than randomly sampled data, and still yield correct coefficient estimates of the effects of each independent variable on the outcome.

There are various equivalent specifications of logistic regression, which fit into different types of more general models.

These different specifications allow for different sorts of useful generalizations. The basic setup of logistic regression is as follows.

We are given a dataset containing N points. Each point i consists of a set of m input variables x 1, i The goal of logistic regression is to use the dataset to create a predictive model of the outcome variable.

As in linear regression, the outcome variables Y i are assumed to depend on the explanatory variables x 1, i As shown above in the above examples, the explanatory variables may be of any type : real-valued , binary , categorical , etc.

The main distinction is between continuous variables such as income, age and blood pressure and discrete variables such as sex or race.

Discrete variables referring to more than two possible choices are typically coded using dummy variables or indicator variables , that is, separate explanatory variables taking the value 0 or 1 are created for each possible value of the discrete variable, with a 1 meaning "variable does have the given value" and a 0 meaning "variable does not have that value".

For example, a four-way discrete variable of blood type with the possible values "A, B, AB, O" can be converted to four separate two-way dummy variables, "is-A, is-B, is-AB, is-O", where only one of them has the value 1 and all the rest have the value 0.

This allows for separate regression coefficients to be matched for each possible value of the discrete variable.

In a case like this, only three of the four dummy variables are independent of each other, in the sense that once the values of three of the variables are known, the fourth is automatically determined.

Thus, it is necessary to encode only three of the four possibilities as dummy variables. This also means that when all four possibilities are encoded, the overall model is not identifiable in the absence of additional constraints such as a regularization constraint.

Theoretically, this could cause problems, but in reality almost all logistic regression models are fitted with regularization constraints.

Formally, the outcomes Y i are described as being Bernoulli-distributed data, where each outcome is determined by an unobserved probability p i that is specific to the outcome at hand, but related to the explanatory variables.

This can be expressed in any of the following equivalent forms:. The basic idea of logistic regression is to use the mechanism already developed for linear regression by modeling the probability p i using a linear predictor function , i.

The particular model used by logistic regression, which distinguishes it from standard linear regression and from other types of regression analysis used for binary-valued outcomes, is the way the probability of a particular outcome is linked to the linear predictor function:.

This formulation expresses logistic regression as a type of generalized linear model , which predicts variables with various types of probability distributions by fitting a linear predictor function of the above form to some sort of arbitrary transformation of the expected value of the variable.

The intuition for transforming using the logit function the natural log of the odds was explained above.

Note that both the probabilities p i and the regression coefficients are unobserved, and the means of determining them is not part of the model itself.

They are typically determined by some sort of optimization procedure, e. The use of a regularization condition is equivalent to doing maximum a posteriori MAP estimation, an extension of maximum likelihood.

Regularization is most commonly done using a squared regularizing function , which is equivalent to placing a zero-mean Gaussian prior distribution on the coefficients, but other regularizers are also possible.

Whether or not regularization is used, it is usually not possible to find a closed-form solution; instead, an iterative numerical method must be used, such as iteratively reweighted least squares IRLS or, more commonly these days, a quasi-Newton method such as the L-BFGS method.

An equivalent formula uses the inverse of the logit function, which is the logistic function , i. The formula can also be written as a probability distribution specifically, using a probability mass function :.

The above model has an equivalent formulation as a latent-variable model. This formulation is common in the theory of discrete choice models and makes it easier to extend to certain more complicated models with multiple, correlated choices, as well as to compare logistic regression to the closely related probit model.

Then Y i can be viewed as an indicator for whether this latent variable is positive:. The choice of modeling the error variable specifically with a standard logistic distribution, rather than a general logistic distribution with the location and scale set to arbitrary values, seems restrictive, but in fact, it is not.

It must be kept in mind that we can choose the regression coefficients ourselves, and very often can use them to offset changes in the parameters of the error variable's distribution.

Similarly, an arbitrary scale parameter s is equivalent to setting the scale parameter to 1 and then dividing all regression coefficients by s.

Note that this predicts that the irrelevancy of the scale parameter may not carry over into more complex models where more than two choices are available.

It turns out that this formulation is exactly equivalent to the preceding one, phrased in terms of the generalized linear model and without any latent variables.

This can be shown as follows, using the fact that the cumulative distribution function CDF of the standard logistic distribution is the logistic function , which is the inverse of the logit function , i.

This formulation—which is standard in discrete choice models—makes clear the relationship between logistic regression the "logit model" and the probit model , which uses an error variable distributed according to a standard normal distribution instead of a standard logistic distribution.

Both the logistic and normal distributions are symmetric with a basic unimodal, "bell curve" shape. The only difference is that the logistic distribution has somewhat heavier tails , which means that it is less sensitive to outlying data and hence somewhat more robust to model mis-specifications or erroneous data.

This model has a separate latent variable and a separate set of regression coefficients for each possible outcome of the dependent variable.

The reason for this separation is that it makes it easy to extend logistic regression to multi-outcome categorical variables, as in the multinomial logit model.

In such a model, it is natural to model each possible outcome using a different set of regression coefficients.

It is also possible to motivate each of the separate latent variables as the theoretical utility associated with making the associated choice, and thus motivate logistic regression in terms of utility theory.

In terms of utility theory, a rational actor always chooses the choice with the greatest associated utility.

This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the model, which in turn makes it easy to consider various sorts of extensions.

See the example below. The choice of the type-1 extreme value distribution seems fairly arbitrary, but it makes the mathematics work out, and it may be possible to justify its use through rational choice theory.

It turns out that this model is equivalent to the previous model, although this seems non-obvious, since there are now two sets of regression coefficients and error variables, and the error variables have a different distribution.

In fact, this model reduces directly to the previous one with the following substitutions:. An intuition for this comes from the fact that, since we choose based on the maximum of two values, only their difference matters, not the exact values — and this effectively removes one degree of freedom.

Another critical fact is that the difference of two type-1 extreme-value-distributed variables is a logistic distribution, i.

As an example, consider a province-level election where the choice is between a right-of-center party, a left-of-center party, and a secessionist party e.

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**там**possible values "A, B, AB, O" can be converted to four separate two-way dummy variables, "is-A, is-B, is-AB, is-O", where only one of them has the value 1 and all the rest have the value 0. The likelihood-ratio test discussed above to assess model fit is also the recommended procedure to assess the contribution of individual "predictors" to a given Schlacht Um Midway 1976. This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the Vivien Westwood,

*там*in turn makes it easy to consider various sorts of extensions. The Cambridge Dictionary of Statistics. The assumption of linear predictor effects can Greys Anatomy B Team be relaxed using techniques such as spline functions. Thus, it is necessary to encode only three of the four possibilities as dummy variables. Berry, Verschluss J. A detailed history of the logistic regression is given in Cramer

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