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In statistics, binomial regression is a regression analysis technique in which the response (often referred to as Y) has a binomial distribution: it is the number of successes in a series of independent Bernoulli trials, where each trial has probability of success . In binomial regression, the probability of a success is related to explanatory variables: the corresponding concept in ordinary regression is to relate the mean value of the unobserved response to explanatory variables.
Binomial regression is closely related to binary regression: if the response is a binary variable (two possible outcomes), then it can be considered as a binomial distribution with trial by considering one of the outcomes as "success" and the other as "failure", counting the outcomes as either 1 or 0: counting a success as 1 success out of 1 trial, and counting a failure as 0 successes out of 1 trial. Binomial regression models are essentially the same as binary choice models, one type of discrete choice model. The primary difference is in the theoretical motivation.
In machine learning, binomial regression is considered a special case of probabilistic classification, and thus a generalization of binary classification.
In one published example of an application of binomial regression,the details were as follows. The observed outcome variable was whether or not a fault occurred in an industrial process. There were two explanatory variables: the first was a simple two-case factor representing whether or not a modified version of the process was used and the second was an ordinary quantitative variable measuring the purity of the material being supplied for the process.
Discrete choice models are motivated using utility theory so as to handle various types of correlated and uncorrelated choices, while binomial regression models are generally described in terms of the generalized linear model, an attempt to generalize various types of linear regression models. As a result, discrete choice models are usually described primarily with a latent variable indicating the "utility" of making a choice, and with randomness introduced through an error variable distributed according to a specific probability distribution. Note that the latent variable itself is not observed, only the actual choice, which is assumed to have been made if the net utility was greater than 0. Binary regression models, however, dispense with both the latent and error variable and assume that the choice itself is a random variable, with a link function that transforms the expected value of the choice variable into a value that is then predicted by the linear predictor. It can be shown that the two are equivalent, at least in the case of binary choice models: the link function corresponds to the quantile function of the distribution of the error variable, and the inverse link function to the cumulative distribution function (CDF) of the error variable. The latent variable has an equivalent if one imagines generating a uniformly distributed number between 0 and 1, subtracting from it the mean (in the form of the linear predictor transformed by the inverse link function), and inverting the sign. One then has a number whose probability of being greater than 0 is the same as the probability of success in the choice variable, and can be thought of as a latent variable indicating whether a 0 or 1 was chosen.
The results are assumed to be binomially distributed.They are often fitted as a generalised linear model where the predicted values μ are the probabilities that any individual event will result in a success. The likelihood of the predictions is then given by
where 1A is the indicator function which takes on the value one when the event A occurs, and zero otherwise: in this formulation, for any given observation yi, only one of the two terms inside the product contributes, according to whether yi=0 or 1. The likelihood function is more fully specified by defining the formal parameters μi as parameterised functions of the explanatory variables: this defines the likelihood in terms of a much reduced number of parameters. Fitting of the model is usually achieved by employing the method of maximum likelihood to determine these parameters. In practice, the use of a formulation as a generalised linear model allows advantage to be taken of certain algorithmic ideas which are applicable across the whole class of more general models but which do not apply to all maximum likelihood problems.
Models used in binomial regression can often be extended to multinomial data.
There are many methods of generating the values of μ in systematic ways that allow for interpretation of the model; they are discussed below.
There is a requirement that the modelling linking the probabilities μ to the explanatory variables should be of a form which only produces values in the range 0 to 1. Many models can be fitted into the form
Here η is an intermediate variable representing a linear combination, containing the regression parameters, of the explanatory variables. The function g is the cumulative distribution function (cdf) of some probability distribution. Usually this probability distribution has a support from minus infinity to plus infinity so that any finite value of η is transformed by the function g to a value inside the range 0 to 1.
In the case of logistic regression, the link function is the log of the odds ratio or logistic function. In the case of probit, the link is the cdf of the normal distribution. The linear probability model is not a proper binomial regression specification because predictions need not be in the range of zero to one; it is sometimes used for this type of data when the probability space is where interpretation occurs or when the analyst lacks sufficient sophistication to fit or calculate approximate linearizations of probabilities for interpretation.
A binary choice model assumes a latent variable Un, the utility (or net benefit) that person n obtains from taking an action (as opposed to not taking the action). The utility the person obtains from taking the action depends on the characteristics of the person, some of which are observed by the researcher and some are not:
where is a set of regression coefficients and is a set of independent variables (also known as "features") describing person n, which may be either discrete "dummy variables" or regular continuous variables. is a random variable specifying "noise" or "error" in the prediction, assumed to be distributed according to some distribution. Normally, if there is a mean or variance parameter in the distribution, it cannot be identified, so the parameters are set to convenient values — by convention usually mean 0, variance 1.
The person takes the action, yn = 1, if Un > 0. The unobserved term, εn, is assumed to have a logistic distribution.
The specification is written succinctly as:
Let us write it slightly differently:
Here we have made the substitution en = −εn. This changes a random variable into a slightly different one, defined over a negated domain. As it happens, the error distributions we usually consider (e.g. logistic distribution, standard normal distribution, standard Student's t-distribution, etc.) are symmetric about 0, and hence the distribution over en is identical to the distribution over εn.
Denote the cumulative distribution function (CDF) of as and the quantile function (inverse CDF) of as
Since is a Bernoulli trial, where we have
Note that this is exactly equivalent to the binomial regression model expressed in the formalism of the generalized linear model.
If i.e. distributed as a standard normal distribution, then
which is exactly a probit model.
If i.e. distributed as a standard logistic distribution with mean 0 and scale parameter 1, then the corresponding quantile function is the logit function, and
which is exactly a logit model.
Note that the two different formalisms — generalized linear models (GLM's) and discrete choice models — are equivalent in the case of simple binary choice models, but can be extended if differing ways:
A latent variable model involving a binomial observed variable Y can be constructed such that Y is related to the latent variable Y* via
The latent variable Y* is then related to a set of regression variables X by the model
This results in a binomial regression model.
The variance of ϵ can not be identified and when it is not of interest is often assumed to be equal to one. If ϵ is normally distributed, then a probit is the appropriate model and if ϵ is log-Weibull distributed, then a logit is appropriate. If ϵ is uniformly distributed, then a linear probability model is appropriate.
In statistics, the logistic model is used to model the probability of a certain class or event existing such as pass/fail, win/lose, alive/dead or healthy/sick. This can be extended to model several classes of events such as determining whether an image contains a cat, dog, lion, etc. Each object being detected in the image would be assigned a probability between 0 and 1, with a sum of one.
In probability theory and statistics, the logistic distribution is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails. The logistic distribution is a special case of the Tukey lambda distribution.
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In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more independent variables. The most common form of regression analysis is linear regression, in which a researcher finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters or estimate the conditional expectation across a broader collection of non-linear models.
In statistics, a confidence region is a multi-dimensional generalization of a confidence interval. It is a set of points in an n-dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem, although other shapes can occur.
In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model.
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