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mid mathbf x alpha mathbf beta mathbf x where α r displaystyle alpha in mathbb r and β r n displaystyle mathbf beta in mathbb r n sometimes this is written more compactly as log e y x θ x displaystyle log operatorname e y mid mathbf x boldsymbol theta mathbf x where x displaystyle mathbf x is now an n 1 dimensional vector consisting of n independent variables concatenated to the number one here θ displaystyle theta is simply β displaystyle beta concatenated to α displaystyle alpha thus when given a poisson regression model θ displaystyle theta and an input vector x displaystyle mathbf x the predicted mean of the associated poisson distribution is given by e y x e θ x displaystyle operatorname e y mid mathbf x e boldsymbol theta mathbf x if y i displaystyle y_ i are independent observations with corresponding values x i displaystyle mathbf x _ i of the predictor variables then θ displaystyle theta can be estimated by maximum likelihood the maximum likelihood estimates lack a closed form expression and must be found by numerical methods the probability surface for maximum likelihood poisson regression is always concave making newton raphson or other gradient based methods appropriate estimation techniques interpretation of coefficients edit suppose we have a model with a single predictor that is n 1 displaystyle n 1 log e y x α β x displaystyle log operatorname e y mid mathbf x alpha beta x suppose we compute the predicted values at point y 2 x 2 displaystyle y_ 2 x_ 2 and y 1 x 1 displaystyle y_ 1 x_ 1 log e y 2 x 2 α β x 2 displaystyle log operatorname e y_ 2 mid x_ 2 alpha beta x_ 2 log e y 1 x 1 α β x 1 displaystyle log operatorname e y_ 1 mid x_ 1 alpha beta x_ 1 by subtracting the first from the second log e y 2 x 2 log e y 1 x 1 β x 2 x 1 displaystyle log operatorname e y_ 2 mid x_ 2 log operatorname e y_ 1 mid x_ 1 beta x_ 2 x_ 1 suppose now that x 2 x 1 1 displaystyle x_ 2 x_ 1 1 we obtain log e y 2 x 2 log e y 1 x 1 β displaystyle log operatorname e y_ 2 mid x_ 2 log operatorname e y_ 1 mid x_ 1 beta so the coefficient of the model is to be interpreted as the increase in the logarithm of the count of the outcome variable when the independent variable increases by 1 by applying the rules of logarithms log e y 2 x 2 e y 1 x 1 β displaystyle log left dfrac operatorname e y_ 2 mid x_ 2 operatorname e y_ 1 mid x_ 1 right beta e y 2 x 2 e y 1 x 1 e β displaystyle dfrac operatorname e y_ 2 mid x_ 2 operatorname e y_ 1 mid x_ 1 e beta e y 2 x 2 e β e y 1 x 1 displaystyle operatorname e y_ 2 mid x_ 2 e beta operatorname e y_ 1 mid x_ 1 that is when the independent variable increases by 1 the outcome variable is multiplied by the exponentiated coefficient the exponentiated coefficient is also called the incidence ratio average partial effect edit often the object of interest is the average partial effect or average marginal effect e y x x displaystyle frac partial e y x partial x which is interpreted as the change in the outcome y displaystyle y for a one unit change in the independent variable x displaystyle x the average partial effect in the poisson model for a continuous x displaystyle x can be shown to be 2 e y x x exp θ x β displaystyle frac partial e y x partial x exp theta mathbb x beta this can be estimated using the coefficient estimates from the poisson model θ α β displaystyle hat theta hat alpha hat beta with the observed values of x displaystyle mathbb x maximum likelihood based parameter estimation edit given a set of parameters θ and an input vector x the mean of the predicted poisson distribution as stated above is given by λ e y x e θ x displaystyle lambda operatorname e y mid x e theta x and thus the poisson distribution s probability mass function is given by p y x θ λ y y e λ e y θ x e e θ x y displaystyle p y mid x theta frac lambda y y e lambda frac e y theta x e e theta x y now suppose we are given a data set consisting of m vectors x i r n 1 i 1 m displaystyle x_ i in mathbb r n 1 i 1 ldots m along with a set of m values y 1 y m n displaystyle y_ 1 ldots y_ m in mathbb n then for a given set of parameters θ the probability of attaining this particular set of data is given by p y 1 y m x 1 x m θ i 1 m e y i θ x i e e θ x i y i displaystyle p y_ 1 ldots y_ m mid x_ 1 ldots x_ m theta prod _ i 1 m frac e y_ i theta x_ i e e theta x_ i y_ i by the method of maximum likelihood we wish to find the set of parameters θ that makes this probability as large as possible to do this the equation is first rewritten as a likelihood function in terms of θ l θ x y i 1 m e y i θ x i e e θ x i y i displaystyle l theta mid x y prod _ i 1 m frac e y_ i theta x_ i e e theta x_ i y_ i note that the expression on the right hand side has not actually changed a formula in this form is typically difficult to work with instead one uses the log likelihood ℓ θ x y log l θ x y i 1 m y i θ x i e θ x i log y i displaystyle ell theta mid x y log l theta mid x y sum _ i 1 m left y_ i theta x_ i e theta x_ i log y_ i right notice that the parameters θ only appear in the first two terms of each term in the summation therefore given that we are only interested in finding the best value for θ we may drop the y i and simply write ℓ θ x y i 1 m y i θ x i e θ x i displaystyle ell theta mid x y sum _ i 1 m left y_ i theta x_ i e theta x_ i right to find a maximum we need to solve an equation ℓ θ x y θ 0 displaystyle frac partial ell theta mid x y partial theta 0 which has no closed form solution however the negative log likelihood ℓ θ x y displaystyle ell theta mid x y is a convex function and so standard convex optimization techniques such as gradient descent can be applied to find the optimal value of θ poisson regression in practice edit poisson regression may be appropriate when the dependent variable is a count for instance of events such as the arrival of a telephone call at a call centre 3 the events must be independent in the sense that the arrival of one call will not make another more or less likely but the probability per unit time of events is understood to be related to covariates such as time of day exposure and offset edit poisson regression may also be appropriate for rate data where the rate is a count of events divided by some measure of that unit s exposure a particular unit of observation 4 for example biologists may count the number of tree species in a forest events would be tree observations exposure would be unit area and rate would be the number of species per unit area demographers may model death rates in geographic areas as the count of deaths divided by person years more generally event rates can be calculated as events per unit time which allows the observation window to vary for each unit in these examples exposure is respectively unit area person years and unit time in poisson regression this is handled as an offset if the rate is count exposure multiplying both sides of the equation by exposure moves it to the right side of the equation when both sides of the equation are then logged the final model contains log exposure as a term that is added to the regression coefficients this logged variable log exposure is called the offset variable and enters on the right hand side of the equation with a parameter estimate for log exposure constrained to 1 log e y x θ x displaystyle log operatorname e y mid x theta x which implies log e y x exposure log e y x log exposure θ x displaystyle log left frac operatorname e y mid x text exposure right log operatorname e y mid x log text exposure theta x log e y x θ x log exposure displaystyle log left operatorname e y mid x right theta x log text exposure offset in the case of a glm in r can be achieved using the offset function glm y offset log exposure x family poisson link log overdispersion and zero inflation edit a characteristic of the poisson distribution is that its mean is equal to its variance in certain circumstances it will be found that the observed variance is greater than the mean this is known as overdispersion and indicates that the model is not appropriate a common reason is the omission of relevant explanatory variables or dependent observations under some circumstances the problem of overdispersion can be solved by using quasi likelihood estimation or a negative binomial distribution instead 5 6 ver hoef and boveng described the difference between quasi poisson also called overdispersion with quasi likelihood and negative binomial equivalent to gamma poisson as follows if e y μ the quasi poisson model assumes var y θμ while the gamma poisson assumes var y μ 1 κμ where θ is the quasi poisson overdispersion parameter and κ is the shape parameter of the negative binomial distribution for both models parameters are estimated using iteratively reweighted least squares for quasi poisson the weights are μ θ for negative binomial the weights are μ 1 κμ with large μ and substantial extra poisson variation the negative binomial weights are capped at 1 κ ver hoef and boveng discussed an example where they selected between the two by plotting mean squared residuals vs the mean 7 another common problem with poisson regression is excess zeros if there are two processes at work one determining whether there are zero events or any events and a poisson process determining how many events there are there will be more zeros than a poisson regression would predict an example would be the distribution of cigarettes smoked in an hour by members of a group where some individuals are non smokers other generalized linear models such as the negative binomial model or zero inflated model may function better in these cases on the contrary underdispersion may pose an issue for parameter estimation 8 use in survival analysis edit poisson regression creates proportional hazards models one class of survival analysis see proportional hazards models for descriptions of cox models extensions edit regularized poisson regression edit when estimating the parameters for poisson regression one typically tries to find values for θ that maximize the likelihood of an expression of the form i 1 m log p y i e θ x i displaystyle sum _ i 1 m log p y_ i e theta x_ i where m is the number of examples in the data set and p y i e θ x i displaystyle p y_ i e theta x_ i is the probability mass function of the poisson distribution with the mean set to e θ x i displaystyle e theta x_ i regularization can be added to this optimization problem by instead maximizing 9 i 1 m log p y i e θ x i λ θ 2 2 displaystyle sum _ i 1 m log p y_ i e theta x_ i lambda left theta right _ 2 2 for some positive constant λ displaystyle lambda this technique similar to ridge regression can reduce overfitting see also edit zero inflated model poisson distribution fixed effect poisson model partial likelihood methods for panel data pooled qmle for poisson models control function econometrics endogeneity in poisson regression references edit nelder j a 1974 log linear models for contingency tables a generalization of classical least squares journal of the royal statistical society series c applied statistics 23 3 pp 323 329 doi 10 2307 2347125 jstor 2347125 wooldridge jeffrey 2010 econometric analysis of cross section and panel data 2nd ed cambridge massachusetts the mit press p 726 greene william h 2003 econometric analysis fifth ed prentice hall pp 740 752 isbn 978 0130661890 frome edward l 1983 the analysis of rates using poisson regression models biometrics 39 3 pp 665 674 doi 10 2307 2531094 jstor 2531094 paternoster r brame r 1997 multiple routes to delinquency a test of developmental and general theories of crime criminology 35 49 84 doi 10 1111 j 1745 9125 1997 tb00870 x eissn 1745 9125 issn 0011 1384 berk r macdonald j 2008 overdispersion and poisson regression journal of quantitative criminology 24 3 269 284 doi 10 1007 s10940 008 9048 4 s2cid 121273486 ver hoef jay m boveng peter l 2007 01 01 quasi poisson vs negative binomial regression how should we model overdispersed count data ecology 88 11 2766 2772 bibcode 2007ecol 88 2766v doi 10 1890 07 0043 1 pmid 18051645 retrieved 2016 09 01 schwarzenegger rafael quigley john walls lesley 23 november 2021 is eliciting dependency worth the effort a study for the multivariate poisson gamma probability model proceedings of the institution of mechanical engineers part o journal of risk and reliability 237 5 5 doi 10 1177 1748006x211059417 perperoglou aris 2011 09 08 fitting survival data with penalized poisson regression statistical methods applications 20 4 springer nature 451 462 doi 10 1007 s10260 011 0172 1 issn 1618 2510 s2cid 10883925 further reading edit cameron a c trivedi p k 1998 regression analysis of count data cambridge university press isbn 978 0 521 63201 0 christensen ronald 1997 log linear models and logistic regression springer texts in statistics second ed new york springer verlag isbn 978 0 387 98247 2 mr 1633357 gouriéroux christian 2000 the econometrics of discrete positive variables the poisson model econometrics of qualitative dependent variables new york cambridge university press pp 270 83 isbn 978 0 521 58985 7 greene william h 2008 models for event counts and duration econometric analysis 8th ed upper saddle river prentice hall pp 906 944 isbn 978 0 13 600383 0 dead link hilbe j m 2007 negative binomial regression cambridge university press isbn 978 0 521 85772 7 jones andrew m et al 2013 models for count data applied health economics london routledge pp 295 341 isbn 978 0 415 67682 3 myers raymond h et al 2010 logistic and poisson regression models generalized linear models with applications in engineering and the sciences second ed new jersey wiley pp 176 183 isbn 978 0 470 45463 3 v t e statistics outline index descriptive statistics continuous data center mean arithmetic arithmetic geometric contraharmonic cubic generalized power geometric harmonic heronian heinz lehmer median mode dispersion average absolute deviation coefficient of variation interquartile range percentile range standard deviation variance shape central limit theorem moments kurtosis l moments skewness count data index of dispersion summary tables contingency table frequency distribution grouped data dependence partial correlation pearson product moment correlation rank correlation kendall s τ spearman s ρ scatter plot graphics bar chart biplot box plot control chart correlogram fan chart forest plot histogram pie chart q q plot radar chart run chart scatter plot stem and leaf display violin plot heatmap scatter plot matrix ecdf plot line chart statistical data processing transformations data transformation log transformation power transform box cox transformation yeo johnson transformation variance stabilizing transformation anscombe transform fisher transformation scaling and normalization feature scaling normalization standardization z score min max normalization unit vector normalization data cleaning data cleaning outlier winsorizing truncation missing data data reduct...