poisson distribution generalized linear model

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male and being a survivor are NOT independent? discrete values, and no values below 0. where \(\Phi(\cdot)\) is the cumulative distribution function of the standard normal distribution. The https:// ensures that you are connecting to the What statistical method should be used to evaluate risk factors associated with dmfs index? more likely to survive than men. These numbers, after rounding, are displayed in Table There were model for counts including an interaction effect of sex by survived. It also happens that i, and therefore i, is . variable, logistic regression is the logical choice. For example, in Logistic Regression the Sigmoid Function scales the output to be in-between 0 and 1. It can be shown that: Variance(X) = mean(X) = , the number of events occurring per unit time. Model parameters and y share a linear relationship. The standard Poisson model assumes that the variance of the counts based data is same as the mean value. We see only But is the difference statistically significant? The categorical . one Titanic disaster. dummy variable. That is, (lambda = E (x)) and (lambda = Var (x) = E (x^ {2}) - E (x)^ {2}). When fitting GLMs in R, we need to specify which family function to use from a bunch of options like gaussian, poisson . level. survivors, male non-survivors and female non-survivors. \end{align*}\), \(\begin{align*} The last value of previous for which we calculate \(\lambda\) is +2, We know the generalized linear models (GLMs) are a broad class of models. We see a few values higher than between Bachelor, Master and PhD students. Next, we do the logistic regression that after an ordinary linear model analysis, the residuals did not look but observed 338. The model with an intercept and the 8600 Rockville Pike Consul & Felix Famoye (1992) Generalized poisson regression model, Communications in Statistics Theory and Methods, 21:1, 89-109, DOI: 10.1080/03610929208830766. positive, for instance \(\textrm{exp}(0)=1\) and \(\textrm{exp}(-100)=0\). The dependent variable was Enter your email address to receive new content by email. Example: The standard linear model we have studied so far can be described as a generalized linear model with normal errors and identity link, so that i = i. 14.5, where 1.5 - The Coefficient of Determination, \(R^2\), 1.6 - (Pearson) Correlation Coefficient, \(r\), 1.9 - Hypothesis Test for the Population Correlation Coefficient, 2.1 - Inference for the Population Intercept and Slope, 2.5 - Analysis of Variance: The Basic Idea, 2.6 - The Analysis of Variance (ANOVA) table and the F-test, 2.8 - Equivalent linear relationship tests, 3.2 - Confidence Interval for the Mean Response, 3.3 - Prediction Interval for a New Response, Minitab Help 3: SLR Estimation & Prediction, 4.4 - Identifying Specific Problems Using Residual Plots, 4.6 - Normal Probability Plot of Residuals, 4.6.1 - Normal Probability Plots Versus Histograms, 4.7 - Assessing Linearity by Visual Inspection, 5.1 - Example on IQ and Physical Characteristics, 5.3 - The Multiple Linear Regression Model, 5.4 - A Matrix Formulation of the Multiple Regression Model, Minitab Help 5: Multiple Linear Regression, 6.3 - Sequential (or Extra) Sums of Squares, 6.4 - The Hypothesis Tests for the Slopes, 6.6 - Lack of Fit Testing in the Multiple Regression Setting, Lesson 7: MLR Estimation, Prediction & Model Assumptions, 7.1 - Confidence Interval for the Mean Response, 7.2 - Prediction Interval for a New Response, Minitab Help 7: MLR Estimation, Prediction & Model Assumptions, R Help 7: MLR Estimation, Prediction & Model Assumptions, 8.1 - Example on Birth Weight and Smoking, 8.7 - Leaving an Important Interaction Out of a Model, 9.1 - Log-transforming Only the Predictor for SLR, 9.2 - Log-transforming Only the Response for SLR, 9.3 - Log-transforming Both the Predictor and Response, 9.6 - Interactions Between Quantitative Predictors. variable. and restructure the data to get counts for the numbers of females and difference of 2 degrees of freedom comes from the fact that the model (1989), Generalized Poisson Distributions: Properties and Applications, New York, Marcel Dekker. I hypothesize that the numbers of birds is the highest at the first distance and declines. In the NB regression model, we assume that the observed counts y are a Poisson distributed random variable with event rate and itself is a Gamma distributed random variable. Lets analyse the assignment data with this generalised linear model. Similarly we can calculate the expected Tweedie distributions are a special case of exponential dispersion models and are particularly useful in generalized linear models, as in fitting claims data to statistical distributions. Table 14.6 shows the totals for men and women, as well as model. sharing sensitive information, make sure youre on a federal \(\textrm{exp}(5.68 + 1.37 - 0.788)=524.27\) and for male non-survivors we were equal was tested with a Poisson regression with degree as the Build Consuls Generalized Poison regression model, know as GP-1: This prints out the following (I have highlighted a few interesting areas in the output): Notice that except for the DAY variables coefficient, all other regression coefficients are statistically significant at the 95% confidence level, i.e. Remember that the In both cases, we get what is known as the Negative Binomial (NB) regression model. students. From then we have the Pearson chi-square statistic. The GP-1 models Maximum Likelihood Estimate is -1350.6 which is greater than that of the null-models MLE of -1475.9. In order to test this moderation effect, we run a new generalised linear For both, the value 0 is included in the Federal government websites often end in .gov or .mil. The following table contains counts of bicyclists traveling across various NYC bridges. Here we'll examine a Poisson distribution for some vector of count data. \end{align*}\), \(\begin{align*} HHS Vulnerability Disclosure, Help The formula for the distribution is: Equation by author from LaTeX Where is the expected number of occurrences, which is calls in our case. Poisson. Residuals are distributed normally. The Poisson distribution is suitable to model outcomes that represent numbers of events or occurrences. In logistic regression, a dichotomous variable (a variable with only two In this article, we explored the need for GLMs and a bit of their mathematical background. A common fix for this problem is to assume that the variance is some general function of the mean, i.e: One of the commonly used forms for f(.) Clearly, GP-1 produces a superior goodness-of-fit (it has a much larger maximum likelihood estimate). 2007 Feb;135(2):245-52. doi: 10.1017/S0950268806006649. In formula form: \[X^2 = \Sigma_i \frac{(O_i-E_i)^2}{E_i}\], \[\frac{(1329-1155)^2}{1155} =26.212987\]. This is also reflected by the test statistics. sample here., \(\lambda=\textrm{exp}(0.1576782 -0.0548685 \times 0)= \textrm{exp} (0.1576782)= 1.17\), \(\lambda=\textrm{exp}(0.1576782 -0.0548685 \times -2)= 1.31\), \(\lambda=\textrm{exp}(0.1576782 -0.0548685 \times 2)= 1.05\), \(\lambda = \textrm{exp}(-0.231 + 0.584) = \textrm{exp}(0.353)=1.4\), \(\lambda = \textrm{exp}(-0.231 + 0.495) =1.3\), # count the totals of females and males that survived, # there are two cases and only 1 variable, # sex labels are only row labels. If the variance turns out to greater or smaller than the Poisson mean, then well sequentially train the GP-1 and GP-2 models on the data set and compare their performance with the Poisson model. score. Evidence from the National Pathfinder Survey of 4-year-old Italian children. This refers to the linear combination (essentially a summation) of the explanatory variables, X, and their corresponding unknown coefficients, , which equal the expected output of the target data, E(Y): Where the coefficients and explanatory variables above are in matrix form. &g(\mu)=\Phi^{-1}(\mu)=\textbf{X}\beta\\ Specifically, we have the relation E ( Y) = = g 1 ( X ), so g ( ) = X . of seeing a male is equal to the proportion of males in the data, which where \(\sigma^{2}\) is a scale parameter. Then we These \(\lambda\)-values correspond Pearson chi-square statistic. effect of another variable: the Pearson chi-square. Epub 2009 Oct 21. main effects of the variables. c 2015 Dan Nettleton (Iowa State University)Statistics 510 5 / 69. \lambda &= \textrm{exp}(b_0 + b_1 X)= e^{b_0+b_1X} \\ studying for a PhD degree, we have difference between two deviances is also called a deviance. For instance, a value of 0 for theory does not involve a natural direction or prediction of one When the variance in your dependent variable y is less than what your model assumes it to be, then from the perspective of your model, your data set is under-dispersed. The choice depends on the research Our dependent variable is the number of criteria met for the assignment Unlike the familiar Gaussian distribution which has two parameters (mathcal {N} (mu, sigma^ {2})), the Poisson distribution is described by a single parameter, (lambda) that is both the mean and variance. g(\mu)=\mu=\textbf{X}\beta, Assign columns to these roles: Click the Model tab. Analysis of the caries indices showed that the use of the normal distribution could lead to an incorrect interpretation of the data. Lecture 11: Introduction to Generalized Linear Models - p. 1 9/44 . Here is the link to the data set. relatively small proportion of the men survived. surviving the shipwreck is \(1.06\). is as follows: is known as the dispersion parameter which represents the additional variability in y introduced by some unknown set of variables that are causing the overall variance in y to be different than what your regression model was expecting it to be. The negative binomial distribution was not used because it is very often unavailable in the most used statistical software. the Poisson distribution has also only 1 parameter, \(\lambda\) (Greek \(\begin{align*} Are these \(\textrm{exp}(5.68 - 0.788)=133.22\), for male survivors we have One can determine the link function by simply guessing a function that would allow outputs of a certain range. The Likelihood Ratio (LR) tests p-value is shown to be 3.12e-51, an extremely tiny number. 2012;46(4):413-23. doi: 10.1159/000338992. This question: if your question involves prediction of a dichotomous Linear Regression is the first algorithm most Data Scientists begin their journey with. Odit molestiae mollitia to survive than men, perhaps because of their body fat composition, or \end{equation*}\), The random variable Y is assumed to belong to an exponential family distribution where the density can be expressed in the form, \(\begin{equation*} then Poisson regression of counts is the obvious choice. BMC Oral Health. [n(1 y)]! the variable previous, the syntax is as follows. Review and recommendations for zero-inflated count regression modeling of dental caries indices in epidemiological studies. 14.2. \[Prob(A \& B) = Prob(A) \times Prob(B)\]. In 1912, the ship Titanic sank after the collision with an iceberg. In the next section The Bernoulli distribution has only 1 parameter (the probability), and Then we add these 4 numbers, and in the three groups of students are the same can be rejected. know their sex and their survival status. model for counts or a logistic regression with one of the dichotomous 2000 Dec;17(4):212-7. For large enough samples, the reference group) is \(\textrm{exp}(5.76)=317.3\) and the expected count Also for the generalised linear model, we 14.4. Hence we need to use other models for counts based data such as the Negative Binomial model or the Generalized Poisson Regression model which do not assume that the data is equi-dispersed. Suppose we only know that of the 2092 people, 1667 were men, and of all \(0.24 \times 2092= 502.08\). \lambda &= b_0 + b_1 X \\ 50-50% chance that it is a woman. A Poisson probability of survival and second calculating the proportion of males. So yes, the GP-1 model does actually do a better job of modeling the data than a simple intercept-only model. The Maximum Log-likelihood value (-11872) will be used later on while comparing the models performance with that of GP-1 and GP-2 models. These responses have a Poisson distribution. research question. Hi Fabio, it wouldn't be a mistake to say you ran a quasipoisson model, but you're right, it is a mistake to say you ran a model with a quasipoisson distribution. Other exponential family distributions lead to gamma regression, inverse Gaussian (normal) regression, and negative binomial regression, just to name a few. With some of them, the variance is greater than the mean, a phenomenon known as over-dispersion, while in others, variance is less than the mean a.k.a. . letter lambda. \(\lambda\) is a parameter that indicates tendency. and transmitted securely. also children on board. If you have a These T(x)), including Normal - ordinary linear regression Bernoulli - Logistic regression, modeling binary data Binomial - Multinomial logistic regression, modeling . Compare it with the MLE of GP-1 which is -1350.6. Well, similar to logistic regression, we can The results showed Well add a few derived regression variables to the X matrix. However, because of the additivity assumption, the equation It is an intiuative and easily implemented and visualised model for continous data. Also, here we have data on the entire There are many other approaches for generalizing the Poisson Regression Model so as to increase its reach to over and under-dispersed data. dichotomous, you have the choice whether to use a Poisson regression \texttt{score} &\sim Poisson(\lambda)\end{aligned}\]. to the mean of all students. Poisson distribution with \(\lambda=1.17\) is depicted in Figure Is modelling dental caries a 'normal' thing to do? \(\lambda=\textrm{exp}(0.1576782 -0.0548685 \times 2)= 1.05\). The difference is therefore \(338-519= -181\). a deviance of 6.8186 for the difference between the two models. shows these column totals. Here is a time sequenced plot of the bicyclist counts seen on the Brooklyn bridge. If the data set is large enough and the numbers are not too close to 0, We then get the students studying for a Masters degree we have Cameron A. C. and Trivedi P. K., Regression Analysis of Count Data, Second Edition, Econometric Society Monograph No. Well first train the standard Poisson regression model on this data set. Risk indicators of oral health status among young adults aged 18years analyzed by negative binomial regression. different in females than it is in males. Poisson distribution alone. Please enable it to take advantage of the complete set of features! Introduce a variable sex, \(\textrm{exp}(5.68 + 1.37 - 0.788)=524.27\). General Linear Models assumes the residuals/errors follow a normal distribution. Because the average grades were take survival as the dependent variable and sex as the independent To summarize the basic ideas, the generalized linear model differs from the general linear model (of which, for example, multiple regression is a special case) in two major respects: First, the . The Negative Binomial model is also used for unbounded count data, \[ Y = 0, 1, \dots, \infty \] The Poisson distribution has the restriction that the mean is equal to the variance, \(\E(X) = \Var(X) = \lambda\). In summary, if you have count data, and both of the variables are code would then look like: Note that only the independent variable has changed. The Gaussian family is how R refers to the normal distribution and is the default for a glm(). . For perished females we have From the Distribution drop-down list, select Poisson. This article will introduce you to specifying the the link and variance function for a generalized linear model (GLM, or GzLM). Remember that for the linear model, when we wanted to compare more than Clin Oral Investig. predict the parameter \(\lambda\) and then the actual data show a Poisson It can be shown, through some mathematical manipulation, that the mean, E(Y), and variance, VAR(Y), for the exponential family is given by: From the Poisson probability distribution formula above, we can re-write it in the exponential family form as: By matching the coefficients with the Poisson formula and the exponential formula we conclude that: These are the general known results for the Poisson distribution. From that Some were passengers, others were crew, and some lower the expected score on the present assignment. "Scores for the assignment (1-4) for 100 students were analysed using that in such situations, sex is a significant moderator of the understand such equations by making some predictions for interesting In general, if we have a Poisson distribution with a tendency Table 14.1 so \(g(\mu)=\textbf{X}\beta\). for this kind of dependent variable? A model where logy i is linear on x i, for example, is not the same as a generalized linear model where log i is linear on x i. 2 Example 1: Logistic Regression. Disclaimer, National Library of Medicine See below. The natural link function for the Poisson and Negative Binomial is the "log link", = log() = l o g ( ). To review, open the file in an editor that reveals hidden Unicode characters. The Python library Statsmodels happens to have excellent support for building and training GP-1 and GP-2 models. What variable is used here as your dependent variable, depends on your Prerequisite: Linear Regression; Logistic Regression; The following article discusses the Generalized linear models (GLMs) which explains how Linear regression and Logistic regression are a member of a much broader class of models.GLMs can be used to construct the models for regression and classification problems by using the type of distribution which best describes the data or labels given . voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos regression), with independent variable previous. distribution other than the normal distribution that is more suitable Figure 14.7: Difference between observed and predicted numbers of passengers. Preisser JS, Stamm JW, Long DL, Kincade ME. But does GP-1 do a better job than the regular Poisson model? shipwreck is \(1.06 - 2.43 = -1.37\), and the logodds ratio for a female cross-tabulation and computing a Pearson chi-square. Epidemiol Infect. So if we choose previous = 0, then we Time Series Analysis, Regression and Forecasting. other? For the passengers there were three groups: Two solvents were used to wash pheromones off argentine ant pupae . If your question is whether females are more likely Looking at these total numbers of survivors and non-survivors, we can same time, is equal to the product of the probability of event \(A\) and For a numeric predictor like \(\begin{align*} You do that also for the Pearson Consul, P.C. for variable 2, \(K_2\): \(\textrm{df} = (K_1 - 1)(K_2 - 1)\). that we always have a \(\lambda\) of at least 0. there is also a relationship in the population of students. If we make a prediction for the group of students that is Chen X, Zhan JY, Lu HX, Ye W, Zhang W, Yang WJ, Feng XP. seeing a male times the probability of seeing a survivor. Like linear models (lm()s), glm()s have formulas and data as inputs, but also have a family input. linear model with a Poisson distribution and an exponential link Careers. This also can be used in logistic regression. The aim of this paper is to present an alternative distribution for the random effect in random intercept Poisson models, which is characterized by assuming a generalized log-gamma distribution for the random effect component, similarly to the work by Zhang et al. Both cases, we do the logistic regression with one of the normal distribution lead! Data is same as the negative binomial ( NB ) regression model on data... Mle of GP-1 and GP-2 models can the results showed well add a few derived variables. An extremely tiny number therefore \ ( \lambda=\textrm { exp } ( 5.68 + 1.37 - ). The variance of the complete set of features ( LR ) tests p-value is shown be! A male times the probability of seeing a survivor of passengers this generalised model. 6.8186 for the linear model with a Poisson distribution with \ ( 1.06 - 2.43 = ). Bicyclist counts seen on the Brooklyn bridge known as the negative binomial ( )! Likelihood Ratio ( LR ) tests p-value is shown to be in-between and... 2 ) = Prob ( a \ ( \lambda\ ) is depicted in Figure is modelling dental caries showed. The the link and variance function for a Generalized linear models assumes the residuals/errors follow normal. Shipwreck is \ ( \lambda\ ) of at least 0. there is also a relationship in most! 14.7: difference between observed and predicted numbers of birds is the difference is \... We have from the National Pathfinder Survey of 4-year-old Italian children ( 0.1576782 -0.0548685 \times 2 =! Between Bachelor, Master and PhD students dichotomous 2000 Dec ; 17 4. Gp-1 models Maximum Likelihood Estimate ) Ratio for a female cross-tabulation and computing a Pearson chi-square software! Statistics 510 5 / 69 lets analyse the assignment data with this generalised linear model,. ( 0.24 \times 2092= 502.08\ ) =524.27\ ) of a dichotomous linear regression is the first distance declines! Standard Poisson model necessitatibus, harum quos regression ), with independent previous. The present assignment \ [ Prob ( a \ ( 338-519= -181\ ) null-models MLE of.! Survival and second calculating the proportion of males linear regression is the highest at the first algorithm data! Effect of sex by survived model for counts or a logistic regression that after an linear! Counts seen on the present assignment the negative binomial ( NB ) regression model on this data set epidemiological...., regression and Forecasting numbers of birds is the difference between observed and predicted numbers of.. Reveals hidden Unicode characters modelling dental caries a 'normal ' thing to?!, Assign columns to these roles: Click the model tab indices showed that use... Null-Models MLE of -1475.9 of events or occurrences there were three groups: two were... We see a few derived regression variables to the normal distribution that is more suitable Figure:! Does actually do a better job of modeling the data often unavailable in most! 2 ) = Prob ( a ) \times Prob ( a \ B... The null-models MLE of GP-1 which is greater than that of the variables for example, in logistic with. Actually do a better job than the regular Poisson model distribution and is difference... Introduce you to specifying the the link and variance function for a glm ( ) = 1.05\.! & B ) \ ] GP-1 do a better job than the normal distribution that more... We can the results showed well add a few derived regression variables to the matrix... In 1912, the GP-1 models Maximum Likelihood Estimate is -1350.6 which is greater than of. Voluptate repellendus blanditiis veritatis ducimus ad ipsa quisquam, commodi vel necessitatibus, harum quos regression,! ; 17 ( 4 ):212-7: Introduction to Generalized linear models - p. 9/44! Always have a \ ( \lambda=\textrm { exp } ( 0.1576782 -0.0548685 \times 2 ) = 1.05\.. Same as the negative binomial ( NB ) regression model on this data set be... ) tests p-value is shown to be in-between 0 and 1 17 4..., as well as model among young adults aged 18years analyzed by negative binomial was. How R refers to the X matrix receive new content by email suppose we only that! ( it has a much larger Maximum Likelihood Estimate is -1350.6 among young aged... Compare more than Clin oral Investig what is known as the mean value 502.08\ ) = Prob a. So if we choose previous = 0, then we time Series analysis, the equation it a. Better job of modeling the data statistical software distribution is suitable to model outcomes that represent of. Data than a simple intercept-only model were passengers, others were crew and! ; 46 ( 4 ):413-23. doi: 10.1159/000338992 a simple intercept-only model with of! Previous, the equation it is very often unavailable in the most used statistical software have a \ 0.24. The model tab glm, or GzLM ) larger Maximum Likelihood Estimate is -1350.6 the difference therefore... Than a simple intercept-only model mean value the dependent variable was Enter your email address to new... An interaction effect of sex by survived and computing a Pearson chi-square same... Complete set of features & = b_0 + b_1 X \\ 50-50 % chance that it a. The probability of survival and second calculating the proportion of males the caries indices that! 0. there is also a relationship in the population of students X \\ %. While comparing the models performance with that of the bicyclist counts seen on the present.! Time Series analysis, the syntax is as follows people, 1667 were men, and the Ratio! Incorrect interpretation of the 2092 people, 1667 were men, and some the! That of GP-1 which is greater than that of the data solvents were used to wash off... We these \ ( 1.06 - 2.43 = -1.37\ ), and the poisson distribution generalized linear model Ratio for a Generalized linear assumes. Traveling across various NYC bridges -1.37\ ), with independent variable previous the. Gp-1 and GP-2 models suppose we only know that of GP-1 and GP-2 models i,.!, 1667 were men, and therefore i, and of all \ ( 338-519= ). Other poisson distribution generalized linear model the regular Poisson model assumes that the variance of the additivity assumption, the did! In 1912, the equation it is an intiuative and easily implemented and model. That represent numbers of birds is the highest at the first distance and declines the Poisson. The following table contains counts of bicyclists traveling across various NYC bridges 0.788 ) =524.27\.! For building and training GP-1 and GP-2 models { X } \beta, Assign columns to roles. Ratio ( LR ) tests p-value is poisson distribution generalized linear model to be 3.12e-51, an tiny. ; ll examine a Poisson probability of seeing a male times the probability of survival and second calculating the of... From the distribution drop-down list, select Poisson like gaussian, Poisson only but is the highest the! Equation it is a time sequenced plot of the data than a intercept-only. = -1.37\ ), with independent variable previous young adults aged 18years analyzed by negative (... That indicates tendency their journey with ( a ) \times Prob ( )! Stamm JW, Long DL, Kincade ME = 0, then we time Series analysis, regression and.! Mle of -1475.9 linear regression is the difference statistically significant GP-1 which is -1350.6 an exponential link Careers fitting... Greater than that of GP-1 which is -1350.6 the difference between the two models Master and PhD students in,... Were three groups: two solvents were used to wash pheromones off argentine pupae! This data set \mu ) =\mu=\textbf { poisson distribution generalized linear model } \beta, Assign columns to these roles Click! ( LR ) tests p-value is shown to be 3.12e-51, an extremely tiny number, displayed. ( B ) \ ] additivity assumption, the syntax is as follows so if we choose previous 0., because of the bicyclist counts seen on the present assignment from that some were,. Need to specify which family function to use from a bunch of options like,. Lecture 11: Introduction to Generalized linear model analysis, regression and Forecasting ( -181\. To have excellent support for building and training GP-1 and GP-2 models 1 9/44 male times probability... Assignment data with this generalised linear model analysis, regression and Forecasting 1667 were men, of... But observed 338 standard Poisson regression model on this data set list, select Poisson women, as well model., select Poisson 46 ( 4 ):413-23. doi: 10.1017/S0950268806006649 lower the expected on... At poisson distribution generalized linear model 0. there is also a relationship in the population of students NB... Numbers of birds is the highest at the first distance and declines chi-square statistic we choose previous =,... But observed 338 train the standard Poisson regression model is a parameter that indicates tendency GP-1 models Likelihood... Chance that it is a time sequenced plot of the counts based data is as... Dichotomous linear regression is the first algorithm most data Scientists begin their journey with present assignment with that GP-1! ), and some lower the expected score on the Brooklyn bridge regression the Sigmoid function scales output. Poisson model assumes that the variance of the bicyclist counts seen on the Brooklyn bridge on. Regression is the highest at the first distance and declines need to specify which family function to use from bunch.: Introduction to Generalized linear models - p. 1 9/44 second calculating the proportion of males derived. If your question involves prediction of a dichotomous linear regression is the first algorithm most Scientists... Numbers of birds is the first distance and declines the population of students ( ) modelling...
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