Generalized Linear Models (Glms)
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An Introduction to Poisson Regression Russ Lavery, K&L Consulting Services, King of Prussia, PA, U.S.A
NESUG 2010 Statistics and Analysis An Animated Guide: An Introduction To Poisson Regression Russ Lavery, K&L Consulting Services, King of Prussia, PA, U.S.A. ABSTRACT: This paper will be a brief introduction to Poisson regression (theory, steps to be followed, complications and interpretation) via a worked example. It is hoped that this will increase motivation towards learning this useful statistical technique. INTRODUCTION: Poisson regression is available in SAS through the GENMOD procedure (generalized modeling). It is appropriate when: 1) the process that generates the conditional Y distributions would, theoretically, be expected to be a Poisson random process and 2) when there is no evidence of overdispersion and 3) when the mean of the marginal distribution is less than ten (preferably less than five and ideally close to one). THE POISSON DISTRIBUTION: The Poison distribution is a discrete Percent of observations where the random variable X is expected distribution and is appropriate for to have the value x, given that the Poisson distribution has a mean modeling counts of observations. of λ= P(X=x, λ ) = (e - λ * λ X) / X! Counts are observed cases, like the 0.4 count of measles cases in cities. You λ can simply model counts if all data 0.35 were collected in the same measuring 0.3 unit (e.g. the same number of days or 0.3 0.5 0.8 same number of square feet). 0.25 λ 1 0.2 3 You can use the Poisson Distribution = 5 for modeling rates (rates are counts 0.15 20 per unit) if the units of collection were 8 different. -
Bayesian Inference Chapter 4: Regression and Hierarchical Models
Bayesian Inference Chapter 4: Regression and Hierarchical Models Conchi Aus´ınand Mike Wiper Department of Statistics Universidad Carlos III de Madrid Master in Business Administration and Quantitative Methods Master in Mathematical Engineering Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 1 / 35 Objective AFM Smith Dennis Lindley We analyze the Bayesian approach to fitting normal and generalized linear models and introduce the Bayesian hierarchical modeling approach. Also, we study the modeling and forecasting of time series. Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 2 / 35 Contents 1 Normal linear models 1.1. ANOVA model 1.2. Simple linear regression model 2 Generalized linear models 3 Hierarchical models 4 Dynamic models Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 3 / 35 Normal linear models A normal linear model is of the following form: y = Xθ + ; 0 where y = (y1;:::; yn) is the observed data, X is a known n × k matrix, called 0 the design matrix, θ = (θ1; : : : ; θk ) is the parameter set and follows a multivariate normal distribution. Usually, it is assumed that: 1 ∼ N 0 ; I : k φ k A simple example of normal linear model is the simple linear regression model T 1 1 ::: 1 where X = and θ = (α; β)T . x1 x2 ::: xn Conchi Aus´ınand Mike Wiper Regression and hierarchical models Masters Programmes 4 / 35 Normal linear models Consider a normal linear model, y = Xθ + . A conjugate prior distribution is a normal-gamma distribution: -
The Hexadecimal Number System and Memory Addressing
C5537_App C_1107_03/16/2005 APPENDIX C The Hexadecimal Number System and Memory Addressing nderstanding the number system and the coding system that computers use to U store data and communicate with each other is fundamental to understanding how computers work. Early attempts to invent an electronic computing device met with disappointing results as long as inventors tried to use the decimal number sys- tem, with the digits 0–9. Then John Atanasoff proposed using a coding system that expressed everything in terms of different sequences of only two numerals: one repre- sented by the presence of a charge and one represented by the absence of a charge. The numbering system that can be supported by the expression of only two numerals is called base 2, or binary; it was invented by Ada Lovelace many years before, using the numerals 0 and 1. Under Atanasoff’s design, all numbers and other characters would be converted to this binary number system, and all storage, comparisons, and arithmetic would be done using it. Even today, this is one of the basic principles of computers. Every character or number entered into a computer is first converted into a series of 0s and 1s. Many coding schemes and techniques have been invented to manipulate these 0s and 1s, called bits for binary digits. The most widespread binary coding scheme for microcomputers, which is recog- nized as the microcomputer standard, is called ASCII (American Standard Code for Information Interchange). (Appendix B lists the binary code for the basic 127- character set.) In ASCII, each character is assigned an 8-bit code called a byte. -
How Prediction Statistics Can Help Us Cope When We Are Shaken, Scared and Irrational
EGU21-15219 https://doi.org/10.5194/egusphere-egu21-15219 EGU General Assembly 2021 © Author(s) 2021. This work is distributed under the Creative Commons Attribution 4.0 License. How prediction statistics can help us cope when we are shaken, scared and irrational Yavor Kamer1, Shyam Nandan2, Stefan Hiemer1, Guy Ouillon3, and Didier Sornette4,5 1RichterX.com, Zürich, Switzerland 2Windeggstrasse 5, 8953 Dietikon, Zurich, Switzerland 3Lithophyse, Nice, France 4Department of Management,Technology and Economics, ETH Zürich, Zürich, Switzerland 5Institute of Risk Analysis, Prediction and Management (Risks-X), SUSTech, Shenzhen, China Nature is scary. You can be sitting at your home and next thing you know you are trapped under the ruble of your own house or sucked into a sinkhole. For millions of years we have been the figurines of this precarious scene and we have found our own ways of dealing with the anxiety. It is natural that we create and consume prophecies, conspiracies and false predictions. Information technologies amplify not only our rational but also irrational deeds. Social media algorithms, tuned to maximize attention, make sure that misinformation spreads much faster than its counterpart. What can we do to minimize the adverse effects of misinformation, especially in the case of earthquakes? One option could be to designate one authoritative institute, set up a big surveillance network and cancel or ban every source of misinformation before it spreads. This might have worked a few centuries ago but not in this day and age. Instead we propose a more inclusive option: embrace all voices and channel them into an actual, prospective earthquake prediction platform (Kamer et al. -
Generalized Linear Models with Poisson Family: Applications in Ecology
UNIVERSITY OF ABOMEY- CALAVI *********** FACULTY OF AGRONOMIC SCIENCES *************** **************** Master Program in Statistics, Major Biostatistics 1st batch Generalized linear models with Poisson family: applications in ecology A thesis submitted to the Faculty of Agronomic Sciences in partial fulfillment of the requirements for the degree of the Master of Sciences in Biostatistics Presented by: LOKONON Enagnon Bruno Supervisor: Pr Romain L. GLELE KAKAÏ, Professor of Biostatistics and Forest estimation Academic year: 2014-2015 UNIVERSITE D’ABOMEY- CALAVI *********** FACULTE DES SCIENCES AGRONOMIQUES *************** ************** Programme de Master en Biostatistiques 1ère Promotion Modèles linéaires généralisés de la famille de Poisson : applications en écologie Mémoire soumis à la Faculté des Sciences Agronomiques pour obtenir le Diplôme de Master recherche en Biostatistiques Présenté par: LOKONON Enagnon Bruno Superviseur: Pr Romain L. GLELE KAKAÏ, Professeur titulaire de Biostatistiques et estimation forestière Année académique: 2014-2015 Certification I certify that this work has been achieved by LOKONON E. Bruno under my entire supervision at the University of Abomey-Calavi (Benin) in order to obtain his Master of Science degree in Biostatistics. Pr Romain L. GLELE KAKAÏ Professor of Biostatistics and Forest estimation i Acknowledgements This research was supported by WAAPP/PPAAO-BENIN (West African Agricultural Productivity Program/ Programme de Productivité Agricole en Afrique de l‟Ouest). This dissertation could only have been possible through the generous contributions of many people. First and foremost, I am grateful to my supervisor Pr Romain L. GLELE KAKAÏ, Professor of Biostatistics and Forest estimation who tirelessly played key role in orientation, scientific writing and mentoring during this research. In particular, I thank him for his prompt availability whenever needed. -
Reliability Engineering: Trends, Strategies and Best Practices
Reliability Engineering: Trends, Strategies and Best Practices WHITE PAPER September 2007 Predictive HCL’s Predictive Engineering encompasses the complete product life-cycle process, from concept to design to prototyping/testing, all the way to manufacturing. This helps in making decisions early Engineering in the design process, perfecting the product – thereby cutting down cycle time and costs, and Think. Design. Perfect! meeting set reliability and quality standards. Reliability Engineering: Trends, Strategies and Best Practices | September 2007 TABLE OF CONTENTS Abstract 3 Importance of reliability engineering in product industry 3 Current trends in reliability engineering 4 Reliability planning – an important task 5 Strength of reliability analysis 6 Is your reliability test plan optimized? 6 Challenges to overcome 7 Outsourcing of a reliability task 7 About HCL 10 © 2007, HCL Technologies. Reproduction Prohibited. This document is protected under Copyright by the Author, all rights reserved. Reliability Engineering: Trends, Strategies and Best Practices | September 2007 Abstract In reliability engineering, product industries now follow a conscious and planned approach to effectively address multiple issues related to product certification, failure returns, customer satisfaction, market competition and product lifecycle cost. Today, reliability professionals face new challenges posed by advanced and complex technology. Expertise and experience are necessary to provide an optimized solution meeting time and cost constraints associated with analysis and testing. In the changed scenario, the reliability approach has also become more objective and result oriented. This is well supported by analysis software. This paper discusses all associated issues including outsourcing of reliability tasks to a professional service provider as an alternate cost-effective option. Importance of reliability engineering in product industry The degree of importance given to the reliability of products varies depending on their criticality. -
Bayesian Methods: Review of Generalized Linear Models
Bayesian Methods: Review of Generalized Linear Models RYAN BAKKER University of Georgia ICPSR Day 2 Bayesian Methods: GLM [1] Likelihood and Maximum Likelihood Principles Likelihood theory is an important part of Bayesian inference: it is how the data enter the model. • The basis is Fisher’s principle: what value of the unknown parameter is “most likely” to have • generated the observed data. Example: flip a coin 10 times, get 5 heads. MLE for p is 0.5. • This is easily the most common and well-understood general estimation process. • Bayesian Methods: GLM [2] Starting details: • – Y is a n k design or observation matrix, θ is a k 1 unknown coefficient vector to be esti- × × mated, we want p(θ Y) (joint sampling distribution or posterior) from p(Y θ) (joint probabil- | | ity function). – Define the likelihood function: n L(θ Y) = p(Y θ) | i| i=1 Y which is no longer on the probability metric. – Our goal is the maximum likelihood value of θ: θˆ : L(θˆ Y) L(θ Y) θ Θ | ≥ | ∀ ∈ where Θ is the class of admissable values for θ. Bayesian Methods: GLM [3] Likelihood and Maximum Likelihood Principles (cont.) Its actually easier to work with the natural log of the likelihood function: • `(θ Y) = log L(θ Y) | | We also find it useful to work with the score function, the first derivative of the log likelihood func- • tion with respect to the parameters of interest: ∂ `˙(θ Y) = `(θ Y) | ∂θ | Setting `˙(θ Y) equal to zero and solving gives the MLE: θˆ, the “most likely” value of θ from the • | parameter space Θ treating the observed data as given. -
Generalized Linear Models
Generalized Linear Models A generalized linear model (GLM) consists of three parts. i) The first part is a random variable giving the conditional distribution of a response Yi given the values of a set of covariates Xij. In the original work on GLM’sby Nelder and Wedderburn (1972) this random variable was a member of an exponential family, but later work has extended beyond this class of random variables. ii) The second part is a linear predictor, i = + 1Xi1 + 2Xi2 + + ··· kXik . iii) The third part is a smooth and invertible link function g(.) which transforms the expected value of the response variable, i = E(Yi) , and is equal to the linear predictor: g(i) = i = + 1Xi1 + 2Xi2 + + kXik. ··· As shown in Tables 15.1 and 15.2, both the general linear model that we have studied extensively and the logistic regression model from Chapter 14 are special cases of this model. One property of members of the exponential family of distributions is that the conditional variance of the response is a function of its mean, (), and possibly a dispersion parameter . The expressions for the variance functions for common members of the exponential family are shown in Table 15.2. Also, for each distribution there is a so-called canonical link function, which simplifies some of the GLM calculations, which is also shown in Table 15.2. Estimation and Testing for GLMs Parameter estimation in GLMs is conducted by the method of maximum likelihood. As with logistic regression models from the last chapter, the generalization of the residual sums of squares from the general linear model is the residual deviance, Dm 2(log Ls log Lm), where Lm is the maximized likelihood for the model of interest, and Ls is the maximized likelihood for a saturated model, which has one parameter per observation and fits the data as well as possible. -
Heteroscedastic Errors
Heteroscedastic Errors ◮ Sometimes plots and/or tests show that the error variances 2 σi = Var(ǫi ) depend on i ◮ Several standard approaches to fixing the problem, depending on the nature of the dependence. ◮ Weighted Least Squares. ◮ Transformation of the response. ◮ Generalized Linear Models. Richard Lockhart STAT 350: Heteroscedastic Errors and GLIM Weighted Least Squares ◮ Suppose variances are known except for a constant factor. 2 2 ◮ That is, σi = σ /wi . ◮ Use weighted least squares. (See Chapter 10 in the text.) ◮ This usually arises realistically in the following situations: ◮ Yi is an average of ni measurements where you know ni . Then wi = ni . 2 ◮ Plots suggest that σi might be proportional to some power of 2 γ γ some covariate: σi = kxi . Then wi = xi− . Richard Lockhart STAT 350: Heteroscedastic Errors and GLIM Variances depending on (mean of) Y ◮ Two standard approaches are available: ◮ Older approach is transformation. ◮ Newer approach is use of generalized linear model; see STAT 402. Richard Lockhart STAT 350: Heteroscedastic Errors and GLIM Transformation ◮ Compute Yi∗ = g(Yi ) for some function g like logarithm or square root. ◮ Then regress Yi∗ on the covariates. ◮ This approach sometimes works for skewed response variables like income; ◮ after transformation we occasionally find the errors are more nearly normal, more homoscedastic and that the model is simpler. ◮ See page 130ff and check under transformations and Box-Cox in the index. Richard Lockhart STAT 350: Heteroscedastic Errors and GLIM Generalized Linear Models ◮ Transformation uses the model T E(g(Yi )) = xi β while generalized linear models use T g(E(Yi )) = xi β ◮ Generally latter approach offers more flexibility. -
Generalized Linear Models
CHAPTER 6 Generalized linear models 6.1 Introduction Generalized linear modeling is a framework for statistical analysis that includes linear and logistic regression as special cases. Linear regression directly predicts continuous data y from a linear predictor Xβ = β0 + X1β1 + + Xkβk.Logistic regression predicts Pr(y =1)forbinarydatafromalinearpredictorwithaninverse-··· logit transformation. A generalized linear model involves: 1. A data vector y =(y1,...,yn) 2. Predictors X and coefficients β,formingalinearpredictorXβ 1 3. A link function g,yieldingavectoroftransformeddataˆy = g− (Xβ)thatare used to model the data 4. A data distribution, p(y yˆ) | 5. Possibly other parameters, such as variances, overdispersions, and cutpoints, involved in the predictors, link function, and data distribution. The options in a generalized linear model are the transformation g and the data distribution p. In linear regression,thetransformationistheidentity(thatis,g(u) u)and • the data distribution is normal, with standard deviation σ estimated from≡ data. 1 1 In logistic regression,thetransformationistheinverse-logit,g− (u)=logit− (u) • (see Figure 5.2a on page 80) and the data distribution is defined by the proba- bility for binary data: Pr(y =1)=y ˆ. This chapter discusses several other classes of generalized linear model, which we list here for convenience: The Poisson model (Section 6.2) is used for count data; that is, where each • data point yi can equal 0, 1, 2, ....Theusualtransformationg used here is the logarithmic, so that g(u)=exp(u)transformsacontinuouslinearpredictorXiβ to a positivey ˆi.ThedatadistributionisPoisson. It is usually a good idea to add a parameter to this model to capture overdis- persion,thatis,variationinthedatabeyondwhatwouldbepredictedfromthe Poisson distribution alone. -
Generalized Linear Models
Generalized Linear Models Advanced Methods for Data Analysis (36-402/36-608) Spring 2014 1 Generalized linear models 1.1 Introduction: two regressions • So far we've seen two canonical settings for regression. Let X 2 Rp be a vector of predictors. In linear regression, we observe Y 2 R, and assume a linear model: T E(Y jX) = β X; for some coefficients β 2 Rp. In logistic regression, we observe Y 2 f0; 1g, and we assume a logistic model (Y = 1jX) log P = βT X: 1 − P(Y = 1jX) • What's the similarity here? Note that in the logistic regression setting, P(Y = 1jX) = E(Y jX). Therefore, in both settings, we are assuming that a transformation of the conditional expec- tation E(Y jX) is a linear function of X, i.e., T g E(Y jX) = β X; for some function g. In linear regression, this transformation was the identity transformation g(u) = u; in logistic regression, it was the logit transformation g(u) = log(u=(1 − u)) • Different transformations might be appropriate for different types of data. E.g., the identity transformation g(u) = u is not really appropriate for logistic regression (why?), and the logit transformation g(u) = log(u=(1 − u)) not appropriate for linear regression (why?), but each is appropriate in their own intended domain • For a third data type, it is entirely possible that transformation neither is really appropriate. What to do then? We think of another transformation g that is in fact appropriate, and this is the basic idea behind a generalized linear model 1.2 Generalized linear models • Given predictors X 2 Rp and an outcome Y , a generalized linear model is defined by three components: a random component, that specifies a distribution for Y jX; a systematic compo- nent, that relates a parameter η to the predictors X; and a link function, that connects the random and systematic components • The random component specifies a distribution for the outcome variable (conditional on X). -
Modelling Binary Outcomes
Modelling Binary Outcomes 01/12/2020 Contents 1 Modelling Binary Outcomes 5 1.1 Cross-tabulation . .5 1.1.1 Measures of Effect . .6 1.1.2 Limitations of Tabulation . .6 1.2 Linear Regression and dichotomous outcomes . .6 1.2.1 Probabilities and Odds . .8 1.3 The Binomial Distribution . .9 1.4 The Logistic Regression Model . 10 1.4.1 Parameter Interpretation . 10 1.5 Logistic Regression in Stata . 11 1.5.1 Using predict after logistic ........................ 13 1.6 Other Possible Models for Proportions . 13 1.6.1 Log-binomial . 14 1.6.2 Other Link Functions . 16 2 Logistic Regression Diagnostics 19 2.1 Goodness of Fit . 19 2.1.1 R2 ........................................ 19 2.1.2 Hosmer-Lemeshow test . 19 2.1.3 ROC Curves . 20 2.2 Assessing Fit of Individual Points . 21 2.3 Problems of separation . 23 3 Logistic Regression Practical 25 3.1 Datasets . 25 3.2 Cross-tabulation and Logistic Regression . 25 3.3 Introducing Continuous Variables . 26 3.4 Goodness of Fit . 27 3.5 Diagnostics . 27 3.6 The CHD Data . 28 3 Contents 4 1 Modelling Binary Outcomes 1.1 Cross-tabulation If we are interested in the association between two binary variables, for example the presence or absence of a given disease and the presence or absence of a given exposure. Then we can simply count the number of subjects with the exposure and the disease; those with the exposure but not the disease, those without the exposure who have the disease and those without the exposure who do not have the disease.