DOCSLIB.ORG

Bayesian Poisson Regression for Crowd Counting

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bayesian Poisson Regression for Crowd Counting Appears in IEEE Int’l Conf. on Computer Vision, Kyoto, 2009. Bayesian Poisson Regression for Crowd Counting Antoni B. Chan Nuno Vasconcelos Department of Computer Science Dept. of Electrical and Computer Engineering City University of Hong Kong University of California, San Diego [email protected] [email protected] Abstract by reducing the confidence when the prediction is far from an integer. In addition, the confidence levels are currently Poisson regression models the noisy output of a count- measured in standard-deviations, which provides little intu- ing function as a Poisson random variable, with a log-mean ition on the reliability of the estimates. A confidence mea- parameter that is a linear function of the input vector. In sure based on posterior probabilities seems more intuitive this work, we analyze Poisson regression in a Bayesian set- for counting numbers. Finally, negative outputs of the GP ting, by introducing a prior distribution on the weights of must be truncated to zero, and it is unclear how this affects the linear function. Since exact inference is analytically un- the optimality of the predictive distribution. obtainable, we derive a closed-form approximation to the One common method of regression for counting num- predictive distribution of the model. We show that the pre- bers is Poisson regression [3], which models the noisy out- dictive distribution can be kernelized, enabling the repre- put of a counting function as a Poisson random variable, sentation of non-linear log-mean functions. We also derive with a log-mean parameter that is a linear function of the an approximate marginal likelihood that can be optimized input vector. This is analogous to standard linear regres- to learn the hyperparameters of the kernel. We then relate sion, except that the mean is modeled as the exponential the proposed approximate Bayesian Poisson regression to of a linear function to ensure non-negative values, and that Gaussian processes. Finally, we present experimental re- the noise model is Poisson because the outputs are count- sults using Bayesian Poisson regression for crowd counting ing numbers. One way of extending Poisson regression to from low-level features. the Bayesian setting is to adopt a hierarchical model, where the log-mean function is modeled with a standard Gaus- 1. Introduction sian process [4, 5, 6]. These solutions, however, have two Recent work [1, 2] on crowd counting using low-level disadvantages. First, because of the lack of conjugacy be- feature regression has shown promise in computer vision. tween the Poisson and the GP, [4, 5, 6] must approximate One advantage with these methods is that they bypass in- inference with Markov-chain Monte Carlo (MCMC), which termediate processing stages, such as people detection or limits these algorithms to small datasets. Second, the hier- tracking, that may be susceptible to problems when the archical model contains two redundant noise sources: 1) crowd is dense. In [1], the scene is segmented into crowds the Poisson-distributed observation noise, and 2) the Gaus- moving in different directions and various low-level fea- sian noise of the GP in the log-mean function. These two tures are extracted from each crowd segment (e.g. informa- noise terms model essentially the same thing: the noise in tion on the shape, edges and texture of the segment). The observing the count. A more parsimonious representation crowd count in each segment is then estimated with a Gaus- would include only the observation noise, while modeling sian process (GP) regression function that maps feature vec- the mean as a deterministic function. tors to the crowd size. Experiments in [1] indicate that the In this work, we analyze the standard Poisson regression counting algorithm is capable of producing accurate counts, model in a Bayesian setting, by adding a Gaussian prior for a wide range of crowd densities. on the weights of the linear log-mean function. Since ex- One problem with the system of [1] is that it uses GP re- act inference is analytically unobtainable, approximate in- gression, which models continuous real-valued functions, ference is still necessary. However, in contrast to previous to predict discrete counting numbers. Because of this work [4, 5, 6], we propose a closed-form approximation to mismatch, regression may not be taking full advantage of Bayesian inference. The contributions of this paper, with Bayesian inference. For example, rounding of the real- respect to Bayesian Poisson regression (BPR), are five-fold: valued predictions is not handled in a principled way, e.g. 1) we derive a closed-form approximation to the predictive 1 distribution for BPR; 2) we kernelize the predictive distri- of f∗ = f(x∗) is obtained by averaging over all possible bution, enabling the representation of non-linear log-mean model parameterizations, with respect to the posterior dis- functions via kernel functions; 3) we derive an approxi- tribution of w [7], mate marginal likelihood function for optimizing the hyper- parameters of the kernel function with Type-II maximum p(f∗|x∗,X,y)= p(f∗|x∗, w)p(w|X, y)dw (4) likelihood; 4) we show that the proposed approximation to BPR is related to a Gaussian process with a special non- G f | 1 xT A−1Xy, xT A−1x . = ( ∗ 2 ∗ ∗ ∗) (5) i.i.d. noise term; 5) finally, we present experimental results σn that show improvement in crowd counting accuracy when using the proposed model. 2.2. Kernelized regression The remainder of this paper is organized as follows. In The predictive distribution in (5) can be rewritten to only Sections 2 and 3, we briefly review Gaussian processes and depend on the inner products between the inputs x i. Hence, Poisson regression. In Section 4, we present the Bayesian the “kernel trick” can be applied to obtain a kernel version framework for Poisson regression, derive a closed-form ap- of the Bayesian linear regression. Consider the model proximation for the predictive distribution and marginal T likelihood, and kernelize the regression model. Finally, f(x)=φ(x) w, (6) in Section 5, we present experimental results on Bayesian Poisson regression for crowd counting. where φ(x) is a high-dimensional feature transformation of d D x from dimension d to D, i.e. φ : R → R , and w ∈ D 2. Gaussian process regression R . Substituting into (5) and applying the matrix inversion lemma, the predictive distribution can be rewritten in terms Gaussian process (GP) regression [7] is a Bayesian treat- T k(x, x )=φ(x) Σpφ(x ) ment for predicting a function value f(x) from the input of the kernel function [7] d vector x ∈ R . Consider the case when f(x) is linear, from p(f∗|x∗,X,y)=G(f∗|µ∗, Σ∗), (7) which we observe a noisy target y, i.e. T where the predictive mean and covariance are f(x)=x w,y= f(x)+, (1) µ kT K σ2 I −1y, w ∈ Rd ∗ = ∗ ( + n ) (8) where is the weight vector of the linear model, T 2 −1 2 Σ∗ = k(x∗, x∗) − k∗ (K + σnI) k∗, (9) and the observation noise is Gaussian, ∼N(0,σn). The Bayesian model assumes a prior distribution on the weight and K is the kernel matrix with entries Kij = k(xi, xj ), vectors, w ∼N(0, Σp), where Σp is the covariance matrix T and k∗ =[k(x∗, x1) ···k(x∗, xN )] . Hence, non-linear of the weight prior. regression is achieved by adopting different positive definite 2.1. Bayesian prediction kernel functions. For example, using a linear kernel, 2 T 2 Let X =[x1, ···xN ] be the matrix of observed input kl(x, x )=θ1(x x +1)+θ2, (10) T vectors xi, and y =[y1 ··· yN ] be the vector of ob- served outputs yi. Bayesian inference on (1) is based on the results in standard Bayesian linear regression, while em- posterior distribution of the weights w, conditioned on the ploying a squared-exponential (RBF) kernel, observed data {X, y}, and is computed with Bayes’ rule, − 1 x−x2 2 θ2 2 k x, x θ e 2 θ , p y|X, w p w r( )= 1 + 3 (11) p w|X, y ( ) ( ) . ( )= p y|X, w p w dw (2) ( ) ( ) yields Bayesian regression for locally smooth, infinitely dif- Since the data-likelihood and weight prior are both Gaus- ferentiable, functions. Finally, a compound kernel, such as sian, (2) is also Gaussian [7], the RBF-RBF kernel, − 1 x−x2 − 1 x−x2 1 −1 −1 2 θ2 2 θ2 2 p w|X, y G w| A Xy,A , krr(x, x )=θ1e 2 + θ3e 4 + θ5, (12) ( )= ( 2 ) (3) σn which contains two RBF functions with different length d − 1 1 2 − 2 2 where G(x|µ, Σ) = (2π) |Σ| exp(− 2 x − µΣ) scales, can simultaneously model both global non-linear is the equation of a multivariate Gaussian distribution, trends and local deviations from the trend. 2 T −1 1 T −1 xΣ = x Σ x, and A = 2 XX +Σp . Finally, θ k x, x σn The hyperparameters of the kernel function ( ) given a novel input vector x∗, the predictive distribution can be learned with Type-II maximum likelihood. The N marginal likelihood of the training data {x i,yi}i=1 is max- overdispersion replaces the Poisson noise with a negative imized with respect to the hyperparameters ([7], Chapter 5) binomial [3], T µ(x)=exp(x β),y∼ NegBin(µ(x),α), (16) p(y|X, θ)= p(y|w,X,θ)p(w|θ)dw (13) where α is the scale parameter of the negative binomial. 1 T −1 1 N = − y K y − log |Ky|− log 2π, (14) y x 2 y 2 2 The likelihood of given an input vector is −1 K K σ2 I Γ(y + α ) α−1 y where y = + n .
Load more

Recommended publications
  • 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.
    [Show full text]
  • Generalized Linear Models (Glms)
    San Jos´eState University Math 261A: Regression Theory & Methods Generalized Linear Models (GLMs) Dr. Guangliang Chen This lecture is based on the following textbook sections: • Chapter 13: 13.1 – 13.3 Outline of this presentation: • What is a GLM? • Logistic regression • Poisson regression Generalized Linear Models (GLMs) What is a GLM? In ordinary linear regression, we assume that the response is a linear function of the regressors plus Gaussian noise: 0 2 y = β0 + β1x1 + ··· + βkxk + ∼ N(x β, σ ) | {z } |{z} linear form x0β N(0,σ2) noise The model can be reformulate in terms of • distribution of the response: y | x ∼ N(µ, σ2), and • dependence of the mean on the predictors: µ = E(y | x) = x0β Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University3/24 Generalized Linear Models (GLMs) beta=(1,2) 5 4 3 β0 + β1x b y 2 y 1 0 −1 0.0 0.2 0.4 0.6 0.8 1.0 x x Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University4/24 Generalized Linear Models (GLMs) Generalized linear models (GLM) extend linear regression by allowing the response variable to have • a general distribution (with mean µ = E(y | x)) and • a mean that depends on the predictors through a link function g: That is, g(µ) = β0x or equivalently, µ = g−1(β0x) Dr. Guangliang Chen | Mathematics & Statistics, San Jos´e State University5/24 Generalized Linear Models (GLMs) In GLM, the response is typically assumed to have a distribution in the exponential family, which is a large class of probability distributions that have pdfs of the form f(x | θ) = a(x)b(θ) exp(c(θ) · T (x)), including • Normal - ordinary linear regression • Bernoulli - Logistic regression, modeling binary data • Binomial - Multinomial logistic regression, modeling general cate- gorical data • Poisson - Poisson regression, modeling count data • Exponential, Gamma - survival analysis Dr.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Bayesian Hierarchical Poisson Regression Model for Overdispersed Count Data
    Bayesian Hierarchical Poisson Regression Model for Overdispersed Count Data Overview This example uses the RANDOM statement in MCMC procedure to fit a Bayesian hierarchical Poisson regression model to overdispersed count data. The RANDOM statement, available in SAS/STAT 9.3 and later, provides a convenient way to specify random effects with substantionally improved performance. Overdispersion occurs when count data appear more dispersed than expected under a reference model. Overdispersion can be caused by positive correlation among the observations, an incorrect model, an in- correct distributional specification, or incorrect variance functions. The example displays how Bayesian hierarchical Poisson regression models are effective in capturing overdispersion and providing a better fit. The SAS source code for this example is available as a text file attachment. In Adobe Acrobat, right-click the icon in the margin and select Save Embedded File to Disk. You can also double-click the icon to open the file immediately. Analysis Count data frequently display overdispersion (more variation than expected from a standard parametric model). Breslow(1984) discusses these types of models and suggests several different ways to model them. Hierarchical Poisson models have been found effective in capturing the overdispersion in data sets with extra Poisson variation. Hierarchical Poisson regression models are expressed as Poisson models with a log link and a normal vari- ance on the mean parameter. More formally, a hierarchical Poisson regression model is written as Yij ij Poisson.ij / j log.ij / Xi ˇ ij D C 2 ij normal.0; / for i 1; :::; n, j 1; :::; J , and y 0; 1; 2; ::: .
    [Show full text]
  • Using Geographically Weighted Poisson Regression for County-Level Crash Modeling in California ⇑ Zhibin Li A,B, , Wei Wang A,1, Pan Liu A,2, John M
    Safety Science 58 (2013) 89–97 Contents lists available at SciVerse ScienceDirect Safety Science journal homepage: www.elsevier.com/locate/ssci Using Geographically Weighted Poisson Regression for county-level crash modeling in California ⇑ Zhibin Li a,b, , Wei Wang a,1, Pan Liu a,2, John M. Bigham b,3, David R. Ragland b,3 a School of Transportation, Southeast University, Si Pai Lou #2, Nanjing 210096, China b Safe Transportation Research and Education Center, Institute of Transportation Studies, University of California, Berkeley, 2614 Dwight Way #7374, Berkeley, CA 94720-7374, United States article info abstract Article history: Development of crash prediction models at the county-level has drawn the interests of state agencies for Received 25 December 2012 forecasting the normal level of traffic safety according to a series of countywide characteristics. A com- Received in revised form 11 March 2013 mon technique for the county-level crash modeling is the generalized linear modeling (GLM) procedure. Accepted 13 April 2013 However, the GLM fails to capture the spatial heterogeneity that exists in the relationship between crash counts and explanatory variables over counties. This study aims to evaluate the use of a Geographically Weighted Poisson Regression (GWPR) to capture these spatially varying relationships in the county-level Keywords: crash data. The performance of a GWPR was compared to a traditional GLM. Fatal crashes and countywide Safety factors including traffic patterns, road network attributes, and socio-demographic characteristics were Crash County-level collected from the 58 counties in California. Results showed that the GWPR was useful in capturing Geographically Weighted Regression the spatially non-stationary relationships between crashes and predicting factors at the county level.
    [Show full text]
  • Generalized Linear Models and Point Count Data: Statistical Considerations for the Design and Analysis of Monitoring Studies
    Generalized Linear Models and Point Count Data: Statistical Considerations for the Design and Analysis of Monitoring Studies Nathaniel E. Seavy,1,2,3 Suhel Quader,1,4 John D. Alexander,2 and C. John Ralph5 ________________________________________ Abstract The success of avian monitoring programs to effec- Key words: Generalized linear models, juniper remov- tively guide management decisions requires that stud- al, monitoring, overdispersion, point count, Poisson. ies be efficiently designed and data be properly ana- lyzed. A complicating factor is that point count surveys often generate data with non-normal distributional pro- perties. In this paper we review methods of dealing with deviations from normal assumptions, and we Introduction focus on the application of generalized linear models (GLMs). We also discuss problems associated with Measuring changes in bird abundance over time and in overdispersion (more variation than expected). In order response to habitat management is widely recognized to evaluate the statistical power of these models to as an important aspect of ecological monitoring detect differences in bird abundance, it is necessary for (Greenwood et al. 1993). The use of long-term avian biologists to identify the effect size they believe is monitoring programs (e.g., the Breeding Bird Survey) biologically significant in their system. We illustrate to identify population trends is a powerful tool for bird one solution to this challenge by discussing the design conservation (Sauer and Droege 1990, Sauer and Link of a monitoring program intended to detect changes in 2002). Short-term studies comparing bird abundance in bird abundance as a result of Western juniper (Juniper- treated and untreated areas are also important because us occidentalis) reduction projects in central Oregon.
    [Show full text]
  • Poisson Regression 1 Poisson Regression
    Poisson regression 1 Poisson regression In statistics, Poisson regression is a form of regression analysis used to model count data and contingency tables. Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. A Poisson regression model is sometimes known as a log-linear model, especially when used to model contingency tables. Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function. Regression models If is a vector of independent variables, then the model takes the form where and . Sometimes this is written more compactly as where x is now an n+1-dimensional vector consisting of n independent variables concatenated to some constant, usually 1. Here θ is simply a concatenated to b. Thus, when given a Poisson regression model θ and an input vector , the predicted mean of the associated Poisson distribution is given by If Y are independent observations with corresponding values x of the predictor variable, then θ can be estimated by i i 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 convex, making Newton-Raphson or other gradient-based methods appropriate estimation techniques. Maximum likelihood-based parameter estimation Given a set of parameters θ and an input vector x, the mean of the predicted Poisson distribution, as stated above, is given by , and thus, the Poisson distribution's probability mass function is given by Now suppose we are given a data set consisting of m vectors , along with a set of m values .
    [Show full text]
  • Advanced Regression Methods Symposium on Updates on Clinical Research Methodology March 18, 2013
    Advanced Regression Methods Symposium on Updates on Clinical Research Methodology March 18, 2013 Lloyd Mancl, PhD Charles Spiekerman, PhD Oral Health Sciences Oral Health Sciences University of Washington University of Washington [email protected] [email protected] Outline • Introduce common regression methods for different types of outcomes – Logistic regression – Multiple linear regression – Cox proportional hazards regression – Poisson or log-linear regression • Uses of regression: – Adjust for confounding – Assess for effect modification or interaction – Account for non-independent outcomes Uses for Multiple regression analysis • Used to adjust for confounding – In observational studies, groups of interest can differ on other variables that may be related to the outcome. – In an RCT, randomization may not result in balanced groups. • Used to assess simultaneously the associations for several explanatory variables, as well as, interactions between variables – In observational studies, you may be interested in how several explanatory variables are related to an outcome. – In an RCT, we are often interested in testing if treatment is modified by another variable (i.e., interaction or moderation). – In designed experiments, we typically test for interactions between the different study factors. • Used to develop a prediction equation – Not commonly used for this purpose; not covered in this workshop. Multiple regression analysis Models the association between one outcome variable, Y, and multiple variables of interest, X1, X2,…,Xk. Multiple Linear
    [Show full text]
  • R-Squared Measures for Count Data Regression Models with Applications to Health Care Utilization
    R-Squared Measures for Count Data Regression Models With Applications to Health Care Utilization A. Colin Cameron Dept. of Economics University of California Davis CA 95616-8578 USA Frank A.G. Windmeijer Dept. of Economics University College London London WC1E 6BT UK April 1995 Journal of Business and Economic Statistics (forthcoming) Abstract R-squared measures of goodness of fit for count data are rarely, if ever, reported in empirical studies or by statistical packages. We propose several R-squared measures based on various definitions of residuals, for the basic Poisson regression model and for more general models such as negative binomial that accommodate overdispersed data. The preferred R-squared is based on the deviance residual. An application to data on health care service utilization measured in counts illustrates the performance and usefulness of the various R-squareds. KEY WORDS: Goodness-of-fit, Poisson regression, negative binomial regression, deviance, deviance residual, Pearson residual. 1. INTRODUCTION R-squared (R 2 ) measures were originally developed for linear regression models with homoscedastic errors. Extensions to models with heteroscedastic errors with known variance were proposed by Buse (1973). Extensions to other models are rare, with the notable exceptions of logit and probit models, see Windmeijer (1994) and the references therein, and tobit models, surveyed by Veall and Zimmermann (1994). In this paper we investigate R 2 measures for Poisson and other related count data regression models. Surprisingly, R 2 is rarely reported in empirical studies or by statistical packages for count data. (For Poisson, exceptions are Merkle and Zimmermann (1992) and the statistical package STATA.
    [Show full text]
  • Making Models with Bayes
    California State University, San Bernardino CSUSB ScholarWorks Electronic Theses, Projects, and Dissertations Office of aduateGr Studies 12-2017 Making Models with Bayes Pilar Olid Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd Part of the Applied Statistics Commons, Multivariate Analysis Commons, Other Applied Mathematics Commons, Other Mathematics Commons, Other Statistics and Probability Commons, Probability Commons, and the Statistical Models Commons Recommended Citation Olid, Pilar, "Making Models with Bayes" (2017). Electronic Theses, Projects, and Dissertations. 593. https://scholarworks.lib.csusb.edu/etd/593 This Thesis is brought to you for free and open access by the Office of aduateGr Studies at CSUSB ScholarWorks. It has been accepted for inclusion in Electronic Theses, Projects, and Dissertations by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Making Models with Bayes A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Pilar Olid December 2017 Making Models with Bayes A Thesis Presented to the Faculty of California State University, San Bernardino by Pilar Olid December 2017 Approved by: Dr. Charles Stanton, Committee Chair Dr. Jeremy Aikin, Committee Member Dr. Yuichiro Kakihara, Committee Member Dr. Charles Stanton, Chair, Department of Mathematics Dr. Corey Dunn, Graduate Coordinator iii Abstract Bayesian statistics is an important approach to modern statistical analyses. It allows us to use our prior knowledge of the unknown parameters to construct a model for our data set. The foundation of Bayesian analysis is Bayes' Rule, which in its proportional form indicates that the posterior is proportional to the prior times the likelihood.
    [Show full text]