Models for Count Data 1 Running Head
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Statistical Approaches for Highly Skewed Data 1
Running head: STATISTICAL APPROACHES FOR HIGHLY SKEWED DATA 1 (in press) Journal of Clinical Child and Adolescent Psychology Statistical Approaches for Highly Skewed Data: Evaluating Relations between Child Maltreatment and Young Adults’ Non-Suicidal Self-injury Ana Gonzalez-Blanks, PhD Jessie M. Bridgewater, MA *Tuppett M. Yates, PhD Department of Psychology University of California, Riverside Acknowledgments: We gratefully acknowledge Dr. Chandra Reynolds who provided generous instruction and consultation throughout the execution of the analyses reported herein. *Please address correspondence to: Tuppett M. Yates, Department of Psychology, University of California, Riverside CA, 92521. Email: [email protected]. STATISTICAL APPROACHES FOR HIGHLY SKEWED DATA 2 Abstract Objective: Clinical phenomena often feature skewed distributions with an overabundance of zeros. Unfortunately, empirical methods for dealing with this violation of distributional assumptions underlying regression are typically discussed in statistical journals with limited translation to applied researchers. Therefore, this investigation compared statistical approaches for addressing highly skewed data as applied to the evaluation of relations between child maltreatment and non-suicidal self-injury (NSSI). Method: College students (N = 2,651; 64.2% female; 85.2% non-White) completed the Child Abuse and Trauma Scale and the Functional Assessment of Self-Mutilation. Statistical models were applied to cross-sectional data to provide illustrative comparisons across predictions to a) raw, highly skewed NSSI outcomes, b) natural log, square-root, and inverse NSSI transformations to reduce skew, c) zero-inflated Poisson (ZIP) and negative-binomial zero- inflated (NBZI) regression models to account for both disproportionate zeros and skewness in the NSSI data, and d) the skew-t distribution to model NSSI skewness. -
Negative Binomial Regression Models and Estimation Methods
Appendix D: Negative Binomial Regression Models and Estimation Methods By Dominique Lord Texas A&M University Byung-Jung Park Korea Transport Institute This appendix presents the characteristics of Negative Binomial regression models and discusses their estimating methods. Probability Density and Likelihood Functions The properties of the negative binomial models with and without spatial intersection are described in the next two sections. Poisson-Gamma Model The Poisson-Gamma model has properties that are very similar to the Poisson model discussed in Appendix C, in which the dependent variable yi is modeled as a Poisson variable with a mean i where the model error is assumed to follow a Gamma distribution. As it names implies, the Poisson-Gamma is a mixture of two distributions and was first derived by Greenwood and Yule (1920). This mixture distribution was developed to account for over-dispersion that is commonly observed in discrete or count data (Lord et al., 2005). It became very popular because the conjugate distribution (same family of functions) has a closed form and leads to the negative binomial distribution. As discussed by Cook (2009), “the name of this distribution comes from applying the binomial theorem with a negative exponent.” There are two major parameterizations that have been proposed and they are known as the NB1 and NB2, the latter one being the most commonly known and utilized. NB2 is therefore described first. Other parameterizations exist, but are not discussed here (see Maher and Summersgill, 1996; Hilbe, 2007). NB2 Model Suppose that we have a series of random counts that follows the Poisson distribution: i e i gyii; (D-1) yi ! where yi is the observed number of counts for i 1, 2, n ; and i is the mean of the Poisson distribution. -
“Multivariate Count Data Generalized Linear Models: Three Approaches Based on the Sarmanov Distribution”
Institut de Recerca en Economia Aplicada Regional i Pública Document de Treball 2017/18 1/25 pág. Research Institute of Applied Economics Working Paper 2017/18 1/25 pág. “Multivariate count data generalized linear models: Three approaches based on the Sarmanov distribution” Catalina Bolancé & Raluca Vernic 4 WEBSITE: www.ub.edu/irea/ • CONTACT: [email protected] The Research Institute of Applied Economics (IREA) in Barcelona was founded in 2005, as a research institute in applied economics. Three consolidated research groups make up the institute: AQR, RISK and GiM, and a large number of members are involved in the Institute. IREA focuses on four priority lines of investigation: (i) the quantitative study of regional and urban economic activity and analysis of regional and local economic policies, (ii) study of public economic activity in markets, particularly in the fields of empirical evaluation of privatization, the regulation and competition in the markets of public services using state of industrial economy, (iii) risk analysis in finance and insurance, and (iv) the development of micro and macro econometrics applied for the analysis of economic activity, particularly for quantitative evaluation of public policies. IREA Working Papers often represent preliminary work and are circulated to encourage discussion. Citation of such a paper should account for its provisional character. For that reason, IREA Working Papers may not be reproduced or distributed without the written consent of the author. A revised version may be available directly from the author. Any opinions expressed here are those of the author(s) and not those of IREA. Research published in this series may include views on policy, but the institute itself takes no institutional policy positions. -
A Comparison of Generalized Linear Models for Insect Count Data
International Journal of Statistics and Analysis. ISSN 2248-9959 Volume 9, Number 1 (2019), pp. 1-9 © Research India Publications http://www.ripublication.com A Comparison of Generalized Linear Models for Insect Count Data S.R Naffees Gowsar1*, M Radha 1, M Nirmala Devi2 1*PG Scholar in Agricultural Statistics, Tamil Nadu Agricultural University, Coimbatore, Tamil Nadu, India. 1Faculty of Agricultural Statistics, Tamil Nadu Agricultural University, Coimbatore, Tamil Nadu, India. 2Faculty of Mathematics, Tamil Nadu Agricultural University, Coimbatore, Tamil Nadu, India. Abstract Statistical models are powerful tools that can capture the essence of many biological systems and investigate ecological patterns associated to ecological stability dependent on endogenous and exogenous factors. Generalized linear model is the flexible generalization of ordinary linear regression, allows for response variables that have error distribution models other than a normal distribution. In order to fit a model for Count, Binary and Proportionate data, transformation of variables can be done and can fit the model using general linear model (GLM). But without transforming the nature of the data, the models can be fitted by using generalized linear model (GzLM). In this study, an attempt has been made to compare the generalized linear regression models for insect count data. The best model has been identified for the data through Vuong test. Keywords: Count data, Regression Models, Criterions, Vuong test. INTRODUCTION: Understanding the type of data before deciding the modelling approach is the foremost thing in data analysis. The predictors and response variables which follow non normal distributions are linearly modelled, it suffers from methodological limitations and statistical properties. -
Measures of Fit for Logistic Regression Paul D
Paper 1485-2014 SAS Global Forum Measures of Fit for Logistic Regression Paul D. Allison, Statistical Horizons LLC and the University of Pennsylvania ABSTRACT One of the most common questions about logistic regression is “How do I know if my model fits the data?” There are many approaches to answering this question, but they generally fall into two categories: measures of predictive power (like R-square) and goodness of fit tests (like the Pearson chi-square). This presentation looks first at R-square measures, arguing that the optional R-squares reported by PROC LOGISTIC might not be optimal. Measures proposed by McFadden and Tjur appear to be more attractive. As for goodness of fit, the popular Hosmer and Lemeshow test is shown to have some serious problems. Several alternatives are considered. INTRODUCTION One of the most frequent questions I get about logistic regression is “How can I tell if my model fits the data?” Often the questioner is expressing a genuine interest in knowing whether a model is a good model or a not-so-good model. But a more common motivation is to convince someone else--a boss, an editor, or a regulator--that the model is OK. There are two very different approaches to answering this question. One is to get a statistic that measures how well you can predict the dependent variable based on the independent variables. I’ll refer to these kinds of statistics as measures of predictive power. Typically, they vary between 0 and 1, with 0 meaning no predictive power whatsoever and 1 meaning perfect predictions. -
Multiple Binomial Regression Models of Learning Style Preferences of Students of Sidhu School, Wilkes University
International Journal of Statistics and Probability; Vol. 7, No. 3; May 2018 ISSN 1927-7032 E-ISSN 1927-7040 Published by Canadian Center of Science and Education Multiple Binomial Regression Models of Learning Style Preferences of Students of Sidhu School, Wilkes University Adekanmbi, D. B1 1 Department of Statistics, Ladoke Akintola University of Technology, Nigeria Correspondence: Adekanmbi, D.B, Department of Statistics, Ladoke Akintola University of Technology, Ogbomoso, Nigeria. Received: November 6, 2017 Accepted: November 27, 2017 Online Published: March 13, 2018 doi:10.5539/ijsp.v7n3p9 URL: https://doi.org/10.5539/ijsp.v7n3p9 Abstract The interest of this study is to explore the relationship between a dichotomous response, learning style preferences by university students of Sidhu School, Wilkes University, as a function of the following predictors: Gender, Age, employment status, cumulative grade point assessment (GPA) and level of study, as in usual generalized linear model. The response variable is the students’ preference for either Behaviorist or Humanist learning style. Four different binomial regression models were fitted to the data. Model A is a logit regression model that fits all the predictors, Model B is a probit model that fits all the predictors, Model C is a logit model with an effect modifier, while Model D is a probit model also with an effect modifier. Models A and B appeared to have performed poorly in fitting the data. Models C and D fit the data well as confirmed by the non-significant chi-square lack of fit with p-values 0.1409 and 0.1408 respectively. Among the four models considered for fitting the data, Model D, the probit model with effect modifier fit best. -
Regression Models for Count Data Jason Brinkley, Abt Associates
SESUG Paper SD135-2018 Regression Models for Count Data Jason Brinkley, Abt Associates ABSTRACT Outcomes in the form of counts are becoming an increasingly popular metric in a wide variety of fields. For example, studying the number of hospital, emergency room, or in-patient doctor’s office visits has been a major focal point for many recent health studies. Many investigators want to know the impact of many different variables on these counts and help describe ways in which interventions or therapies might bring those numbers down. Traditional least squares regression was the primary mechanism for studying this type of data for decades. However, alternative methods were developed some time ago that are far superior for dealing with this type of data. The focus of this paper is to illustrate how count regression models can outperform traditional methods while utilizing the data in a more appropriate manner. Poisson Regression and Negative Binomial Regression are popular techniques when the data are overdispersed and using Zero-Inflated techniques for data with many more zeroes than is expected under traditional count regression models. These examples are applied to studies with real data. INTRODUCTION Outcomes in the form of counts are becoming an increasingly popular metric in a wide variety of fields. In the health sciences examples include number of hospitalizations, chronic conditions, medications, and so on. Many investigators want to know the impact of many different variables on these counts and help describe ways in which interventions or therapies might bring those numbers down. Standard methods (regression, t-tests, ANOVA) are useful for some count data studies. -
Shrinkage Improves Estimation of Microbial Associations Under Di↵Erent Normalization Methods 1 2 1,3,4, 5,6,7, Michelle Badri , Zachary D
bioRxiv preprint doi: https://doi.org/10.1101/406264; this version posted April 4, 2020. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY-NC 4.0 International license. Shrinkage improves estimation of microbial associations under di↵erent normalization methods 1 2 1,3,4, 5,6,7, Michelle Badri , Zachary D. Kurtz , Richard Bonneau ⇤, and Christian L. Muller¨ ⇤ 1Department of Biology, New York University, New York, 10012 NY, USA 2Lodo Therapeutics, New York, 10016 NY, USA 3Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, 10010 NY, USA 4Computer Science Department, Courant Institute, New York, 10012 NY, USA 5Center for Computational Mathematics, Flatiron Institute, Simons Foundation, New York, 10010 NY, USA 6Institute of Computational Biology, Helmholtz Zentrum M¨unchen, Neuherberg, Germany 7Department of Statistics, Ludwig-Maximilians-Universit¨at M¨unchen, Munich, Germany ⇤correspondence to: [email protected], cmueller@flatironinstitute.org ABSTRACT INTRODUCTION Consistent estimation of associations in microbial genomic Recent advances in microbial amplicon and metagenomic survey count data is fundamental to microbiome research. sequencing as well as large-scale data collection efforts Technical limitations, including compositionality, low sample provide samples across different microbial habitats that are sizes, and technical variability, obstruct standard -
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. -
Lecture 3 Residual Analysis + Generalized Linear Models
Lecture 3 Residual Analysis + Generalized Linear Models Colin Rundel 1/23/2017 1 Residual Analysis 2 Atmospheric CO2 (ppm) from Mauna Loa 360 co2 350 1988 1992 1996 date 3 360 co2 350 1988 1992 1996 date 2.5 0.0 resid −2.5 1988 1992 1996 date Where to start? Well, it looks like stuff is going up on average … 4 360 co2 350 1988 1992 1996 date 2.5 0.0 resid −2.5 1988 1992 1996 date Where to start? Well, it looks like stuff is going up on average … 4 2.5 0.0 resid −2.5 1988 1992 1996 date 1.0 0.5 0.0 resid2 −0.5 −1.0 1988 1992 1996 date and then? Well there is some periodicity lets add the month … 5 2.5 0.0 resid −2.5 1988 1992 1996 date 1.0 0.5 0.0 resid2 −0.5 −1.0 1988 1992 1996 date and then? Well there is some periodicity lets add the month … 5 1.0 0.5 0.0 resid2 −0.5 −1.0 1988 1992 1996 date 0.8 0.4 0.0 resid3 −0.4 −0.8 1988 1992 1996 date and then and then? Maybe there is some different effect by year … 6 1.0 0.5 0.0 resid2 −0.5 −1.0 1988 1992 1996 date 0.8 0.4 0.0 resid3 −0.4 −0.8 1988 1992 1996 date and then and then? Maybe there is some different effect by year … 6 Too much (lm = lm(co2~date + month + as.factor(year), data=co2_df)) ## ## Call: ## lm(formula = co2 ~ date + month + as.factor(year), data = co2_df) ## ## Coefficients: ## (Intercept) date monthAug ## -2.645e+03 1.508e+00 -4.177e+00 ## monthDec monthFeb monthJan ## -3.612e+00 -2.008e+00 -2.705e+00 ## monthJul monthJun monthMar ## -2.035e+00 -3.251e-01 -1.227e+00 ## monthMay monthNov monthOct ## 4.821e-01 -4.838e+00 -6.135e+00 ## monthSep as.factor(year)1986 as.factor(year)1987 ## -6.064e+00 -2.585e-01 9.722e-03 ## as.factor(year)1988 as.factor(year)1989 as.factor(year)1990 ## 1.065e+00 9.978e-01 7.726e-01 ## as.factor(year)1991 as.factor(year)1992 as.factor(year)1993 ## 7.067e-01 1.236e-02 -7.911e-01 ## as.factor(year)1994 as.factor(year)1995 as.factor(year)1996 ## -4.146e-01 1.119e-01 3.768e-01 ## as.factor(year)1997 7 ## NA 360 co2 350 1988 1992 1996 date 8 Generalized Linear Models 9 Background A generalized linear model has three key components: 1. -
The Overlooked Potential of Generalized Linear Models in Astronomy, I: Binomial Regression
Astronomy and Computing 12 (2015) 21–32 Contents lists available at ScienceDirect Astronomy and Computing journal homepage: www.elsevier.com/locate/ascom Full length article The overlooked potential of Generalized Linear Models in astronomy, I: Binomial regression R.S. de Souza a,∗, E. Cameron b, M. Killedar c, J. Hilbe d,e, R. Vilalta f, U. Maio g,h, V. Biffi i, B. Ciardi j, J.D. Riggs k, for the COIN collaboration a MTA Eötvös University, EIRSA ``Lendulet'' Astrophysics Research Group, Budapest 1117, Hungary b Department of Zoology, University of Oxford, Tinbergen Building, South Parks Road, Oxford, OX1 3PS, United Kingdom c Universitäts-Sternwarte München, Scheinerstrasse 1, D-81679, München, Germany d Arizona State University, 873701, Tempe, AZ 85287-3701, USA e Jet Propulsion Laboratory, 4800 Oak Grove Dr., Pasadena, CA 91109, USA f Department of Computer Science, University of Houston, 4800 Calhoun Rd., Houston TX 77204-3010, USA g INAF — Osservatorio Astronomico di Trieste, via G. Tiepolo 11, 34135 Trieste, Italy h Leibniz Institute for Astrophysics, An der Sternwarte 16, 14482 Potsdam, Germany i SISSA — Scuola Internazionale Superiore di Studi Avanzati, Via Bonomea 265, 34136 Trieste, Italy j Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany k Northwestern University, Evanston, IL, 60208, USA article info a b s t r a c t Article history: Revealing hidden patterns in astronomical data is often the path to fundamental scientific breakthroughs; Received 26 September 2014 meanwhile the complexity of scientific enquiry increases as more subtle relationships are sought. Con- Accepted 2 April 2015 temporary data analysis problems often elude the capabilities of classical statistical techniques, suggest- Available online 29 April 2015 ing the use of cutting edge statistical methods. -
Negative Binomial Regression Second Edition, Cambridge University Press
Negative Binomial Regression Second edition, Cambridge University Press Joseph M Hilbe ERRATA & COMMENTS as of 12 January, 2012 Note: Data sets and software code can be downloaded from: http://works.bepress.com/joseph_hilbe/ ERRATA P. xv: The inset following the 1st paragraph on page, replace Stata bookstore (Stata files/commands) http://www.Stata.com/bookstore/nbr2.html with Data sets, software code, and electronic Extensions to the text can be downloaded from: http://works.bepress.com/joseph_hilbe/ p 17 Table 2.2. second to last line in table, final term should read, prop.r=FALSE, not pror.r. Read as CrossTable(survived, age, prop.t=FALSE, prop.r=FALSE, prop.c=FALSE, prop.chisq=FALSE) p 18 Equation 2.3. The second or middle term should have (implied) product signs, not division. Read as: p. 20 Table 2.4. The final three lines can be reduced to one line: irr*rse. Revise Table 2.4 to appear as: Table 2.4: R – Poisson model with robust standard errors ==================================================== titanic$class <- relevel(factor(titanic$class), ref=3) tit3 <- glm(survived ~ factor(class), family=poisson, data=titanic) irr <- exp(coef(tit3)) # vector of IRRs library("sandwich") rse <- sqrt(diag(vcovHC(tit3, type="HC0"))) # coef robust SEs irr*rse # IRR robust SEs ==================================================== P 22 Table 2.5. The final three lines of the code may be condensed to one line, as it was in Table 2.4 above. Replace ec2 <- c(irr2) rs2 <- c(rse2) ec2 * rs2 with irr2 * rse2 P 27 The interpretation of the model (in italics) can better be expressed as: “the odds of a child surviving on the Titanic is nearly twice that of adults.” The line below the italicized interpretation should be amended to that this section in the book should read as: ----------------------------------------------------------------------------------------------- The odds of a child surviving on the Titanic is nearly twice that of adults.