Principles of Regression Analysis NJ Gogtay, SP Deshpande, UM Thatte

Total Page:16

File Type:pdf, Size:1020Kb

Principles of Regression Analysis NJ Gogtay, SP Deshpande, UM Thatte 48 Journal of The Association of Physicians of India ■ Vol. 65 ■ April 2017 STATISTICS FOR RESEARCHERS Principles of Regression Analysis NJ Gogtay, SP Deshpande, UM Thatte Introduction relationship between one or more the “dependent” variable. In the independent variables and the second example, since CAD is hile correlation analysis helps dependent variable. A correlation associated with both diabetes and Win identifying associations analysis with a scatter plot and hypertension, CAD would become or relationships between two a regression line1 is however a the dependent variable and diabetes variables,1 the regression technique prerequisite to regression and and hypertension would be the two or regression analysis is used to both analyses are often carried out independent variables. If you will “model” this relationship so as together. notice, diabetes in one example is to be able to predict what will a dependent variable and in the happen in a real-world setting. Let Dependent and other, an independent variable. us understand this with something Independent Variables Thus, what is a dependent and that happens in daily life. As what is an independent variable children, we are often told by our An important first step before needs to be defined à priori by the parents that education is a vital carrying out a regression analysis researcher before carrying out the part of our lives and a “good” is to understand the concept regression analysis. The choice education will help us get “good of independent and dependent of the variables would in turn be jobs” and thus “good wages”! If variables. An independent variable, defined by the research question we were to explore the relationship as the name suggests is a “stand and the hypothesis being explored. between education and wages, alone” variable, and one that Independent variables are often we could think up two questions remains unaffected by other called predictor variables or – a) Does a relationship exist variables that are measured in a exogenous variables and dependent between education and wages? And study. The “dependent” variable is variables are called prognostic or a related but more interesting and the one that is usually of interest to endogenous variables. important question b) for every extra the researcher and alters in response year spent in education, do wages to change/s in the independent variable. Types of Regression commensurately increase [and if Let us understand this concept Essentially in medical research, yes, by how much?]? The latter with the following two research there are three common types of question is moving into the realm of questions – 1) “As age increases, regression analyses that are used prediction. Regression attempts to does the risk of developing diabetes viz., linear, logistic regression and answer these and similar questions as measured by HbA1C or blood Cox regression. These are chosen regarding relationships between sugar levels increase? Or b) “Given depending on the type of variables variables. that a patient has both diabetes and that we are dealing with (Table 1). hypertension, what is his risk of Cox regression is a special type of Correlation Versus developing coronary artery disease regression analysis that is applied Regression- [CAD]”? In the former example, to survival or “time to event “data we are exploring the relationship Understanding the and will be discussed in detail in between age and diabetes and in the next article in the series. Distinction the latter, the relationship between Linear regression can be simple The difference between two variables with a third variable– i.e., diabetes and hypertension linear or multiple linear regression correlation analysis and regression while Logistic regression could be lies in the fact that the former focuses with CAD. In the first example, age would be the independent Polynomial in certain cases (Table on the strength and direction of 1). the relationship between two or variable while diabetes, which is more variables without making any dependent on age would become The type of regression analysis assumptions about one variable being independent and the other dependent [see below], but regression analysis Department of Clinical Pharmacology, Seth GS Medical College and KEM Hospital, Mumbai, Maharashtra Received: 07.01.2017; Accepted: 15.01.2017 assumes a dependence or causal Journal of The Association of Physicians of India ■ Vol. 65 ■ April 2017 49 Table 1: Types of regression duration of surgery, extent of Type of regression Dependent variable Independent variable Relationship between blood loss and Hb levels at and its nature and its nature variables day 7, it is intuitive that all Simple linear One, continuous, One, continuous, Linear are correlated amongst each normally distributed normally distributed other and therefore will lead to Multiple linear One, continuous Two or more, may Linear erroneous results. What should be continuous or be taken as the independent categorical variable is only the duration of Logistic One, binary Two or more, may Need not be linear be continuous or surgery. categorical • The sample size should be Polynomial (logistic) Non-binary Two or more, may Need not be linear adequate [multinomial] be continuous or categorical If the dependent variable is Cox or proportional Time to an event Two or more, may Is rarely linear non-binary and has more than hazards regression be continuous or two possibilities, we use the categorical multinomial or polynomial to be used in a given situation is be quantitative or qualitative. logistic regression. primarily driven by the following Both the independent variables three metrics here could be expressed either Step Wise Regression • Number and nature of as continuous data (blood Stepwise regression is an independent variable/s pressure or HbA1C values) automated tool that can be used or qualitative data (presence • Number and nature of the in the exploratory stages of model or absence of diabetes as dependent variable/s building to identify a useful subset defined by the ADA 2016 or of predictor variables. The process • Shape of the regression line hypertension as defined by JNC systematically adds the most 3,4 A. Linear regression: Linear VIII). A linear relationship significant variable or removes regression is the most basic should exist between the the least significant variable during and commonly used regression dependent and independent each step. The idea behind using technique and is of two types variables. such automated tools is to maximize viz. simple and multiple B. Logistic regression: This type the power of prediction with a regression. You can use Simple of regression analysis is used minimum number of independent linear regression when there when the dependent variable is variables. is a single dependent and a binary in nature. For example, if single independent variable the outcome of interest is death Steps in Conducting a (e.g. for the research question in a cancer study, any patient Regression Analysis described above “As age in the study can have only one increases, does the risk of of two possible outcomes- dead Regression analysis is done in developing diabetes increase or alive. The impact of one or 3 steps: as measured by HbA1C or more predictor variables on this 1. Analyzing the correlation blood sugar levels?”. Both the binary variable is assessed. The [strength and directionality of variables must be continuous predictor variables can be either the data] 2 (quantitative data ) and the line quantitative or qualitative. 2. Fitting the regression or least describing the relationship is a Unlike linear regression, this squares line, and straight line (linear). type of regression does not Multiple linear regression on require a linear relationship 3. Evaluating the validity and the other hand can be used between the predictor and usefulness of the model. when we have one continuous dependent variables. Step 1: This has been described in 1 dependent variable and two or For logistic regression to be the article on correlation analysis more independent variables, meaningful, the following Step 2: Fitting the regression line for example when we want to criteria must be met/satisfied Conventionally, in mathematics, answer the second question • The independent variables the equation of a line is given by mentioned above, “Given that must not be correlated the formula “y = mx+c”, where, a patient has both diabetes and amongst each other e.g. if m is the “slope” or gradient of hypertension, what is his risk the dependent variable is the line and “c” is where the line of developing coronary artery presence (or absence) of post- “cuts” the y-axis, also called as disease (CAD)”. Importantly, operative wound infection and the “intercept”, (Figure 1). In the independent variables could the independent variables are regression, this equation is given by 50 Journal of The Association of Physicians of India ■ Vol. 65 ■ April 2017 Regression Equa�on Table 2: Age and Blood pressure in women (n=30) Sr. No. Age (years) Systolic blood HbA1C (%) HbA1C (Y) = B0 + B1*age (X). pressure (mm Hg) 1 22 131 Y-axis B1 2 23 128 B0 3 24 116 4 27 106 5 28 114 6 29 123 X - axis Age (years) 7 30 117 8 32 122 9 35 99 Fig. 1: Regression analysis exploring the relationship between Age and HbA1C 10 35 121 A scatter plot and regression line of age 11 40 147 12 41 139 versus systolic blood pressure y = 0.889x + 94.61 13 41 171 2 R = 0.292 14 46 137 200 15 47 111 16 48 115 150 17 49 133 18 49 128 100 19 50 183 SBP 20 51 130 50 21 51 133 22 51 144 23 52 128 0 24 54 105 0 10 20 30 40 50 60 70 80 25 56 145 26 57 141 Age 27 58 153 Fig. 2: Scatter plot and regression Line 28 59 157 29 63 155 30 67 176 y=B0 +B1x which are the notations of the model commonly used by statisticians.
Recommended publications
  • A Simple Censored Median Regression Estimator
    Statistica Sinica 16(2006), 1043-1058 A SIMPLE CENSORED MEDIAN REGRESSION ESTIMATOR Lingzhi Zhou The Hong Kong University of Science and Technology Abstract: Ying, Jung and Wei (1995) proposed an estimation procedure for the censored median regression model that regresses the median of the survival time, or its transform, on the covariates. The procedure requires solving complicated nonlinear equations and thus can be very difficult to implement in practice, es- pecially when there are multiple covariates. Moreover, the asymptotic covariance matrix of the estimator involves the density of the errors that cannot be esti- mated reliably. In this paper, we propose a new estimator for the censored median regression model. Our estimation procedure involves solving some convex min- imization problems and can be easily implemented through linear programming (Koenker and D'Orey (1987)). In addition, a resampling method is presented for estimating the covariance matrix of the new estimator. Numerical studies indi- cate the superiority of the finite sample performance of our estimator over that in Ying, Jung and Wei (1995). Key words and phrases: Censoring, convexity, LAD, resampling. 1. Introduction The accelerated failure time (AFT) model, which relates the logarithm of the survival time to covariates, is an attractive alternative to the popular Cox (1972) proportional hazards model due to its ease of interpretation. The model assumes that the failure time T , or some monotonic transformation of it, is linearly related to the covariate vector Z 0 Ti = β0Zi + "i; i = 1; : : : ; n: (1.1) Under censoring, we only observe Yi = min(Ti; Ci), where Ci are censoring times, and Ti and Ci are independent conditional on Zi.
    [Show full text]
  • Cluster Analysis, a Powerful Tool for Data Analysis in Education
    International Statistical Institute, 56th Session, 2007: Rita Vasconcelos, Mßrcia Baptista Cluster Analysis, a powerful tool for data analysis in Education Vasconcelos, Rita Universidade da Madeira, Department of Mathematics and Engeneering Caminho da Penteada 9000-390 Funchal, Portugal E-mail: [email protected] Baptista, Márcia Direcção Regional de Saúde Pública Rua das Pretas 9000 Funchal, Portugal E-mail: [email protected] 1. Introduction A database was created after an inquiry to 14-15 - year old students, which was developed with the purpose of identifying the factors that could socially and pedagogically frame the results in Mathematics. The data was collected in eight schools in Funchal (Madeira Island), and we performed a Cluster Analysis as a first multivariate statistical approach to this database. We also developed a logistic regression analysis, as the study was carried out as a contribution to explain the success/failure in Mathematics. As a final step, the responses of both statistical analysis were studied. 2. Cluster Analysis approach The questions that arise when we try to frame socially and pedagogically the results in Mathematics of 14-15 - year old students, are concerned with the types of decisive factors in those results. It is somehow underlying our objectives to classify the students according to the factors understood by us as being decisive in students’ results. This is exactly the aim of Cluster Analysis. The hierarchical solution that can be observed in the dendogram presented in the next page, suggests that we should consider the 3 following clusters, since the distances increase substantially after it: Variables in Cluster1: mother qualifications; father qualifications; student’s results in Mathematics as classified by the school teacher; student’s results in the exam of Mathematics; time spent studying.
    [Show full text]
  • The Detection of Heteroscedasticity in Regression Models for Psychological Data
    Psychological Test and Assessment Modeling, Volume 58, 2016 (4), 567-592 The detection of heteroscedasticity in regression models for psychological data Andreas G. Klein1, Carla Gerhard2, Rebecca D. Büchner2, Stefan Diestel3 & Karin Schermelleh-Engel2 Abstract One assumption of multiple regression analysis is homoscedasticity of errors. Heteroscedasticity, as often found in psychological or behavioral data, may result from misspecification due to overlooked nonlinear predictor terms or to unobserved predictors not included in the model. Although methods exist to test for heteroscedasticity, they require a parametric model for specifying the structure of heteroscedasticity. The aim of this article is to propose a simple measure of heteroscedasticity, which does not need a parametric model and is able to detect omitted nonlinear terms. This measure utilizes the dispersion of the squared regression residuals. Simulation studies show that the measure performs satisfactorily with regard to Type I error rates and power when sample size and effect size are large enough. It outperforms the Breusch-Pagan test when a nonlinear term is omitted in the analysis model. We also demonstrate the performance of the measure using a data set from industrial psychology. Keywords: Heteroscedasticity, Monte Carlo study, regression, interaction effect, quadratic effect 1Correspondence concerning this article should be addressed to: Prof. Dr. Andreas G. Klein, Department of Psychology, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 6, 60629 Frankfurt; email: [email protected] 2Goethe University Frankfurt 3International School of Management Dortmund Leipniz-Research Centre for Working Environment and Human Factors 568 A. G. Klein, C. Gerhard, R. D. Büchner, S. Diestel & K. Schermelleh-Engel Introduction One of the standard assumptions underlying a linear model is that the errors are inde- pendently identically distributed (i.i.d.).
    [Show full text]
  • Testing Hypotheses for Differences Between Linear Regression Lines
    United States Department of Testing Hypotheses for Differences Agriculture Between Linear Regression Lines Forest Service Stanley J. Zarnoch Southern Research Station e-Research Note SRS–17 May 2009 Abstract The general linear model (GLM) is often used to test for differences between means, as for example when the Five hypotheses are identified for testing differences between simple linear analysis of variance is used in data analysis for designed regression lines. The distinctions between these hypotheses are based on a priori assumptions and illustrated with full and reduced models. The studies. However, the GLM has a much wider application contrast approach is presented as an easy and complete method for testing and can also be used to compare regression lines. The for overall differences between the regressions and for making pairwise GLM allows for the use of both qualitative and quantitative comparisons. Use of the Bonferroni adjustment to ensure a desired predictor variables. Quantitative variables are continuous experimentwise type I error rate is emphasized. SAS software is used to illustrate application of these concepts to an artificial simulated dataset. The values such as diameter at breast height, tree height, SAS code is provided for each of the five hypotheses and for the contrasts age, or temperature. However, many predictor variables for the general test and all possible specific tests. in the natural resources field are qualitative. Examples of qualitative variables include tree class (dominant, Keywords: Contrasts, dummy variables, F-test, intercept, SAS, slope, test of conditional error. codominant); species (loblolly, slash, longleaf); and cover type (clearcut, shelterwood, seed tree). Fraedrich and others (1994) compared linear regressions of slash pine cone specific gravity and moisture content Introduction on collection date between families of slash pine grown in a seed orchard.
    [Show full text]
  • 5. Dummy-Variable Regression and Analysis of Variance
    Sociology 740 John Fox Lecture Notes 5. Dummy-Variable Regression and Analysis of Variance Copyright © 2014 by John Fox Dummy-Variable Regression and Analysis of Variance 1 1. Introduction I One of the limitations of multiple-regression analysis is that it accommo- dates only quantitative explanatory variables. I Dummy-variable regressors can be used to incorporate qualitative explanatory variables into a linear model, substantially expanding the range of application of regression analysis. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 2 2. Goals: I To show how dummy regessors can be used to represent the categories of a qualitative explanatory variable in a regression model. I To introduce the concept of interaction between explanatory variables, and to show how interactions can be incorporated into a regression model by forming interaction regressors. I To introduce the principle of marginality, which serves as a guide to constructing and testing terms in complex linear models. I To show how incremental I -testsareemployedtotesttermsindummy regression models. I To show how analysis-of-variance models can be handled using dummy variables. c 2014 by John Fox Sociology 740 ° Dummy-Variable Regression and Analysis of Variance 3 3. A Dichotomous Explanatory Variable I The simplest case: one dichotomous and one quantitative explanatory variable. I Assumptions: Relationships are additive — the partial effect of each explanatory • variable is the same regardless of the specific value at which the other explanatory variable is held constant. The other assumptions of the regression model hold. • I The motivation for including a qualitative explanatory variable is the same as for including an additional quantitative explanatory variable: to account more fully for the response variable, by making the errors • smaller; and to avoid a biased assessment of the impact of an explanatory variable, • as a consequence of omitting another explanatory variables that is relatedtoit.
    [Show full text]
  • Simple Linear Regression: Straight Line Regression Between an Outcome Variable (Y ) and a Single Explanatory Or Predictor Variable (X)
    1 Introduction to Regression \Regression" is a generic term for statistical methods that attempt to fit a model to data, in order to quantify the relationship between the dependent (outcome) variable and the predictor (independent) variable(s). Assuming it fits the data reasonable well, the estimated model may then be used either to merely describe the relationship between the two groups of variables (explanatory), or to predict new values (prediction). There are many types of regression models, here are a few most common to epidemiology: Simple Linear Regression: Straight line regression between an outcome variable (Y ) and a single explanatory or predictor variable (X). E(Y ) = α + β × X Multiple Linear Regression: Same as Simple Linear Regression, but now with possibly multiple explanatory or predictor variables. E(Y ) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: A special case is polynomial regression. 2 3 E(Y ) = α + β1 × X + β2 × X + β3 × X + ::: Generalized Linear Model: Same as Multiple Linear Regression, but with a possibly transformed Y variable. This introduces considerable flexibil- ity, as non-linear and non-normal situations can be easily handled. G(E(Y )) = α + β1 × X1 + β2 × X2 + β3 × X3 + ::: In general, the transformation function G(Y ) can take any form, but a few forms are especially common: • Taking G(Y ) = logit(Y ) describes a logistic regression model: E(Y ) log( ) = α + β × X + β × X + β × X + ::: 1 − E(Y ) 1 1 2 2 3 3 2 • Taking G(Y ) = log(Y ) is also very common, leading to Poisson regression for count data, and other so called \log-linear" models.
    [Show full text]
  • Chapter 2 Simple Linear Regression Analysis the Simple
    Chapter 2 Simple Linear Regression Analysis The simple linear regression model We consider the modelling between the dependent and one independent variable. When there is only one independent variable in the linear regression model, the model is generally termed as a simple linear regression model. When there are more than one independent variables in the model, then the linear model is termed as the multiple linear regression model. The linear model Consider a simple linear regression model yX01 where y is termed as the dependent or study variable and X is termed as the independent or explanatory variable. The terms 0 and 1 are the parameters of the model. The parameter 0 is termed as an intercept term, and the parameter 1 is termed as the slope parameter. These parameters are usually called as regression coefficients. The unobservable error component accounts for the failure of data to lie on the straight line and represents the difference between the true and observed realization of y . There can be several reasons for such difference, e.g., the effect of all deleted variables in the model, variables may be qualitative, inherent randomness in the observations etc. We assume that is observed as independent and identically distributed random variable with mean zero and constant variance 2 . Later, we will additionally assume that is normally distributed. The independent variables are viewed as controlled by the experimenter, so it is considered as non-stochastic whereas y is viewed as a random variable with Ey()01 X and Var() y 2 . Sometimes X can also be a random variable.
    [Show full text]
  • Pubh 7405: REGRESSION ANALYSIS
    PubH 7405: REGRESSION ANALYSIS Review #2: Simple Correlation & Regression COURSE INFORMATION • Course Information are at address: www.biostat.umn.edu/~chap/pubh7405 • On each class web page, there is a brief version of the lecture for the day – the part with “formulas”; you can review, preview, or both – and as often as you like. • Follow Reading & Homework assignments at the end of the page when & if applicable. OFFICE HOURS • Instructor’s scheduled office hours: 1:15 to 2:15 Monday & Wednesday, in A441 Mayo Building • Other times are available by appointment • When really needed, can just drop in and interrupt me; could call before coming – making sure that I’m in. • I’m at a research facility on Fridays. Variables • A variable represents a characteristic or a class of measurement. It takes on different values on different subjects/persons. Examples include weight, height, race, sex, SBP, etc. The observed values, also called “observations,” form items of a data set. • Depending on the scale of measurement, we have different types of data. There are “observed variables” (Height, Weight, etc… each takes on different values on different subjects/person) and there are “calculated variables” (Sample Mean, Sample Proportion, etc… each is a “statistic” and each takes on different values on different samples). The Standard Deviation of a calculated variable is called the Standard Error of that variable/statistic. A “variable” – sample mean, sample standard deviation, etc… included – is like a “function”; when you apply it to a target element in its domain, the result is a “number”. For example, “height” is a variable and “the height of Mrs.
    [Show full text]
  • Tests of Hypotheses Using Statistics
    Tests of Hypotheses Using Statistics Adam Massey¤and Steven J. Millery Mathematics Department Brown University Providence, RI 02912 Abstract We present the various methods of hypothesis testing that one typically encounters in a mathematical statistics course. The focus will be on conditions for using each test, the hypothesis tested by each test, and the appropriate (and inappropriate) ways of using each test. We conclude by summarizing the di®erent tests (what conditions must be met to use them, what the test statistic is, and what the critical region is). Contents 1 Types of Hypotheses and Test Statistics 2 1.1 Introduction . 2 1.2 Types of Hypotheses . 3 1.3 Types of Statistics . 3 2 z-Tests and t-Tests 5 2.1 Testing Means I: Large Sample Size or Known Variance . 5 2.2 Testing Means II: Small Sample Size and Unknown Variance . 9 3 Testing the Variance 12 4 Testing Proportions 13 4.1 Testing Proportions I: One Proportion . 13 4.2 Testing Proportions II: K Proportions . 15 4.3 Testing r £ c Contingency Tables . 17 4.4 Incomplete r £ c Contingency Tables Tables . 18 5 Normal Regression Analysis 19 6 Non-parametric Tests 21 6.1 Tests of Signs . 21 6.2 Tests of Ranked Signs . 22 6.3 Tests Based on Runs . 23 ¤E-mail: [email protected] yE-mail: [email protected] 1 7 Summary 26 7.1 z-tests . 26 7.2 t-tests . 27 7.3 Tests comparing means . 27 7.4 Variance Test . 28 7.5 Proportions . 28 7.6 Contingency Tables .
    [Show full text]
  • Correlation and Regression Analysis
    OIC ACCREDITATION CERTIFICATION PROGRAMME FOR OFFICIAL STATISTICS Correlation and Regression Analysis TEXTBOOK ORGANISATION OF ISLAMIC COOPERATION STATISTICAL ECONOMIC AND SOCIAL RESEARCH AND TRAINING CENTRE FOR ISLAMIC COUNTRIES OIC ACCREDITATION CERTIFICATION PROGRAMME FOR OFFICIAL STATISTICS Correlation and Regression Analysis TEXTBOOK {{Dr. Mohamed Ahmed Zaid}} ORGANISATION OF ISLAMIC COOPERATION STATISTICAL ECONOMIC AND SOCIAL RESEARCH AND TRAINING CENTRE FOR ISLAMIC COUNTRIES © 2015 The Statistical, Economic and Social Research and Training Centre for Islamic Countries (SESRIC) Kudüs Cad. No: 9, Diplomatik Site, 06450 Oran, Ankara – Turkey Telephone +90 – 312 – 468 6172 Internet www.sesric.org E-mail [email protected] The material presented in this publication is copyrighted. The authors give the permission to view, copy download, and print the material presented that these materials are not going to be reused, on whatsoever condition, for commercial purposes. For permission to reproduce or reprint any part of this publication, please send a request with complete information to the Publication Department of SESRIC. All queries on rights and licenses should be addressed to the Statistics Department, SESRIC, at the aforementioned address. DISCLAIMER: Any views or opinions presented in this document are solely those of the author(s) and do not reflect the views of SESRIC. ISBN: xxx-xxx-xxxx-xx-x Cover design by Publication Department, SESRIC. For additional information, contact Statistics Department, SESRIC. i CONTENTS Acronyms
    [Show full text]
  • Profiling Heteroscedasticity in Linear Regression Models
    358 The Canadian Journal of Statistics Vol. 43, No. 3, 2015, Pages 358–377 La revue canadienne de statistique Profiling heteroscedasticity in linear regression models Qian M. ZHOU1*, Peter X.-K. SONG2 and Mary E. THOMPSON3 1Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, British Columbia, Canada 2Department of Biostatistics, University of Michigan, Ann Arbor, Michigan, U.S.A. 3Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Ontario, Canada Key words and phrases: Heteroscedasticity; hybrid test; information ratio; linear regression models; model- based estimators; sandwich estimators; screening; weighted least squares. MSC 2010: Primary 62J05; secondary 62J20 Abstract: Diagnostics for heteroscedasticity in linear regression models have been intensively investigated in the literature. However, limited attention has been paid on how to identify covariates associated with heteroscedastic error variances. This problem is critical in correctly modelling the variance structure in weighted least squares estimation, which leads to improved estimation efficiency. We propose covariate- specific statistics based on information ratios formed as comparisons between the model-based and sandwich variance estimators. A two-step diagnostic procedure is established, first to detect heteroscedasticity in error variances, and then to identify covariates the error variance structure might depend on. This proposed method is generalized to accommodate practical complications, such as when covariates associated with the heteroscedastic variances might not be associated with the mean structure of the response variable, or when strong correlation is present amongst covariates. The performance of the proposed method is assessed via a simulation study and is illustrated through a data analysis in which we show the importance of correct identification of covariates associated with the variance structure in estimation and inference.
    [Show full text]
  • Regression Analysis
    CALIFORNIA STATE UNIVERSITY, SACRAMENTO College of Business Administration NOTES FOR DATA ANALYSIS [Eleventh Edition] Manfred W. Hopfe, Ph.D. Stanley A. Taylor, Ph.D. Last Revised July 2008 by Min Li, Ph.D. 1 NOTES FOR DATA ANALYSIS [Tenth Edition] As stated in previous editions, the topics presented in this publication, which we have produced to assist our students, have been heavily influenced by the Making Statistics More Effective in Schools of Business Conferences held throughout the United States. The first conference was held at the University of Chicago in 1986. The School of Business Administration at California State University, Sacramento, hosted the tenth annual conference June 15-17, 1995. Most recent conferences were held at Babson College, (June 1999) and Syracuse University (June 2000). As with any publication in its developmental stages, there will be errors. If you find any errors, we ask for your feedback since this is a dynamic publication we continually revise. Throughout the semester you will be provided additional handouts to supplement the material in this book. Manfred W. Hopfe, Ph.D. Stanley A. Taylor, Ph.D. Carmichael, California August 2000 2 TABLE OF CONTENTS REVIEW................................................................................................................................................................ 5 Statistics vs. Parameters.................................................................................................................................... 5 Mean and Variance ..........................................................................................................................................
    [Show full text]