Applied Linear Regression Models Pdf
Magnhild Mongolo
"Applied Linear Statistical Models" is not a formal class at ETSU, but the material here might overlap some with the Statistical Methods sequence (STAT 5710 and 5720).The catalogue description for Statistical Methods 1 (STAT 5710) is: "Population and samples, probability distributions, estimation and testing, regression and correlation analysis, and diagnostic methods for assessing the assumptions of such models." The prerequisites are Linear Algebra (MATH 2010) and Elementary Statistics (MATH 2050 or equivalent).The catalogue description for Statistical Methods 2 (STAT 5710) is: "Generalized linear models, analysis of variance and regression on dummy variables, analysis of covariance, random and mixed effects models, and repeated measures." The prerequisites are Statistical Methods 1 (STAT 5710) or permission of instructor.These descriptions are based on the ETSU 2020-21 Graduate Catalog.
Copies of the classnotes are on the internet in PDF format as given below. The notes and supplements may contain hyperlinks to posted webpages; the links appear in red fonts. The "Proofs of Theorems" files were prepared in Beamer. The "Printout of Proofs" are printable PDF files of the Beamer slides without the pauses. These notes have not been classroom tested and may have typographical errors.
STAT-S 519 A Gentle Introduction to Statistics in R (3cr.) P: Consent of instructor. Introduces basic concepts of probability and statistical inference through careful study of several important procedures. Topics include discrete and continuous random variables, estimation, hypothesis testing, 1- and 2-sample location problems, ANOVA and simple linear regression. Assignments involve applying probability models and/or statistical methods to practical situations and/or actual data.
STAT-S 520 Introduction to Statistics (3 cr.) P: MATH M212, M301, M303, or the equivalent. Basic concepts of data analysis and statistical inference, applied to 1-sample and 2-sample location problems, the analysis of variance, and linear regression. Probability models and statistical methods applied to practical situations and actual data sets from various disciplines. Elementary statistical theory, including the plug-in principle, maximum likelihood, and the method of least squares.
STAT-S 530 Biometry (3 cr.) P: Consent of instructor. Applied general linear models course designed for graduate students with an interest in ecology, evolution, and behavior. Topics include: t-tests and nonparametric analogs, linear regression, ANOVA and non-linear analogs, logistic regression.
STAT-S 580 Introduction to Regression Models and Nonparametrics (3 cr.) P: STAT-S 519 or STAT-S 520, or consent of instructor. This course is a sequel to STAT-S 519, covering exploratory data analysis (EDA), regression models, and nonparametric statistics. Students will learn a wider range of statistical tools for data analysis than they encounter in an introductory statistics course. Enrollment in this online course is restricted to students in IU's online MS in Data Science degree program. Credit not given for both STAT-S 580 and STAT-S 625.
STAT-S 610 Introduction Statistical Computing (3 cr.) P: STAT S520, or consent of instructor. Introduction to R from a software engineering perspective and an introduction to algorithms commonly used in applied statistics. Optimization algorithms including gradient descent, stochastic gradient descent, the EM algorithm, and topics in convex optimization. Stochastic algorithms including rejection sampling, Metropolis-Hastings, and Gibbs sampling.
STAT-S 621 Fundamentals of Statistical Methods and Theory I (3cr.). Part 1 of a 2-semester sequence on fundamental concepts, principles, and techniques of probabilistic/statistical inference. Random variables and probability distributions, expectation, limit laws, parametric vs nonparametric inference, Bayesian vs frequentist inference, point estimation, hypothesis testing, confidence sets, simulation.
STAT-S 622 Fundamentals of Statistical Methods and Theory II (3cr.) Part 2 of a 2-semester sequence on fundamental concepts, principles, and techniques of probabilisitic/statistical inference. Hypothesis testing, Bayesian inference, regression, causal inference, classification, stochastic processes, simulation methods.
STAT-S 625 Nonparametric Theory and Data Analysis (3 cr.) P: Two statistics courses at the graduate level, or consent of instructor. Survey of methods for statistical inference that do not rely on parametric probability models. Statistical functionals, bootstrapping, empirical likelihood. Nonparametric density and curve estimation. Rank and permutation tests.
STAT-S 692 Internship in Statistical Consulting (3 cr.) P: STAT S690 and permission of Director of Graduate Studies. One semester internship at the Indiana Statistical Consulting Center (ISCC). Students work on actual consulting problems under the direct supervision of professional statisticians.
Applied Linear Regression, Third Edition has been thoroughly updated to help students master the theory and applications of linear regression modeling. Focusing on model building, assessing fit and reliability, and drawing conclusions, the text demonstrates how to develop estimation, confidence, and testing procedures primarily through the use of least squares regression. To facilitate quick learning, the Third Edition stresses the use of graphical methods in an effort to find appropriate models and to better understand them. In that spirit, most analyses and homework problems use graphs for the discovery of structure as well as for the summarization of results.
With its focus on graphical methods and analysis, coupled with many practical examples and exercises, this is an excellent textbook for upper-level undergraduates and graduate students, who will quickly learn how to use linear regression analysis techniques to solve and gain insight into real-life problems.
In statistics, linear regression is a statistical model which estimates the linear relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression.[1] This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.[2] If the explanatory variables are measured with error then errors-in-variables models are required, also known as measurement error models.
In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data. Such models are called linear models.[3] Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile is used. Like all forms of regression analysis, linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis.
Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications.[4] This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because the statistical properties of the resulting estimators are easier to determine.
Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the "lack of fit" in some other norm (as with least absolute deviations regression), or by minimizing a penalized version of the least squares cost function as in ridge regression (L2-norm penalty) and lasso (L1-norm penalty). Use of the Mean Squared Error(MSE) as the cost on a dataset that has many large outliers, can result in a model that fits the outliers more than the true data due to the higher importance assigned by MSE to large errors. So, cost functions that are robust to outliers should be used if the dataset has many large outliers. Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous.
Consider a situation where a small ball is being tossed up in the air and then we measure its heights of ascent hi at various moments in time ti. Physics tells us that, ignoring the drag, the relationship can be modeled as
Standard linear regression models with standard estimation techniques make a number of assumptions about the predictor variables, the response variables and their relationship. Numerous extensions have been developed that allow each of these assumptions to be relaxed (i.e. reduced to a weaker form), and in some cases eliminated entirely. Generally these extensions make the estimation procedure more complex and time-consuming, and may also require more data in order to produce an equally precise model.[citation needed]
Violations of these assumptions can result in biased estimations of β, biased standard errors, untrustworthy confidence intervals and significance tests. Beyond these assumptions, several other statistical properties of the data strongly influence the performance of different estimation methods:
Care must be taken when interpreting regression results, as some of the regressors may not allow for marginal changes (such as dummy variables, or the intercept term), while others cannot be held fixed (recall the example from the introduction: it would be impossible to "hold ti fixed" and at the same time change the value of ti2).
c80f0f1006