Well-developed and understood properties
|
|
- Robert Mason
- 5 years ago
- Views:
Transcription
1 1 INTRODUCTION TO LINEAR MODELS 1 THE CLASSICAL LINEAR MODEL Most commonly used statistical models Flexible models Well-developed and understood properties Ease of interpretation Building block for more general models 1 General Linear Model 2 Generalized Linear Model 3 Generalized Estimating Equations 4 Generalized Linear Mixed Model, etc 5 Heirarchical Generalized Linear Mixed Model, etc EXAMPLES: 1 EXAMPLE 1 Simple linear regression Objective: Relate weight to blood pressure Consider a random sample of n individuals The i-th patient has weight x i and blood pressure Y i (i = 1,2,,n) where Y i is the response variable, x i is a regressor variable, Y i = β 0 + x i + ε i, β 0, are regression coefficients unknown model parameters to be estimated, ε i is an error term
2 2 1 INTRODUCTION TO LINEAR MODELS x y Figure 1 y= x + 3*x^2 y= *x 2 EXAMPLE 2 Polynomial regression Objective: Same as EXAMPLE 1 Y i = β 0 + x i + β 2 x 2 i + ε i Is this still a linear model? 3 EXAMPLE 3 Multiple linear regression Objective: Relate blood pressure to weight and age For the i-th patient, x i1 = is weight and x i2 = is age Y i = β 0 + x i1 + β 2 x i2 + β 3 x i3 + ε i where x i3 = x i1 x i2 is the weight-by-age interaction term
3 1 INTRODUCTION TO LINEAR MODELS 3 4 EXAMPLE 4 Data Transformations Seber & Lee, Example 12 The Law of Gravity The Inverse Square Law states that the force of gravity F between two bodies a distance D apart is given by F = c D β Question: By transforming variables, how can this be viewed as a linear regression model for the paramter β? where Y i = log(f i ), x i = log(d i ), β 0 = log(c), = β, ε i is an error term Y i = β 0 + x i + ε i, Seber states and from experimental data we can estimate β and test whether β = 2
4 4 1 INTRODUCTION TO LINEAR MODELS MATRIX REPRESENTATION OF LINEAR MODELS The general linear model in matrix form: Y 1 x 10 x 11 x 12 x 1,p 1 Y 2 = x 20 x 21 x 22 x 2,p 1 Y n x n0 x n1 x n2 x n,p 1 Equivalent shorthand form: Y = Xβ + ε β 0 β 2 β p 1 Y (n 1) is the response vector + X (n p) is the design (or model or regression) matrix β (p 1) is the vector of regression coefficients (model parameters) ε (n 1) is the error vector (mean 0) ε 1 ε 2 ε n (a) Example 1 Y 1 Y 2 Y n 1 x 1 1 x 2 = 1 x n ( β0 ) + ε 1 ε 2 ε n
5 1 INTRODUCTION TO LINEAR MODELS 5 (b) Example 2 Y 1 Y 2 Y n 1 x 1 x x 2 x 2 2 = 1 x n x 2 n β 0 β 2 + ε 1 ε 2 ε n (c) Example 3 Y 1 1 x 11 x 12 x 13 Y 2 = 1 x 21 x 22 x 23 Y n 1 x n1 x n2 x n3 β 0 β 2 β 3 + ε 1 ε 2 ε n (d) Example 4 log(f 1 ) log(f 2 ) = log(f n ) 1 log(d 1 ) 1 log(d 2 ) 1 log(d n ) ( β0 ) + ε 1 ε 2 ε n
6 6 1 INTRODUCTION TO LINEAR MODELS (e) EXAMPLE 5 One-way analysis of variance Objective: Compare two treatments for blood pressure Consider random samples of J individuals taking one of two blood pressure medications Y ij is the blood pressure for individual j from treatment group i Y ij = µ + α i + ε ij µ = overall mean blood pressure, α i = effect on blood pressure for treatment i (i = 1,2), ε ij = error term for subject j receiving treatment i Alternate Model Representation: Y ij = µ + α 1 I 1 + α 2 I 2 + ε ij where I 1 is an indicator variable for membership in treatment group 1 and I 2 is an indicator variable for membership in treatment group 2 (f) EXAMPLE 6 Two-way analysis of variance Objective: Same as EXAMPLE 5, but we now also consider a patient s sex Y ijk = µ + α i + β j + ε ijk µ = overall mean blood pressure, α i = effect on blood pressure for treatment i (i = 1,2), β j = effect on blood pressure for sex j (j = 1,2), ε ijk = error term for subject k of sex j receiving tmnt i Alternate Model Representation: Y ij = µ + α 1 I 1 + α 2 I J 1 + β 2 J 2 + ε ijk where I 1 is an indicator variable for membership in treatment group 1, I 2 is an indicator variable for membership in treatment group 2, J 1 indicates sex 1, and J 2 indicates sex 2
7 1 INTRODUCTION TO LINEAR MODELS 7 (g) EXAMPLE 7 Analysis of covariance Objective: Same as EXAMPLE 5, controlling for age Y ij = µ + α i + β(x ij x ) + ε ij µ = overall mean blood pressure, α i = effect on blood pressure for treatment i (i = 1,2), β = slope parameter, x ij = age of subject j receiving treatment i, x = overall mean age, ε ij = error term for subject j receiving treatment i Note: The alternative model representations for these ANOVA and AN- COVA models make it clear that these are linear models Let s continue with matrix representation of these models 5 Example 5 Y 11 Y 1J Y 21 Y 2J = µ α 1 α 2 + ε 11 ε 1J ε 21 ε 2J
8 8 1 INTRODUCTION TO LINEAR MODELS 6 Example 6 Y 111 Y 11K Y 121 Y 12K Y 211 Y 21K Y 221 Y 22K = µ α 1 α 2 β 2 + ε 111 ε 11K ε 121 ε 12K ε 211 ε 21K ε 221 ε 22K 7 Example 7 Y 11 Y 1J Y 21 Y 2J = (x 11 x ) (x 1J x ) (x 21 x ) (x 2J x ) µ α 1 α 2 β + ε 11 ε 1J ε 21 ε 2J
9 1 INTRODUCTION TO LINEAR MODELS 9 In Summary: Linear models have the form Y = Xβ + ε, where Y n 1 response vector, X n p model matrix, β p 1 vector of unknown regression parameters, ε n 1 mean zero random error vector Notes: 1 Usually x i0 = 1 for all i That is, usually there is an intercept β 0 in the model and the first column of the design matrix X is all 1 s 2 x i0,x i1,,x i,p 1 are called the predictor variables or regressor variables or the covariates They are the data 3 Linear Model means the model is linear in the unknown regression coefficients β 0,,,β p 1 4 Instead of matrices, we can write the model in terms of vectors: p 1 Y = β j x j + ε, j=0 where x j = (x 1j,x 2j,,x nj ) (Also note that we will use the convention in this course that a vector is a column vector) 5 ε is the random part of the model Right now we just assume that E(ε) = 0 Later, we will make more assumptions about the distribution of ε 6 Y is random because ε is random Y inherits randomness from ε 7 We can thus evaluate p 1 E[Y] = E[ j=0 p 1 β j x j + ε] = E[ j=0 p 1 β j x j ] + E[ε] = β j x j 8 The vector E[Y] is a linear combination of the x j 9 E[Y] span(x 0,x 1,,x p 1 ) Ω, j=0
10 10 1 INTRODUCTION TO LINEAR MODELS FACT: We obtain least squares estimators (LSE s) of the β j, denoted ˆβ j, by projecting Y onto Ω Note that Y is n-dimensional and Ω has dimension p The projection of Y onto Ω is denoted Ŷ In Class Exercise: Model Y i = βx i + ε i (linear regression through the origin) Two datasets: 1 {(x,y) = (2,1),(0,1)} 2 {(x,y) = (1,2),(1/2,2)} For each dataset: 1 Write out the model with vectors and matrices (using the data) 2 Show the vectors y and x on a graph 3 Identify (on your graph) Ω, 4 Plot ŷ = Proj Ω (y), 5 Plot ˆε = y ŷ, identify Ω 6 What is the dimension of Ω? What is the dimension of Ω? 7 Also make a scatterplot of the data and sketch the least squares line 8 Alternatively, if we fit the model Y i = β 0 + x i + ε i, how do your answers to the previous questions change?
11 1 INTRODUCTION TO LINEAR MODELS 11 2 Dataset 1 2 Dataset 2 dimension dimension dimension 1 Scatterplot dimension 1 Scatterplot y y x x
3. For a given dataset and linear model, what do you think is true about least squares estimates? Is Ŷ always unique? Yes. Is ˆβ always unique? No.
7. LEAST SQUARES ESTIMATION 1 EXERCISE: Least-Squares Estimation and Uniqueness of Estimates 1. For n real numbers a 1,...,a n, what value of a minimizes the sum of squared distances from a to each of
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 47 Outline 1 Regression 2 Simple Linear regression 3 Basic concepts in regression 4 How to estimate unknown parameters 5 Properties of Least Squares Estimators:
More informationCategorical Predictor Variables
Categorical Predictor Variables We often wish to use categorical (or qualitative) variables as covariates in a regression model. For binary variables (taking on only 2 values, e.g. sex), it is relatively
More informationNeed for Several Predictor Variables
Multiple regression One of the most widely used tools in statistical analysis Matrix expressions for multiple regression are the same as for simple linear regression Need for Several Predictor Variables
More informationSummer School in Statistics for Astronomers V June 1 - June 6, Regression. Mosuk Chow Statistics Department Penn State University.
Summer School in Statistics for Astronomers V June 1 - June 6, 2009 Regression Mosuk Chow Statistics Department Penn State University. Adapted from notes prepared by RL Karandikar Mean and variance Recall
More informationBIOS 2083: Linear Models
BIOS 2083: Linear Models Abdus S Wahed September 2, 2009 Chapter 0 2 Chapter 1 Introduction to linear models 1.1 Linear Models: Definition and Examples Example 1.1.1. Estimating the mean of a N(μ, σ 2
More informationLecture 4 Multiple linear regression
Lecture 4 Multiple linear regression BIOST 515 January 15, 2004 Outline 1 Motivation for the multiple regression model Multiple regression in matrix notation Least squares estimation of model parameters
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More information11 Hypothesis Testing
28 11 Hypothesis Testing 111 Introduction Suppose we want to test the hypothesis: H : A q p β p 1 q 1 In terms of the rows of A this can be written as a 1 a q β, ie a i β for each row of A (here a i denotes
More informationLecture 2. The Simple Linear Regression Model: Matrix Approach
Lecture 2 The Simple Linear Regression Model: Matrix Approach Matrix algebra Matrix representation of simple linear regression model 1 Vectors and Matrices Where it is necessary to consider a distribution
More informationLecture 14 Simple Linear Regression
Lecture 4 Simple Linear Regression Ordinary Least Squares (OLS) Consider the following simple linear regression model where, for each unit i, Y i is the dependent variable (response). X i is the independent
More informationScatter plot of data from the study. Linear Regression
1 2 Linear Regression Scatter plot of data from the study. Consider a study to relate birthweight to the estriol level of pregnant women. The data is below. i Weight (g / 100) i Weight (g / 100) 1 7 25
More informationMAT2377. Rafa l Kulik. Version 2015/November/26. Rafa l Kulik
MAT2377 Rafa l Kulik Version 2015/November/26 Rafa l Kulik Bivariate data and scatterplot Data: Hydrocarbon level (x) and Oxygen level (y): x: 0.99, 1.02, 1.15, 1.29, 1.46, 1.36, 0.87, 1.23, 1.55, 1.40,
More informationProblems. Suppose both models are fitted to the same data. Show that SS Res, A SS Res, B
Simple Linear Regression 35 Problems 1 Consider a set of data (x i, y i ), i =1, 2,,n, and the following two regression models: y i = β 0 + β 1 x i + ε, (i =1, 2,,n), Model A y i = γ 0 + γ 1 x i + γ 2
More informationStatistical View of Least Squares
May 23, 2006 Purpose of Regression Some Examples Least Squares Purpose of Regression Purpose of Regression Some Examples Least Squares Suppose we have two variables x and y Purpose of Regression Some Examples
More informationINTRODUCING LINEAR REGRESSION MODELS Response or Dependent variable y
INTRODUCING LINEAR REGRESSION MODELS Response or Dependent variable y Predictor or Independent variable x Model with error: for i = 1,..., n, y i = α + βx i + ε i ε i : independent errors (sampling, measurement,
More informationRegression Analysis Chapter 2 Simple Linear Regression
Regression Analysis Chapter 2 Simple Linear Regression Dr. Bisher Mamoun Iqelan biqelan@iugaza.edu.ps Department of Mathematics The Islamic University of Gaza 2010-2011, Semester 2 Dr. Bisher M. Iqelan
More information4 Multiple Linear Regression
4 Multiple Linear Regression 4. The Model Definition 4.. random variable Y fits a Multiple Linear Regression Model, iff there exist β, β,..., β k R so that for all (x, x 2,..., x k ) R k where ε N (, σ
More informationMultivariate Regression
Multivariate Regression The so-called supervised learning problem is the following: we want to approximate the random variable Y with an appropriate function of the random variables X 1,..., X p with the
More information14 Multiple Linear Regression
B.Sc./Cert./M.Sc. Qualif. - Statistics: Theory and Practice 14 Multiple Linear Regression 14.1 The multiple linear regression model In simple linear regression, the response variable y is expressed in
More informationLecture 19 Multiple (Linear) Regression
Lecture 19 Multiple (Linear) Regression Thais Paiva STA 111 - Summer 2013 Term II August 1, 2013 1 / 30 Thais Paiva STA 111 - Summer 2013 Term II Lecture 19, 08/01/2013 Lecture Plan 1 Multiple regression
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationSTAT5044: Regression and Anova. Inyoung Kim
STAT5044: Regression and Anova Inyoung Kim 2 / 51 Outline 1 Matrix Expression 2 Linear and quadratic forms 3 Properties of quadratic form 4 Properties of estimates 5 Distributional properties 3 / 51 Matrix
More informationMA 575 Linear Models: Cedric E. Ginestet, Boston University Mixed Effects Estimation, Residuals Diagnostics Week 11, Lecture 1
MA 575 Linear Models: Cedric E Ginestet, Boston University Mixed Effects Estimation, Residuals Diagnostics Week 11, Lecture 1 1 Within-group Correlation Let us recall the simple two-level hierarchical
More informationSimple Linear Regression
Simple Linear Regression In simple linear regression we are concerned about the relationship between two variables, X and Y. There are two components to such a relationship. 1. The strength of the relationship.
More informationSimple Linear Regression
Simple Linear Regression September 24, 2008 Reading HH 8, GIll 4 Simple Linear Regression p.1/20 Problem Data: Observe pairs (Y i,x i ),i = 1,...n Response or dependent variable Y Predictor or independent
More informationMultivariate Regression (Chapter 10)
Multivariate Regression (Chapter 10) This week we ll cover multivariate regression and maybe a bit of canonical correlation. Today we ll mostly review univariate multivariate regression. With multivariate
More informationChapter 14 Simple Linear Regression (A)
Chapter 14 Simple Linear Regression (A) 1. Characteristics Managerial decisions often are based on the relationship between two or more variables. can be used to develop an equation showing how the variables
More informationLinear Regression. In this problem sheet, we consider the problem of linear regression with p predictors and one intercept,
Linear Regression In this problem sheet, we consider the problem of linear regression with p predictors and one intercept, y = Xβ + ɛ, where y t = (y 1,..., y n ) is the column vector of target values,
More informationWeighted Least Squares
Weighted Least Squares The standard linear model assumes that Var(ε i ) = σ 2 for i = 1,..., n. As we have seen, however, there are instances where Var(Y X = x i ) = Var(ε i ) = σ2 w i. Here w 1,..., w
More informationChapter 5 Matrix Approach to Simple Linear Regression
STAT 525 SPRING 2018 Chapter 5 Matrix Approach to Simple Linear Regression Professor Min Zhang Matrix Collection of elements arranged in rows and columns Elements will be numbers or symbols For example:
More information3 Multiple Linear Regression
3 Multiple Linear Regression 3.1 The Model Essentially, all models are wrong, but some are useful. Quote by George E.P. Box. Models are supposed to be exact descriptions of the population, but that is
More informationCh 2: Simple Linear Regression
Ch 2: Simple Linear Regression 1. Simple Linear Regression Model A simple regression model with a single regressor x is y = β 0 + β 1 x + ɛ, where we assume that the error ɛ is independent random component
More informationMultiple Linear Regression
Multiple Linear Regression University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html 1 / 42 Passenger car mileage Consider the carmpg dataset taken from
More informationSimple linear regression
Simple linear regression Prof. Giuseppe Verlato Unit of Epidemiology & Medical Statistics, Dept. of Diagnostics & Public Health, University of Verona Statistics with two variables two nominal variables:
More informationStatistical View of Least Squares
Basic Ideas Some Examples Least Squares May 22, 2007 Basic Ideas Simple Linear Regression Basic Ideas Some Examples Least Squares Suppose we have two variables x and y Basic Ideas Simple Linear Regression
More informationMachine Learning. Regression Case Part 1. Linear Regression. Machine Learning: Regression Case.
.. Cal Poly CSC 566 Advanced Data Mining Alexander Dekhtyar.. Machine Learning. Regression Case Part 1. Linear Regression Machine Learning: Regression Case. Dataset. Consider a collection of features X
More informationData Analysis and Statistical Methods Statistics 651
y 1 2 3 4 5 6 7 x Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Lecture 32 Suhasini Subba Rao Previous lecture We are interested in whether a dependent
More informationSimple and Multiple Linear Regression
Sta. 113 Chapter 12 and 13 of Devore March 12, 2010 Table of contents 1 Simple Linear Regression 2 Model Simple Linear Regression A simple linear regression model is given by Y = β 0 + β 1 x + ɛ where
More informationMultiple Linear Regression
Multiple Linear Regression Simple linear regression tries to fit a simple line between two variables Y and X. If X is linearly related to Y this explains some of the variability in Y. In most cases, there
More informationCLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS
HEALTH ECONOMICS, VOL. 6: 439 443 (1997) HEALTH ECONOMICS LETTERS CLUSTER EFFECTS AND SIMULTANEITY IN MULTILEVEL MODELS RICHARD BLUNDELL 1 AND FRANK WINDMEIJER 2 * 1 Department of Economics, University
More informationLinear Regression. 1 Introduction. 2 Least Squares
Linear Regression 1 Introduction It is often interesting to study the effect of a variable on a response. In ANOVA, the response is a continuous variable and the variables are discrete / categorical. What
More informationBasic Distributional Assumptions of the Linear Model: 1. The errors are unbiased: E[ε] = The errors are uncorrelated with common variance:
8. PROPERTIES OF LEAST SQUARES ESTIMATES 1 Basic Distributional Assumptions of the Linear Model: 1. The errors are unbiased: E[ε] = 0. 2. The errors are uncorrelated with common variance: These assumptions
More informationCh 3: Multiple Linear Regression
Ch 3: Multiple Linear Regression 1. Multiple Linear Regression Model Multiple regression model has more than one regressor. For example, we have one response variable and two regressor variables: 1. delivery
More information. a m1 a mn. a 1 a 2 a = a n
Biostat 140655, 2008: Matrix Algebra Review 1 Definition: An m n matrix, A m n, is a rectangular array of real numbers with m rows and n columns Element in the i th row and the j th column is denoted by
More informationECON 3150/4150 spring term 2014: Exercise set for the first seminar and DIY exercises for the first few weeks of the course
ECON 3150/4150 spring term 2014: Exercise set for the first seminar and DIY exercises for the first few weeks of the course Ragnar Nymoen 13 January 2014. Exercise set to seminar 1 (week 6, 3-7 Feb) This
More informationMatrices and vectors A matrix is a rectangular array of numbers. Here s an example: A =
Matrices and vectors A matrix is a rectangular array of numbers Here s an example: 23 14 17 A = 225 0 2 This matrix has dimensions 2 3 The number of rows is first, then the number of columns We can write
More informationReference: Davidson and MacKinnon Ch 2. In particular page
RNy, econ460 autumn 03 Lecture note Reference: Davidson and MacKinnon Ch. In particular page 57-8. Projection matrices The matrix M I X(X X) X () is often called the residual maker. That nickname is easy
More informationBusiness Statistics. Tommaso Proietti. Linear Regression. DEF - Università di Roma 'Tor Vergata'
Business Statistics Tommaso Proietti DEF - Università di Roma 'Tor Vergata' Linear Regression Specication Let Y be a univariate quantitative response variable. We model Y as follows: Y = f(x) + ε where
More informationSimple Linear Regression
Simple Linear Regression ST 430/514 Recall: A regression model describes how a dependent variable (or response) Y is affected, on average, by one or more independent variables (or factors, or covariates)
More informationRegression diagnostics
Regression diagnostics Kerby Shedden Department of Statistics, University of Michigan November 5, 018 1 / 6 Motivation When working with a linear model with design matrix X, the conventional linear model
More informationACOVA and Interactions
Chapter 15 ACOVA and Interactions Analysis of covariance (ACOVA) incorporates one or more regression variables into an analysis of variance. As such, we can think of it as analogous to the two-way ANOVA
More informationMa 3/103: Lecture 24 Linear Regression I: Estimation
Ma 3/103: Lecture 24 Linear Regression I: Estimation March 3, 2017 KC Border Linear Regression I March 3, 2017 1 / 32 Regression analysis Regression analysis Estimate and test E(Y X) = f (X). f is the
More informationTMA4255 Applied Statistics V2016 (5)
TMA4255 Applied Statistics V2016 (5) Part 2: Regression Simple linear regression [11.1-11.4] Sum of squares [11.5] Anna Marie Holand To be lectured: January 26, 2016 wiki.math.ntnu.no/tma4255/2016v/start
More informationBIOSTATISTICS NURS 3324
Simple Linear Regression and Correlation Introduction Previously, our attention has been focused on one variable which we designated by x. Frequently, it is desirable to learn something about the relationship
More informationLecture 6: Linear models and Gauss-Markov theorem
Lecture 6: Linear models and Gauss-Markov theorem Linear model setting Results in simple linear regression can be extended to the following general linear model with independently observed response variables
More informationMeasuring the fit of the model - SSR
Measuring the fit of the model - SSR Once we ve determined our estimated regression line, we d like to know how well the model fits. How far/close are the observations to the fitted line? One way to do
More informationLecture 6: Linear Regression
Lecture 6: Linear Regression Reading: Sections 3.1-3 STATS 202: Data mining and analysis Jonathan Taylor, 10/5 Slide credits: Sergio Bacallado 1 / 30 Simple linear regression Model: y i = β 0 + β 1 x i
More informationChapter 1: Linear Regression with One Predictor Variable also known as: Simple Linear Regression Bivariate Linear Regression
BSTT523: Kutner et al., Chapter 1 1 Chapter 1: Linear Regression with One Predictor Variable also known as: Simple Linear Regression Bivariate Linear Regression Introduction: Functional relation between
More informationMATH 644: Regression Analysis Methods
MATH 644: Regression Analysis Methods FINAL EXAM Fall, 2012 INSTRUCTIONS TO STUDENTS: 1. This test contains SIX questions. It comprises ELEVEN printed pages. 2. Answer ALL questions for a total of 100
More informationSTAT5044: Regression and Anova
STAT5044: Regression and Anova Inyoung Kim 1 / 25 Outline 1 Multiple Linear Regression 2 / 25 Basic Idea An extra sum of squares: the marginal reduction in the error sum of squares when one or several
More informationSTAT 540: Data Analysis and Regression
STAT 540: Data Analysis and Regression Wen Zhou http://www.stat.colostate.edu/~riczw/ Email: riczw@stat.colostate.edu Department of Statistics Colorado State University Fall 205 W. Zhou (Colorado State
More informationMa 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA
Ma 3/103: Lecture 25 Linear Regression II: Hypothesis Testing and ANOVA March 6, 2017 KC Border Linear Regression II March 6, 2017 1 / 44 1 OLS estimator 2 Restricted regression 3 Errors in variables 4
More informationTopic 7 - Matrix Approach to Simple Linear Regression. Outline. Matrix. Matrix. Review of Matrices. Regression model in matrix form
Topic 7 - Matrix Approach to Simple Linear Regression Review of Matrices Outline Regression model in matrix form - Fall 03 Calculations using matrices Topic 7 Matrix Collection of elements arranged in
More informationOutline. Remedial Measures) Extra Sums of Squares Standardized Version of the Multiple Regression Model
Outline 1 Multiple Linear Regression (Estimation, Inference, Diagnostics and Remedial Measures) 2 Special Topics for Multiple Regression Extra Sums of Squares Standardized Version of the Multiple Regression
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini June 9, 2018 1 / 56 Table of Contents Multiple linear regression Linear model setup Estimation of β Geometric interpretation Estimation of σ 2 Hat matrix Gram matrix
More informationx 21 x 22 x 23 f X 1 X 2 X 3 ε
Chapter 2 Estimation 2.1 Example Let s start with an example. Suppose that Y is the fuel consumption of a particular model of car in m.p.g. Suppose that the predictors are 1. X 1 the weight of the car
More informationMultiple Regression. Dr. Frank Wood. Frank Wood, Linear Regression Models Lecture 12, Slide 1
Multiple Regression Dr. Frank Wood Frank Wood, fwood@stat.columbia.edu Linear Regression Models Lecture 12, Slide 1 Review: Matrix Regression Estimation We can solve this equation (if the inverse of X
More informationMatrix Approach to Simple Linear Regression: An Overview
Matrix Approach to Simple Linear Regression: An Overview Aspects of matrices that you should know: Definition of a matrix Addition/subtraction/multiplication of matrices Symmetric/diagonal/identity matrix
More informationChapter 14. Linear least squares
Serik Sagitov, Chalmers and GU, March 5, 2018 Chapter 14 Linear least squares 1 Simple linear regression model A linear model for the random response Y = Y (x) to an independent variable X = x For a given
More informationLatent Growth Models 1
1 We will use the dataset bp3, which has diastolic blood pressure measurements at four time points for 256 patients undergoing three types of blood pressure medication. These are our observed variables:
More information4.7 Confidence and Prediction Intervals
4.7 Confidence and Prediction Intervals Instead of conducting tests we could find confidence intervals for a regression coefficient, or a set of regression coefficient, or for the mean of the response
More informationSTAT 100C: Linear models
STAT 100C: Linear models Arash A. Amini April 27, 2018 1 / 1 Table of Contents 2 / 1 Linear Algebra Review Read 3.1 and 3.2 from text. 1. Fundamental subspace (rank-nullity, etc.) Im(X ) = ker(x T ) R
More informationLecture 20: Linear model, the LSE, and UMVUE
Lecture 20: Linear model, the LSE, and UMVUE Linear Models One of the most useful statistical models is X i = β τ Z i + ε i, i = 1,...,n, where X i is the ith observation and is often called the ith response;
More informationSimple Linear Regression Analysis
LINEAR REGRESSION ANALYSIS MODULE II Lecture - 6 Simple Linear Regression Analysis Dr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur Prediction of values of study
More informationBayesian Linear Regression
Bayesian Linear Regression Sudipto Banerjee 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. September 15, 2010 1 Linear regression models: a Bayesian perspective
More informationStatistical Thinking in Biomedical Research Session #3 Statistical Modeling
Statistical Thinking in Biomedical Research Session #3 Statistical Modeling Lily Wang, PhD Department of Biostatistics (modified from notes by J.Patrie, R.Abbott, U of Virginia and WD Dupont, Vanderbilt
More informationPh.D. Qualifying Exam Friday Saturday, January 3 4, 2014
Ph.D. Qualifying Exam Friday Saturday, January 3 4, 2014 Put your solution to each problem on a separate sheet of paper. Problem 1. (5166) Assume that two random samples {x i } and {y i } are independently
More informationLecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε,
2. REVIEW OF LINEAR ALGEBRA 1 Lecture 1 Review: Linear models have the form (in matrix notation) Y = Xβ + ε, where Y n 1 response vector and X n p is the model matrix (or design matrix ) with one row for
More informationExample: Poisondata. 22s:152 Applied Linear Regression. Chapter 8: ANOVA
s:5 Applied Linear Regression Chapter 8: ANOVA Two-way ANOVA Used to compare populations means when the populations are classified by two factors (or categorical variables) For example sex and occupation
More informationApplied Econometrics (QEM)
Applied Econometrics (QEM) The Simple Linear Regression Model based on Prinicples of Econometrics Jakub Mućk Department of Quantitative Economics Jakub Mućk Applied Econometrics (QEM) Meeting #2 The Simple
More informationGeneral Linear Model (Chapter 4)
General Linear Model (Chapter 4) Outcome variable is considered continuous Simple linear regression Scatterplots OLS is BLUE under basic assumptions MSE estimates residual variance testing regression coefficients
More informationFIRST MIDTERM EXAM ECON 7801 SPRING 2001
FIRST MIDTERM EXAM ECON 780 SPRING 200 ECONOMICS DEPARTMENT, UNIVERSITY OF UTAH Problem 2 points Let y be a n-vector (It may be a vector of observations of a random variable y, but it does not matter how
More informationFinal Review. Yang Feng. Yang Feng (Columbia University) Final Review 1 / 58
Final Review Yang Feng http://www.stat.columbia.edu/~yangfeng Yang Feng (Columbia University) Final Review 1 / 58 Outline 1 Multiple Linear Regression (Estimation, Inference) 2 Special Topics for Multiple
More informationBusiness Statistics. Lecture 10: Correlation and Linear Regression
Business Statistics Lecture 10: Correlation and Linear Regression Scatterplot A scatterplot shows the relationship between two quantitative variables measured on the same individuals. It displays the Form
More informationIntroduction to Econometrics Midterm Examination Fall 2005 Answer Key
Introduction to Econometrics Midterm Examination Fall 2005 Answer Key Please answer all of the questions and show your work Clearly indicate your final answer to each question If you think a question is
More informationThe Gauss-Markov Model. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics / 61
The Gauss-Markov Model Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 611 1 / 61 Recall that Cov(u, v) = E((u E(u))(v E(v))) = E(uv) E(u)E(v) Var(u) = Cov(u, u) = E(u E(u)) 2 = E(u 2
More informationChapter 2 Multiple Regression I (Part 1)
Chapter 2 Multiple Regression I (Part 1) 1 Regression several predictor variables The response Y depends on several predictor variables X 1,, X p response {}}{ Y predictor variables {}}{ X 1, X 2,, X p
More informationECON 5350 Class Notes Functional Form and Structural Change
ECON 5350 Class Notes Functional Form and Structural Change 1 Introduction Although OLS is considered a linear estimator, it does not mean that the relationship between Y and X needs to be linear. In this
More informationCorrelation and Regression Theory 1) Multivariate Statistics
Correlation and Regression Theory 1) Multivariate Statistics What is a multivariate data set? How to statistically analyze this data set? Is there any kind of relationship between different variables in
More informationSimple Linear Regression
9-1 l Chapter 9 l Simple Linear Regression 9.1 Simple Linear Regression 9.2 Scatter Diagram 9.3 Graphical Method for Determining Regression 9.4 Least Square Method 9.5 Correlation Coefficient and Coefficient
More informationChaper 5: Matrix Approach to Simple Linear Regression. Matrix: A m by n matrix B is a grid of numbers with m rows and n columns. B = b 11 b m1 ...
Chaper 5: Matrix Approach to Simple Linear Regression Matrix: A m by n matrix B is a grid of numbers with m rows and n columns B = b 11 b 1n b m1 b mn Element b ik is from the ith row and kth column A
More informationUNIVERSITY OF MASSACHUSETTS. Department of Mathematics and Statistics. Basic Exam - Applied Statistics. Tuesday, January 17, 2017
UNIVERSITY OF MASSACHUSETTS Department of Mathematics and Statistics Basic Exam - Applied Statistics Tuesday, January 17, 2017 Work all problems 60 points are needed to pass at the Masters Level and 75
More informationLecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)
Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model
More informationLAB 5 INSTRUCTIONS LINEAR REGRESSION AND CORRELATION
LAB 5 INSTRUCTIONS LINEAR REGRESSION AND CORRELATION In this lab you will learn how to use Excel to display the relationship between two quantitative variables, measure the strength and direction of the
More informationSTAT 350: Geometry of Least Squares
The Geometry of Least Squares Mathematical Basics Inner / dot product: a and b column vectors a b = a T b = a i b i a b a T b = 0 Matrix Product: A is r s B is s t (AB) rt = s A rs B st Partitioned Matrices
More informationStatistical Techniques II EXST7015 Simple Linear Regression
Statistical Techniques II EXST7015 Simple Linear Regression 03a_SLR 1 Y - the dependent variable 35 30 25 The objective Given points plotted on two coordinates, Y and X, find the best line to fit the data.
More informationAnalysing data: regression and correlation S6 and S7
Basic medical statistics for clinical and experimental research Analysing data: regression and correlation S6 and S7 K. Jozwiak k.jozwiak@nki.nl 2 / 49 Correlation So far we have looked at the association
More information36-707: Regression Analysis Homework Solutions. Homework 3
36-707: Regression Analysis Homework Solutions Homework 3 Fall 2012 Problem 1 Y i = βx i + ɛ i, i {1, 2,..., n}. (a) Find the LS estimator of β: RSS = Σ n i=1(y i βx i ) 2 RSS β = Σ n i=1( 2X i )(Y i βx
More information