Module 10: Mixed model theory II: Tests and confidence intervals

Size: px
Start display at page:

Download "Module 10: Mixed model theory II: Tests and confidence intervals"

Transcription

1 /MIXED LINEAR MODELS PREPARED BY THE STATISTICS GROUPS AT IMM, DTU AND KU-LIFE Module 10: Mixed model theory II: Tests and confidence intervals 10.1 Notes Summary of the first theory module Testing fixed effects Confidence intervals of fixed effects The estimate and the contrast statements Test for random effects parameters Confidence intervals for random effects parameters Notes The first theory module described how a mixed model is defined and how the model parameters in a mixed model are estimated from observed data. This module describes how the tests for the fixed effects are computed (typically represented in an ANOVA table), and how to construct confidence intervals Summary of the first theory module Recall from the first theory module that any linear normal mixed model, can be expressed as: y N(Xβ, V), Here X is the design matrix for the fixed effects part of the model, β is the fixed effects parameters, and V is the covariance matrix. The covariance matrix V is specified via the random effects in the model and the additional R matrix, but that is not important here /Mixed Linear Models Last modified August 23, 2011

2 Module 10: Mixed model theory II: Tests and confidence intervals Testing fixed effects Typically the hypothesis of interest can be expressed as some linear combination of the model parameters: L β = c where L is a matrix, or a column vector with the same number of rows as there are elements in β. c is a constant and quite often zero. Consider the following example: In a one way ANOVA model with three treatments the fixed effects parameter vector would be β = (µ, α 1, α 2, α 3 ). The test for similar effect of treatment 1 and treatment 2 can be expressed as: ( ) }{{} L µ α 1 α 2 α 3 = 0 which is the same as α 1 α 2 = 0. The hypothesis that all three treatments have the same effect can similarly be expressed as: ( ) µ α α 2 = 0 }{{} α L 3 where the L matrix express that α 1 α 2 = 0 and α 1 α 3 = 0, which is the same as all three being equal. Not every hypothesis that can be expressed as a linear combination of the parameters are meaningful. Consider again the one way ANOVA example with parameters β = (µ, α 1, α 2, α 3 ). The hypothesis α 1 = 0 is not meaningful for this model. This is not obvious right away, but consider the fixed part of the model with arbitrary α 1, and with α 1 = 0: E(y) = µ + α 1 α 2 and E(y) = µ + α 3 The model with zero in place of α 1 can provide exactly the same predictions in each treatment group, as the model with arbitrary α 1. If for instance α 1 = 3 in the first case, then setting µ = µ + 3, α 2 = α 2 3 and α 3 = α 3 3 will give the same predictions in the second case. In other words the two models are identical and comparing them with a statistical test is meaningless. To avoid this and similar situations the following definition is given: Definition: A linear combination of the fixed effects model parameters L β is said to be estimable if and only if there is a vector λ such that λ X = L. 0 α 2 α 3

3 Module 10: Mixed model theory II: Tests and confidence intervals 3 In the following it is assumed that the hypothesis in question is estimable. This is not a restriction as all meaningful hypothesis are estimable. The estimate of the linear combination of model parameters L β is L β. The estimate of β is known from the first theory module, so: L β = L (X V 1 X) 1 X V 1 y Applying the rule cov(ax) = Acov(x)A from the fist theory module, and doing few matrix calculations show that the covariance of L β is L (X V 1 X) 1 L, and the mean is Lβ. This all amounts to: If the hypothesis L β = c is true, then: L β N(L β, L (X V 1 X) 1 L) (L β c) N(0, L (X V 1 X) 1 L) Now the distribution is described, and the so called Wald test can be constructed by: W = (L β c) (L (X V 1 X) 1 L) 1 (L β c) The Wald test can be thought of as the squared difference from the hypothesis divided by its variance. W has an approximate χ 2 df 1 distribution with degrees of freedom df 1 equal to the number of parameters eliminated by the hypothesis, which is the same as the rank of L. This asymptotic result is based on the assumption that the variance V is known without error, but V is estimated from the observations, and not known. A better approximation can be archived by using the Wald F test: F = W df 1 in combination with Satterthwaite s approximation. In this case Satterthwaite s approximation supply an estimate of the denominator degrees of freedom df 2 (assuming that F is F df1,df 2 distributed). The P value for the hypothesis L β = c is computed as: P L β=c = P (F df1,df 2 F ) If the /ddfm=satterth option is specified on proc mixed, then all the tests in the ANOVA table for the fixed effects are computed this way Confidence intervals of fixed effects Confidence intervals based on the approximative t distribution, can be applied for linear combinations of the fixed effects. When a single fixed effect parameter or a

4 Module 10: Mixed model theory II: Tests and confidence intervals 4 single estimable linear combination of fixed effect parameters is considered, the L matrix has only one column, and the 95% confidence interval become: L β = L β ± t0.975,df L (X V 1 X) 1 L Here the covariance matrix V is not known, but based variance parameter estimates. The only problem remaining is to determine the appropriate degrees of freedom df. Once again Satterthwaite s approximation is recommended. The following section will illustrate how to compute these confidence intervals in SAS The estimate and the contrast statements A linear combination of fixed effects parameters can be specified directly in SAS proc mixed. These are specified in terms of the L matrix. Consider for instance a one way ANOVA model with five treatments and an additional random block effect: y i = µ + α(treatment i ) + b(block i ) + ε i where b(block i ) N(0, σ 2 b ) and ε i N(0, σ 2 ). The SAS code for this model could look something like: proc mixed; class treatment block; model y = treatment/ddfm=satterth; random block; estimate tmt1-tmt2 treatment /cl; run; The estimate statement has three arguments. The first argument tmt1-tmt2 is a user defined label and is only used to recognize the estimate in the comprehensive SAS output. The second argument treatment is the name a variable (factor or covariate). The third argument specify one number for each level of the variable. These numbers specify the linear combination by multiplying each to the corresponding parameter estimate and adding it all together. The example above corresponds to: 1 α 1 + ( 1) α α α α 5 = α 1 α 2 which is the comparison of the two first treatments. The added /tt to the estimate statement prints the confidence interval from the previous section in the output. The estimate statement can also be used to compute linear combinations of parameters from more than one variable. For instance to estimate the mean value in the first treatment group including the intercept term, the following estimate statement would do it:

5 Module 10: Mixed model theory II: Tests and confidence intervals 5 estimate Mean of tmt1 int 1 treatment /cl; The estimate statement can only handle the case where the resulting linear combination is a single number (L is a single column). For comparison of several treatments in one test the very similar contrast statement is needed. To test if the first three treatments have the same effect α 1 = α 2 = α 3 the following statement can be used: contrast tmt1=tmt2=tmt3 treatment , treatment ; The contrast statement does not compute confidence intervals and estimates of the different linear combinations, so the estimate statement is not dispensable Test for random effects parameters The restricted/residual likelihood ratio test can be used to test the significance a random effects parameter. The likelihood ratio test is used to compare two models A and B, where B is a sub model of A. Here the model including some variance parameter (model A), and the model without this variance parameter (model B) is to be compared. Using the test consists of two steps: 1) Compute the two negative restricted/residual log-likelihood values (l (A) re and l (B) re ) by running both models. 2) Compute the test statistic: G A B = 2l re (B) 2l (A) re Asymptotically G A B follows a χ 2 1 distribution. (One degree of freedom, because one variance parameter is tested when comparing A to B) Confidence intervals for random effects parameters The confidence interval for a given variance parameter is based on the assumption that the estimate of the variance parameter σ b 2 σ2 is approximately df χ2 df distributed. This is true in balanced (and other nice ) cases. A consequence of this is that the confidence interval takes the form: df σ b 2 < σ χ 2 b 2 < df σ2 b, 0.025;df χ ;df but with the degrees of freedom df still undetermined. The task is to choose the df such that the corresponding χ 2 distribution matches the distribution of the estimate. The (theoretical) variance of σ2 b ( ) σ 2 var b df χ2 df = 2σ4 b df df χ2 df is:

6 Module 10: Mixed model theory II: Tests and confidence intervals 6 The actual variance of the of the parameter can be estimated from the curvature of the negative log likelihood function l. By matching the estimated actual variance of the estimator to the variance of the desired distribution, and solving the equation: var( σ 2 b ) = 2σ4 b df the following estimate of the degrees of freedom is obtained, after plugging in the estimated variance: df = 2 σ4 b var( σ b 2) This way of approximating the degrees of freedom is a special case of Satterthwaite s approximation, which has been used frequently in this course. To get these confidence intervals computed in proc mixed, the option cl must be added to the mixed procedure, like: proc mixed cl; class treatment block; model y = treatment/ddfm=satterth; random block; run; Notice the first line.

Fixed vs. Random Effects

Fixed vs. Random Effects Statistics 203: Introduction to Regression and Analysis of Variance Fixed vs. Random Effects Jonathan Taylor - p. 1/19 Today s class Implications for Random effects. One-way random effects ANOVA. Two-way

More information

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means

Lesson 1: Comparison of Population Means Part c: Comparison of Two- Means Lesson : Comparison of Population Means Part c: Comparison of Two- Means Welcome to lesson c. This third lesson of lesson will discuss hypothesis testing for two independent means. Steps in Hypothesis

More information

Multiple Hypothesis Testing: The F-test

Multiple Hypothesis Testing: The F-test Multiple Hypothesis Testing: The F-test Matt Blackwell December 3, 2008 1 A bit of review When moving into the matrix version of linear regression, it is easy to lose sight of the big picture and get lost

More information

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam

Hypothesis Testing Level I Quantitative Methods. IFT Notes for the CFA exam Hypothesis Testing 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 3 2. Hypothesis Testing... 3 3. Hypothesis Tests Concerning the Mean... 10 4. Hypothesis Tests

More information

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

More information

Factorial Analysis of Variance

Factorial Analysis of Variance Chapter 560 Factorial Analysis of Variance Introduction A common task in research is to compare the average response across levels of one or more factor variables. Examples of factor variables are income

More information

1. The maximum likelihood principle 2. Properties of maximum-likelihood estimates

1. The maximum likelihood principle 2. Properties of maximum-likelihood estimates The maximum-likelihood method Volker Blobel University of Hamburg March 2005 1. The maximum likelihood principle 2. Properties of maximum-likelihood estimates Keys during display: enter = next page; =

More information

SAS Software to Fit the Generalized Linear Model

SAS Software to Fit the Generalized Linear Model SAS Software to Fit the Generalized Linear Model Gordon Johnston, SAS Institute Inc., Cary, NC Abstract In recent years, the class of generalized linear models has gained popularity as a statistical modeling

More information

Chapter 6: Multivariate Cointegration Analysis

Chapter 6: Multivariate Cointegration Analysis Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration

More information

PASS Sample Size Software

PASS Sample Size Software Chapter 250 Introduction The Chi-square test is often used to test whether sets of frequencies or proportions follow certain patterns. The two most common instances are tests of goodness of fit using multinomial

More information

Regression, least squares

Regression, least squares Regression, least squares Joe Felsenstein Department of Genome Sciences and Department of Biology Regression, least squares p.1/24 Fitting a straight line X Two distinct cases: The X values are chosen

More information

Full Factorial Design of Experiments

Full Factorial Design of Experiments Full Factorial Design of Experiments 0 Module Objectives Module Objectives By the end of this module, the participant will: Generate a full factorial design Look for factor interactions Develop coded orthogonal

More information

Inferences About Differences Between Means Edpsy 580

Inferences About Differences Between Means Edpsy 580 Inferences About Differences Between Means Edpsy 580 Carolyn J. Anderson Department of Educational Psychology University of Illinois at Urbana-Champaign Inferences About Differences Between Means Slide

More information

2 Sample t-test (unequal sample sizes and unequal variances)

2 Sample t-test (unequal sample sizes and unequal variances) Variations of the t-test: Sample tail Sample t-test (unequal sample sizes and unequal variances) Like the last example, below we have ceramic sherd thickness measurements (in cm) of two samples representing

More information

SIMPLE REGRESSION ANALYSIS

SIMPLE REGRESSION ANALYSIS SIMPLE REGRESSION ANALYSIS Introduction. Regression analysis is used when two or more variables are thought to be systematically connected by a linear relationship. In simple regression, we have only two

More information

Estimation and Inference in Cointegration Models Economics 582

Estimation and Inference in Cointegration Models Economics 582 Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there

More information

Examination 110 Probability and Statistics Examination

Examination 110 Probability and Statistics Examination Examination 0 Probability and Statistics Examination Sample Examination Questions The Probability and Statistics Examination consists of 5 multiple-choice test questions. The test is a three-hour examination

More information

Multivariate Normal Distribution

Multivariate Normal Distribution Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

More information

STATISTICA Formula Guide: Logistic Regression. Table of Contents

STATISTICA Formula Guide: Logistic Regression. Table of Contents : Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

More information

Probability and Statistics Lecture 9: 1 and 2-Sample Estimation

Probability and Statistics Lecture 9: 1 and 2-Sample Estimation Probability and Statistics Lecture 9: 1 and -Sample Estimation to accompany Probability and Statistics for Engineers and Scientists Fatih Cavdur Introduction A statistic θ is said to be an unbiased estimator

More information

The Delta Method and Applications

The Delta Method and Applications Chapter 5 The Delta Method and Applications 5.1 Linear approximations of functions In the simplest form of the central limit theorem, Theorem 4.18, we consider a sequence X 1, X,... of independent and

More information

Statistiek (WISB361)

Statistiek (WISB361) Statistiek (WISB361) Final exam June 29, 2015 Schrijf uw naam op elk in te leveren vel. Schrijf ook uw studentnummer op blad 1. The maximum number of points is 100. Points distribution: 23 20 20 20 17

More information

Simple Linear Regression, Scatterplots, and Bivariate Correlation

Simple Linear Regression, Scatterplots, and Bivariate Correlation 1 Simple Linear Regression, Scatterplots, and Bivariate Correlation This section covers procedures for testing the association between two continuous variables using the SPSS Regression and Correlate analyses.

More information

Notes on Applied Linear Regression

Notes on Applied Linear Regression Notes on Applied Linear Regression Jamie DeCoster Department of Social Psychology Free University Amsterdam Van der Boechorststraat 1 1081 BT Amsterdam The Netherlands phone: +31 (0)20 444-8935 email:

More information

Hedonism example. Our questions in the last session. Our questions in this session

Hedonism example. Our questions in the last session. Our questions in this session Random Slope Models Hedonism example Our questions in the last session Do differences between countries in hedonism remain after controlling for individual age? How much of the variation in hedonism is

More information

" Y. Notation and Equations for Regression Lecture 11/4. Notation:

 Y. Notation and Equations for Regression Lecture 11/4. Notation: Notation: Notation and Equations for Regression Lecture 11/4 m: The number of predictor variables in a regression Xi: One of multiple predictor variables. The subscript i represents any number from 1 through

More information

Ordinal Regression. Chapter

Ordinal Regression. Chapter Ordinal Regression Chapter 4 Many variables of interest are ordinal. That is, you can rank the values, but the real distance between categories is unknown. Diseases are graded on scales from least severe

More information

Regression III: Advanced Methods

Regression III: Advanced Methods Lecture 16: Generalized Additive Models Regression III: Advanced Methods Bill Jacoby Michigan State University http://polisci.msu.edu/jacoby/icpsr/regress3 Goals of the Lecture Introduce Additive Models

More information

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems

ELEC-E8104 Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Stochastics models and estimation, Lecture 3b: Linear Estimation in Static Systems Minimum Mean Square Error (MMSE) MMSE estimation of Gaussian random vectors Linear MMSE estimator for arbitrarily distributed

More information

Logit Models for Binary Data

Logit Models for Binary Data Chapter 3 Logit Models for Binary Data We now turn our attention to regression models for dichotomous data, including logistic regression and probit analysis. These models are appropriate when the response

More information

Lecture 12: Generalized Linear Models for Binary Data

Lecture 12: Generalized Linear Models for Binary Data Lecture 12: Generalized Linear Models for Binary Data Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of

More information

One-Way ANOVA using SPSS 11.0. SPSS ANOVA procedures found in the Compare Means analyses. Specifically, we demonstrate

One-Way ANOVA using SPSS 11.0. SPSS ANOVA procedures found in the Compare Means analyses. Specifically, we demonstrate 1 One-Way ANOVA using SPSS 11.0 This section covers steps for testing the difference between three or more group means using the SPSS ANOVA procedures found in the Compare Means analyses. Specifically,

More information

Statistical Inference

Statistical Inference Statistical Inference Idea: Estimate parameters of the population distribution using data. How: Use the sampling distribution of sample statistics and methods based on what would happen if we used this

More information

Outline. Correlation & Regression, III. Review. Relationship between r and regression

Outline. Correlation & Regression, III. Review. Relationship between r and regression Outline Correlation & Regression, III 9.07 4/6/004 Relationship between correlation and regression, along with notes on the correlation coefficient Effect size, and the meaning of r Other kinds of correlation

More information

Models for Count Data With Overdispersion

Models for Count Data With Overdispersion Models for Count Data With Overdispersion Germán Rodríguez November 6, 2013 Abstract This addendum to the WWS 509 notes covers extra-poisson variation and the negative binomial model, with brief appearances

More information

t-tests and F-tests in regression

t-tests and F-tests in regression t-tests and F-tests in regression Johan A. Elkink University College Dublin 5 April 2012 Johan A. Elkink (UCD) t and F-tests 5 April 2012 1 / 25 Outline 1 Simple linear regression Model Variance and R

More information

Introduction to Stata

Introduction to Stata Introduction to Stata September 23, 2014 Stata is one of a few statistical analysis programs that social scientists use. Stata is in the mid-range of how easy it is to use. Other options include SPSS,

More information

SAS Syntax and Output for Data Manipulation:

SAS Syntax and Output for Data Manipulation: Psyc 944 Example 5 page 1 Practice with Fixed and Random Effects of Time in Modeling Within-Person Change The models for this example come from Hoffman (in preparation) chapter 5. We will be examining

More information

Chapter 5 Analysis of variance SPSS Analysis of variance

Chapter 5 Analysis of variance SPSS Analysis of variance Chapter 5 Analysis of variance SPSS Analysis of variance Data file used: gss.sav How to get there: Analyze Compare Means One-way ANOVA To test the null hypothesis that several population means are equal,

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

Instructions for Using the Calculator for Statistics

Instructions for Using the Calculator for Statistics Descriptive Statistics Entering Data General Statistics mean, median, stdev, quartiles, etc Five Number Summary Box Plot with Outliers Histogram Distributions: Normal, Student t Area under a normal curve

More information

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

More information

TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL

TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL TESTING THE ONE-PART FRACTIONAL RESPONSE MODEL AGAINST AN ALTERNATIVE TWO-PART MODEL HARALD OBERHOFER AND MICHAEL PFAFFERMAYR WORKING PAPER NO. 2011-01 Testing the One-Part Fractional Response Model against

More information

Multiple Linear Regression. Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables.

Multiple Linear Regression. Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables. 1 Multiple Linear Regression Basic Concepts Multiple linear regression is the extension of simple linear regression to the case of two or more independent variables. In simple linear regression, we had

More information

HYPOTHESIS TESTS AND MODEL SELECTION Q

HYPOTHESIS TESTS AND MODEL SELECTION Q 5 HYPOTHESIS TESTS AND MODEL SELECTION Q 5.1 INTRODUCTION The linear regression model is used for three major purposes: estimation and prediction, which were the subjects of the previous chapter, and hypothesis

More information

Least Squares Estimation

Least Squares Estimation Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN-13: 978-0-470-86080-9 ISBN-10: 0-470-86080-4 Editors Brian S Everitt & David

More information

Lecture 5 Hypothesis Testing in Multiple Linear Regression

Lecture 5 Hypothesis Testing in Multiple Linear Regression Lecture 5 Hypothesis Testing in Multiple Linear Regression BIOST 515 January 20, 2004 Types of tests 1 Overall test Test for addition of a single variable Test for addition of a group of variables Overall

More information

Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

More information

HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009

HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 HURDLE AND SELECTION MODELS Jeff Wooldridge Michigan State University BGSE/IZA Course in Microeconometrics July 2009 1. Introduction 2. A General Formulation 3. Truncated Normal Hurdle Model 4. Lognormal

More information

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as

LOGISTIC REGRESSION. Nitin R Patel. where the dependent variable, y, is binary (for convenience we often code these values as LOGISTIC REGRESSION Nitin R Patel Logistic regression extends the ideas of multiple linear regression to the situation where the dependent variable, y, is binary (for convenience we often code these values

More information

L10: Probability, statistics, and estimation theory

L10: Probability, statistics, and estimation theory L10: Probability, statistics, and estimation theory Review of probability theory Bayes theorem Statistics and the Normal distribution Least Squares Error estimation Maximum Likelihood estimation Bayesian

More information

What is the interpretation of R 2?

What is the interpretation of R 2? What is the interpretation of R 2? Karl G. Jöreskog October 2, 1999 Consider a regression equation between a dependent variable y and a set of explanatory variables x'=(x 1, x 2,..., x q ): or in matrix

More information

Survey, Statistics and Psychometrics Core Research Facility University of Nebraska-Lincoln. Log-Rank Test for More Than Two Groups

Survey, Statistics and Psychometrics Core Research Facility University of Nebraska-Lincoln. Log-Rank Test for More Than Two Groups Survey, Statistics and Psychometrics Core Research Facility University of Nebraska-Lincoln Log-Rank Test for More Than Two Groups Prepared by Harlan Sayles (SRAM) Revised by Julia Soulakova (Statistics)

More information

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne

Applied Statistics. J. Blanchet and J. Wadsworth. Institute of Mathematics, Analysis, and Applications EPF Lausanne Applied Statistics J. Blanchet and J. Wadsworth Institute of Mathematics, Analysis, and Applications EPF Lausanne An MSc Course for Applied Mathematicians, Fall 2012 Outline 1 Model Comparison 2 Model

More information

One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

More information

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations

The Phase Plane. Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations The Phase Plane Phase portraits; type and stability classifications of equilibrium solutions of systems of differential equations Phase Portraits of Linear Systems Consider a systems of linear differential

More information

Chapter 15. Mixed Models. 15.1 Overview. A flexible approach to correlated data.

Chapter 15. Mixed Models. 15.1 Overview. A flexible approach to correlated data. Chapter 15 Mixed Models A flexible approach to correlated data. 15.1 Overview Correlated data arise frequently in statistical analyses. This may be due to grouping of subjects, e.g., students within classrooms,

More information

1. Table A1: Descriptive statistics and measurement model estimates a

1. Table A1: Descriptive statistics and measurement model estimates a Online Appendix Contents: 1. Table A1: Descriptive statistics and measurement model estimates 2. Results of alternative measurement model and structural model specifications 3. Discussion of the effects

More information

Statistical Inference and t-tests

Statistical Inference and t-tests 1 Statistical Inference and t-tests Objectives Evaluate the difference between a sample mean and a target value using a one-sample t-test. Evaluate the difference between a sample mean and a target value

More information

Chapter 10. Verification and Validation of Simulation Models Prof. Dr. Mesut Güneş Ch. 10 Verification and Validation of Simulation Models

Chapter 10. Verification and Validation of Simulation Models Prof. Dr. Mesut Güneş Ch. 10 Verification and Validation of Simulation Models Chapter 10 Verification and Validation of Simulation Models 10.1 Contents Model-Building, Verification, and Validation Verification of Simulation Models Calibration and Validation 10.2 Purpose & Overview

More information

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo

Matrix Norms. Tom Lyche. September 28, Centre of Mathematics for Applications, Department of Informatics, University of Oslo Matrix Norms Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 28, 2009 Matrix Norms We consider matrix norms on (C m,n, C). All results holds for

More information

Overview Classes. 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7)

Overview Classes. 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7) Overview Classes 12-3 Logistic regression (5) 19-3 Building and applying logistic regression (6) 26-3 Generalizations of logistic regression (7) 2-4 Loglinear models (8) 5-4 15-17 hrs; 5B02 Building and

More information

ln(p/(1-p)) = α +β*age35plus, where p is the probability or odds of drinking

ln(p/(1-p)) = α +β*age35plus, where p is the probability or odds of drinking Dummy Coding for Dummies Kathryn Martin, Maternal, Child and Adolescent Health Program, California Department of Public Health ABSTRACT There are a number of ways to incorporate categorical variables into

More information

Statistics in Retail Finance. Chapter 2: Statistical models of default

Statistics in Retail Finance. Chapter 2: Statistical models of default Statistics in Retail Finance 1 Overview > We consider how to build statistical models of default, or delinquency, and how such models are traditionally used for credit application scoring and decision

More information

Basic Statistics. Probability and Confidence Intervals

Basic Statistics. Probability and Confidence Intervals Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

More information

Instrumental Variables & 2SLS

Instrumental Variables & 2SLS Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

More information

Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements

Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements Don t forget the degrees of freedom: evaluating uncertainty from small numbers of repeated measurements Blair Hall b.hall@irl.cri.nz Talk given via internet to the 35 th ANAMET Meeting, October 20, 2011.

More information

Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine

Multivariate Analysis of Variance. The general purpose of multivariate analysis of variance (MANOVA) is to determine 2 - Manova 4.3.05 25 Multivariate Analysis of Variance What Multivariate Analysis of Variance is The general purpose of multivariate analysis of variance (MANOVA) is to determine whether multiple levels

More information

Let s explore SAS Proc T-Test

Let s explore SAS Proc T-Test Let s explore SAS Proc T-Test Ana Yankovsky Research Statistical Analyst Screening Programs, AHS Ana.Yankovsky@albertahealthservices.ca Goals of the presentation: 1. Look at the structure of Proc TTEST;

More information

1.5 Oneway Analysis of Variance

1.5 Oneway Analysis of Variance Statistics: Rosie Cornish. 200. 1.5 Oneway Analysis of Variance 1 Introduction Oneway analysis of variance (ANOVA) is used to compare several means. This method is often used in scientific or medical experiments

More information

One-Way Analysis of Variance

One-Way Analysis of Variance One-Way Analysis of Variance Note: Much of the math here is tedious but straightforward. We ll skim over it in class but you should be sure to ask questions if you don t understand it. I. Overview A. We

More information

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013

Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit STRUCTURAL EQUATION MODELING 2013 Indices of Model Fit A recommended minimal set of fit indices that should be reported and interpreted when reporting the results of SEM analyses:

More information

hp calculators HP 50g Trend Lines The STAT menu Trend Lines Practice predicting the future using trend lines

hp calculators HP 50g Trend Lines The STAT menu Trend Lines Practice predicting the future using trend lines The STAT menu Trend Lines Practice predicting the future using trend lines The STAT menu The Statistics menu is accessed from the ORANGE shifted function of the 5 key by pressing Ù. When pressed, a CHOOSE

More information

Statistics - Written Examination MEC Students - BOVISA

Statistics - Written Examination MEC Students - BOVISA Statistics - Written Examination MEC Students - BOVISA Prof.ssa A. Guglielmi 26.0.2 All rights reserved. Legal action will be taken against infringement. Reproduction is prohibited without prior consent.

More information

Hypothesis Testing (unknown σ)

Hypothesis Testing (unknown σ) Hypothesis Testing (unknown σ) Business Statistics Recall: Plan for Today Null and Alternative Hypotheses Types of errors: type I, type II Types of correct decisions: type A, type B Level of Significance

More information

Poisson Models for Count Data

Poisson Models for Count Data Chapter 4 Poisson Models for Count Data In this chapter we study log-linear models for count data under the assumption of a Poisson error structure. These models have many applications, not only to the

More information

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.

MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1. MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column

More information

Mantel Permutation Tests

Mantel Permutation Tests PERMUTATION TESTS Mantel Permutation Tests Basic Idea: In some experiments a test of treatment effects may be of interest where the null hypothesis is that the different populations are actually from the

More information

Statistical Models in R

Statistical Models in R Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Structure of models in R Model Assessment (Part IA) Anova

More information

Specifications for this HLM2 run

Specifications for this HLM2 run One way ANOVA model 1. How much do U.S. high schools vary in their mean mathematics achievement? 2. What is the reliability of each school s sample mean as an estimate of its true population mean? 3. Do

More information

Linear Dependence Tests

Linear Dependence Tests Linear Dependence Tests The book omits a few key tests for checking the linear dependence of vectors. These short notes discuss these tests, as well as the reasoning behind them. Our first test checks

More information

Instrumental Variables & 2SLS

Instrumental Variables & 2SLS Instrumental Variables & 2SLS y 1 = β 0 + β 1 y 2 + β 2 z 1 +... β k z k + u y 2 = π 0 + π 1 z k+1 + π 2 z 1 +... π k z k + v Economics 20 - Prof. Schuetze 1 Why Use Instrumental Variables? Instrumental

More information

Linear Models for Continuous Data

Linear Models for Continuous Data Chapter 2 Linear Models for Continuous Data The starting point in our exploration of statistical models in social research will be the classical linear model. Stops along the way include multiple linear

More information

Lecture #2 Overview. Basic IRT Concepts, Models, and Assumptions. Lecture #2 ICPSR Item Response Theory Workshop

Lecture #2 Overview. Basic IRT Concepts, Models, and Assumptions. Lecture #2 ICPSR Item Response Theory Workshop Basic IRT Concepts, Models, and Assumptions Lecture #2 ICPSR Item Response Theory Workshop Lecture #2: 1of 64 Lecture #2 Overview Background of IRT and how it differs from CFA Creating a scale An introduction

More information

Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares

Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation

More information

Statistics and research

Statistics and research Statistics and research Usaneya Perngparn Chitlada Areesantichai Drug Dependence Research Center (WHOCC for Research and Training in Drug Dependence) College of Public Health Sciences Chulolongkorn University,

More information

K-Means Clustering. Clustering and Classification Lecture 8

K-Means Clustering. Clustering and Classification Lecture 8 K-Means Clustering Clustering and Lecture 8 Today s Class K-means clustering: What it is How it works What it assumes Pitfalls of the method (locally optimal results) 2 From Last Time If you recall the

More information

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression

Data Mining and Data Warehousing. Henryk Maciejewski. Data Mining Predictive modelling: regression Data Mining and Data Warehousing Henryk Maciejewski Data Mining Predictive modelling: regression Algorithms for Predictive Modelling Contents Regression Classification Auxiliary topics: Estimation of prediction

More information

VI. Introduction to Logistic Regression

VI. Introduction to Logistic Regression VI. Introduction to Logistic Regression We turn our attention now to the topic of modeling a categorical outcome as a function of (possibly) several factors. The framework of generalized linear models

More information

Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217

Part 3. Comparing Groups. Chapter 7 Comparing Paired Groups 189. Chapter 8 Comparing Two Independent Groups 217 Part 3 Comparing Groups Chapter 7 Comparing Paired Groups 189 Chapter 8 Comparing Two Independent Groups 217 Chapter 9 Comparing More Than Two Groups 257 188 Elementary Statistics Using SAS Chapter 7 Comparing

More information

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA)

INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) INTERPRETING THE ONE-WAY ANALYSIS OF VARIANCE (ANOVA) As with other parametric statistics, we begin the one-way ANOVA with a test of the underlying assumptions. Our first assumption is the assumption of

More information

Technical report. in SPSS AN INTRODUCTION TO THE MIXED PROCEDURE

Technical report. in SPSS AN INTRODUCTION TO THE MIXED PROCEDURE Linear mixedeffects modeling in SPSS AN INTRODUCTION TO THE MIXED PROCEDURE Table of contents Introduction................................................................3 Data preparation for MIXED...................................................3

More information

Logistic (RLOGIST) Example #1

Logistic (RLOGIST) Example #1 Logistic (RLOGIST) Example #1 SUDAAN Statements and Results Illustrated EFFECTS RFORMAT, RLABEL REFLEVEL EXP option on MODEL statement Hosmer-Lemeshow Test Input Data Set(s): BRFWGT.SAS7bdat Example Using

More information

Lecture 14: GLM Estimation and Logistic Regression

Lecture 14: GLM Estimation and Logistic Regression Lecture 14: GLM Estimation and Logistic Regression Dipankar Bandyopadhyay, Ph.D. BMTRY 711: Analysis of Categorical Data Spring 2011 Division of Biostatistics and Epidemiology Medical University of South

More information

5. Ordinal regression: cumulative categories proportional odds. 6. Ordinal regression: comparison to single reference generalized logits

5. Ordinal regression: cumulative categories proportional odds. 6. Ordinal regression: comparison to single reference generalized logits Lecture 23 1. Logistic regression with binary response 2. Proc Logistic and its surprises 3. quadratic model 4. Hosmer-Lemeshow test for lack of fit 5. Ordinal regression: cumulative categories proportional

More information

Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212.

Variance of OLS Estimators and Hypothesis Testing. Randomness in the model. GM assumptions. Notes. Notes. Notes. Charlie Gibbons ARE 212. Variance of OLS Estimators and Hypothesis Testing Charlie Gibbons ARE 212 Spring 2011 Randomness in the model Considering the model what is random? Y = X β + ɛ, β is a parameter and not random, X may be

More information

SUGI 29 Statistics and Data Analysis

SUGI 29 Statistics and Data Analysis Paper 194-29 Head of the CLASS: Impress your colleagues with a superior understanding of the CLASS statement in PROC LOGISTIC Michelle L. Pritchard and David J. Pasta Ovation Research Group, San Francisco,

More information

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not. Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

Standard errors of marginal effects in the heteroskedastic probit model

Standard errors of marginal effects in the heteroskedastic probit model Standard errors of marginal effects in the heteroskedastic probit model Thomas Cornelißen Discussion Paper No. 320 August 2005 ISSN: 0949 9962 Abstract In non-linear regression models, such as the heteroskedastic

More information