SAS Procedures Inference about the Line ffl model statement in proc reg has many options ffl To construct confidence intervals use alpha=, clm, cli, c

Size: px
Start display at page:

Download "SAS Procedures Inference about the Line ffl model statement in proc reg has many options ffl To construct confidence intervals use alpha=, clm, cli, c"

Transcription

1 Inference About the Slope ffl As with all estimates, ^fi1 subject to sampling var ffl Because Y jx _ Normal, the estimate ^fi1 _ Normal A linear combination of indep Normals is Normal Simple Linear Regression Applied Regression and Other Multivariable Methods Sections , Thm: If Y i _ N(μ i;ff i), then X X qx L = c iy i _ N c iμ i; c 2 i ff 2 i Can write ^fi 1 as linear combination of E's ffl Standard error of ^fi1 = S Y jx S x p n 1 ffl Use std error to form CI or test hypothesis ffl Degrees of freedom n 2 Confidence Int : ^fi1 ± t n 2;ff=2 Hypothesis Test: T = (^fi1 fi 0 1 )= Inference About the Slope Inference About the Intercept ffl Want small for inference ffl In situation where X is under experimental control If S X made large! small Can increase S X by increasing dispersion of X Can also increase n to decrease, increase df ffl Sometimes interested in intercept fi 0 ffl Standard error of ^fi0 S^fi0 = S Y jx s 1 n + X 2 (n 1)S 2 X ffl In situation where X is under experimental control If S X made large! small S^fi0 Increase S X by increasing dispersion ffl Test H 0 : fi 1 = 0 to see if linear association ffl Does X help explain Y through a linear model? Rejecting does not mean linear model is best" Not rejecting doesn't mean X unimportant See page 56 for examples If X close to zero! small S^fi0 ffl Can also increase n to decrease, increase df ffl Test H 0 : fi 0 = 0 to see if line goes through origin Does not test linear model fit Really only meaningful if X around zero

2 SAS Procedures Inference about the Line ffl model statement in proc reg has many options ffl To construct confidence intervals use alpha=, clm, cli, clb model sbp=age / clb alpha=.01; /* Form 99% CI for parameters */ Dependent Variable: sbp Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Corrected Total Root MSE R-Square Dependent Mean Adj R-Sq Coeff Var Parameter Estimates Parameter Standard Variable DF Estimate Error t Value Pr > t Intercept <.0001 age <.0001 Parameter Estimates Variable DF 99% Confidence Limits Intercept age ffl Line describes the mean population response for X ffl Predicted mean at X = X 0 is ^μ Y jx0 = ^fi 0 + ^fi 1X 0 ffl Standard error of ^μ Y jx0 is S Y jxr 1 (X0 X)2 + n (n 1)S 2 X ffl New predicted observation at X = X 0 is ^Y X0 = ^fi 0 + ^fi 1X 0 ffl Standard error of ^Y X0 is S Y jx r 1+ 1 (X0 X)2 + n (n 1)S 2 X ffl New obs doesn't have to fall on line! bigger var ffl Recall Y X0 = μ Y jx0 + E and Var(E)=S 2 Y jx 5-5 SAS Procedures Interpolation vs Extrapolation ffl Must use caution in interpretation of ^Y X0, ^μ Y jx0 ffl If X 0 within range of observed X's! interpolation ffl If X 0 outside range of observed X's! extrapolation ffl Extrapolation should be avoided No assurances still linear outside range of data Example: Fish activity and Water Temp ffl Can also construct confidence/prediction bands ffl Prediction bands wider than confidence bands ffl Most narrow at X 5-6 model sbp=age /cli clm; /* Confidence int for i=indiv m=mean */ plot sbp*age / conf pred; /* Create plot with conf and pred bands */ Dependent Variable: sbp Output Statistics Dep Var Predicted Std Error Obs sbp Value Mean Predict 95% CL Mean

3 Output Statistics Obs 95% CL Predict Residual Sum of Residuals 0 Sum of Squared Residuals Predicted Residual SS (PRESS) Diagnostics Regression Diagnostics ffl Will study more procedures throughout semester ffl These focus on simple linear regression ffl Assumptions 1 Model is correct (linearity) 2 Independent observations 3 Errors normally distributed 4 Constant variance Y i = ^μ YijX i + (Y i ^μ YijX i) Y i = ^Y i + ^E i observed = predicted + residual ffl Diagnostics will use predicted and residual values 5-10 ffl Normality Histogram/Boxplot of residuals Normal probability plot / QQ plot Shapiro-Wilks/Kolmogorov-Smirnov Test ffl Variance Plot ^E i vs ^Y i (residual plot) Bartlett's or Levene's Test (provided repeat X i obs) ffl Independence Plot ^E i vs time/space Runs test/durbin-watson Test ffl Outliers Is it influential? With and without analysis Formal tests (e.g. standardized residuals) Investigate why result may occur, don't try to eliminate 5-11

4 Normality Assumption ffl Histogram/Boxplot Is histogram of residuals bell-shaped? Is boxplot/histogram symmetric? ffl Normal Probability/QQ Plot Ordered residuals vs cumulative normal probs Is it approximately linear? Constant Variance ffl Often experiments with non-constant variance ffl Size of residual associated with predicted value ffl Residual plot Plot ^E i vs ^Y i Is the range constant for different levels of ^Y i ffl Bartlett's and Levene's Test More formal test Compares pooled var with sample variances Bartlett sensitive to Normality assumption 5-12 Independence ffl Plot of the residuals over time Is there a drift or pattern as trials proceed? ffl Plot residuals versus relevant variables Often variables omitted from analysis Experimental conditions (e.g., temp) May result in inclusion of factor in next exp ffl Durbin-Watson or Runs Test DW model statement option Assumes observations presented in time order Runs tests look at number of pos/neg residuals in a row 5-13 The UNIVARIATE Procedure Variable: res (Residual) SAS Procedures model sbp=age; plot r.*nqq.; /* Generate QQ Plot */ plot r.*p.; /* Generate residual Plot */ output out=fit r=res p=pred; proc gplot; /* Generate Residual Plot */ plot res*pred /vref=0 frame; proc univariate normal pctdef=4; /* Check Normality of Residuals */ var res; histogram res / normal kernel (L=2); qqplot res / normal (L=1 mu=est sigma=est); 5-14 Moments N 30 Sum Weights 30 Mean 0 Sum Observations 0 Std Deviation Variance Skewness Kurtosis Uncorrected SS Corrected SS Coeff Variation. Std Error Mean Basic Statistical Measures Location Variability Mean Std Deviation Median Variance Mode. Range Interquartile Range Tests for Normality Test --Statistic p Value Shapiro-Wilk W Pr < W < Kolmogorov-Smirnov D Pr > D < Cramer-von Mises W-Sq Pr > W-Sq < Anderson-Darling A-Sq Pr > A-Sq < Quantiles (Definition 4) Quantile Estimate 100% Max % % % % Q % Median % Q % % % % Min

5 Extreme Observations Lowest Highest----- Value Obs Value Obs Goodness-of-Fit Tests for Normal Distribution Test ---Statistic p Value----- Kolmogorov-Smirnov D Pr > D <0.010 Cramer-von Mises W-Sq Pr > W-Sq <0.005 Anderson-Darling A-Sq Pr > A-Sq <0.005 Quantiles for Normal Distribution Quantile Percent Observed Estimated

unadjusted model for baseline cholesterol 22:31 Monday, April 19,

unadjusted model for baseline cholesterol 22:31 Monday, April 19, unadjusted model for baseline cholesterol 22:31 Monday, April 19, 2004 1 Class Level Information Class Levels Values TRETGRP 3 3 4 5 SEX 2 0 1 Number of observations 916 unadjusted model for baseline cholesterol

More information

Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3-1 through 3-3

Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3-1 through 3-3 Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3-1 through 3-3 Page 1 Tensile Strength Experiment Investigate the tensile strength of a new synthetic fiber. The factor is the weight percent

More information

Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3.1 through 3.3

Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3.1 through 3.3 Lecture 3. Experiments with a Single Factor: ANOVA Montgomery 3.1 through 3.3 Fall, 2013 Page 1 Tensile Strength Experiment Investigate the tensile strength of a new synthetic fiber. The factor is the

More information

Comparison of a Population Means

Comparison of a Population Means Analysis of Variance Interested in comparing Several treatments Several levels of one treatment Comparison of a Population Means Could do numerous two-sample t-tests but... ANOVA provides method of joint

More information

Chapter 8 (More on Assumptions for the Simple Linear Regression)

Chapter 8 (More on Assumptions for the Simple Linear Regression) EXST3201 Chapter 8b Geaghan Fall 2005: Page 1 Chapter 8 (More on Assumptions for the Simple Linear Regression) Your textbook considers the following assumptions: Linearity This is not something I usually

More information

Lecture 11: Simple Linear Regression

Lecture 11: Simple Linear Regression Lecture 11: Simple Linear Regression Readings: Sections 3.1-3.3, 11.1-11.3 Apr 17, 2009 In linear regression, we examine the association between two quantitative variables. Number of beers that you drink

More information

Answer Keys to Homework#10

Answer Keys to Homework#10 Answer Keys to Homework#10 Problem 1 Use either restricted or unrestricted mixed models. Problem 2 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean

More information

BE640 Intermediate Biostatistics 2. Regression and Correlation. Simple Linear Regression Software: SAS. Emergency Calls to the New York Auto Club

BE640 Intermediate Biostatistics 2. Regression and Correlation. Simple Linear Regression Software: SAS. Emergency Calls to the New York Auto Club BE640 Intermediate Biostatistics 2. Regression and Correlation Simple Linear Regression Software: SAS Emergency Calls to the New York Auto Club Source: Chatterjee, S; Handcock MS and Simonoff JS A Casebook

More information

Assignment 9 Answer Keys

Assignment 9 Answer Keys Assignment 9 Answer Keys Problem 1 (a) First, the respective means for the 8 level combinations are listed in the following table A B C Mean 26.00 + 34.67 + 39.67 + + 49.33 + 42.33 + + 37.67 + + 54.67

More information

Single Factor Experiments

Single Factor Experiments Single Factor Experiments Bruce A Craig Department of Statistics Purdue University STAT 514 Topic 4 1 Analysis of Variance Suppose you are interested in comparing either a different treatments a levels

More information

EXST7015: Estimating tree weights from other morphometric variables Raw data print

EXST7015: Estimating tree weights from other morphometric variables Raw data print Simple Linear Regression SAS example Page 1 1 ********************************************; 2 *** Data from Freund & Wilson (1993) ***; 3 *** TABLE 8.24 : ESTIMATING TREE WEIGHTS ***; 4 ********************************************;

More information

Handout 1: Predicting GPA from SAT

Handout 1: Predicting GPA from SAT Handout 1: Predicting GPA from SAT appsrv01.srv.cquest.utoronto.ca> appsrv01.srv.cquest.utoronto.ca> ls Desktop grades.data grades.sas oldstuff sasuser.800 appsrv01.srv.cquest.utoronto.ca> cat grades.data

More information

Overview Scatter Plot Example

Overview Scatter Plot Example Overview Topic 22 - Linear Regression and Correlation STAT 5 Professor Bruce Craig Consider one population but two variables For each sampling unit observe X and Y Assume linear relationship between variables

More information

COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 26, 2005, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTION

COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 26, 2005, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTION COMPREHENSIVE WRITTEN EXAMINATION, PAPER III FRIDAY AUGUST 26, 2005, 9:00 A.M. 1:00 P.M. STATISTICS 174 QUESTION Answer all parts. Closed book, calculators allowed. It is important to show all working,

More information

Lecture 4. Checking Model Adequacy

Lecture 4. Checking Model Adequacy Lecture 4. Checking Model Adequacy Montgomery: 3-4, 15-1.1 Page 1 Model Checking and Diagnostics Model Assumptions 1 Model is correct 2 Independent observations 3 Errors normally distributed 4 Constant

More information

1) Answer the following questions as true (T) or false (F) by circling the appropriate letter.

1) Answer the following questions as true (T) or false (F) by circling the appropriate letter. 1) Answer the following questions as true (T) or false (F) by circling the appropriate letter. T F T F T F a) Variance estimates should always be positive, but covariance estimates can be either positive

More information

Topic 14: Inference in Multiple Regression

Topic 14: Inference in Multiple Regression Topic 14: Inference in Multiple Regression Outline Review multiple linear regression Inference of regression coefficients Application to book example Inference of mean Application to book example Inference

More information

General Linear Model (Chapter 4)

General 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 information

Topic 2. Chapter 3: Diagnostics and Remedial Measures

Topic 2. Chapter 3: Diagnostics and Remedial Measures Topic Overview This topic will cover Regression Diagnostics Remedial Measures Statistics 512: Applied Linear Models Some other Miscellaneous Topics Topic 2 Chapter 3: Diagnostics and Remedial Measures

More information

5.3 Three-Stage Nested Design Example

5.3 Three-Stage Nested Design Example 5.3 Three-Stage Nested Design Example A researcher designs an experiment to study the of a metal alloy. A three-stage nested design was conducted that included Two alloy chemistry compositions. Three ovens

More information

Outline. Topic 20 - Diagnostics and Remedies. Residuals. Overview. Diagnostics Plots Residual checks Formal Tests. STAT Fall 2013

Outline. Topic 20 - Diagnostics and Remedies. Residuals. Overview. Diagnostics Plots Residual checks Formal Tests. STAT Fall 2013 Topic 20 - Diagnostics and Remedies - Fall 2013 Diagnostics Plots Residual checks Formal Tests Remedial Measures Outline Topic 20 2 General assumptions Overview Normally distributed error terms Independent

More information

STOR 455 STATISTICAL METHODS I

STOR 455 STATISTICAL METHODS I STOR 455 STATISTICAL METHODS I Jan Hannig Mul9variate Regression Y=X β + ε X is a regression matrix, β is a vector of parameters and ε are independent N(0,σ) Es9mated parameters b=(x X) - 1 X Y Predicted

More information

This is a Randomized Block Design (RBD) with a single factor treatment arrangement (2 levels) which are fixed.

This is a Randomized Block Design (RBD) with a single factor treatment arrangement (2 levels) which are fixed. EXST3201 Chapter 13c Geaghan Fall 2005: Page 1 Linear Models Y ij = µ + βi + τ j + βτij + εijk This is a Randomized Block Design (RBD) with a single factor treatment arrangement (2 levels) which are fixed.

More information

IES 612/STA 4-573/STA Winter 2008 Week 1--IES 612-STA STA doc

IES 612/STA 4-573/STA Winter 2008 Week 1--IES 612-STA STA doc IES 612/STA 4-573/STA 4-576 Winter 2008 Week 1--IES 612-STA 4-573-STA 4-576.doc Review Notes: [OL] = Ott & Longnecker Statistical Methods and Data Analysis, 5 th edition. [Handouts based on notes prepared

More information

Chapter 1 Linear Regression with One Predictor

Chapter 1 Linear Regression with One Predictor STAT 525 FALL 2018 Chapter 1 Linear Regression with One Predictor Professor Min Zhang Goals of Regression Analysis Serve three purposes Describes an association between X and Y In some applications, the

More information

Stat 427/527: Advanced Data Analysis I

Stat 427/527: Advanced Data Analysis I Stat 427/527: Advanced Data Analysis I Review of Chapters 1-4 Sep, 2017 1 / 18 Concepts you need to know/interpret Numerical summaries: measures of center (mean, median, mode) measures of spread (sample

More information

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2

Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Lecture 2: Basic Concepts and Simple Comparative Experiments Montgomery: Chapter 2 Fall, 2013 Page 1 Random Variable and Probability Distribution Discrete random variable Y : Finite possible values {y

More information

Lecture 3: Inference in SLR

Lecture 3: Inference in SLR Lecture 3: Inference in SLR STAT 51 Spring 011 Background Reading KNNL:.1.6 3-1 Topic Overview This topic will cover: Review of hypothesis testing Inference about 1 Inference about 0 Confidence Intervals

More information

Introduction to Design and Analysis of Experiments with the SAS System (Stat 7010 Lecture Notes)

Introduction to Design and Analysis of Experiments with the SAS System (Stat 7010 Lecture Notes) Introduction to Design and Analysis of Experiments with the SAS System (Stat 7010 Lecture Notes) Asheber Abebe Discrete and Statistical Sciences Auburn University Contents 1 Completely Randomized Design

More information

Lecture 12 Inference in MLR

Lecture 12 Inference in MLR Lecture 12 Inference in MLR STAT 512 Spring 2011 Background Reading KNNL: 6.6-6.7 12-1 Topic Overview Review MLR Model Inference about Regression Parameters Estimation of Mean Response Prediction 12-2

More information

Introduction to Regression

Introduction to Regression Introduction to Regression Using Mult Lin Regression Derived variables Many alternative models Which model to choose? Model Criticism Modelling Objective Model Details Data and Residuals Assumptions 1

More information

Biological Applications of ANOVA - Examples and Readings

Biological Applications of ANOVA - Examples and Readings BIO 575 Biological Applications of ANOVA - Winter Quarter 2010 Page 1 ANOVA Pac Biological Applications of ANOVA - Examples and Readings One-factor Model I (Fixed Effects) This is the same example for

More information

ST505/S697R: Fall Homework 2 Solution.

ST505/S697R: Fall Homework 2 Solution. ST505/S69R: Fall 2012. Homework 2 Solution. 1. 1a; problem 1.22 Below is the summary information (edited) from the regression (using R output); code at end of solution as is code and output for SAS. a)

More information

3 Variables: Cyberloafing Conscientiousness Age

3 Variables: Cyberloafing Conscientiousness Age title 'Cyberloafing, Mike Sage'; run; PROC CORR data=sage; var Cyberloafing Conscientiousness Age; run; quit; The CORR Procedure 3 Variables: Cyberloafing Conscientiousness Age Simple Statistics Variable

More information

Statistics 512: Solution to Homework#11. Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat).

Statistics 512: Solution to Homework#11. Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat). Statistics 512: Solution to Homework#11 Problems 1-3 refer to the soybean sausage dataset of Problem 20.8 (ch21pr08.dat). 1. Perform the two-way ANOVA without interaction for this model. Use the results

More information

Lecture notes on Regression & SAS example demonstration

Lecture notes on Regression & SAS example demonstration Regression & Correlation (p. 215) When two variables are measured on a single experimental unit, the resulting data are called bivariate data. You can describe each variable individually, and you can also

More information

Table 1: Fish Biomass data set on 26 streams

Table 1: Fish Biomass data set on 26 streams Math 221: Multiple Regression S. K. Hyde Chapter 27 (Moore, 5th Ed.) The following data set contains observations on the fish biomass of 26 streams. The potential regressors from which we wish to explain

More information

Failure Time of System due to the Hot Electron Effect

Failure Time of System due to the Hot Electron Effect of System due to the Hot Electron Effect 1 * exresist; 2 option ls=120 ps=75 nocenter nodate; 3 title of System due to the Hot Electron Effect ; 4 * TIME = failure time (hours) of a system due to drift

More information

PubH 7405: REGRESSION ANALYSIS SLR: DIAGNOSTICS & REMEDIES

PubH 7405: REGRESSION ANALYSIS SLR: DIAGNOSTICS & REMEDIES PubH 7405: REGRESSION ANALYSIS SLR: DIAGNOSTICS & REMEDIES Normal Error RegressionModel : Y = β 0 + β ε N(0,σ 2 1 x ) + ε The Model has several parts: Normal Distribution, Linear Mean, Constant Variance,

More information

Outline. Review regression diagnostics Remedial measures Weighted regression Ridge regression Robust regression Bootstrapping

Outline. Review regression diagnostics Remedial measures Weighted regression Ridge regression Robust regression Bootstrapping Topic 19: Remedies Outline Review regression diagnostics Remedial measures Weighted regression Ridge regression Robust regression Bootstrapping Regression Diagnostics Summary Check normality of the residuals

More information

Lecture 10: 2 k Factorial Design Montgomery: Chapter 6

Lecture 10: 2 k Factorial Design Montgomery: Chapter 6 Lecture 10: 2 k Factorial Design Montgomery: Chapter 6 Page 1 2 k Factorial Design Involving k factors Each factor has two levels (often labeled + and ) Factor screening experiment (preliminary study)

More information

STAT 3A03 Applied Regression Analysis With SAS Fall 2017

STAT 3A03 Applied Regression Analysis With SAS Fall 2017 STAT 3A03 Applied Regression Analysis With SAS Fall 2017 Assignment 5 Solution Set Q. 1 a The code that I used and the output is as follows PROC GLM DataS3A3.Wool plotsnone; Class Amp Len Load; Model CyclesAmp

More information

Lecture 1 Linear Regression with One Predictor Variable.p2

Lecture 1 Linear Regression with One Predictor Variable.p2 Lecture Linear Regression with One Predictor Variablep - Basics - Meaning of regression parameters p - β - the slope of the regression line -it indicates the change in mean of the probability distn of

More information

T-test: means of Spock's judge versus all other judges 1 12:10 Wednesday, January 5, judge1 N Mean Std Dev Std Err Minimum Maximum

T-test: means of Spock's judge versus all other judges 1 12:10 Wednesday, January 5, judge1 N Mean Std Dev Std Err Minimum Maximum T-test: means of Spock's judge versus all other judges 1 The TTEST Procedure Variable: pcwomen judge1 N Mean Std Dev Std Err Minimum Maximum OTHER 37 29.4919 7.4308 1.2216 16.5000 48.9000 SPOCKS 9 14.6222

More information

STA 302 H1F / 1001 HF Fall 2007 Test 1 October 24, 2007

STA 302 H1F / 1001 HF Fall 2007 Test 1 October 24, 2007 STA 302 H1F / 1001 HF Fall 2007 Test 1 October 24, 2007 LAST NAME: SOLUTIONS FIRST NAME: STUDENT NUMBER: ENROLLED IN: (circle one) STA 302 STA 1001 INSTRUCTIONS: Time: 90 minutes Aids allowed: calculator.

More information

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression Correlation and Simple Linear Regression Sasivimol Rattanasiri, Ph.D Section for Clinical Epidemiology and Biostatistics Ramathibodi Hospital, Mahidol University E-mail: sasivimol.rat@mahidol.ac.th 1 Outline

More information

Chapter 2 Inferences in Simple Linear Regression

Chapter 2 Inferences in Simple Linear Regression STAT 525 SPRING 2018 Chapter 2 Inferences in Simple Linear Regression Professor Min Zhang Testing for Linear Relationship Term β 1 X i defines linear relationship Will then test H 0 : β 1 = 0 Test requires

More information

ANALYSIS OF VARIANCE OF BALANCED DAIRY SCIENCE DATA USING SAS

ANALYSIS OF VARIANCE OF BALANCED DAIRY SCIENCE DATA USING SAS ANALYSIS OF VARIANCE OF BALANCED DAIRY SCIENCE DATA USING SAS Ravinder Malhotra and Vipul Sharma National Dairy Research Institute, Karnal-132001 The most common use of statistics in dairy science is testing

More information

Statistics for exp. medical researchers Regression and Correlation

Statistics for exp. medical researchers Regression and Correlation Faculty of Health Sciences Regression analysis Statistics for exp. medical researchers Regression and Correlation Lene Theil Skovgaard Sept. 28, 2015 Linear regression, Estimation and Testing Confidence

More information

Booklet of Code and Output for STAC32 Final Exam

Booklet of Code and Output for STAC32 Final Exam Booklet of Code and Output for STAC32 Final Exam December 8, 2014 List of Figures in this document by page: List of Figures 1 Popcorn data............................. 2 2 MDs by city, with normal quantile

More information

Scenarios Where Utilizing a Spline Model in Developing a Regression Model Is Appropriate

Scenarios Where Utilizing a Spline Model in Developing a Regression Model Is Appropriate Paper 1760-2014 Scenarios Where Utilizing a Spline Model in Developing a Regression Model Is Appropriate Ning Huang, University of Southern California ABSTRACT Linear regression has been a widely used

More information

Week 7.1--IES 612-STA STA doc

Week 7.1--IES 612-STA STA doc Week 7.1--IES 612-STA 4-573-STA 4-576.doc IES 612/STA 4-576 Winter 2009 ANOVA MODELS model adequacy aka RESIDUAL ANALYSIS Numeric data samples from t populations obtained Assume Y ij ~ independent N(μ

More information

3rd Quartile. 1st Quartile) Minimum

3rd Quartile. 1st Quartile) Minimum EXST7034 - Regression Techniques Page 1 Regression diagnostics dependent variable Y3 There are a number of graphic representations which will help with problem detection and which can be used to obtain

More information

One-Way Analysis of Variance (ANOVA) There are two key differences regarding the explanatory variable X.

One-Way Analysis of Variance (ANOVA) There are two key differences regarding the explanatory variable X. One-Way Analysis of Variance (ANOVA) Also called single factor ANOVA. The response variable Y is continuous (same as in regression). There are two key differences regarding the explanatory variable X.

More information

Business Statistics. Lecture 10: Course Review

Business Statistics. Lecture 10: Course Review Business Statistics Lecture 10: Course Review 1 Descriptive Statistics for Continuous Data Numerical Summaries Location: mean, median Spread or variability: variance, standard deviation, range, percentiles,

More information

Lecture 18: Simple Linear Regression

Lecture 18: Simple Linear Regression Lecture 18: Simple Linear Regression BIOS 553 Department of Biostatistics University of Michigan Fall 2004 The Correlation Coefficient: r The correlation coefficient (r) is a number that measures the strength

More information

a. The least squares estimators of intercept and slope are (from JMP output): b 0 = 6.25 b 1 =

a. The least squares estimators of intercept and slope are (from JMP output): b 0 = 6.25 b 1 = Stat 28 Fall 2004 Key to Homework Exercise.10 a. There is evidence of a linear trend: winning times appear to decrease with year. A straight-line model for predicting winning times based on year is: Winning

More information

STAT5044: Regression and Anova

STAT5044: Regression and Anova STAT5044: Regression and Anova Inyoung Kim 1 / 49 Outline 1 How to check assumptions 2 / 49 Assumption Linearity: scatter plot, residual plot Randomness: Run test, Durbin-Watson test when the data can

More information

Module 6: Model Diagnostics

Module 6: Model Diagnostics St@tmaster 02429/MIXED LINEAR MODELS PREPARED BY THE STATISTICS GROUPS AT IMM, DTU AND KU-LIFE Module 6: Model Diagnostics 6.1 Introduction............................... 1 6.2 Linear model diagnostics........................

More information

One-way ANOVA Model Assumptions

One-way ANOVA Model Assumptions One-way ANOVA Model Assumptions STAT:5201 Week 4: Lecture 1 1 / 31 One-way ANOVA: Model Assumptions Consider the single factor model: Y ij = µ + α }{{} i ij iid with ɛ ij N(0, σ 2 ) mean structure random

More information

CHAPTER 2 SIMPLE LINEAR REGRESSION

CHAPTER 2 SIMPLE LINEAR REGRESSION CHAPTER 2 SIMPLE LINEAR REGRESSION 1 Examples: 1. Amherst, MA, annual mean temperatures, 1836 1997 2. Summer mean temperatures in Mount Airy (NC) and Charleston (SC), 1948 1996 Scatterplots outliers? influential

More information

Density Temp vs Ratio. temp

Density Temp vs Ratio. temp Temp Ratio Density 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Density 0.0 0.2 0.4 0.6 0.8 1.0 1. (a) 170 175 180 185 temp 1.0 1.5 2.0 2.5 3.0 ratio The histogram shows that the temperature measures have two peaks,

More information

Analysis of Variance. Source DF Squares Square F Value Pr > F. Model <.0001 Error Corrected Total

Analysis of Variance. Source DF Squares Square F Value Pr > F. Model <.0001 Error Corrected Total Math 221: Linear Regression and Prediction Intervals S. K. Hyde Chapter 23 (Moore, 5th Ed.) (Neter, Kutner, Nachsheim, and Wasserman) The Toluca Company manufactures refrigeration equipment as well as

More information

PLS205!! Lab 9!! March 6, Topic 13: Covariance Analysis

PLS205!! Lab 9!! March 6, Topic 13: Covariance Analysis PLS205!! Lab 9!! March 6, 2014 Topic 13: Covariance Analysis Covariable as a tool for increasing precision Carrying out a full ANCOVA Testing ANOVA assumptions Happiness! Covariable as a Tool for Increasing

More information

SAS Commands. General Plan. Output. Construct scatterplot / interaction plot. Run full model

SAS Commands. General Plan. Output. Construct scatterplot / interaction plot. Run full model Topic 23 - Unequal Replication Data Model Outline - Fall 2013 Parameter Estimates Inference Topic 23 2 Example Page 954 Data for Two Factor ANOVA Y is the response variable Factor A has levels i = 1, 2,...,

More information

Lecture 12: 2 k Factorial Design Montgomery: Chapter 6

Lecture 12: 2 k Factorial Design Montgomery: Chapter 6 Lecture 12: 2 k Factorial Design Montgomery: Chapter 6 1 Lecture 12 Page 1 2 k Factorial Design Involvingk factors: each has two levels (often labeled+and ) Very useful design for preliminary study Can

More information

Analysis of variance and regression. April 17, Contents Comparison of several groups One-way ANOVA. Two-way ANOVA Interaction Model checking

Analysis of variance and regression. April 17, Contents Comparison of several groups One-way ANOVA. Two-way ANOVA Interaction Model checking Analysis of variance and regression Contents Comparison of several groups One-way ANOVA April 7, 008 Two-way ANOVA Interaction Model checking ANOVA, April 008 Comparison of or more groups Julie Lyng Forman,

More information

Objectives Simple linear regression. Statistical model for linear regression. Estimating the regression parameters

Objectives Simple linear regression. Statistical model for linear regression. Estimating the regression parameters Objectives 10.1 Simple linear regression Statistical model for linear regression Estimating the regression parameters Confidence interval for regression parameters Significance test for the slope Confidence

More information

STATISTICS 479 Exam II (100 points)

STATISTICS 479 Exam II (100 points) Name STATISTICS 79 Exam II (1 points) 1. A SAS data set was created using the following input statement: Answer parts(a) to (e) below. input State $ City $ Pop199 Income Housing Electric; (a) () Give the

More information

df=degrees of freedom = n - 1

df=degrees of freedom = n - 1 One sample t-test test of the mean Assumptions: Independent, random samples Approximately normal distribution (from intro class: σ is unknown, need to calculate and use s (sample standard deviation)) Hypotheses:

More information

Outline. Analysis of Variance. Acknowledgements. Comparison of 2 or more groups. Comparison of serveral groups

Outline. Analysis of Variance. Acknowledgements. Comparison of 2 or more groups. Comparison of serveral groups Outline Analysis of Variance Analysis of variance and regression course http://staff.pubhealth.ku.dk/~lts/regression10_2/index.html Comparison of serveral groups Model checking Marc Andersen, mja@statgroup.dk

More information

Math 3330: Solution to midterm Exam

Math 3330: Solution to midterm Exam Math 3330: Solution to midterm Exam Question 1: (14 marks) Suppose the regression model is y i = β 0 + β 1 x i + ε i, i = 1,, n, where ε i are iid Normal distribution N(0, σ 2 ). a. (2 marks) Compute the

More information

Statistics 512: Applied Linear Models. Topic 1

Statistics 512: Applied Linear Models. Topic 1 Topic Overview This topic will cover Course Overview & Policies SAS Statistics 512: Applied Linear Models Topic 1 KNNL Chapter 1 (emphasis on Sections 1.3, 1.6, and 1.7; much should be review) Simple linear

More information

STAT 350. Assignment 4

STAT 350. Assignment 4 STAT 350 Assignment 4 1. For the Mileage data in assignment 3 conduct a residual analysis and report your findings. I used the full model for this since my answers to assignment 3 suggested we needed the

More information

EXST Regression Techniques Page 1. We can also test the hypothesis H :" œ 0 versus H :"

EXST Regression Techniques Page 1. We can also test the hypothesis H : œ 0 versus H : EXST704 - Regression Techniques Page 1 Using F tests instead of t-tests We can also test the hypothesis H :" œ 0 versus H :" Á 0 with an F test.! " " " F œ MSRegression MSError This test is mathematically

More information

Nature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals. Regression Output. Conditions for inference.

Nature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals. Regression Output. Conditions for inference. Understanding regression output from software Nature vs. nurture? Lecture 18 - Regression: Inference, Outliers, and Intervals In 1966 Cyril Burt published a paper called The genetic determination of differences

More information

Lecture 11 Multiple Linear Regression

Lecture 11 Multiple Linear Regression Lecture 11 Multiple Linear Regression STAT 512 Spring 2011 Background Reading KNNL: 6.1-6.5 11-1 Topic Overview Review: Multiple Linear Regression (MLR) Computer Science Case Study 11-2 Multiple Regression

More information

Regression: Main Ideas Setting: Quantitative outcome with a quantitative explanatory variable. Example, cont.

Regression: Main Ideas Setting: Quantitative outcome with a quantitative explanatory variable. Example, cont. TCELL 9/4/205 36-309/749 Experimental Design for Behavioral and Social Sciences Simple Regression Example Male black wheatear birds carry stones to the nest as a form of sexual display. Soler et al. wanted

More information

Statistics for exp. medical researchers Comparison of groups, T-tests and ANOVA

Statistics for exp. medical researchers Comparison of groups, T-tests and ANOVA Faculty of Health Sciences Outline Statistics for exp. medical researchers Comparison of groups, T-tests and ANOVA Lene Theil Skovgaard Sept. 14, 2015 Paired comparisons: tests and confidence intervals

More information

ANOVA: Analysis of Variation

ANOVA: Analysis of Variation ANOVA: Analysis of Variation The basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative variables depend on which group (given by categorical

More information

Sociology 6Z03 Review II

Sociology 6Z03 Review II Sociology 6Z03 Review II John Fox McMaster University Fall 2016 John Fox (McMaster University) Sociology 6Z03 Review II Fall 2016 1 / 35 Outline: Review II Probability Part I Sampling Distributions Probability

More information

STAT 350: Summer Semester Midterm 1: Solutions

STAT 350: Summer Semester Midterm 1: Solutions Name: Student Number: STAT 350: Summer Semester 2008 Midterm 1: Solutions 9 June 2008 Instructor: Richard Lockhart Instructions: This is an open book test. You may use notes, text, other books and a calculator.

More information

Lecture 7: Latin Square and Related Design

Lecture 7: Latin Square and Related Design Lecture 7: Latin Square and Related Design Montgomery: Section 4.2-4.3 Page 1 Automobile Emission Experiment Four cars and four drivers are employed in a study for possible differences between four gasoline

More information

Confidence Intervals, Testing and ANOVA Summary

Confidence Intervals, Testing and ANOVA Summary Confidence Intervals, Testing and ANOVA Summary 1 One Sample Tests 1.1 One Sample z test: Mean (σ known) Let X 1,, X n a r.s. from N(µ, σ) or n > 30. Let The test statistic is H 0 : µ = µ 0. z = x µ 0

More information

36-309/749 Experimental Design for Behavioral and Social Sciences. Sep. 22, 2015 Lecture 4: Linear Regression

36-309/749 Experimental Design for Behavioral and Social Sciences. Sep. 22, 2015 Lecture 4: Linear Regression 36-309/749 Experimental Design for Behavioral and Social Sciences Sep. 22, 2015 Lecture 4: Linear Regression TCELL Simple Regression Example Male black wheatear birds carry stones to the nest as a form

More information

Forestry 430 Advanced Biometrics and FRST 533 Problems in Statistical Methods Course Materials 2010

Forestry 430 Advanced Biometrics and FRST 533 Problems in Statistical Methods Course Materials 2010 Forestr 430 Advanced Biometrics and FRST 533 Problems in Statistical Methods Course Materials 00 Instructor: Dr. Valerie LeMa, Forest Sciences 039, 604-8-4770, EMAIL: Valerie.LeMa@ubc.ca Course Objectives

More information

Analysis of variance. April 16, Contents Comparison of several groups

Analysis of variance. April 16, Contents Comparison of several groups Contents Comparison of several groups Analysis of variance April 16, 2009 One-way ANOVA Two-way ANOVA Interaction Model checking Acknowledgement for use of presentation Julie Lyng Forman, Dept. of Biostatistics

More information

Lecture 9: Factorial Design Montgomery: chapter 5

Lecture 9: Factorial Design Montgomery: chapter 5 Lecture 9: Factorial Design Montgomery: chapter 5 Page 1 Examples Example I. Two factors (A, B) each with two levels (, +) Page 2 Three Data for Example I Ex.I-Data 1 A B + + 27,33 51,51 18,22 39,41 EX.I-Data

More information

The Model Building Process Part I: Checking Model Assumptions Best Practice

The Model Building Process Part I: Checking Model Assumptions Best Practice The Model Building Process Part I: Checking Model Assumptions Best Practice Authored by: Sarah Burke, PhD 31 July 2017 The goal of the STAT T&E COE is to assist in developing rigorous, defensible test

More information

Analysis of variance. April 16, 2009

Analysis of variance. April 16, 2009 Analysis of variance April 16, 2009 Contents Comparison of several groups One-way ANOVA Two-way ANOVA Interaction Model checking Acknowledgement for use of presentation Julie Lyng Forman, Dept. of Biostatistics

More information

The Model Building Process Part I: Checking Model Assumptions Best Practice (Version 1.1)

The Model Building Process Part I: Checking Model Assumptions Best Practice (Version 1.1) The Model Building Process Part I: Checking Model Assumptions Best Practice (Version 1.1) Authored by: Sarah Burke, PhD Version 1: 31 July 2017 Version 1.1: 24 October 2017 The goal of the STAT T&E COE

More information

Outline. Analysis of Variance. Comparison of 2 or more groups. Acknowledgements. Comparison of serveral groups

Outline. Analysis of Variance. Comparison of 2 or more groups. Acknowledgements. Comparison of serveral groups Outline Analysis of Variance Analysis of variance and regression course http://staff.pubhealth.ku.dk/~jufo/varianceregressionf2011.html Comparison of serveral groups Model checking Marc Andersen, mja@statgroup.dk

More information

Residuals from regression on original data 1

Residuals from regression on original data 1 Residuals from regression on original data 1 Obs a b n i y 1 1 1 3 1 1 2 1 1 3 2 2 3 1 1 3 3 3 4 1 2 3 1 4 5 1 2 3 2 5 6 1 2 3 3 6 7 1 3 3 1 7 8 1 3 3 2 8 9 1 3 3 3 9 10 2 1 3 1 10 11 2 1 3 2 11 12 2 1

More information

Analysis of Variance

Analysis of Variance 1 / 70 Analysis of Variance Analysis of variance and regression course http://staff.pubhealth.ku.dk/~lts/regression11_2 Marc Andersen, mja@statgroup.dk Analysis of variance and regression for health researchers,

More information

Chapter 11 : State SAT scores for 1982 Data Listing

Chapter 11 : State SAT scores for 1982 Data Listing EXST3201 Chapter 12a Geaghan Fall 2005: Page 1 Chapter 12 : Variable selection An example: State SAT scores In 1982 there was concern for scores of the Scholastic Aptitude Test (SAT) scores that varied

More information

SPECIAL TOPICS IN REGRESSION ANALYSIS

SPECIAL TOPICS IN REGRESSION ANALYSIS 1 SPECIAL TOPICS IN REGRESSION ANALYSIS Representing Nominal Scales in Regression Analysis There are several ways in which a set of G qualitative distinctions on some variable of interest can be represented

More information

Course Information Text:

Course Information Text: Course Information Text: Special reprint of Applied Linear Statistical Models, 5th edition by Kutner, Neter, Nachtsheim, and Li, 2012. Recommended: Applied Statistics and the SAS Programming Language,

More information

Chapter 1: Linear Regression with One Predictor Variable also known as: Simple Linear Regression Bivariate Linear Regression

Chapter 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 information

Overview. Prerequisites

Overview. Prerequisites Overview Introduction Practicalities Review of basic ideas Peter Dalgaard Department of Biostatistics University of Copenhagen Structure of the course The normal distribution t tests Determining the size

More information

Chapter 6 Multiple Regression

Chapter 6 Multiple Regression STAT 525 FALL 2018 Chapter 6 Multiple Regression Professor Min Zhang The Data and Model Still have single response variable Y Now have multiple explanatory variables Examples: Blood Pressure vs Age, Weight,

More information