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
|
|
- Betty Nelson
- 6 years ago
- Views:
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, 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 informationLecture 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 informationLecture 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 informationComparison 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 informationChapter 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 informationLecture 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 informationAnswer 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 informationBE640 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 informationAssignment 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 informationSingle 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 informationEXST7015: 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 informationHandout 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 informationOverview 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 informationCOMPREHENSIVE 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 informationLecture 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 information1) 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 informationTopic 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 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 informationTopic 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 information5.3 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 informationOutline. 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 informationSTOR 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 informationThis 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 informationIES 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 informationChapter 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 informationStat 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 informationLecture 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 informationLecture 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 informationIntroduction 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 informationLecture 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 informationIntroduction 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 informationBiological 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 informationST505/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 information3 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 informationStatistics 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 informationLecture 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 informationTable 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 informationFailure 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 informationPubH 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 informationOutline. 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 informationLecture 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 informationSTAT 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 informationLecture 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 informationT-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 informationSTA 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 informationCorrelation 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 informationChapter 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 informationANALYSIS 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 informationStatistics 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 informationBooklet 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 informationScenarios 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 informationWeek 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 information3rd 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 informationOne-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 informationBusiness 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 informationLecture 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 informationa. 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 informationSTAT5044: 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 informationModule 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 informationOne-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 informationCHAPTER 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 informationDensity 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 informationAnalysis 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 informationPLS205!! 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 informationSAS 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 informationLecture 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 informationAnalysis 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 informationObjectives 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 informationSTATISTICS 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 informationdf=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 informationOutline. 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 informationMath 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 informationStatistics 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 informationSTAT 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 informationEXST 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 informationNature 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 informationLecture 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 informationRegression: 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 informationStatistics 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 informationANOVA: 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 informationSociology 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 informationSTAT 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 informationLecture 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 informationConfidence 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 information36-309/749 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 informationForestry 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 informationAnalysis 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 informationLecture 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 informationThe 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 informationAnalysis 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 informationThe 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 informationOutline. 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 informationResiduals 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 informationAnalysis 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 informationChapter 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 informationSPECIAL 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 informationCourse 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 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 informationOverview. 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 informationChapter 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