The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15

Size: px
Start display at page:

Download "The Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15"

Transcription

1 The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5

2 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the Analyss of Covarance Example ANCOVA Analyss

3 THE ANALYSIS OF COVARIANCE

4 The Analyss of Covarance In a completely randomzed desgn subjects are randomly assgned to the expermental treatments The completely randomzed desgn s relatvely defcent n power If there s a varable avalable before the start of experment that s reasonably correlated wth the dependent varable abe(e,co (.e., control o varable, abe,concomtant varable, or covarate; e.g., ntellgence, grade pont average, etc.), we may ether employ blockng or statstcal ttt adjustment t

5 The Analyss of Covarance The randomzed blocks desgn ncludes groups of homogeneous subjects drawn from respectve blocks Advantages of the randomzed blocks desgn are: Blockng helps to equate the treatment groups before the start of the experment more effectvely than s accomplshed n the completely l randomzed ddesgn The power s ncreased because smallererrorerror term usually assocated wth the blockng desgn Interactons can be assessed

6 The Analyss of Covarance Dsadvantages may nclude: There wll be cost of ntroducng the blockng factor It may be dffcult to fnd blockng factors that are hghly correlated ltdwth the dependent d varable bl Loss of power may occur f a poorly correlated blockng p y p y g factor s used

7 The Analyss of Covarance The analyss of covarance reduces expermental error by statstcal, rather than expermental, means Subjects arefrstmeasured on the concomtant varable called the covarate whch conssts of some relevant ablty or characterstcs Subjects are then randomly assgned to the treatment j y g group wthout regard for ther scores on the covarate

8 The Analyss of Covarance The analyss of covarance refnes estmates of expermental error and uses the adjust treatment effects for any dfferences between the treatment groups that exsted before the expermental treatments were admnstered

9 Today s Example Data We wll rehash data from Secton.5 A researcher was studyng the effects of nstructonal materal on how well college students learn basc concepts n statstcs Two nstructonal materal groups (a 2) Pretest gven General quanttatve ablty (now called the covarate X) DV Scores on a test on basc statstcs test (Y)

10 The Data Group a Group a 2 X Y X Y Mean:

11 Our Data As a Plot

12 COVARIANCE AND LINEAR REGRESSION

13 Covarance and Lnear Regresson The correlaton coeffcent between two varables X and Y s: r XY xy Where the covarance between X and Y s: ( X X )( Y Y ) s s x s y N N The standard devaton of The standard devaton of X s: Y s: s x N ( X X ) 2 N s XY s Y N ( ) 2 Y Y N

14 Wth Our Data Set The correlaton coeffcent between two varables X and Y s: xy *3.055 Where the covarance between X and Y s: ( X X )( Y Y ) s 0.66 r XY s 5. sxs 667 XY y N The standard devaton of The standard devaton of X s: Y s: s x N ( X X ) 2 N s Y N N ( ) 2 Y Y N 3.055

15 Other Useful Terms (Wll Help n ANCOVA) We may defne the sum of products: SP And the sums of squares: 2 XY N ( X X )( Y Y ) 85 N SS ( ) X X X 8 N SS ( ) Y Y Y 40 Consequently: 2 r XY SP SS X XY SS Y 85 8*

16 THE LINEAR REGRESSION EQUATION

17 The Lnear Regresson Equaton The lnear regresson lne relatng the dependent varable Y to the covarate X s: Y β 0 + β X + And the predcton equaton for s: Where: ˆ + E Y b0 b X b 0 Y b X S SPXY S X SS X Y b rxy

18 Wth Our Data The lnear regresson lne relatng the dependent varable Y to the covarate X s: Y β 0 + β X + And the predcton equaton for s: Where: Yˆ b. 720X E 0 + b X.403+ SY SPXY b0 Y b X.403 b rxy. 720 S SS X X

19 Regresson Hypothess Tests In regresson, the key hypothess test we are worred about s that for the slope, p, b H 0 : β 0 H : β 0 Usng these hypotheses, and our method for calculatng sums of squares from Chapter 4 (the GLM chapter), we can then create an ANOVA lke table for the regresson. We need: H0 H SSun exp SS unexp

20 Regresson Sums of Squares H Recall that SS 0exp un came from calculatng the sums of squares where the predcted value came from the lnear model when H 0 was true It was treatment + error Inregresson the correspondng predcted valuesare: are: Ths comes from: df un Yˆ b 0 Y b Y b X Y 0* X 0 H 0 The exp comes from the number of observatons (here 6) mnus the number of parameters () so H 0 df un exp 5 Y

21 Regresson Sums of Squares H Recall that SS exp un came from calculatng the sums of squares where the predcted value came from the lnear model when H was true It was error In regresson the correspondng predcted values are: ˆ + Y b 0 b X H The df un exp comes from the number of observatons (6) mnus the number of parameters (2), so H df un exp 4

22 Resdual Varaton and the Lnear Model Another way of lookng at Sums of Squares: The sum of the squared devaton from the mean s the Sums of Squares Total: T N SSY ( ) 2 H 0 Y Y SS exp SS Thesum of the squared resduals fromthe regresson lne s: N SS Re gresson SSY X un ( ) 2 ˆ H H Y Y SS SS Fnally, the sum of squares for error s: N 2 ˆ H ( ) Y Y SS SS Error un exp 0 unexp unexp

23 Puttng t Together n an ANOVA Table: See Excel worksheet on ELC for computatons

24 Resdual Varaton and the Lnear Model The amount of the Y varablty that can be attrbuted to the regresson equaton s: SS regresson SSY SSY X The squared correlaton coeffcent (or what we call the effect sze eta squared) s: r SS 2 regresson XY SSY.437

25 Now wth SPSS

26 THE ANALYSIS OF COVARIANCE

27 The Analyss of Covarance Now that we know a lttle about regresson, we can use t to mprove power to detect group dfferences n an ANOVA lke expermental desgn ANCOVA tests for dfferences between groups by comparng a descrpton of the data based on a sngle regresson lne to one based on lnes wth the same slope and dfferent ntercepts for each group

28 In the Begnnng There was the ANOVA Model Recall our GLM representaton of ANOVA: The H model: The H 0 model: Y μ + α + j T j E j Y μ + j μ T E j ANCOVA adds n the covarate X to the model

29 The Analyss of Covarance and the General Lnear Model For the analyss of covarance, the alternatvehypothess model s: Y X + j β0 + α j + β j E j And the null hypothess model (testng the effect of the treatment means) s: Y β 0 + β X + j j E j

30 Decomposng ANCOVA If the H model holds n ANCOVA, t means that the ntercepts of the regresson lne dffer across groups: For our 2 group example that means, for group : For group 2: ( β ) 0 + α + X E Y β + ( β ) 0 + α 2 + β X 2 E 2 Y β + 2

31 Does t? Hmm H 0 Model same ntercepts H Model Dff. Intercepts

32 ...Its a Queston of Statstcal Sgnfcance Usng our knowledge of GLM, we could form SS for each of the possble categores or we could use SPSS: Covarate (X the score on the general quanttatve test) Treatment group (A)

33 Comparng ANOVA and ANCOVA ANOVA ANCOVA

34 The Dfference? Adjusted Means For ANCOVA, the adjusted mean (or Least Squares Mean) s: ' Y b ( X X ) Y j T

35 What about the Rest of ANOVA All of the rest of ANOVA happens on the adjusted means lke contrasts or post hoc tests Both of whch are made easy n SPSS The desgn can be extended to nclude more factors and more covarates Of course, thngs get more tedous to understand as you add more varables Th k t l th t f t f tl The key s to realze that f your covarate sgnfcantly predcts Y (above and beyond the group varable), you wll have more power to detect group dfferences

36 ASSUMPTIONS UNDERLYING THE ANALYSIS OF COVARIANCE

37 Assumptons Underlyng the Analyss of Covarance Three assumptons n addton to the usual analyss of varance assumptons are:. The assumpton of lnear regresson: The devatons from regresson are normally and ndependently dstrbuted n the populaton, wth means of zero and homogeneous varances Hard to check defntvely 2. The assumpton of homogeneous group regresson coeffcents: The wthn groupsregressoncoeffcent regresson s actually an average of the regresson coeffcents for the respectve treatment groups Can be checked by addng an nteracton term between the covarate and group varable 3. The exact measurement of the covarate: The covarate s measured wthout error Ubqutous term

38 Checkng the Homogenety of Slopes Assumpton You can easly check the homogenety of slopes assumpton by testng the nteracton between your group IV and the covarate The H model becomes: j β 0 + α j + j ( β ) X j E j Y β X + αβ + If true, t mples, for group : ( β ) [ ( ) ] 0 + α + β + X E Y αβ + And for group 2: ( β ) [ ( ) ] 0 + α 2 + β + X 2 E 2 Y αβ j

39 In SPSS Does t meet our assumpton of homogenety of slopes?

40 Wrappng Up Today s class covered a method for controllng for mportant varables n an experment: ANCOVA ANCOVA s a general technque that adds addtonal (contnuous/quanttatve) varables to a model and adjusts for the values of such varables Any ANOVA desgn can nclude such varables

41 Up Next In lab tonght: How to do ANCOVA n SPSS Homework: Assgned n the mornng, due next week before class Readng for next week: Chapter 6: Wthn Subject Desgns Only two lectures left!

Simple Linear Regression and Correlation

Simple Linear Regression and Correlation Smple Lnear Regresson and Correlaton In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable The meanng of the regresson coeffcents

More information

Randomized Complete Block Designs (RCBD)

Randomized Complete Block Designs (RCBD) ANOVA (III) 1 Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn that we recently learned. In ths desgn, blocks of expermental unts

More information

Introduction to Regression

Introduction to Regression Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. -Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. -

More information

Regression. REGRESSION I: One Independent Variable. Review. I made a quick data set using my calculator. Let s look at a quick example

Regression. REGRESSION I: One Independent Variable. Review. I made a quick data set using my calculator. Let s look at a quick example REGRESSION I: One Independent Varable Tom Ilvento FREC Regresson We are lookng at the relatonshp between two or more varables One s called the Dependent Varable (Y), whch s to be predcted (Def. p53) The

More information

β = to determine whether there was a significant relationship

β = to determine whether there was a significant relationship Topc 4. Multple Regresson (Ch. 9) ) Smple Lnear Regresson Revsted Recall our smple lnear regresson model was y = α + βx+ ε () where ε ~ IN (0, σ ). Accordng to ths model, an observaton y s formed from

More information

Multiple Regression 2

Multiple Regression 2 Overvew Multple Regresson Dr Tom Ilvento Department of Food and Resource Economcs We wll contnue our dscusson of Multple Regresson wth a specal focus on lookng at Resduals Resdual Analyss looks at the

More information

Regression 1 LINEAR REGRESSION

Regression 1 LINEAR REGRESSION Regresson LINEAR REGRESSION Defn: a Scatterplot shows the relatonshp between two quanttatve varables measured on the same populaton unts. The values of the response varable are plotted on the Y-axs and

More information

Analysis of Variance (ANOVA) Chapter 15

Analysis of Variance (ANOVA) Chapter 15 Analyss of Varance (ANOVA) Chapter 15 Ratonale Many varables create varaton n the dstrbuton of a quanttatve response varable. The regresson analyss provdes one approach for modelng and studyng varaton

More information

Chapter 10 Dummy Variable Models

Chapter 10 Dummy Variable Models Chapter ummy Varable Models In general, the explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, the varables lke temperature, dstance, age etc. are quanttatve

More information

Topic 8. Analysis of Variance (Ch. 12)

Topic 8. Analysis of Variance (Ch. 12) Topc 8. Analyss of Varance (Ch. 1) 1) One-way analyss of varance (ANOVA) 1.1 General ratonale We studed the comparson of two means n the last chapter, and found that we could use the t and/or z statstcs

More information

Completely Randomized Design

Completely Randomized Design Lnear Model for One acr CRD Completely Randomzed Desgn Homogeneous Subjects or Expermental Unts re ssgned Randomly Treatments Let y j be the j-th sample observaton from the populaton, y j = μ + τ + ε j

More information

SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

More information

Chapter 6 Bivariate Correlation & Regression

Chapter 6 Bivariate Correlation & Regression Chapter 6 Bvarate Correlaton & Regresson 6.1 Scatterplots and Regresson Lnes 6. Estmatng a Lnear Regresson Equaton 6.3 R-Square and Correlaton 6.4 Sgnfcance Tests for Regresson Parameters Scatterplot:

More information

Questions that we may have about the variables

Questions that we may have about the variables Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

More information

b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei

b) The mean of the fitted (predicted) values of Y is equal to the mean of the Y values: c) The residuals of the regression line sum up to zero: = ei Mathematcal Propertes of the Least Squares Regresson The least squares regresson lne obeys certan mathematcal propertes whch are useful to know n practce. The followng propertes can be establshed algebracally:

More information

Chapter 4. Linear regression and correlation analysis

Chapter 4. Linear regression and correlation analysis 4.. Lnear regresson Lnear regresson s appled to nfer the relatonshp between the dependent varable and the regressors, usng a set of sample data. Predcton can be made accordng to the ftted model. Generally,

More information

Multiple Regression. SPSS output. Multiple Regression Multiple Regression Model:

Multiple Regression. SPSS output. Multiple Regression Multiple Regression Model: Weght Multple Regresson Multple Regresson Relatng a response (dependent, nput) y to a set of eplanatory (ndependent, output, predctor) varables,,,. A technue for modelng the relatonshp between one response

More information

Ordinary Least Squares (OLS) Estimation of the Simple CLRM. 1. The Nature of the Estimation Problem

Ordinary Least Squares (OLS) Estimation of the Simple CLRM. 1. The Nature of the Estimation Problem ECOOMICS 35* -- OTE ECO 35* -- OTE Ordnary Least Squares (OLS) Estmaton of the Smple CLRM. The ature of the Estmaton Problem Ths note derves the Ordnary Least Squares (OLS) coeffcent estmators for the

More information

Lecture 19 Topic 13: Analysis of Covariance (ANCOVA), Part I [ST&D, chapter 17]

Lecture 19 Topic 13: Analysis of Covariance (ANCOVA), Part I [ST&D, chapter 17] Lecture 19 Topc 13: Analyss of Covarance (ANCOVA), Part I [ST&D, chapter 17] Suppose that the response varable (Y) s lnearly related to some other contnuous varable (X) that the expermenter cannot control

More information

One-Way Analysis of Variance (ANOVA)

One-Way Analysis of Variance (ANOVA) One-ay Analyss of Varance (ANOVA) e are examnng dfferences between populaton means 1,,, k of k dfferent populatons where k s usually bgger than. Our goal s to determne whether there s any dfference between

More information

Lecture 10: Linear Regression Approach, Assumptions and Diagnostics

Lecture 10: Linear Regression Approach, Assumptions and Diagnostics Approach to Modelng I Lecture 1: Lnear Regresson Approach, Assumptons and Dagnostcs Sandy Eckel seckel@jhsph.edu 8 May 8 General approach for most statstcal modelng: Defne the populaton of nterest State

More information

Sample ANOVA problem from last week

Sample ANOVA problem from last week Sample ANOVA problem from last week A natonwde car company wants to study the relatonshp between commsson rates and number of cars sold per week. Commsson rates dffer across markets. They comple the followng

More information

Multiple Regression. SPSS output. Multiple Regression. α, β 1, β 2, β 3,..., β q in the model can all be estimated by least square estimators:

Multiple Regression. SPSS output. Multiple Regression. α, β 1, β 2, β 3,..., β q in the model can all be estimated by least square estimators: Multple Regresson Relatng a response (dependent, nput) y to a set of explanatory (ndependent, output, predctor) varables x, x,, x. A technue for modelng the relatonshp between one response varable wth

More information

Chapter 12 Simple Linear Regression

Chapter 12 Simple Linear Regression Chapter 1 Smple Lnear Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Smple Lnear Regresson Model The equaton that descrbes how

More information

June 9, Correlation and Simple Linear Regression Analysis

June 9, Correlation and Simple Linear Regression Analysis June 9, 2004 Correlaton and Smple Lnear Regresson Analyss Bvarate Data We wll see how to explore the relatonshp between two contnuous varables. X= Number of years n the job, Y=Number of unts produced X=

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Week 8 Lecture: Regression Models for Quantitative & Qualitative Predictor

Week 8 Lecture: Regression Models for Quantitative & Qualitative Predictor Week 8 Lecture: Regresson Models for Quanttatve & Qualtatve Predctor Varables (Chapter 8 Polynomal Regresson Polynomal regresson models are commonly used to model quadratc and hgher power responses, manly

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

Lecture 9. The multiple Classical Linear Regression model

Lecture 9. The multiple Classical Linear Regression model Lecture 9. The multple Classcal Lnear Regresson model Often we want to estmate the relaton between a dependent varable Y and say K ndependent varables X,, X K K. Even wth K explanatory varables, the lnear

More information

THE TITANIC SHIPWRECK: WHO WAS

THE TITANIC SHIPWRECK: WHO WAS THE TITANIC SHIPWRECK: WHO WAS MOST LIKELY TO SURVIVE? A STATISTICAL ANALYSIS Ths paper examnes the probablty of survvng the Ttanc shpwreck usng lmted dependent varable regresson analyss. Ths appled analyss

More information

Modeling known-fate data on survival binomials & logistic regression. # Survived. Ppn Survived

Modeling known-fate data on survival binomials & logistic regression. # Survived. Ppn Survived Modelng known-fate data on survval bnomals & logstc regresson Consder the followng dataset: Week # Females # Ded # Survved Ppn Survved (p*(-p))/n se-hat 48 47.979.42.2 2 47 2 45.957.87.29 3 4 2 39.95.3.34

More information

Appendix 5: Excel Data Analysis Worksheet

Appendix 5: Excel Data Analysis Worksheet 65 Appendx 5: Excel Data Analyss Worksheet Locaton: Chemstry Computer Labs n Schlundt Hall Several of our experments wll use the Excel spread sheet program to assst n data analyss. Least squares lnear

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

Week 3: Residual Analysis (Chapter 3)

Week 3: Residual Analysis (Chapter 3) Week 3: Resdual Analyss (Chapter 3) Propertes of Resduals Here s some propertes of resduals, some of whch you learned n prevous lectures:. Defnton of Resdual: e = Y Ŷ. ε = Y E( Y ) 3. ε ~ NID( 0 σ ), 4.

More information

Chapter 5 Student Lecture Notes 5-1

Chapter 5 Student Lecture Notes 5-1 Chapter 5 Student Lecture Notes 5- Chapter Goals QM5: Busness Statstcs Chapter 5 Analyss of Varance () After completng ths chapter, you should be able to: Recognze stuatons n whch to use analyss of varance

More information

Math 10. Math 10 Part 8 Slides. Maurice Geraghty Characteristics of the Chi- Square Distribution. Part 8

Math 10. Math 10 Part 8 Slides. Maurice Geraghty Characteristics of the Chi- Square Distribution. Part 8 14- Characterstcs of the Ch- Square Dstrbuton Math 10 M Geraghty Part 8 Ch-square and ANOVA tests The major characterstcs of the chsquare dstrbuton are: It s postvely skewed It s non-negatve It s based

More information

A Multiple Linear Regression Model to Predict the Student s Final Grade in a Mathematics Class ABSTRACT

A Multiple Linear Regression Model to Predict the Student s Final Grade in a Mathematics Class ABSTRACT A Multple Lnear Regresson Model to Predct the Student s Fnal Grade n a Mathematcs Class Dr. M. Shakl Department of Mathematcs Mam-Dade College, Haleah Campus 78 West 49th Street Haleah, Florda, USA E-mal:

More information

Homework 1: The two-variable linear model Answer key to analytical questions

Homework 1: The two-variable linear model Answer key to analytical questions Econ471: Appled Econometrcs Walter Sosa-Escudero Homework 1: The two-varable lnear model Answer key to analytcal questons I) Excercses 1. Consder the data shown n the next table, on consumpton C and ncome

More information

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES

CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES CHAPTER 5 RELATIONSHIPS BETWEEN QUANTITATIVE VARIABLES In ths chapter, we wll learn how to descrbe the relatonshp between two quanttatve varables. Remember (from Chapter 2) that the terms quanttatve varable

More information

Analysis of Covariance

Analysis of Covariance Chapter 551 Analyss of Covarance Introducton A common tas n research s to compare the averages of two or more populatons (groups). We mght want to compare the ncome level of two regons, the ntrogen content

More information

ST 524 NCSU - Fall 2008 Completely Randomized Design with Subsampling. Completely Randomized Design with subsamples

ST 524 NCSU - Fall 2008 Completely Randomized Design with Subsampling. Completely Randomized Design with subsamples ST 5 NCSU - Fall 008 Completely Randomzed Desgn wth subsamples Example (ST&D, p 59) Experment to analyze the effect of hours of daylght and nght temperature on the stem growth of mnt plants: 6 treatments

More information

STA305 Winter 2010 Notes on Nested Designs

STA305 Winter 2010 Notes on Nested Designs Nested Desgns Example STA305 Wnter 010 Notes on Nested Desgns A large restaurant chan uses 4 supplers of cookng ol. The owners have decded to test whether volume of ol receved s exactly what was ordered.

More information

Part 1: quick summary 5. Part 2: understanding the basics of ANOVA 8

Part 1: quick summary 5. Part 2: understanding the basics of ANOVA 8 Statstcs Rudolf N. Cardnal Graduate-level statstcs for psychology and neuroscence NOV n practce, and complex NOV desgns Verson of May 4 Part : quck summary 5. Overvew of ths document 5. Background knowledge

More information

Regression Analysis Tutorial 183 LECTURE / DISCUSSION. Weighted Least Squares

Regression Analysis Tutorial 183 LECTURE / DISCUSSION. Weighted Least Squares Regresson Analyss Tutoral 183 LETURE / DISUSSION Weghted Least Squares Econometrcs Laboratory Unversty of alforna at Berkeley 22-26 March 1999 Regresson Analyss Tutoral 184 Introducton In a regresson problem

More information

The response or dependent variable is the response of interest, the variable we want to predict, and is usually denoted by y.

The response or dependent variable is the response of interest, the variable we want to predict, and is usually denoted by y. 1 DISPLAYING THE RELATIONSHIP DEFINITIONS: Studes are often conducted to attempt to show that some eplanatory varable causes the values of some response varable to occur. The response or dependent varable

More information

Module 11 Correlation and Regression

Module 11 Correlation and Regression Module Correlaton and Regresson. Correlaton Analyss.. Seral or Auto-Correlaton.. Cross-Correlaton..3 Spurous correlaton..4 Inferences on Correlaton Coeffcents..5 Kendall s Rank Correlaton Test. Regresson

More information

I. One more assumption about the error: Part III, normality, it is that Assumption 5: the errors are normally distributed.

I. One more assumption about the error: Part III, normality, it is that Assumption 5: the errors are normally distributed. Econ 388 R. Butler 4 revsons Lecture 7 I. One more assumpton about the error: Part III, normalty, t s that Assumpton 5: the errors are normally dstrbuted. The regresson model can be convenently summarzed

More information

Chapter 16 One-way Analysis of Variance

Chapter 16 One-way Analysis of Variance Chapter 16 One-way Analyss of Varance I am assumng that most people would prefer to see the solutons to these problems as computer prntout. (I wll use SPSS for consstency.) 16.1 Analyss of Eysenck s data:

More information

POPULATION Random Variables X, Y: numerical Definition: Population Linear Correlation Coefficient of X, Y. σ XY σ X σ Y. ρ = SAMPLE, size n

POPULATION Random Variables X, Y: numerical Definition: Population Linear Correlation Coefficient of X, Y. σ XY σ X σ Y. ρ = SAMPLE, size n Ismor Fscher, 8/12/2008 Stat 541 / 7-5 7.2 Lnear Correlaton and Regresson POPULATION Random Varables X, Y: numercal Defnton: Populaton Lnear Correlaton Coeffcent of X, Y ρ = σ XY σ X σ Y FACT: 1 ρ +1 SAMPLE,

More information

Section 13.5 Test of Hypothesis for Population Proportion

Section 13.5 Test of Hypothesis for Population Proportion Secton 135 Test of Hypothess for Populaton Proporton As dscussed n Secton 114-115, nference about the probablty of success n ndependent Bernoull trals s same as the mean of ndependent observatons on bnary

More information

Lecture 21 Factor Effects Model

Lecture 21 Factor Effects Model Lecture 21 Factor Effects Model STAT 512 Sprng 2011 Background Readng KNNL: 16.7, 16.10-16.11 21-1 Topc Overvew Factor Effects Model for Sngle Factor ANOVA Cash Offers example 21-2 Revew: Cell Means Model

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

5.2 Least-Squares Fit to a Straight Line

5.2 Least-Squares Fit to a Straight Line 5. Least-Squares Ft to a Straght Lne total probablty and ch-square mnmzng ch-square for a straght lne solvng the determnants example least-squares ft weghted least-squares wth an example least-squares

More information

The Two Variable Linear Model

The Two Variable Linear Model Chapter 1 The Two Varable Lnear Model 1.1 The Basc Lnear Model The goal of ths secton s to buld a smple model for the non-exact relatonshp between two varables Y and X, related by some economc theory.

More information

Sample Exam for EC 306 BRIEF SKETCHES OF ANSWERS GIVEN IN BOLD

Sample Exam for EC 306 BRIEF SKETCHES OF ANSWERS GIVEN IN BOLD Sample Exam for EC 306 BRIEF SKETCHES OF ASWERS GIVE I BOLD Ths exam contans four questons. Please do all questons. 1. Emprcal Practce (0%) In a study of costs n the bankng ndustry, data has been collected

More information

Accelerated Failure Time Model (II)

Accelerated Failure Time Model (II) Accelerated Falure Tme Model (II) Purpose:. Introducton to the accelerated falure tme model 2. Estmatng parametrc regresson models wth PROC LIFEREG n SAS Data descrpton: The data conssts of 432 males nmates

More information

Econometrics Deriving the OLS estimator in the univariate and bivariate cases

Econometrics Deriving the OLS estimator in the univariate and bivariate cases Econometrcs Dervng the OLS estmator n the unvarate and bvarate cases athanel Hggns nhggns@jhu.edu Prmer otaton Here are some bts of notaton that you wll need to understand the dervatons below. They are

More information

Logistic Regression. Basic Idea: Lecture 13: Introduction to. Logistic Regression. Remember the Binomial Distribution? Review of the Binomial Model

Logistic Regression. Basic Idea: Lecture 13: Introduction to. Logistic Regression. Remember the Binomial Distribution? Review of the Binomial Model Logstc Regresson Lecture 3: Introducton to Logstc Regresson Sandy Eckel seckel@jhsph.edu 3 May 28 Basc Idea: Logstc regresson s the type of regresson we use for a response varable (Y) that follows a bnomal

More information

Assumptions for linear regression. STK4900/ Lecture 5. Program

Assumptions for linear regression. STK4900/ Lecture 5. Program STK4900/9900 - Lecture 5 Program 1. Checkng model assumptons Lnearty Equal varances Normalty Influental observatons Importance of model assumptons. Selecton of predctors Forward and backward selecton Crtera

More information

9.1 The Cumulative Sum Control Chart

9.1 The Cumulative Sum Control Chart Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Combined 5 2 cv F Test for Comparing Supervised Classification Learning Algorithms

Combined 5 2 cv F Test for Comparing Supervised Classification Learning Algorithms NOTE Communcated by Thomas Detterch Combned 5 2 cv F Test for Comparng Supervsed Classfcaton Learnng Algorthms Ethem Alpaydın IDIAP, CP 592 CH-1920 Martgny, Swtzerland and Department of Computer Engneerng,

More information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Sldes Prepared JOHN S. LOUCKS St. Edward s Unverst Slde Chapter 4 Smple Lnear Regresson Smple Lnear Regresson Model Least Squares Method Coeffcent of Determnaton Model Assumptons Testng for Sgnfcance Usng

More information

Homework Solution (#4)

Homework Solution (#4) Chapter 0: 4, 9,9,4 and 4 Homework Soluton (#4 Chapter 0. The Analyss of Varance 0.4 (new edton: Page 40; old edton: Page 4 3 ( 5.9. 08, so ( 4.39 5.9 +... + (.3 5.9 ] (.08 x Ix SST 9.9 49.95. 3 SSTr 8[

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

Economic Interpretation of Regression. Theory and Applications

Economic Interpretation of Regression. Theory and Applications Economc Interpretaton of Regresson Theor and Applcatons Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve

More information

Gregory Carey 1998 Math & Regression - 1. The Mathematical Theory of Regression

Gregory Carey 1998 Math & Regression - 1. The Mathematical Theory of Regression Gregory Carey 1998 Math & Regresson - 1 The Mathematcal Theory of Regresson The basc formulaton for multple regresson s dagramed n Fgure 1. Here, there are a seres of predctor varables, termed x 1, x,...

More information

Chapter 18 Seemingly Unrelated Regression Equations Models

Chapter 18 Seemingly Unrelated Regression Equations Models Chapter 8 Seemngly Unrelated Regresson Equatons Models A basc nature of multple regresson model s that t descrbes the behavour of a partcular study varable based on a set of explanatory varables When the

More information

The Probit Model. Alexander Spermann. SoSe 2009

The Probit Model. Alexander Spermann. SoSe 2009 The Probt Model Aleander Spermann Unversty of Freburg SoSe 009 Course outlne. Notaton and statstcal foundatons. Introducton to the Probt model 3. Applcaton 4. Coeffcents and margnal effects 5. Goodness-of-ft

More information

The topics in this section concern with the second course objective. Correlation is a linear relation between two random variables.

The topics in this section concern with the second course objective. Correlation is a linear relation between two random variables. 4.1 Correlaton The topcs n ths secton concern wth the second course objectve. Correlaton s a lnear relaton between two random varables. Note that the term relaton used n ths secton means connecton or relatonshp

More information

STATISTICAL DATA ANALYSIS IN EXCEL

STATISTICAL DATA ANALYSIS IN EXCEL Mcroarray Center STATISTICAL DATA ANALYSIS IN EXCEL Lecture 6 Some Advanced Topcs Dr. Petr Nazarov 14-01-013 petr.nazarov@crp-sante.lu Statstcal data analyss n Ecel. 6. Some advanced topcs Correcton for

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Chi-Square Analyses. Goodness of Fit Often, the first introduction to Chi-square is the Goodness-of-Fit model. Consider the following problem:

Chi-Square Analyses. Goodness of Fit Often, the first introduction to Chi-square is the Goodness-of-Fit model. Consider the following problem: Ch-Square Analyses NCCTM Oct. 10, 003 Ch-Square Analyses Ch-square analyss s one of the more complcated analyses n the AP Statstcs currculum. The complcaton s a result of several features: 1) the pluralty

More information

Chapter 15 Multiple Regression

Chapter 15 Multiple Regression Chapter 5 Multple Regresson In chapter 9, we consdered one dependent varable (Y) and one predctor (regressor or ndependent varable) (X) and predcted Y based on X only, whch also known as the smple lnear

More information

The use of negative controls to detect confounding and other sources of error in experimental and observational science

The use of negative controls to detect confounding and other sources of error in experimental and observational science The use of negatve controls to detect confoundng and other sources of error n expermental and observatonal scence Marc Lpstch Erc Tchetgen Tchetgen Ted Cohen eppendx. Use of negatve control outcomes to

More information

CHAPTER 7 THE TWO-VARIABLE REGRESSION MODEL: HYPOTHESIS TESTING

CHAPTER 7 THE TWO-VARIABLE REGRESSION MODEL: HYPOTHESIS TESTING CHAPTER 7 THE TWO-VARIABLE REGRESSION MODEL: HYPOTHESIS TESTING QUESTIONS 7.1. (a) In the regresson contet, the method of least squares estmates the regresson parameters n such a way that the sum of the

More information

14.74 Lecture 5: The long run impact of a childhood intervention A randomized evaluation

14.74 Lecture 5: The long run impact of a childhood intervention A randomized evaluation 14.74 Lecture 5: The long run mpact of a chldhood nterventon A randomzed evaluaton Prof. Esther Duflo February 16, 2010 Last lecture, we saw that one mpact of nutrton whch may be a source of a nutrton-based

More information

Study on CET4 Marks in China s Graded English Teaching

Study on CET4 Marks in China s Graded English Teaching Study on CET4 Marks n Chna s Graded Englsh Teachng CHE We College of Foregn Studes, Shandong Insttute of Busness and Technology, P.R.Chna, 264005 Abstract: Ths paper deploys Logt model, and decomposes

More information

Comparison of Several Univariate Normality Tests Regarding Type I Error Rate and Power of the Test in Simulation based Small Samples

Comparison of Several Univariate Normality Tests Regarding Type I Error Rate and Power of the Test in Simulation based Small Samples Journal of Appled Scence Research (5): 96-300, 006 006, INSInet Publcaton Comparson of Several Unvarate Normalty Tests Regardng Type I Error Rate and Power of the Test n Smulaton based Small Samples Sddk

More information

Multiple Linear Regression

Multiple Linear Regression Multple Lnear Regresson Heteroskedastcty Heteroskedastcty Multple Lnear Regresson: Assumptons Assumpton MLR. (Lnearty n parameters) Assumpton MLR. (Random Samplng from the populaton) We have a random sample:

More information

SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA

SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands E-mal: e.lagendjk@tnw.tudelft.nl

More information

ECON 546: Themes in Econometrics Lab. Exercise 2

ECON 546: Themes in Econometrics Lab. Exercise 2 Unversty of Vctora Department of Economcs ECON 546: Themes n Econometrcs Lab. Exercse Introducton The purpose of ths lab. exercse s to show you how to use EVews to estmate the parameters of a regresson

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

Logistic Regression. Sample StatFolio: logistic.sgp

Logistic Regression. Sample StatFolio: logistic.sgp Logstc Regresson Summary The Logstc Regresson procedure s desgned to ft a regresson model n whch the dependent varable Y characterzes an event wth only two possble outcomes. Two types of data may be modeled:.

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

The Analysis of Variance

The Analysis of Variance The Analyss of Varance Introducton Researchers often perform experments to compare two treatments, for example, two dfferent fertlzers, machnes, methods or materals. The objectves are to determne whether

More information

7. Analysis of Variance (ANOVA)

7. Analysis of Variance (ANOVA) 1 7. Analyss of Varance (ANOVA) 2 7.1 An overvew of ANOVA What s ANOVA? 3 ANOVA refers to statstcal models and assocated procedures, n whch the observed varance s parttoned nto components due to dfferent

More information

Systems of Particles

Systems of Particles Physcs 53 Systems of Partcles Everythng should be as smple as t s, but not smpler. Albert Ensten Overvew An object of ordnary sze whch we call a macroscopc system contans a huge number of atoms or molecules.

More information

Fuzzy Hypothesis Testing Of Anova Model with Fuzzy Data

Fuzzy Hypothesis Testing Of Anova Model with Fuzzy Data Internatonal Journal of odern Engneerng Research (IJER) wwwjmercom Vol, Issue4, July-Aug 0 pp-95-956 ISSN: 49-6645 Fuzzy Hypothess Testng Of Anova odel wth Fuzzy Data D Kalpanaprya P Pan Department of

More information

Direct and indirect effects in dummy variable regression

Direct and indirect effects in dummy variable regression mercan Journal of Theoretcal and ppled Statstcs 214; 3(2): 44-48 Publshed onlne February 2, 214 (http://www.scencepublshnggroup.com/j/ajtas) do: 1.11648/j.ajtas.21432.13 Drect and ndrect effects n dummy

More information

Statistical Methods to Develop Rating Models

Statistical Methods to Develop Rating Models Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and

More information

Economics 375: Introduction to Econometrics Homework #3 This homework is due on Tuesday, May 3 rd.

Economics 375: Introduction to Econometrics Homework #3 This homework is due on Tuesday, May 3 rd. Economcs 375: Introducton to Econometrcs Homework #3 Ths homework s due on Tuesday, May 3 rd. One tool to ad n understandng econometrcs s the Monte Carlo experment. A Monte Carlo experment allows a researcher

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

7 ANALYSIS OF VARIANCE (ANOVA)

7 ANALYSIS OF VARIANCE (ANOVA) 7 ANALYSIS OF VARIANCE (ANOVA) Chapter 7 Analyss of Varance (Anova) Objectves After studyng ths chapter you should apprecate the need for analysng data from more than two samples; understand the underlyng

More information

Multivariate EWMA Control Chart

Multivariate EWMA Control Chart Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

Exponentially Weighted Moving Average (EWMA) Charts

Exponentially Weighted Moving Average (EWMA) Charts Chapter 249 Exponentally Weghted Movng Average (EWMA) Charts Introducton Ths procedure generates exponentally weghted movng average (EWMA) control charts for varables. Charts for the mean and for the varablty

More information

Chapter 7 Random-Number Generation

Chapter 7 Random-Number Generation Chapter 7 Random-Number Generaton Banks, Carson, Nelson & Ncol Dscrete-Event System Smulaton Purpose & Overvew Dscuss the generaton of random numbers. Introduce the subsequent testng for randomness: Frequency

More information