DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

 To view this video please enable JavaScript, and consider upgrading to a web browser that supports HTML5 video
Save this PDF as:

Size: px
Start display at page:

Download "DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9"

Transcription

1 DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression

2 So far in this course, we have looked at statistical models for continuous response (or dependent) variables). We have dealt with: Response variable continuous Explanatory variables 1 continuous Method Regression continuous 1 categorical continuous 2 categorical dummy variable regression / ANOVA table independent samples t-test Dummy variable regression/ ANOVA table. Test of interaction posible 2

3 We have used similar techniques in each of these cases. We have fitted a model with one or more explanatory variables, looked at their parameter estimates and used the t-values to determine whether terms are significant. We continue with continuous covariates, but now look at the situation where there is one categorical and one continuous variable. This type of model has a special name analysis of covariance. We will continue to develop the idea of an interaction term, introducing it into the model and testing its significance. We will, however, be using the same type of approach. 3

4 An example of ANCOVA analysis of covariance BLOCK DESIGNS A sample of 24 children was randomly chosen from the 5 th grade of a state primary school in Sydney. Each child was assigned to one of two experimental groups. The children had to complete four of the 3x3 squared designs in the block design subtest of the Weschler Intelligence Scale for Children (WISC). Children in the first group were told to start with a row of three blocks (the ROW group), and children in the second group were told to start with a corner of three blocks (the CORNER group). The total time to complete the task was then measured. Before the experiment began, the extent of each child s field dependence was measured by using the Embedded Figures Test (EFT), which measures the extent to which subjects can abstract the logical structure of a problem from its context. High scores correspond to high field dependence. 1 We therefore have one continuous explanatory variable EFT, and one categorical variable GROUP. An analysis with mixed types of independent variables is sometimes known as Analysis of Covaraince (ANCOVA) 1 The data is reported in Aitkin,M. Anderson, D., Francis, B. and Hinde, J.(1987) Statistical modelling in GLIM, Oxford University Press 4

5 Row group: time: eft: Corner group: time: eft: The data is stored in blockdesign.dat We read in the data to the dataframe block. block=read.table( blockdesign.dat, header=t) block\$groupf=factor(block\$group) We define two subsets of data a subset for the row group (group==0) and a subset for the corner group (group==1) row=block[block.group==0, ] corner=block[block.group==1, ] The object block can be thought of as a matrix we are selecting all rows for which group=1, and selecting all columns. names(row); summary(row) We can first plot the data. 5

6 plot(block\$eft,block\$time,type="n", xlab="eft score", ylab="time to complete") points(row\$eft,row\$time,pch=1,col=2) points(corner\$eft,corner\$time,pch=2,col=4) legend(20, 750, c("row group","corner group"), pch=c(1,2), col=c(2,4)) time to complete row group corner group We can see that there appears to be a strong relationship between TIME and EFT, and perhaps the corner group is taking less time. We explore this using statistical modelling. We can consider a number of different models. EFT score 6

7 If we fit a model with no explanatory variables, then we have a null model this says that there is no relationship between the time taken and either EFT or the experimental GROUP. m1=lm(block\$time~1) anova(m1) Analysis of Variance Table Response: block\$time Df Sum Sq Mean Sq F value Pr(>F) Residuals > summary(m1) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-13 *** The model is ESTIMATED TIME=384.3 The t-test on the estimate of the intercept is simply indicating whether there is evidence that the intercept estimate is different from zero. 7

8 lines(block\$eft, m1\$fitted.values) title("null model - mean only") null model - mean only time to complete row group corner group Sums of squares = Degrees of freedom = EFT score 8

9 We can then add either EFT or GROUP as explanatory variables. Suppose we add EFT first of all. The model is then EFT there is no group effect, but there is a linear relationship between TIME and EFT. m2=lm(time~ eft, data=block) anova(m2) Response: time Df Sum Sq Mean Sq F value Pr(>F) eft ** Residuals > summary(m2) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-06 *** eft ** The model now becomes ESTIMATED TIME= EFT Thus, for every unit increase of EFT, the TIME to complete the task increases by 2 seconds. Under this mode, the experimental group makes no difference. 9

10 repeating earlier graphics commands, then lines(block\$eft, m2\$fitted.values) title("model: EFT") model: EFT time to complete row group corner group SS= df= EFT score 10

11 We can then add in the group effect. This main effects model is the following model: TIME= B0 +B1 EFT+B2 GROUP If GROUP is defined to be GROUP=0 (Row group) GROUP=1 (Corner group) then this model becomes: TIME=B0+ B1 EFT (Row group) TIME=B0+B2 + B1 EFT (Corner group) The estimate of the group effect B2 is simply the difference between the fitted line for the row group and the fitted line for the corner group. m3=lm(time~ eft+groupf, data=block) anova(m3) Response: time Df Sum Sq Mean Sq F value Pr(>F) eft ** groupf Residuals

12 summary(m3) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) e-06 *** eft ** groupf The estimate of the GROUP effect is the corner group is estimated to be performing over 44 seconds faster than the row group. However, the GROUP effect is not significant. The slope of the two lines are the same. The estimates are ESTIMATED TIME = EFT (Row group) = EFT (Corner group) 12

13 repeating earlier graphics commands, then lines( row\$eft,m3\$fitted.values[block\$groupf==0],col=2, lty=1) lines(corner\$eft,m3\$fitted.values[block\$groupf==1],col=4, lty=2) title("model: eft+group") SS= df=21 The fitted lines of this model are parallel. However, the slope in the row group might be different from the slope of the corner group. 13

14 As with the 2-way ANOVA model last week, this model is known as an interaction model. We want to specify an interaction between groupf and eft- this is saying that the effect of groupf depends on the level of eft. This is a two-way interaction- an interaction between two terms in the model. More complex interaction terms can be fitted. We fit the model using m4=lm(time~ eft+groupf+eft:groupf, data=block) anova(m4) Response: time Df Sum Sq Mean Sq F value Pr(>F) eft ** groupf eft:groupf Residuals

15 summary(m4) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) ** eft ** groupf eft:groupf We now see the fitted model is ESTIMATED TIME = EFT (Row group) = EFT (Corner group) The slope of the corner group regression line is estimated to be the group also has a higher intercept. The regression lines show an increasing difference between the estimated times as EFT increases. However, this difference is not statistically significant. We display the plot, as before 15

16 lines( row\$eft,m4\$fitted.values[block\$groupf==0],col=2, lty=1) lines(corner\$eft,m4\$fitted.values[block\$groupf==1],col=4, lty=2) title("model: eft+group+eft:group") model: eft+group+eft:group time to complete row group corner group SS= df= EFT score 16

17 Model simplification There are two strategies. AIC method. We have already seen this method. We can construct a table of AIC values for each model and choose the model with the lowest value. model AIC value eft eft+group eft+group+eft:group From this analysis we choose the model eft as being the best. However, the interaction model has an AIC value which is very close to the minimum and might be an alternative. 17

18 Classic backward elimination: When we examine the significance level of the interaction term, we find that the difference in slopes of is not statistically significant from zero (p=0.13). We reject this model in favour of the main effects model already discussed. The main effects model can also be simplified as there is no evidence that the group effect is significantly different from zero (p=0.33). Our final model is one where there is no group difference and with eft the only single predictor of time. In this example, the two methods give the same result. In general, the AIC method is less parsimonious, and more likely to retain terms in the model. Backward elimination at the 5% level (p=0.05) will be more likely to exclude terms and is a more conservative method. 18

19 Constructing an Analysis of Variance table. We can construct an analysis of variance table by examining the Sums of Squares given for each model, and differencing them.. We start at the most complex model, and use the sums of squares form this model as our estimate of residual variation. Thus the residual sums of squares is on 20 df. We now work backwards, removing the most complex terms first.. So the sums of squares due to the interaction is on degrees of freedom. Sums of squares due to SS df SS df EFT GROUP given EFT INTERACTION Sum of Squares df Mean F-value p-value square EFT GROUP given EFT Interaction Residual

20 Multiple linear regression In simple linear regression, we attempt to explain the dependent variable in terms of an intercept and a single independent variable which is continuous. Multiple linear regression extends these techniques to allow for several independent continuous variables. Least squares is again used to determine the unstandardised coefficients (the Bs) in the more general equation: Dependent = Intercept + B1 x Independent + B2 x Independent +... Variable (or B0) Variable 1 Variable 2 20

21 Example: professor s salary data University students often complain that universities reward professors for research but not for teaching, and argue that professors react to this situation by devoting more time and energy to research publications and less time and energy to classroom activities. Professors counter that research and teaching go hand in hand; more research makes better teachers. A US student organization decided to investigate part of the issue. They randomly selected 50 psychology professors in their area. The students recorded the salaries of the professors, their average teaching evaluations, and the total number of articles published in their careers. salary salary (in 1000\$) evalulation average teaching evaluation (10 point scale) articles number of articles published All variables can be treated as continuous ( although there is an argument for categorising evaluation what is it?) We therefore use multiple regression to investigate the relationship between SALARY, teaching quality and research output. 21

22 We look at the correlation between the three variables: profs=read.table( profs.dat,header=t) cor(profs) salary evaluation articles salary evaluation articles There is a strong correlation between every pair of variables. Research is associated with good teaching, and both are correlated with salaries. We could plot each variable against each other to obtain a graphical picture of each relationship pair. We fit a linear model to investigate further. m1=lm(salary~evaluation+articles,data=profs) summary(lm) 22

23 Estimate Std. Error t value Pr(> t ) (Intercept) e-07 *** evaluation articles e-10 *** We look at the first part of the output. This is familiar. We see that the variable evaluation (once articles is in the model) is not significant. However the variable articles is significant. This suggests that we can remove articles from the model. The number of articles appears to determine salary, but the teaching evaluation score does not provide an additional component. Salary is not determined by teaching quality. We look to exclude evaluation from the model But first we look at the other part of the output. 23

24 Residual standard error: on 47 degrees of freedom Multiple R-Squared: , Adjusted R-squared: F-statistic: 60.7 on 2 and 47 DF, p-value: 9.453e-14 The Multiple R-squared is a measure of the proportion of variance explained by the model. We can find from anova calculations that the sums of squares from the null ( constant mean) model is sums of squares from the articles+evaluation model is the proportion of variance explained is 1 (2308.8/8272.0) = The adjusted R-squared provides an unbiased estimate of what the proportion of variance explained might be in the population data. 24

25 > m2=lm(salary~articles,data=profs) > summary(m2) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** articles e-15 *** Multiple R-Squared: , Adjusted R-squared: The estimate for articles remains very close to the value in the previous output. The multiple R-squared and adjusted r-squared are close to the previous values. Our final model is estimated salary = x (numberof articles) How well does this model fit? 25

26 plot(profs\$articles, profs\$salary, xlab="number of articles", ylab="salary") lines(profs\$articles, m2\$fitted.values) salary The straight line seems to fit the data well, and the observed points are well scattered around the line. The next lecture wil look at diagnostics for regression models and how to test the assumptions of the linear model more carefully. The model says that every increase in one published article is associated with a estimated salary increase of \$1,112 dollars number of articles 26

27 Assumptions of linear regression. a) effects are linear each covariate has a linear rather than a non-linear relationship to the dependent variable. b) The residuals have a Normal distribution. c) There is constant variance. The variance does not depend on the covariates or on the mean. d) The observations are independent. Important this is part of the design. In the experiment above, the observations would not be independent if one child copied from another. 27

28 Using SPSS In SPSS, we use the SPSS Linear Regression procedure in ENTER mode (all the independent variables are entered into the model) with SALARY as the Dependent Variable and EVALUATI and ARTICLES as Independent Variables, giving the following (edited) summary results: Model 1 Model Summary Adjusted Std. Error of R R Square R Square the Estimate.849 a a. Predictors: (Constant), articles, evaluati Model 1 (Constant) evaluati articles a. Dependent Variable: salary Unstandardized Coefficients Coefficients a Standardized Coefficients B Std. Error Beta t Sig

29 3D plots A simple 3D scatterplot can be carried out by library(lattice) cloud(\$profs\$salary~profs\$articles+profs\$evaluation) More complex interactive plots can be carried out if you have the ability to install software. 29

30 Different types of multiple regression The preceding example showed an example of regression for exploration and prediction. We wanted to explore the data set to build a suitable statistical model for salary from the teaching evaluation and research publications. This is akin to separating out the structure of the data from the noise. The regression analysis tells us which independent (or predictor) variables are not needed for the structure- and can therefore be considered as part of the noise. However, another reason to carry out multiple regression is to CONTROL for the effect of other variables. We may be interested in the association between levels of a hormone and an aggression score. However, our data is observational, not experimental. Observational data means that we survey people and then measuring both the hormone level and the aggression score. Thus we cannot randomly assign individuals to low or high hormone levels our data will be unbalanced and will depend on who we survey. Additionally, we might imagine that other variables affect aggression score in particular age and gender. We need to control for the effect of these variables in our analysis. 30

Multiple Linear Regression

Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is

The scatterplot indicates a positive linear relationship between waist size and body fat percentage:

STAT E-150 Statistical Methods Multiple Regression Three percent of a man's body is essential fat, which is necessary for a healthy body. However, too much body fat can be dangerous. For men between the

SPSS Guide: Regression Analysis

SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar

Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

Regression in ANOVA. James H. Steiger. Department of Psychology and Human Development Vanderbilt University

Regression in ANOVA James H. Steiger Department of Psychology and Human Development Vanderbilt University James H. Steiger (Vanderbilt University) 1 / 30 Regression in ANOVA 1 Introduction 2 Basic Linear

Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear.

Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.

12/31/2016. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2

PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Understand linear regression with a single predictor Understand how we assess the fit of a regression model Total Sum of Squares

Simple Linear Regression One Binary Categorical Independent Variable

Simple Linear Regression Does sex influence mean GCSE score? In order to answer the question posed above, we want to run a linear regression of sgcseptsnew against sgender, which is a binary categorical

12-1 Multiple Linear Regression Models

12-1.1 Introduction Many applications of regression analysis involve situations in which there are more than one regressor variable. A regression model that contains more than one regressor variable is

1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

, then the form of the model is given by: which comprises a deterministic component involving the three regression coefficients (

Multiple regression Introduction Multiple regression is a logical extension of the principles of simple linear regression to situations in which there are several predictor variables. For instance if we

Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

Part 2: Analysis of Relationship Between Two Variables

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

Testing for Lack of Fit

Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit

Practice 3 SPSS. Partially based on Notes from the University of Reading:

Practice 3 SPSS Partially based on Notes from the University of Reading: http://www.reading.ac.uk Simple Linear Regression A simple linear regression model is fitted when you want to investigate whether

Psychology 205: Research Methods in Psychology

Psychology 205: Research Methods in Psychology Using R to analyze the data for study 2 Department of Psychology Northwestern University Evanston, Illinois USA November, 2012 1 / 38 Outline 1 Getting ready

Chapter Seven. Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS

Chapter Seven Multiple regression An introduction to multiple regression Performing a multiple regression on SPSS Section : An introduction to multiple regression WHAT IS MULTIPLE REGRESSION? Multiple

Bivariate Analysis. Correlation. Correlation. Pearson's Correlation Coefficient. Variable 1. Variable 2

Bivariate Analysis Variable 2 LEVELS >2 LEVELS COTIUOUS Correlation Used when you measure two continuous variables. Variable 2 2 LEVELS X 2 >2 LEVELS X 2 COTIUOUS t-test X 2 X 2 AOVA (F-test) t-test AOVA

0.1 Multiple Regression Models

0.1 Multiple Regression Models We will introduce the multiple Regression model as a mean of relating one numerical response variable y to two or more independent (or predictor variables. We will see different

Linear Models in STATA and ANOVA

Session 4 Linear Models in STATA and ANOVA Page Strengths of Linear Relationships 4-2 A Note on Non-Linear Relationships 4-4 Multiple Linear Regression 4-5 Removal of Variables 4-8 Independent Samples

E(y i ) = x T i β. yield of the refined product as a percentage of crude specific gravity vapour pressure ASTM 10% point ASTM end point in degrees F

Random and Mixed Effects Models (Ch. 10) Random effects models are very useful when the observations are sampled in a highly structured way. The basic idea is that the error associated with any linear,

MULTIPLE REGRESSION WITH CATEGORICAL DATA

DEPARTMENT OF POLITICAL SCIENCE AND INTERNATIONAL RELATIONS Posc/Uapp 86 MULTIPLE REGRESSION WITH CATEGORICAL DATA I. AGENDA: A. Multiple regression with categorical variables. Coding schemes. Interpreting

1/27/2013. PSY 512: Advanced Statistics for Psychological and Behavioral Research 2

PSY 512: Advanced Statistics for Psychological and Behavioral Research 2 Introduce moderated multiple regression Continuous predictor continuous predictor Continuous predictor categorical predictor Understand

Correlation and Simple Linear Regression

Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a

Comparing Nested Models

Comparing Nested Models ST 430/514 Two models are nested if one model contains all the terms of the other, and at least one additional term. The larger model is the complete (or full) model, and the smaller

NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates

ANOVA. February 12, 2015

ANOVA February 12, 2015 1 ANOVA models Last time, we discussed the use of categorical variables in multivariate regression. Often, these are encoded as indicator columns in the design matrix. In [1]: %%R

Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

SAS Software to Fit the Generalized Linear Model

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

Simple Linear Regression in SPSS STAT 314

Simple Linear Regression in SPSS STAT 314 1. Ten Corvettes between 1 and 6 years old were randomly selected from last year s sales records in Virginia Beach, Virginia. The following data were obtained,

Module 5: Multiple Regression Analysis

Using Statistical Data Using to Make Statistical Decisions: Data Multiple to Make Regression Decisions Analysis Page 1 Module 5: Multiple Regression Analysis Tom Ilvento, University of Delaware, College

Elementary Statistics Sample Exam #3

Elementary Statistics Sample Exam #3 Instructions. No books or telephones. Only the supplied calculators are allowed. The exam is worth 100 points. 1. A chi square goodness of fit test is considered to

Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

ANNOTATED OUTPUT--SPSS Simple Linear (OLS) Regression

Simple Linear (OLS) Regression Regression is a method for studying the relationship of a dependent variable and one or more independent variables. Simple Linear Regression tells you the amount of variance

Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

Regression III: Dummy Variable Regression

Regression III: Dummy Variable Regression Tom Ilvento FREC 408 Linear Regression Assumptions about the error term Mean of Probability Distribution of the Error term is zero Probability Distribution of

Regression in SPSS. Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology

Regression in SPSS Workshop offered by the Mississippi Center for Supercomputing Research and the UM Office of Information Technology John P. Bentley Department of Pharmacy Administration University of

SPSS: Descriptive and Inferential Statistics. For Windows

For Windows August 2012 Table of Contents Section 1: Summarizing Data...3 1.1 Descriptive Statistics...3 Section 2: Inferential Statistics... 10 2.1 Chi-Square Test... 10 2.2 T tests... 11 2.3 Correlation...

Residuals. Residuals = ª Department of ISM, University of Alabama, ST 260, M23 Residuals & Minitab. ^ e i = y i - y i

A continuation of regression analysis Lesson Objectives Continue to build on regression analysis. Learn how residual plots help identify problems with the analysis. M23-1 M23-2 Example 1: continued Case

Week 5: Multiple Linear Regression

BUS41100 Applied Regression Analysis Week 5: Multiple Linear Regression Parameter estimation and inference, forecasting, diagnostics, dummy variables Robert B. Gramacy The University of Chicago Booth School

CHAPTER 2 AND 10: Least Squares Regression

CHAPTER 2 AND 0: Least Squares Regression In chapter 2 and 0 we will be looking at the relationship between two quantitative variables measured on the same individual. General Procedure:. Make a scatterplot

Stat 411/511 ANOVA & REGRESSION. Charlotte Wickham. stat511.cwick.co.nz. Nov 31st 2015

Stat 411/511 ANOVA & REGRESSION Nov 31st 2015 Charlotte Wickham stat511.cwick.co.nz This week Today: Lack of fit F-test Weds: Review email me topics, otherwise I ll go over some of last year s final exam

Linear Regression in SPSS

Linear Regression in SPSS Data: mangunkill.sav Goals: Examine relation between number of handguns registered (nhandgun) and number of man killed (mankill) checking Predict number of man killed using number

Multiple Linear Regression

Multiple Linear Regression Simple Linear Regression Regression equation for a line (population): y = β 0 + β 1 x + β 0 : point where the line intercepts y-axis β 1 : slope of the line : error in estimating

5. Linear Regression

5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

Chapter 11: Two Variable Regression Analysis

Department of Mathematics Izmir University of Economics Week 14-15 2014-2015 In this chapter, we will focus on linear models and extend our analysis to relationships between variables, the definitions

Multiple Regression. Page 24

Multiple Regression Multiple regression is an extension of simple (bi-variate) regression. The goal of multiple regression is to enable a researcher to assess the relationship between a dependent (predicted)

Final Exam Practice Problem Answers The following data set consists of data gathered from 77 popular breakfast cereals. The variables in the data set are as follows: Brand: The brand name of the cereal

SELF-TEST: SIMPLE REGRESSION

ECO 22000 McRAE SELF-TEST: SIMPLE REGRESSION Note: Those questions indicated with an (N) are unlikely to appear in this form on an in-class examination, but you should be able to describe the procedures

Regression. Name: Class: Date: Multiple Choice Identify the choice that best completes the statement or answers the question.

Class: Date: Regression Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Given the least squares regression line y8 = 5 2x: a. the relationship between

The general form of the PROC GLM statement is

Linear Regression Analysis using PROC GLM Regression analysis is a statistical method of obtaining an equation that represents a linear relationship between two variables (simple linear regression), or

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 > catch

Multiple Regression in SPSS STAT 314

Multiple Regression in SPSS STAT 314 I. The accompanying data is on y = profit margin of savings and loan companies in a given year, x 1 = net revenues in that year, and x 2 = number of savings and loan

Using R for Linear Regression

Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional

7. Tests of association and Linear Regression

7. Tests of association and Linear Regression In this chapter we consider 1. Tests of Association for 2 qualitative variables. 2. Measures of the strength of linear association between 2 quantitative variables.

2SLS HATCO SPSS and SHAZAM Example. by Eddie Oczkowski. August X9: Usage Level (how much of the firm s total product is purchased from HATCO).

2SLS HATCO SPSS and SHAZAM Example by Eddie Oczkowski August 200 This example illustrates how to use SPSS to estimate and evaluate a 2SLS latent variable model. The bulk of the example relates to SPSS,

Chapter 23. Inferences for Regression

Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

An analysis method for a quantitative outcome and two categorical explanatory variables.

Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ

STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material

E205 Final: Version B

Name: Class: Date: E205 Final: Version B Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The owner of a local nightclub has recently surveyed a random

ID X Y

Dale Berger SPSS Step-by-Step Regression Introduction: MRC01 This step-by-step example shows how to enter data into SPSS and conduct a simple regression analysis to develop an equation to predict from.

e = random error, assumed to be normally distributed with mean 0 and standard deviation σ

1 Linear Regression 1.1 Simple Linear Regression Model The linear regression model is applied if we want to model a numeric response variable and its dependency on at least one numeric factor variable.

where b is the slope of the line and a is the intercept i.e. where the line cuts the y axis.

Least Squares Introduction We have mentioned that one should not always conclude that because two variables are correlated that one variable is causing the other to behave a certain way. However, sometimes

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

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

Statistical Modelling in Stata 5: Linear Models

Statistical Modelling in Stata 5: Linear Models Mark Lunt Arthritis Research UK Centre for Excellence in Epidemiology University of Manchester 08/11/2016 Structure This Week What is a linear model? How

Chapter 10. Analysis of Covariance. 10.1 Multiple regression

Chapter 10 Analysis of Covariance An analysis procedure for looking at group effects on a continuous outcome when some other continuous explanatory variable also has an effect on the outcome. This chapter

Regression step-by-step using Microsoft Excel

Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression

5. Multiple regression

5. Multiple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/5 QBUS6840 Predictive Analytics 5. Multiple regression 2/39 Outline Introduction to multiple linear regression Some useful

STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used

Homework 11. Part 1. Name: Score: / null

Name: Score: / Homework 11 Part 1 null 1 For which of the following correlations would the data points be clustered most closely around a straight line? A. r = 0.50 B. r = -0.80 C. r = 0.10 D. There is

10. Analysis of Longitudinal Studies Repeat-measures analysis

Research Methods II 99 10. Analysis of Longitudinal Studies Repeat-measures analysis This chapter builds on the concepts and methods described in Chapters 7 and 8 of Mother and Child Health: Research methods.

KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

Simple Linear Regression

Inference for Regression Simple Linear Regression IPS Chapter 10.1 2009 W.H. Freeman and Company Objectives (IPS Chapter 10.1) Simple linear regression Statistical model for linear regression Estimating

Factors affecting online sales

Factors affecting online sales Table of contents Summary... 1 Research questions... 1 The dataset... 2 Descriptive statistics: The exploratory stage... 3 Confidence intervals... 4 Hypothesis tests... 4

Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1. 1. Introduction p. 2. 2. Statistical Methods Used p. 5. 3. 10 and under Males p.

Sydney Roberts Predicting Age Group Swimmers 50 Freestyle Time 1 Table of Contents 1. Introduction p. 2 2. Statistical Methods Used p. 5 3. 10 and under Males p. 8 4. 11 and up Males p. 10 5. 10 and under

psyc3010 lecture 8 standard and hierarchical multiple regression last week: correlation and regression Next week: moderated regression

psyc3010 lecture 8 standard and hierarchical multiple regression last week: correlation and regression Next week: moderated regression 1 last week this week last week we revised correlation & regression

EPS 625 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM

EPS 6 ANALYSIS OF COVARIANCE (ANCOVA) EXAMPLE USING THE GENERAL LINEAR MODEL PROGRAM ANCOVA One Continuous Dependent Variable (DVD Rating) Interest Rating in DVD One Categorical/Discrete Independent Variable

Simple Linear Regression, Scatterplots, and Bivariate Correlation

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

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION

HYPOTHESIS TESTING: CONFIDENCE INTERVALS, T-TESTS, ANOVAS, AND REGRESSION HOD 2990 10 November 2010 Lecture Background This is a lightning speed summary of introductory statistical methods for senior undergraduate

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

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

We extended the additive model in two variables to the interaction model by adding a third term to the equation.

Quadratic Models We extended the additive model in two variables to the interaction model by adding a third term to the equation. Similarly, we can extend the linear model in one variable to the quadratic

Using SPSS for Multiple Regression. UDP 520 Lab 7 Lin Lin December 4 th, 2007

Using SPSS for Multiple Regression UDP 520 Lab 7 Lin Lin December 4 th, 2007 Step 1 Define Research Question What factors are associated with BMI? Predict BMI. Step 2 Conceptualizing Problem (Theory) Individual

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association

Premaster Statistics Tutorial 4 Full solutions

Premaster Statistics Tutorial 4 Full solutions Regression analysis Q1 (based on Doane & Seward, 4/E, 12.7) a. Interpret the slope of the fitted regression = 125,000 + 150. b. What is the prediction for

Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

Stat 412/512 CASE INFLUENCE STATISTICS. Charlotte Wickham. stat512.cwick.co.nz. Feb 2 2015

Stat 412/512 CASE INFLUENCE STATISTICS Feb 2 2015 Charlotte Wickham stat512.cwick.co.nz Regression in your field See website. You may complete this assignment in pairs. Find a journal article in your field

Interpreting Multiple Regression

Fall Semester, 2001 Statistics 621 Lecture 5 Robert Stine 1 Preliminaries Interpreting Multiple Regression Project and assignments Hope to have some further information on project soon. Due date for Assignment

CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur

Regression Analysis Prof. Soumen Maity Department of Mathematics Indian Institute of Technology, Kharagpur Lecture - 7 Multiple Linear Regression (Contd.) This is my second lecture on Multiple Linear Regression

Simple Linear Regression Chapter 11

Simple Linear Regression Chapter 11 Rationale Frequently decision-making situations require modeling of relationships among business variables. For instance, the amount of sale of a product may be related

1. ε is normally distributed with a mean of 0 2. the variance, σ 2, is constant 3. All pairs of error terms are uncorrelated

STAT E-150 Statistical Methods Residual Analysis; Data Transformations The validity of the inference methods (hypothesis testing, confidence intervals, and prediction intervals) depends on the error term,

Using Minitab for Regression Analysis: An extended example

Using Minitab for Regression Analysis: An extended example The following example uses data from another text on fertilizer application and crop yield, and is intended to show how Minitab can be used to

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups

One-Way Analysis of Variance: A Guide to Testing Differences Between Multiple Groups In analysis of variance, the main research question is whether the sample means are from different populations. The

Predictor Coef StDev T P Constant 970667056 616256122 1.58 0.154 X 0.00293 0.06163 0.05 0.963. S = 0.5597 R-Sq = 0.0% R-Sq(adj) = 0.

Statistical analysis using Microsoft Excel Microsoft Excel spreadsheets have become somewhat of a standard for data storage, at least for smaller data sets. This, along with the program often being packaged

APPLICATION OF LINEAR REGRESSION MODEL FOR POISSON DISTRIBUTION IN FORECASTING

APPLICATION OF LINEAR REGRESSION MODEL FOR POISSON DISTRIBUTION IN FORECASTING Sulaimon Mutiu O. Department of Statistics & Mathematics Moshood Abiola Polytechnic, Abeokuta, Ogun State, Nigeria. Abstract

For example, enter the following data in three COLUMNS in a new View window.

Statistics with Statview - 18 Paired t-test A paired t-test compares two groups of measurements when the data in the two groups are in some way paired between the groups (e.g., before and after on the

Main Effects and Interactions

Main Effects & Interactions page 1 Main Effects and Interactions So far, we ve talked about studies in which there is just one independent variable, such as violence of television program. You might randomly