# Module 3: Multiple Regression Concepts

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Contents Module 3: Multiple Regression Concepts Fiona Steele 1 Centre for Multilevel Modelling...4 What is Multiple Regression?... 4 Motivation... 4 Conditioning... 4 Data for multiple regression analysis... 5 to Dataset C3.1.1 Examining data graphically... 7 C3.1.2 The linear regression model... 9 C3.1.3 C3.1.4 The fitted regression line Explained and unexplained variance and R-squared C3.1.5 Hypothesis testing C3.1.6 Model checking C3.2 Comparing Groups: Regression with a Single Categorical Explanatory Variable C3.2.1 Comparing two groups C3.2.2 Comparing more than two groups C3.2.3 Comparing a large number of groups C3.3 Regression with More than One Explanatory Variable (Multiple Regression) C3.3.1 Statistical control C3.3.2 The multiple regression model C3.3.3 C3.3.4 Using multiple regression to model a non-linear relationship Adding further predictors C3.4 Interaction Effects C3.4.1 Model with fixed slopes across groups C3.4.2 Fitting separate models for each group C3.4.3 Allowing for varying slopes in a pooled analysis: interaction effects C3.4.4 C3.4.5 Testing for interaction effects Another example: allowing age effects to be different in different countries C3.5 Checking Model Assumptions in Multiple Regression C3.5.1 Checking the normality assumption C3.5.2 C3.5.3 Checking the homoskedasticity assumption Outliers All of the sections within this module have online quizzes for you to test your understanding. To find the quizzes: EXAMPLE From within the LEMMA learning environment Go down to the section for Module 3: Multilevel Modelling Click "3.1 Regression with a Single Continuous Explanatory Variable" to open Lesson 3.1 Click Q 1 to open the first question All of the sections within this module have practicals so you can learn how to perform this kind of analysis in MLwiN or other software packages. To find the practicals: EXAMPLE From within the LEMMA learning environment Go down to the section for Module 3: Multiple Regression, then Either Click "3.1 Regression with a Single Continuous Explanatory Variable" to open Lesson 3.1 Click Or Click Print all Module 3 MLwiN Practicals 1 With additional material from Kelvyn Jones. Comments from Sacha Brostoff, Jon Rasbash and Rebecca Pillinger on an earlier draft are gratefully acknowledged. 1 2

2 Pre-requisites Understanding of types of variables (continuous vs. categorical variables, dependent and explanatory); covered in Module 1. Correlation between variables Confidence intervals around estimates Hypothesis testing, p-values Independent samples t-test for comparing the means of two groups Online resources: What is Multiple Regression? Multiple regression is a technique used to study the relationship between an outcome variable and a set of explanatory or predictor variables. Motivation To illustrate the ideas of multiple regression, we will consider a research problem of assessing the evidence for gender discrimination in legal firms. Statistical modelling can provide the following: A quantitative assessment of the size of the effect; e.g. the difference in salary between women and men is 5000 per annum; A quantitative assessment after taking account of other variables; e.g. a female worker earns 6500 less after taking account of years of experience. This conditioning on other variables distinguishes multiple regression modelling from simple testing for differences analyses. A measure of uncertainty for the size of the effect; e.g. we can be 95% confident that the female-male difference in salary in the population from which our sample was drawn is likely to lie between 4500 and We can use regression modelling in different modes: 1) as description (what is the average salary for men and women?), 2) as part of causal inference (does being female result in a lower salary?), and 3) for prediction ( what happens if questions). Conditioning The key feature that distinguishes multiple regression from simple regression is that more than one predictor variable is involved. Even if we are interested in the effect of just one variable (gender) on another (salary) we need to take account of other variables as they may compromise the results. We can recognise three distinct cases where it is important to control or adjust for the effects of other variables: i) Inflation of a relationship when not taking into account extraneous variables. For example, a substantial gender effect could be reduced after taking account of type of employment. This is because jobs that are characterized by poor pay (e.g. in the service sector) have a predominantly female labour force. ii) Suppression of a relationship. An apparent small gender gap could increase when account is taken of years of employment; women having longer service and poorer pay. 3 4

3 iii) No confounding. The original relationship remains substantially unaltered when account is taken of other variables. Note, however, that there may be unmeasured confounders. Data for multiple regression analysis Statistical analysis requires a quantifiable outcome measure (dependent variable) to assess the effects of discrimination. Possibilities include the following, differentiated by the nature of the measurement: a continuous measure of salary, a binary indicator of whether an employee was promoted or not, a three-category indicator of promotion (promoted, not promoted, not even considered), a count of the number of times rejected for promotion, the length of time that it has taken to gain promotion. All of these outcomes can be analysed using regression analysis, but different techniques are required for different scales of measurement. The term multiple regression is usually applied when the dependent variable is measured on a continuous scale. A dichotomous dependent variable can be analysed using logistic regression and multinomial logistic and ordinal regression can be applied to nominal and ordinal dependent variables respectively. There are also methods for handling counts (Poisson regression) and time-to-event data (event history analysis or survival analysis). These techniques will be described in later Modules. The explanatory variables may also have different scales of measurement. For example, gender is a binary categorical variable; ethnicity is categorical with more than two categories; education might be measured on an ordinal scale (e.g. <11, 11-13, and >16 years of education); years of employment could be measured on a continuous scale. Multiple regression can handle all of these types of explanatory variable, and we will consider examples of both continuous and categorical variables in this Module. to Dataset The ideas of multiple regression will be introduced using data from the 2002 European Social Surveys (ESS). Measures of ten human values have been constructed for 20 countries in the European Union. According to value theory, values are defined as desirable, trans-situational goals that serve as guiding principles in people s lives. Further details on value theory and how it is operationalised in the ESS can be found on the ESS education net ( We will study one of the ten values, hedonism, defined as the pleasure and sensuous gratification for oneself. The measure we use is based on responses to the question How much like you is this person? : He (sic) seeks every chance he can to have fun. It is important to him to do things that give him pleasure. Having a good time is important to him. He likes to spoil himself. A respondent s own values are inferred from their self-reported similarity to a person with the above descriptions. Each of the two items is rated on a 6-point scale (from very much like me to not like me at all ). The mean of these ratings is calculated for each individual. The mean of the two hedonism items is then adjusted for individual differences in scale use 2 by subtracting the mean of all value items (a total of 21 are used to measure the 10 values). These centred scores recognise that the 10 values function as a system rather than independently. The centred hedonism score is interpreted as a measure of the relative importance of hedonism to an individual in their whole value system. The scores on the hedonism variable range from to 2.90, where higher scores indicate more hedonistic beliefs. We consider three countries France, Germany and the UK with a total sample size of That is, we use a subsample of the original data. Hedonism is taken as the outcome variable in our analysis. We consider three explanatory variables: Age in years Gender (coded 0 for male and 1 for female) Country (coded 1 for the UK, 2 for Germany and 3 for France) Years of education. An extract of the data is given below. Respondent Hedonism Age Gender Country Education Some individuals will tend to select responses from one side of the scale ( very much like me ) for any item, while others will select from the other side ( not like me at all ). If we ignore these differences in response tendency we might incorrectly infer that the first type of individual believes that all values are important, while the second believes that all values are unimportant. 5 6

4 C3.1 Regression with a Single Continuous Explanatory Variable We will begin with a description of simple linear regression for studying the relationship between a pair of continuous variables, which we denote by Y and X. Simple regression is also commonly known as bivariate regression because only two variables are involved. Y is the outcome variable (also called a response or dependent variable) X is the explanatory variable (also called a predictor or independent variable). C3.1.1 Examining data graphically For a regression analysis the distribution of the explanatory variable is unimportant, but it is sensible to look at descriptive statistics for any variables that we analyse to check for unusual values. We will first consider age as an explanatory variable for hedonism. The age range in our sample is 14 to 98 years with a mean of 46.7 and standard deviation of Relationship between X and Y In its simplest form, a regression analysis assumes that the relationship between X and Y is linear, i.e. that it can be reasonably approximated by a straight line. If the relationship is nonlinear, it may be possible to transform one of the variables to make the relationship linear or the regression model can be modified (see C3.3.3). The relationship between two variables can be viewed in a scatterplot. A scatterplot can also reveal outliers. Before carrying out a regression analysis, it is important to look at your data first. There are various assumptions made when we fit a regression model, which we will consider later, but there are two checks that should always be carried out before fitting any models: i) examine the distribution of the variables and check that the values are all valid, and ii) look at the nature of the relationship between X and Y. Distribution of Y We can examine the distribution of a continuous variable using a histogram. At this stage, we are checking that the values appear reasonable. Are there any outliers, i.e. observations outside the general pattern? Are there any values of - 99 in the data that should be declared as missing values? We also look at the shape of the distribution: is it a symmetrical bell-shaped distribution (normal), or is it skewed? Although it is the residuals 3 that are assumed to be normally distributed in a multiple regression model, rather than the dependent variable, a skewed Y will often produce skewed residuals. If the residuals turn out to be nonnormal, it may be possible to transform Y to obtain a normally distributed variable. For example, a positively skewed distribution (with a long tail to the right) will often look more symmetrical after taking logarithms. Figure 3.1 shows the distribution of the hedonism scores. It appears approximately normal with no obvious outliers. The mean of the hedonism score is and the standard deviation is Distribution of X Figure 3.1. Histogram of hedonism Figure 3.2 shows a scatterplot of hedonism versus age, where the size of the plotting symbol is proportional to the number of respondents represented by a particular data point. Also shown is what is commonly called the line of best fit, which we will come back to in a moment. The scatterplot shows a negative relationship: as age increases then hedonism decreases. The Pearson correlation coefficient for the linear relationship is The residual for each observation is the difference between the observed value of Y and the value of Y predicted by the model. See C3.1.2 for further details. 7 8

5 This document is only the first few pages of the full version. To see the complete document please go to learning materials and register: The course is completely free. We ask for a few details about yourself for our research purposes only. We will not give any details to any other organisation unless it is with your express permission.

### Module 2: Introduction to Quantitative Data Analysis

Module 2: Introduction to Quantitative Data Analysis Contents Antony Fielding 1 University of Birmingham & Centre for Multilevel Modelling Rebecca Pillinger Centre for Multilevel Modelling Introduction...

### Class 6: Chapter 12. Key Ideas. Explanatory Design. Correlational Designs

Class 6: Chapter 12 Correlational Designs l 1 Key Ideas Explanatory and predictor designs Characteristics of correlational research Scatterplots and calculating associations Steps in conducting a correlational

### 11/20/2014. Correlational research is used to describe the relationship between two or more naturally occurring variables.

Correlational research is used to describe the relationship between two or more naturally occurring variables. Is age related to political conservativism? Are highly extraverted people less afraid of rejection

### Fairfield Public Schools

Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

### Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

### 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) -

### Statistics 2014 Scoring Guidelines

AP Statistics 2014 Scoring Guidelines College Board, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks of the College Board. AP Central is the official online home

### Semester 1 Statistics Short courses

Semester 1 Statistics Short courses Course: STAA0001 Basic Statistics Blackboard Site: STAA0001 Dates: Sat. March 12 th and Sat. April 30 th (9 am 5 pm) Assumed Knowledge: None Course Description Statistical

### 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

### If the Shoe Fits! Overview of Lesson GAISE Components Common Core State Standards for Mathematical Practice

If the Shoe Fits! Overview of Lesson In this activity, students explore and use hypothetical data collected on student shoe print lengths, height, and gender in order to help develop a tentative description

### II. DISTRIBUTIONS distribution normal distribution. standard scores

Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

### Organizing Your Approach to a Data Analysis

Biost/Stat 578 B: Data Analysis Emerson, September 29, 2003 Handout #1 Organizing Your Approach to a Data Analysis The general theme should be to maximize thinking about the data analysis and to minimize

### Outline of Topics. Statistical Methods I. Types of Data. Descriptive Statistics

Statistical Methods I Tamekia L. Jones, Ph.D. (tjones@cog.ufl.edu) Research Assistant Professor Children s Oncology Group Statistics & Data Center Department of Biostatistics Colleges of Medicine and Public

### Inferential Statistics

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

### Chapter 8. Linear Regression. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 8 Linear Regression Copyright 2012, 2008, 2005 Pearson Education, Inc. Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King

### Module 2 Project Maths Development Team Draft (Version 2)

5 Week Modular Course in Statistics & Probability Strand 1 Module 2 Analysing Data Numerically Measures of Central Tendency Mean Median Mode Measures of Spread Range Standard Deviation Inter-Quartile Range

### Point-Biserial and Biserial Correlations

Chapter 302 Point-Biserial and Biserial Correlations Introduction This procedure calculates estimates, confidence intervals, and hypothesis tests for both the point-biserial and the biserial correlations.

### 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

### 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

### AP Statistics 2001 Solutions and Scoring Guidelines

AP Statistics 2001 Solutions and Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use

### 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,

### AMS7: WEEK 8. CLASS 1. Correlation Monday May 18th, 2015

AMS7: WEEK 8. CLASS 1 Correlation Monday May 18th, 2015 Type of Data and objectives of the analysis Paired sample data (Bivariate data) Determine whether there is an association between two variables This

### Yiming Peng, Department of Statistics. February 12, 2013

Regression Analysis Using JMP Yiming Peng, Department of Statistics February 12, 2013 2 Presentation and Data http://www.lisa.stat.vt.edu Short Courses Regression Analysis Using JMP Download Data to Desktop

### Final Review Sheet. Mod 2: Distributions for Quantitative Data

Things to Remember from this Module: Final Review Sheet Mod : Distributions for Quantitative Data How to calculate and write sentences to explain the Mean, Median, Mode, IQR, Range, Standard Deviation,

### An example ANOVA situation. 1-Way ANOVA. Some notation for ANOVA. Are these differences significant? Example (Treating Blisters)

An example ANOVA situation Example (Treating Blisters) 1-Way ANOVA MATH 143 Department of Mathematics and Statistics Calvin College Subjects: 25 patients with blisters Treatments: Treatment A, Treatment

### Diagrams and Graphs of Statistical Data

Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in

BIOSTATISTICS QUIZ ANSWERS 1. When you read scientific literature, do you know whether the statistical tests that were used were appropriate and why they were used? a. Always b. Mostly c. Rarely d. Never

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### MTH 140 Statistics Videos

MTH 140 Statistics Videos Chapter 1 Picturing Distributions with Graphs Individuals and Variables Categorical Variables: Pie Charts and Bar Graphs Categorical Variables: Pie Charts and Bar Graphs Quantitative

### DEPARTMENT OF HEALTH AND HUMAN SCIENCES HS900 RESEARCH METHODS

DEPARTMENT OF HEALTH AND HUMAN SCIENCES HS900 RESEARCH METHODS Using SPSS Session 2 Topics addressed today: 1. Recoding data missing values, collapsing categories 2. Making a simple scale 3. Standardisation

### 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

### AP Statistics: Syllabus 3

AP Statistics: Syllabus 3 Scoring Components SC1 The course provides instruction in exploring data. 4 SC2 The course provides instruction in sampling. 5 SC3 The course provides instruction in experimentation.

### List of Examples. Examples 319

Examples 319 List of Examples DiMaggio and Mantle. 6 Weed seeds. 6, 23, 37, 38 Vole reproduction. 7, 24, 37 Wooly bear caterpillar cocoons. 7 Homophone confusion and Alzheimer s disease. 8 Gear tooth strength.

### , 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

### 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.

### Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality

Quantitative Data Analysis: Choosing a statistical test Prepared by the Office of Planning, Assessment, Research and Quality 1 To help choose which type of quantitative data analysis to use either before

### 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

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### SOCI Homework 6 Key

SOCI 252-002 Homework 6 Key Professor François Nielsen Chapter 27 2. (pg. 702 drug use) a) The percentage of 9th graders in these countries who have used other drugs is estimated to have increased 0.615%

### Statistics courses often teach the two-sample t-test, linear regression, and analysis of variance

2 Making Connections: The Two-Sample t-test, Regression, and ANOVA In theory, there s no difference between theory and practice. In practice, there is. Yogi Berra 1 Statistics courses often teach the two-sample

### AP Statistics 2012 Scoring Guidelines

AP Statistics 2012 Scoring Guidelines The College Board The College Board is a mission-driven not-for-profit organization that connects students to college success and opportunity. Founded in 1900, the

### Module 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling

Module 5: Introduction to Multilevel Modelling SPSS Practicals Chris Charlton 1 Centre for Multilevel Modelling Pre-requisites Modules 1-4 Contents P5.1 Comparing Groups using Multilevel Modelling... 4

### 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

### Statistics and research

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

### Elements of statistics (MATH0487-1)

Elements of statistics (MATH0487-1) Prof. Dr. Dr. K. Van Steen University of Liège, Belgium December 10, 2012 Introduction to Statistics Basic Probability Revisited Sampling Exploratory Data Analysis -

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

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

### Analysis of Covariance

Analysis of Covariance 1. Introduction The Analysis of Covariance (generally known as ANCOVA) is a technique that sits between analysis of variance and regression analysis. It has a number of purposes

### Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

### 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

### Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

### The aspect of the data that we want to describe/measure is the degree of linear relationship between and The statistic r describes/measures the degree

PS 511: Advanced Statistics for Psychological and Behavioral Research 1 Both examine linear (straight line) relationships Correlation works with a pair of scores One score on each of two variables ( and

### Algebra 1. ELG HS.S.2: Summarize, represent, and interpret data on two categorical and quantitative variables

Vertical Progression: 7 th Grade 8 th Grade Algebra 2 Draw informal comparative inferences about two populations. o 7.SP.4 Use measures of center and measures of variability for numerical data from random

### Introduction to Statistics and Quantitative Research Methods

Introduction to Statistics and Quantitative Research Methods Purpose of Presentation To aid in the understanding of basic statistics, including terminology, common terms, and common statistical methods.

### 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

### Statistics. Measurement. Scales of Measurement 7/18/2012

Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does

### Descriptive Statistics

Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

### A full analysis example Multiple correlations Partial correlations

A full analysis example Multiple correlations Partial correlations New Dataset: Confidence This is a dataset taken of the confidence scales of 41 employees some years ago using 4 facets of confidence (Physical,

### AP Statistics 2000 Scoring Guidelines

AP Statistics 000 Scoring Guidelines The materials included in these files are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought

### Simple Predictive Analytics Curtis Seare

Using Excel to Solve Business Problems: Simple Predictive Analytics Curtis Seare Copyright: Vault Analytics July 2010 Contents Section I: Background Information Why use Predictive Analytics? How to use

### F. Farrokhyar, MPhil, PhD, PDoc

Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

### Relationship of two variables

Relationship of two variables A correlation exists between two variables when the values of one are somehow associated with the values of the other in some way. Scatter Plot (or Scatter Diagram) A plot

### In most situations involving two quantitative variables, a scatterplot is the appropriate visual display.

In most situations involving two quantitative variables, a scatterplot is the appropriate visual display. Remember our assumption that the observations in our dataset be independent. If individuals appear

### Good luck! BUSINESS STATISTICS FINAL EXAM INSTRUCTIONS. Name:

Glo bal Leadership M BA BUSINESS STATISTICS FINAL EXAM Name: INSTRUCTIONS 1. Do not open this exam until instructed to do so. 2. Be sure to fill in your name before starting the exam. 3. You have two hours

### 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

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

### Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk

Analysing Questionnaires using Minitab (for SPSS queries contact -) Graham.Currell@uwe.ac.uk Structure As a starting point it is useful to consider a basic questionnaire as containing three main sections:

### Case l: Watchdog, Inc: Keeping hospitals honest

Case l: Watchdog, Inc: Keeping hospitals honest Insurance companies hire independent firms like Watchdog, Inc to review their hospital bills. They pay Watchdog a percentage of the overcharges it finds.

### AP Statistics 1998 Scoring Guidelines

AP Statistics 1998 Scoring Guidelines These materials are intended for non-commercial use by AP teachers for course and exam preparation; permission for any other use must be sought from the Advanced Placement

### We are often interested in the relationship between two variables. Do people with more years of full-time education earn higher salaries?

Statistics: Correlation Richard Buxton. 2008. 1 Introduction We are often interested in the relationship between two variables. Do people with more years of full-time education earn higher salaries? Do

### Pearson s correlation

Pearson s correlation Introduction Often several quantitative variables are measured on each member of a sample. If we consider a pair of such variables, it is frequently of interest to establish if there

### STATISTICS 8 CHAPTERS 1 TO 6, SAMPLE MULTIPLE CHOICE QUESTIONS

STATISTICS 8 CHAPTERS 1 TO 6, SAMPLE MULTIPLE CHOICE QUESTIONS Correct answers are in bold italics.. This scenario applies to Questions 1 and 2: A study was done to compare the lung capacity of coal miners

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### Week 1. Exploratory Data Analysis

Week 1 Exploratory Data Analysis Practicalities This course ST903 has students from both the MSc in Financial Mathematics and the MSc in Statistics. Two lectures and one seminar/tutorial per week. Exam

### 4. Describing Bivariate Data

4. Describing Bivariate Data A. Introduction to Bivariate Data B. Values of the Pearson Correlation C. Properties of Pearson's r D. Computing Pearson's r E. Variance Sum Law II F. Exercises A dataset with

### Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam. Software Profiling Seminar, Statistics 101

Dr. Peter Tröger Hasso Plattner Institute, University of Potsdam Software Profiling Seminar, 2013 Statistics 101 Descriptive Statistics Population Object Object Object Sample numerical description Object

### 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

### LEARNING OBJECTIVES SCALES OF MEASUREMENT: A REVIEW SCALES OF MEASUREMENT: A REVIEW DESCRIBING RESULTS DESCRIBING RESULTS 8/14/2016

UNDERSTANDING RESEARCH RESULTS: DESCRIPTION AND CORRELATION LEARNING OBJECTIVES Contrast three ways of describing results: Comparing group percentages Correlating scores Comparing group means Describe

### What is correlational research?

Key Ideas Purpose and use of correlational designs How correlational research developed Types of correlational designs Key characteristics of correlational designs Procedures used in correlational studies

### Chapter 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

### Epidemiology-Biostatistics Exam Exam 2, 2001 PRINT YOUR LEGAL NAME:

Epidemiology-Biostatistics Exam Exam 2, 2001 PRINT YOUR LEGAL NAME: Instructions: This exam is 30% of your course grade. The maximum number of points for the course is 1,000; hence this exam is worth 300

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### Quantitative Research Methods II. Vera E. Troeger Office: Office Hours: by appointment

Quantitative Research Methods II Vera E. Troeger Office: 0.67 E-mail: v.e.troeger@warwick.ac.uk Office Hours: by appointment Quantitative Data Analysis Descriptive statistics: description of central variables

### Module 9: Nonparametric Tests. The Applied Research Center

Module 9: Nonparametric Tests The Applied Research Center Module 9 Overview } Nonparametric Tests } Parametric vs. Nonparametric Tests } Restrictions of Nonparametric Tests } One-Sample Chi-Square Test

### How to interpret scientific & statistical graphs

How to interpret scientific & statistical graphs Theresa A Scott, MS Department of Biostatistics theresa.scott@vanderbilt.edu http://biostat.mc.vanderbilt.edu/theresascott 1 A brief introduction Graphics:

### 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

### Chapter 14: Analyzing Relationships Between Variables

Chapter Outlines for: Frey, L., Botan, C., & Kreps, G. (1999). Investigating communication: An introduction to research methods. (2nd ed.) Boston: Allyn & Bacon. Chapter 14: Analyzing Relationships Between

### Once saved, if the file was zipped you will need to unzip it. For the files that I will be posting you need to change the preferences.

1 Commands in JMP and Statcrunch Below are a set of commands in JMP and Statcrunch which facilitate a basic statistical analysis. The first part concerns commands in JMP, the second part is for analysis

### SPSS Multivariable Linear Models and Logistic Regression

1 SPSS Multivariable Linear Models and Logistic Regression Multivariable Models Single continuous outcome (dependent variable), one main exposure (independent) variable, and one or more potential confounders

### Introduction to Statistics for Computer Science Projects

Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

### DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS

DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 110 012 seema@iasri.res.in 1. Descriptive Statistics Statistics

### Silvermine House Steenberg Office Park, Tokai 7945 Cape Town, South Africa Telephone: +27 21 702 4666 www.spss-sa.com

SPSS-SA Silvermine House Steenberg Office Park, Tokai 7945 Cape Town, South Africa Telephone: +27 21 702 4666 www.spss-sa.com SPSS-SA Training Brochure 2009 TABLE OF CONTENTS 1 SPSS TRAINING COURSES FOCUSING

### 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...

### business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

### Chapter 16 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 16 Multiple Choice Questions (The answers are provided after the last question.) 1. Which of the following symbols represents a population parameter? a. SD b. σ c. r d. 0 2. If you drew all possible

### 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

### SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011

SPSS ADVANCED ANALYSIS WENDIANN SETHI SPRING 2011 Statistical techniques to be covered Explore relationships among variables Correlation Regression/Multiple regression Logistic regression Factor analysis