Correlation and Regression
|
|
- Hortense Barton
- 1 years ago
- Views:
Transcription
1 Correlation and Regression Scatterplots Correlation Explanatory and response variables Simple linear regression General Principles of Data Analysis First plot the data, then add numerical summaries Look for overall patterns and deviations from those patterns When overall pattern is regular, use a compact mathematical model to describe it 1
2 General Principles of Data Analysis Plot your data To understand the data, always start with a series of graphs Interpret what you see Look for overall pattern and deviations from that pattern Numerical summary? Choose an appropriate measure to describe the pattern and deviation Mathematical model? If the pattern is regular, summarize the data in a compact mathematical model Univariate Data Analysis Plot your data For continuous variable, use stemplot, histogram For categorical variable, use bar chart, dot plot Interpret what you see Describe distribution in terms of shape, center, spread, outliers Numerical summary? Select measure of central tendency and spread: mean, standard deviation, five-number summary Mathematical model? If appropriate, use Normal distribution as model for overall pattern 2
3 Bivariate Data Analysis Plot your data For two quantitative variables, use a scatterplot Interpret what you see Describe the direction, form, and strength of the relationship Numerical summary? If pattern is roughly linear, summarize with correlation, means, and standard deviations Mathematical model? Regression gives a compact model of overall pattern, if relationship is roughly linear Explanatory and Response Variables Response variable measures outcome of a study Explanatory variable explains or influences change in response variable Response variables often called dependent variables, explanatory independent Independent and dependent have other meanings in statistics, so prefer to avoid But remember that calling one variable explanatory and other response doesn t necessarily imply cause 3
4 Scatterplot Scatterplot shows relationship between two quantitative variables measured on same cases By convention, explanatory variable on horizontal axis, response on vertical axis Each case appears as point fixed by values of both variables Wine Consumption and Heart Attacks Some evidence that drinking moderate amounts of wine my help prevent heart attacks We have the following data from 19 countries Yearly wine consumption (liters of alcohol from wine, per person) Yearly deaths (per 100,000 people) from heart disease In Stata, use -twoway scatter- Interpret a scatterplot Overall pattern and striking deviations Form, direction, and strength Look for values that fall outside overall pattern of relationship (outlier) 4
5 Heart Disease Deaths (per 100,000 people) Alcohol with Wine (liters per person per year) Measuring Linear Association: Correlation Correlation measures direction and strength of linear relationship between two quantitative variables Usually written as r In Stata, use -correlate- or -pwcorr- Find and interpret correlation for wine and heart disease example 5
6 Using and Interpreting Correlation Ranges from -1 to 1; values closer to 1 indicate stronger linear relationship Positive values indicate positive association Does not distinguish between explanatory and response variables Requires that both variables be quantitative It has no unit of measurement because it uses standardized values, it is scale free Measures strength only of linear relationships Like mean and standard deviation, strongly affected by outlying observations Least-Squares Regression Line Correlation measures direction and strength of linear relationships A regression line summarizes relationship between explanatory, x, and response variable, y We can use regression line to predict value of y for a given value of x These predictions have error, called residuals The least-squares regression line of y is the line that minimizes residuals 6
7 Heart disease death rate (per 100,000) A good line for prediction makes these distances small Vertical distances from least-squares line are residuals Alcohol with Wine (liters per person per year) In Stata, obtain this graph with twoway (lfit heartdis alcohol) (scatter heartdis alcohol) Regression in Stata regress regress heartdis heartdis alcohol alcohol Source Source SS SS df df MS MS Number Number of of obs obs F( F( 1, 1, 17) 17) Model Model Prob Prob > > F F Residual Residual R-squared R-squared Adj Adj R-squared R-squared Total Total Root Root MSE MSE slope, b heartdis heartdis Coef. Coef. Std. Std. Err. Err. t t P>t P>t [95% [95% Conf. Conf. Interval] Interval] alcohol alcohol _cons _cons y-intercept, a y ^ a + bx Estimated heart disease death rate (-22.97)(Per capita alcohol consumption) 7
8 Using and Interpreting Regression Distinction between explanatory and response variables is essential in regression Regression of y on x regression of x on y There is a close connection between correlation and slope of least-squares line b r(s y / s x ) Change of 1 sd in x corresponds to a change of r standard deviations in y Square of correlation, r 2, is fraction of variation in values of y explained by x (PRE) Conditions for Inference Observations are independent True relationship is linear Standard deviation of response about the true line is same everywhere Response varies normally about true regression line Analysis of residuals is key to diagnosing violations of these conditions 8
9 Residuals-versus-Fitted Plot Residuals Fitted values In Stata, obtain this plot after regress with rvfplot, yline(0) Residuals-versus-Predictor Plot Residuals Alcohol with Wine (liters per person per year) In Stata, obtain this plot after regress with rvpplot alcohol, yline(0) 9
10 95% Confidence Interval for Least-Squares Line Heart disease deaths per 100,000 people Alcohol with Wine (liters per person per year) Obtain this graph with twoway (lfitci heartdis alcohol) (scatter heartdis alcohol) 10
MULTIPLE REGRESSION EXAMPLE
MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if
In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a
Math 143 Inference on Regression 1 Review of Linear Regression In Chapter 2, we used linear regression to describe linear relationships. The setting for this is a bivariate data set (i.e., a list of cases/subjects
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
Chapter 2: Looking at Data Relationships (Part 1)
Chapter 2: Looking at Data Relationships (Part 1) Dr. Nahid Sultana Chapter 2: Looking at Data Relationships 2.1: Scatterplots 2.2: Correlation 2.3: Least-Squares Regression 2.5: Data Analysis for Two-Way
Regression in Stata. Alicia Doyle Lynch Harvard-MIT Data Center (HMDC)
Regression in Stata Alicia Doyle Lynch Harvard-MIT Data Center (HMDC) Documents for Today Find class materials at: http://libraries.mit.edu/guides/subjects/data/ training/workshops.html Several formats
Inference for Regression
Simple Linear Regression Inference for Regression The simple linear regression model Estimating regression parameters; Confidence intervals and significance tests for regression parameters Inference about
12.1 Inference for Linear Regression
12.1 Inference for Linear Regression Least Squares Regression Line y = a + bx You might want to refresh your memory of LSR lines by reviewing Chapter 3! 1 Sample Distribution of b p740 Shape Center Spread
Relationships Between Two Variables: Scatterplots and Correlation
Relationships Between Two Variables: Scatterplots and Correlation Example: Consider the population of cars manufactured in the U.S. What is the relationship (1) between engine size and horsepower? (2)
REGRESSION LINES IN STATA
REGRESSION LINES IN STATA THOMAS ELLIOTT 1. Introduction to Regression Regression analysis is about eploring linear relationships between a dependent variable and one or more independent variables. Regression
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.
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
College Education Matters for Happier Marriages and Higher Salaries ----Evidence from State Level Data in the US
College Education Matters for Happier Marriages and Higher Salaries ----Evidence from State Level Data in the US Anonymous Authors: SH, AL, YM Contact TF: Kevin Rader Abstract It is a general consensus
Technology Step-by-Step Using StatCrunch
Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate
IAPRI Quantitative Analysis Capacity Building Series. Multiple regression analysis & interpreting results
IAPRI Quantitative Analysis Capacity Building Series Multiple regression analysis & interpreting results How important is R-squared? R-squared Published in Agricultural Economics 0.45 Best article of the
How Do We Test Multiple Regression Coefficients?
How Do We Test Multiple Regression Coefficients? Suppose you have constructed a multiple linear regression model and you have a specific hypothesis to test which involves more than one regression coefficient.
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
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
Stata Walkthrough 4: Regression, Prediction, and Forecasting
Stata Walkthrough 4: Regression, Prediction, and Forecasting Over drinks the other evening, my neighbor told me about his 25-year-old nephew, who is dating a 35-year-old woman. God, I can t see them getting
Regression Analysis. Data Calculations Output
Regression Analysis In an attempt to find answers to questions such as those posed above, empirical labour economists use a useful tool called regression analysis. Regression analysis is essentially a
MODEL I: DRINK REGRESSED ON GPA & MALE, WITHOUT CENTERING
Interpreting Interaction Effects; Interaction Effects and Centering Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 20, 2015 Models with interaction effects
Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software
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
August 2012 EXAMINATIONS Solution Part I
August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,
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
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
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
A correlation exists between two variables when one of them is related to the other in some way.
Lecture #10 Chapter 10 Correlation and Regression The main focus of this chapter is to form inferences based on sample data that come in pairs. Given such paired sample data, we want to determine whether
Interaction effects between continuous variables (Optional)
Interaction effects between continuous variables (Optional) Richard Williams, University of Notre Dame, http://www.nd.edu/~rwilliam/ Last revised February 0, 05 This is a very brief overview of this somewhat
Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation
Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage
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
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.
Introduction to Stata
Introduction to Stata September 23, 2014 Stata is one of a few statistical analysis programs that social scientists use. Stata is in the mid-range of how easy it is to use. Other options include SPSS,
Chapter 10. Key Ideas Correlation, Correlation Coefficient (r),
Chapter 0 Key Ideas Correlation, Correlation Coefficient (r), Section 0-: Overview We have already explored the basics of describing single variable data sets. However, when two quantitative variables
2. Simple Linear Regression
Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according
Summarizing and Displaying Categorical Data
Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency
Nonlinear Regression Functions. SW Ch 8 1/54/
Nonlinear Regression Functions SW Ch 8 1/54/ The TestScore STR relation looks linear (maybe) SW Ch 8 2/54/ But the TestScore Income relation looks nonlinear... SW Ch 8 3/54/ Nonlinear Regression General
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
Lesson Lesson Outline Outline
Lesson 15 Linear Regression Lesson 15 Outline Review correlation analysis Dependent and Independent variables Least Squares Regression line Calculating l the slope Calculating the Intercept Residuals and
Statistics 104 Final Project A Culture of Debt: A Study of Credit Card Spending in America TF: Kevin Rader Anonymous Students: LD, MH, IW, MY
Statistics 104 Final Project A Culture of Debt: A Study of Credit Card Spending in America TF: Kevin Rader Anonymous Students: LD, MH, IW, MY ABSTRACT: This project attempted to determine the relationship
ECON 142 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE #2
University of California, Berkeley Prof. Ken Chay Department of Economics Fall Semester, 005 ECON 14 SKETCH OF SOLUTIONS FOR APPLIED EXERCISE # Question 1: a. Below are the scatter plots of hourly wages
The importance of graphing the data: Anscombe s regression examples
The importance of graphing the data: Anscombe s regression examples Bruce Weaver Northern Health Research Conference Nipissing University, North Bay May 30-31, 2008 B. Weaver, NHRC 2008 1 The Objective
C2.1. (i) (5 marks) The average participation rate is , the average match rate is summ prate
BOSTON COLLEGE Department of Economics EC 228 01 Econometric Methods Fall 2008, Prof. Baum, Ms. Phillips (tutor), Mr. Dmitriev (grader) Problem Set 2 Due at classtime, Thursday 2 Oct 2008 2.4 (i)(5 marks)
The leverage statistic, h, also called the hat-value, is available to identify cases which influence the regression model more than others.
Outliers Outliers are data points which lie outside the general linear pattern of which the midline is the regression line. A rule of thumb is that outliers are points whose standardized residual is greater
Algebra I: Lesson 5-4 (5074) SAS Curriculum Pathways
Two-Variable Quantitative Data: Lesson Summary with Examples Bivariate data involves two quantitative variables and deals with relationships between those variables. By plotting bivariate data as ordered
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
SPSS for Exploratory Data Analysis Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav)
Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer
Linear Regression Models with Logarithmic Transformations
Linear Regression Models with Logarithmic Transformations Kenneth Benoit Methodology Institute London School of Economics kbenoit@lse.ac.uk March 17, 2011 1 Logarithmic transformations of variables Considering
POL 681: Stata for Regression Analysis
1 Getting Started POL 681: Stata for Regression Analysis When using Stata for regression analysis, or anything else for that matter, the first thing you have to do is make sure Stata has data upon which
Linear Regression with One Regressor
Linear Regression with One Regressor Michael Ash Lecture 10 Analogy to the Mean True parameter µ Y β 0 and β 1 Meaning Central tendency Intercept and slope E(Y ) E(Y X ) = β 0 + β 1 X Data Y i (X i, Y
Linear Regression. Chapter 5. Prediction via Regression Line Number of new birds and Percent returning. Least Squares
Linear Regression Chapter 5 Regression Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects
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.
, has mean A) 0.3. B) the smaller of 0.8 and 0.5. C) 0.15. D) which cannot be determined without knowing the sample results.
BA 275 Review Problems - Week 9 (11/20/06-11/24/06) CD Lessons: 69, 70, 16-20 Textbook: pp. 520-528, 111-124, 133-141 An SRS of size 100 is taken from a population having proportion 0.8 of successes. An
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
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
AP STATISTICS REVIEW (YMS Chapters 1-8)
AP STATISTICS REVIEW (YMS Chapters 1-8) Exploring Data (Chapter 1) Categorical Data nominal scale, names e.g. male/female or eye color or breeds of dogs Quantitative Data rational scale (can +,,, with
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
ST 311 Evening Problem Session Solutions Week 11
1. p. 175, Question 32 (Modules 10.1-10.4) [Learning Objectives J1, J3, J9, J11-14, J17] Since 1980, average mortgage rates have fluctuated from a low of under 6% to a high of over 14%. Is there a relationship
Lecture 15. Endogeneity & Instrumental Variable Estimation
Lecture 15. Endogeneity & Instrumental Variable Estimation Saw that measurement error (on right hand side) means that OLS will be biased (biased toward zero) Potential solution to endogeneity instrumental
South Carolina College- and Career-Ready (SCCCR) Probability and Statistics
South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)
Addressing Alternative. Multiple Regression. 17.871 Spring 2012
Addressing Alternative Explanations: Multiple Regression 17.871 Spring 2012 1 Did Clinton hurt Gore example Did Clinton hurt Gore in the 2000 election? Treatment is not liking Bill Clinton 2 Bivariate
DATA INTERPRETATION AND STATISTICS
PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE
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
Marginal Effects for Continuous Variables Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 21, 2015
Marginal Effects for Continuous Variables Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised February 21, 2015 References: Long 1997, Long and Freese 2003 & 2006 & 2014,
Data Analysis Methodology 1
Data Analysis Methodology 1 Suppose you inherited the database in Table 1.1 and needed to find out what could be learned from it fast. Say your boss entered your office and said, Here s some software project
Correlation Regression
Correlation Regression Διατμηματικό ΠΜΣ Επαγγελματική και Περιβαλλοντική Υγεία-Διαχείριση και Οικονομική Αποτίμηση Δημήτρης Φουσκάκης Correlation To assess whether two variables are linearly associated,,
Chapter 9. Section Correlation
Chapter 9 Section 9.1 - Correlation Objectives: Introduce linear correlation, independent and dependent variables, and the types of correlation Find a correlation coefficient Test a population correlation
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
STAT 350 Practice Final Exam Solution (Spring 2015)
PART 1: Multiple Choice Questions: 1) A study was conducted to compare five different training programs for improving endurance. Forty subjects were randomly divided into five groups of eight subjects
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
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
2013 MBA Jump Start Program. Statistics Module Part 3
2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just
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
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
Department of Economics Session 2012/2013. EC352 Econometric Methods. Solutions to Exercises from Week 10 + 0.0077 (0.052)
Department of Economics Session 2012/2013 University of Essex Spring Term Dr Gordon Kemp EC352 Econometric Methods Solutions to Exercises from Week 10 1 Problem 13.7 This exercise refers back to Equation
Introductory Statistics Notes
Introductory Statistics Notes Jamie DeCoster Department of Psychology University of Alabama 348 Gordon Palmer Hall Box 870348 Tuscaloosa, AL 35487-0348 Phone: (205) 348-4431 Fax: (205) 348-8648 August
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
UNDERSTANDING MULTIPLE REGRESSION
UNDERSTANDING Multiple regression analysis (MRA) is any of several related statistical methods for evaluating the effects of more than one independent (or predictor) variable on a dependent (or outcome)
Example: Boats and Manatees
Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant
Failure to take the sampling scheme into account can lead to inaccurate point estimates and/or flawed estimates of the standard errors.
Analyzing Complex Survey Data: Some key issues to be aware of Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 24, 2015 Rather than repeat material that is
Mathematics. Probability and Statistics Curriculum Guide. Revised 2010
Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction
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) -
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
Elementary Statistics. Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination
Scatter Plot, Regression Line, Linear Correlation Coefficient, and Coefficient of Determination What is a Scatter Plot? A Scatter Plot is a plot of ordered pairs (x, y) where the horizontal axis is used
Department of Economics, Session 2012/2013. EC352 Econometric Methods. Exercises from Week 03
Department of Economics, Session 01/013 University of Essex, Autumn Term Dr Gordon Kemp EC35 Econometric Methods Exercises from Week 03 1 Problem P3.11 The following equation describes the median housing
The Simple Linear Regression Model: Specification and Estimation
Chapter 3 The Simple Linear Regression Model: Specification and Estimation 3.1 An Economic Model Suppose that we are interested in studying the relationship between household income and expenditure on
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.
Econ 371 Problem Set #3 Answer Sheet
Econ 371 Problem Set #3 Answer Sheet 4.1 In this question, you are told that a OLS regression analysis of third grade test scores as a function of class size yields the following estimated model. T estscore
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
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:
, 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
vs. relative cumulative frequency
Variable - what we are measuring Quantitative - numerical where mathematical operations make sense. These have UNITS Categorical - puts individuals into categories Numbers don't always mean Quantitative...
Chapter 2 Summarizing and Graphing Data
Chapter 2 Summarizing and Graphing Data 2-1 Review and Preview 2-2 Frequency Distributions 2-3 Histograms 2-4 Graphs that Enlighten and Graphs that Deceive Preview Characteristics of Data 1. Center: A
AP Statistics Section :12.2 Transforming to Achieve Linearity
AP Statistics Section :12.2 Transforming to Achieve Linearity In Chapter 3, we learned how to analyze relationships between two quantitative variables that showed a linear pattern. When two-variable data
2. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of 2006 model motor vehicles.
Math 1530-017 Exam 1 February 19, 2009 Name Student Number E There are five possible responses to each of the following multiple choice questions. There is only on BEST answer. Be sure to read all possible
Homework 8 Solutions
Math 17, Section 2 Spring 2011 Homework 8 Solutions Assignment Chapter 7: 7.36, 7.40 Chapter 8: 8.14, 8.16, 8.28, 8.36 (a-d), 8.38, 8.62 Chapter 9: 9.4, 9.14 Chapter 7 7.36] a) A scatterplot is given below.
Scatter Plots with Error Bars
Chapter 165 Scatter Plots with Error Bars Introduction The procedure extends the capability of the basic scatter plot by allowing you to plot the variability in Y and X corresponding to each point. Each
Econ 371 Problem Set #3 Answer Sheet
Econ 371 Problem Set #3 Answer Sheet 4.3 In this question, you are told that a OLS regression analysis of average weekly earnings yields the following estimated model. AW E = 696.7 + 9.6 Age, R 2 = 0.023,
Minitab Guide. This packet contains: A Friendly Guide to Minitab. Minitab Step-By-Step
Minitab Guide This packet contains: A Friendly Guide to Minitab An introduction to Minitab; including basic Minitab functions, how to create sets of data, and how to create and edit graphs of different
AP * Statistics Review. Linear Regression
AP * Statistics Review Linear Regression Teacher Packet Advanced Placement and AP are registered trademark of the College Entrance Examination Board. The College Board was not involved in the production
MAT 12O ELEMENTARY STATISTICS I
LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 12O ELEMENTARY STATISTICS I 3 Lecture Hours, 1 Lab Hour, 3 Credits Pre-Requisite:
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