Regression, Correlation. Research Theoretical empirical Usually combination of the two.

Slides:



Advertisements
Similar presentations
Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Curve Fitting ~ Least Squares Regression Chapter.
Advertisements

Regression Analysis Module 3. Regression Regression is the attempt to explain the variation in a dependent variable using the variation in independent.
Regression Greg C Elvers.
Learning Objectives Copyright © 2004 John Wiley & Sons, Inc. Bivariate Correlation and Regression CHAPTER Thirteen.
Probabilistic & Statistical Techniques Eng. Tamer Eshtawi First Semester Eng. Tamer Eshtawi First Semester
Chapter 10 Curve Fitting and Regression Analysis
Correlation and Regression
1 Simple Linear Regression and Correlation The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES Assessing the model –T-tests –R-square.
Simple Regression. Major Questions Given an economic model involving a relationship between two economic variables, how do we go about specifying the.
Chapter 10 Regression. Defining Regression Simple linear regression features one independent variable and one dependent variable, as in correlation the.
Overview Correlation Regression -Definition
Chapter 15 (Ch. 13 in 2nd Can.) Association Between Variables Measured at the Interval-Ratio Level: Bivariate Correlation and Regression.
1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc. Summarizing Bivariate Data Introduction to Linear Regression.
9. SIMPLE LINEAR REGESSION AND CORRELATION
Bivariate Regression CJ 526 Statistical Analysis in Criminal Justice.
REGRESSION What is Regression? What is the Regression Equation? What is the Least-Squares Solution? How is Regression Based on Correlation? What are the.
Correlation and Regression Analysis
The Simple Regression Model
Math 227 Elementary Statistics Math 227 Elementary Statistics Sullivan, 4 th ed.
Business Statistics - QBM117 Least squares regression.
REGRESSION Predict future scores on Y based on measured scores on X Predictions are based on a correlation from a sample where both X and Y were measured.
PSY 307 – Statistics for the Behavioral Sciences Chapter 7 – Regression.
Correlation and Regression Analysis
Chapter 2 – Simple Linear Regression - How. Here is a perfect scenario of what we want reality to look like for simple linear regression. Our two variables.
Review Regression and Pearson’s R SPSS Demo
McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 13 Linear Regression and Correlation.
Chapter 8: Bivariate Regression and Correlation
Chapters 8, 9, 10 Least Squares Regression Line Fitting a Line to Bivariate Data.
Chapter 12 Correlation and Regression Part III: Additional Hypothesis Tests Renee R. Ha, Ph.D. James C. Ha, Ph.D Integrative Statistics for the Social.
Introduction to Linear Regression and Correlation Analysis
ASSOCIATION BETWEEN INTERVAL-RATIO VARIABLES
Chapter 13 Statistics © 2008 Pearson Addison-Wesley. All rights reserved.
Chapter 6 & 7 Linear Regression & Correlation
Agenda Review Association for Nominal/Ordinal Data –  2 Based Measures, PRE measures Introduce Association Measures for I-R data –Regression, Pearson’s.
Chapter 3 concepts/objectives Define and describe density curves Measure position using percentiles Measure position using z-scores Describe Normal distributions.
Simple Linear Regression One reason for assessing correlation is to identify a variable that could be used to predict another variable If that is your.
© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 13-6 Regression and Correlation.
© 2014 by Pearson Higher Education, Inc Upper Saddle River, New Jersey All Rights Reserved HLTH 300 Biostatistics for Public Health Practice, Raul.
1.6 Linear Regression & the Correlation Coefficient.
Linear Regression Least Squares Method: the Meaning of r 2.
Section 5.2: Linear Regression: Fitting a Line to Bivariate Data.
Chapter 3 Section 3.1 Examining Relationships. Continue to ask the preliminary questions familiar from Chapter 1 and 2 What individuals do the data describe?
BIOL 582 Lecture Set 11 Bivariate Data Correlation Regression.
Examining Relationships in Quantitative Research
Warsaw Summer School 2015, OSU Study Abroad Program Regression.
Chapters 8 & 9 Linear Regression & Regression Wisdom.
Chapter 11 Correlation and Simple Linear Regression Statistics for Business (Econ) 1.
11/23/2015Slide 1 Using a combination of tables and plots from SPSS plus spreadsheets from Excel, we will show the linkage between correlation and linear.
Examining Bivariate Data Unit 3 – Statistics. Some Vocabulary Response aka Dependent Variable –Measures an outcome of a study Explanatory aka Independent.
Chapter 4 Summary Scatter diagrams of data pairs (x, y) are useful in helping us determine visually if there is any relation between x and y values and,
Correlation and Regression: The Need to Knows Correlation is a statistical technique: tells you if scores on variable X are related to scores on variable.
MARE 250 Dr. Jason Turner Linear Regression. Linear regression investigates and models the linear relationship between a response (Y) and predictor(s)
CHAPTER 5 CORRELATION & LINEAR REGRESSION. GOAL : Understand and interpret the terms dependent variable and independent variable. Draw a scatter diagram.
1 Data Analysis Linear Regression Data Analysis Linear Regression Ernesto A. Diaz Department of Mathematics Redwood High School.
Linear Regression Day 1 – (pg )
Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 3 Association: Contingency, Correlation, and Regression Section 3.3 Predicting the Outcome.
^ y = a + bx Stats Chapter 5 - Least Squares Regression
Linear Prediction Correlation can be used to make predictions – Values on X can be used to predict values on Y – Stronger relationships between X and Y.
Regression Analysis. 1. To comprehend the nature of correlation analysis. 2. To understand bivariate regression analysis. 3. To become aware of the coefficient.
1 Simple Linear Regression and Correlation Least Squares Method The Model Estimating the Coefficients EXAMPLE 1: USED CAR SALES.
Regression Analysis Deterministic model No chance of an error in calculating y for a given x Probabilistic model chance of an error First order linear.
Simple Linear Regression The Coefficients of Correlation and Determination Two Quantitative Variables x variable – independent variable or explanatory.
Chapters 8 Linear Regression. Correlation and Regression Correlation = linear relationship between two variables. Summarize relationship with line. Called.
Chapter 14 Introduction to Regression Analysis. Objectives Regression Analysis Uses of Regression Analysis Method of Least Squares Difference between.
Describing Bivariate Relationships. Bivariate Relationships When exploring/describing a bivariate (x,y) relationship: Determine the Explanatory and Response.
^ y = a + bx Stats Chapter 5 - Least Squares Regression
CHAPTER 3 Describing Relationships
Least Squares Method: the Meaning of r2
Least-Squares Regression
Ch 4.1 & 4.2 Two dimensions concept
Presentation transcript:

Regression, Correlation

Research Theoretical empirical Usually combination of the two

Proportionality Constant (who cares?) Evaporation is a function of temperature. Evaporation is proportional to temperature (as temperature goes up so does evaporation) E  T But this just tell us they are related to one another qualitatively How can they be quantified?

Regression Equations Foundation of empirical equations Most hydrology is to chaotic to model based on physics alone Therefore most hydrologic equations are empirical (and therefore regression equations)

Regression Calculation of the regression line is straightforward. The best-fit line has the form y = bx + a, where b is the slope of the line and a is the y-intercept.

“Best Fit” Method of Least Squares –The best curve minimizes the sum of the squares of the vertical distance from a point to the curve. The vertical distances are usually called "error" (because the curve differs from the data by that amount) and so this method minimizes the "squared error."

Problem 1: Outliers and Extremes

Problem 2: Only Linear Relationships

How do we establish if we are successful Questions: –Is the model useful? –When is it better to use the average vs. a model?

Correlation Coefficient r is always between -1 and 1 r = 0 means no correlation r = 1 is perfect positive correlation r = -1 is perfect negative positive correlation r 2 is the percent variation explained by the linear correlation

Explained variation and Unexplained variation

Red line represents predicted values Green line is the average Black dots are observed values Green lines represent predicted-average Blue lines represent observed minus the predicted

Total Varation Total variation is given by the following equation: average predicted Total variation is related to the data’s spread.

Explained variation Explained variation is the average predicted Explained variation is fundamental to the spread of the data and is therefore “explained.” If your data have a high standard deviation, then you will expect your model reflect this as well.

Unexplained Variation Unexplained variation in the model is given by equation: average predicted You want unexplained variation to be low. In other words, your predicted values should be close to your observed values. Otherwise why are you bothering people with your model?

Summary of Variation High Total variation does not reflect poorly on your model High explained variation does not reflect poorly on your model High unexplained variation means your observed and predicted do not match…which makes us wonder why you have a model at all

Regression Coefficeint Pearson divided up variation as follows:

Pearson Correlation Coefficient If observed and predicted have same spread about the mean (or difference from the mean) the correlation coefficient will be close to one using the following formula:

R 2 values The average of your data is approximately 0.5 so your model needs to be higher than this 0.7 is acceptable in most empirical models and suggests the model is better than using the average 0.9 and greater is ideal

Standard Error of Estimate (S e ) V = n-p-1 where n is the number of data points, p is the number of unknowns (number of x’s…usually one) V = n-2 for bivariate models (p=1, one x)

Standard Error of Estimate (S e ) The standard error of the estimate is a measure of the accuracy of predictions made with a regression line More sensitive to sample size, generally as sample size increases, standard error decreases Physical indicator or error and has same units as criterion variable (aka y)