Introduction to Correlation and Regression

Introduction toIntroduction to

Correlation and RegressionCorrelation and Regression

Rahul JainRahul Jain

OutlineOutline Introduction Introduction

Linear CorrelationLinear Correlation

Regression Regression Simple Linear Simple Linear

Regression Regression Model/FormulasModel/Formulas

Outline continuedOutline continued ApplicationsApplications

Real-life ApplicationsReal-life Applications Practice ProblemsPractice Problems

Internet Resources Internet Resources Applets Applets Data SourcesData Sources

CorrelationCorrelation Correlation Correlation

A measure of association between A measure of association between two numerical variables.two numerical variables.

Example (positive correlation)Example (positive correlation) Typically, in the summer as the Typically, in the summer as the

temperature increases people are temperature increases people are thirstier.thirstier.

Specific Example Specific Example

For seven For seven random summer random summer days, a person days, a person recorded the recorded the temperaturetemperature and and their their water water consumptionconsumption, , during a three-hour during a three-hour period spent period spent outside.  outside.  

Temperature (F)

Water Consumption

(ounces)

75 16

83 20

85  25

85 27

92 32

97 48

99 48

How would you describe the graph?How would you describe the graph?

How “strong” is the linear relationship?How “strong” is the linear relationship?

Measuring the RelationshipMeasuring the Relationship

Pearson’s Sample Pearson’s Sample Correlation Coefficient, Correlation Coefficient, rr

measures the measures the directiondirection and the and the strengthstrength of the linear association of the linear association

between two numerical paired between two numerical paired variables.variables.

Direction of AssociationDirection of Association

Positive CorrelationPositive Correlation Negative CorrelationNegative Correlation

Strength of Linear AssociationStrength of Linear Association

r value Interpretation

1perfect positive linear

relationship

0 no linear relationship

-1perfect negative linear

relationship

Strength of Linear AssociationStrength of Linear Association

Other Strengths of AssociationOther Strengths of Association

r value Interpretation

0.9 strong association

0.5 moderate association

0.25 weak association

Other Strengths of AssociationOther Strengths of Association

FormulaFormula

    = the sum       n = number of paired items

xi = input variableyi

= output variable 

x = x-bar = mean of x’s

y = y-bar = mean of y’s

sx= standard deviation of x’s

sy= standard deviation of y’s

403.7 15

Coefficient of Determination: RCoefficient of Determination: R22 A quantification of the significance of

estimated model is denoted by R2.

R2 > 85% = significant model R2 < 85% = model is perceived as

inadequate Low R2 will suggest a need for additional

predictors for modeling the mean of Y

bxay ˆ

RegressionRegression

RegressionRegression

Specific statistical methodsSpecific statistical methods for for finding the “line of best fit” for one finding the “line of best fit” for one response (dependent) numerical response (dependent) numerical variable based on one or more variable based on one or more explanatory (independent) explanatory (independent) variables.variables.

Curve Fitting vs. RegressionCurve Fitting vs. Regression

RegressionRegression

Includes using statistical methods Includes using statistical methods to assess the "goodness of fit" of to assess the "goodness of fit" of the model.  (ex. Correlation the model.  (ex. Correlation Coefficient)Coefficient)

Regression: 3 Main PurposesRegression: 3 Main Purposes

To describeTo describe (or model) (or model)

To predictTo predict ( (or estimate) or estimate)

To controlTo control (or administer) (or administer)

Simple Linear RegressionSimple Linear Regression

Statistical method for findingStatistical method for finding the “line of best fit” the “line of best fit”

for one response (dependent) for one response (dependent) numerical variable numerical variable

based on one explanatory based on one explanatory (independent) variable.  (independent) variable.  

Least Squares RegressionLeast Squares Regression GOAL GOAL - -

minimize the minimize the sum of the sum of the square of square of the errors of the errors of the data the data points.points.

This minimizes the This minimizes the Mean Square ErrorMean Square Error

ExampleExample

Plan an outdoor party.Plan an outdoor party.

EstimateEstimate number of soft drinks to buy number of soft drinks to buy per person, based on how hot the per person, based on how hot the weather is.weather is.

Use Temperature/Water data and Use Temperature/Water data and regressionregression..

Steps to Reaching a SolutionSteps to Reaching a Solution Draw a scatterplot of the data.Draw a scatterplot of the data.

Steps to Reaching a SolutionSteps to Reaching a Solution Draw a scatterplot of the data.Draw a scatterplot of the data. Visually, consider the strength of the Visually, consider the strength of the

linear relationship.linear relationship.

Steps to Reaching a SolutionSteps to Reaching a Solution Draw a scatterplot of the data.Draw a scatterplot of the data. Visually, consider the strength of the Visually, consider the strength of the

linear relationship.linear relationship. If the relationship appears relatively If the relationship appears relatively

strong, find the correlation coefficient strong, find the correlation coefficient as a numerical verification.as a numerical verification.

Steps to Reaching a SolutionSteps to Reaching a Solution Draw a scatterplot of the data.Draw a scatterplot of the data. Visually, consider the strength of the Visually, consider the strength of the

linear relationship.linear relationship. If the relationship appears relatively If the relationship appears relatively

strong, find the correlation coefficient strong, find the correlation coefficient as a numerical verification.as a numerical verification.

If the correlation is still relatively If the correlation is still relatively strong, then find the simple linear strong, then find the simple linear regression line.regression line.

Interpreting and VisualizingInterpreting and Visualizing Interpreting the result: Interpreting the result:

y = ax + by = ax + b

The valueThe value ofof aa is the is the slopeslope The value of The value of bb is the is the y-intercepty-intercept rr is the is the correlation coefficientcorrelation coefficient rr22 is the is the coefficient of determinationcoefficient of determination

5. Interpreting and Visualizing5. Interpreting and Visualizing Write down the equation of the Write down the equation of the

line in slope intercept form. line in slope intercept form. Press Press Y=Y= and enter the equation and enter the equation

under Y1. (Clear all other under Y1. (Clear all other equations.) equations.) 

Press Press GRAPHGRAPH and the line will and the line will be graphed through the data be graphed through the data points.points.

Questions ???Questions ???

Interpretation in ContextInterpretation in Context

Regression Equation: Regression Equation:

y=1.5*x - 96.9y=1.5*x - 96.9

Water Consumption = Water Consumption = 1.5*Temperature - 96.9 1.5*Temperature - 96.9

Interpretation in ContextInterpretation in Context

Slope = 1.5 (ounces)/(degrees F)Slope = 1.5 (ounces)/(degrees F)

for each 1 degree F increase in for each 1 degree F increase in temperature, you expect an increase temperature, you expect an increase of 1.5 ounces of water drank.of 1.5 ounces of water drank.

Interpretation in ContextInterpretation in Context

y-intercept = -96.9y-intercept = -96.9

For this example, For this example, when the temperature is 0 degrees F, when the temperature is 0 degrees F, then a person would drink about -97 then a person would drink about -97 ounces of water. ounces of water.

That does not make any sense! That does not make any sense! Our model is not applicable for x=0.  Our model is not applicable for x=0.  

Prediction ExamplePrediction Example

Predict Predict the amount of the amount of water a person would drink when the water a person would drink when the temperature is temperature is 95 degrees F.95 degrees F.

Solution:Solution: Substitute the value of x=95 Substitute the value of x=95 (degrees F) into the regression equation (degrees F) into the regression equation and solve for y (water consumption).and solve for y (water consumption).

If x=95, y=1.5*95 - 96.9 = If x=95, y=1.5*95 - 96.9 = 45.6 ounces.  45.6 ounces.  

Simple Linear Regression ModelSimple Linear Regression Model

The model for The model for simple linear regression issimple linear regression is

There are mathematical assumptions There are mathematical assumptions

behind the concepts thatbehind the concepts that we are covering today.we are covering today.

FormulasFormulas

Prediction Equation: Prediction Equation:

Standard error of the estimateStandard error of the estimate

Real Life ApplicationsReal Life Applications

Cost Estimating for Future Space Cost Estimating for Future Space Flight Vehicles (Multiple Flight Vehicles (Multiple

Regression)Regression)

Real Life ApplicationsReal Life Applications Estimating Seasonal Sales for Estimating Seasonal Sales for

Department Stores (Periodic)Department Stores (Periodic)

Real Life ApplicationsReal Life Applications Predicting Student Grades Based Predicting Student Grades Based

on Time Spent Studyingon Time Spent Studying

Real Life ApplicationsReal Life Applications

. . .. . .

What ideas can you think of?What ideas can you think of?

What ideas can you think of that What ideas can you think of that your students will relate to?your students will relate to?

Practice ProblemsPractice Problems

Is there any correlation between Is there any correlation between shoe size and height? shoe size and height?

Does gender make a difference Does gender make a difference in this analysis?in this analysis?

Practice ProblemsPractice Problems Can the number of points Can the number of points

scored in a basketball game be scored in a basketball game be predicted by predicted by The time a player plays in The time a player plays in

the game?the game?

By the player’s height?By the player’s height?

Idea modified from Steven King, Aiken, Idea modified from Steven King, Aiken, SC. NCTM presentation 1997.)SC. NCTM presentation 1997.)

ResourcesResources Data Analysis and StatisticsData Analysis and Statistics. .

Curriculum and Evaluation Curriculum and Evaluation Standards for School Standards for School Mathematics.  Addenda Series, Mathematics.  Addenda Series, Grades 9-12.  NCTM. 1992.Grades 9-12.  NCTM. 1992.

Data and Story LibraryData and Story Library.  .  Internet Website.   Internet Website.   http://lib.stat.cmu.edu/DASL/ http://lib.stat.cmu.edu/DASL/ 2001. 2001. 

Internet ResourcesInternet Resources CorrelationCorrelation

Guessing CorrelationsGuessing Correlations - An - An interactive site that allows you to interactive site that allows you to try to match correlation coefficients try to match correlation coefficients to scatterplots. University of Illinois, to scatterplots. University of Illinois, Urbanna Champaign Statistics Urbanna Champaign Statistics Program. Program. http://www.stat.uiuc.edu/~stat100/jhttp://www.stat.uiuc.edu/~stat100/java/guess/GCApplet.htmlava/guess/GCApplet.html

Internet ResourcesInternet Resources

RegressionRegression Effects of adding an Effects of adding an

OutlierOutlier. .

W. West, University of South W. West, University of South Carolina. Carolina.

http://www.stat.sc.edu/~west/http://www.stat.sc.edu/~west/javahtml/Regression.htmljavahtml/Regression.html

Internet ResourcesInternet Resources RegressionRegression

Estimate the Regression LineEstimate the Regression Line. . Compare the mean square error Compare the mean square error from different regression lines. Can from different regression lines. Can you find the minimum mean square you find the minimum mean square error? Rice University Virtual error? Rice University Virtual Statistics Lab. Statistics Lab. http://www.ruf.rice.edu/~lane/stat_sihttp://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.htmlm/reg_by_eye/index.html

Internet Resources: Data SetsInternet Resources: Data Sets Data and Story Library. Data and Story Library.

Excellent source for small data sets. Excellent source for small data sets. Search for specific statistical methods Search for specific statistical methods (e.g. boxplots, regression) or for data (e.g. boxplots, regression) or for data concerning a specific field of interest concerning a specific field of interest (e.g. health, environment, sports). (e.g. health, environment, sports). http://lib.stat.cmu.edu/DASL/http://lib.stat.cmu.edu/DASL/

Internet ResourcesInternet Resources OtherOther

Statistics Applets. Using Web Statistics Applets. Using Web Applets to Assist in Statistics Applets to Assist in Statistics Instruction. Robin Lock, St. Instruction. Robin Lock, St. Lawrence University. Lawrence University. http://it.stlawu.edu/~rlock/maa99http://it.stlawu.edu/~rlock/maa99//

Internet ResourcesInternet Resources OtherOther

Ten Websites Every Statistics Ten Websites Every Statistics Instructor Should Bookmark. Instructor Should Bookmark. Robin Lock, St. Lawrence Robin Lock, St. Lawrence University. University. http://it.stlawu.edu/~rlock/10sithttp://it.stlawu.edu/~rlock/10sites.htmles.html

For More Information…For More Information…

On-line version of this presentationOn-line version of this presentationhttp://www.mtsu.edu/~statshttp://www.mtsu.edu/~stats

/corregpres/index.html/corregpres/index.html

More information about regressionMore information about regressionVisit Visit STATS @ MTSUSTATS @ MTSU web site web site

http://www.mtsu.edu/~statshttp://www.mtsu.edu/~stats