Ch 3 – Examining RelationshipsCh 3 – Examining Relationships

YMS – 3.1YMS – 3.1

ScatterplotsScatterplots

Some VocabularySome Vocabulary

Response Variable Response Variable – Measures an outcome of a studyMeasures an outcome of a study– AKA dependent variableAKA dependent variable

Explanatory Variable Explanatory Variable – Attempts to explain the observed outcomesAttempts to explain the observed outcomes– AKA independent variableAKA independent variable

Scatterplot Scatterplot – Shows the relationship between two quantitative Shows the relationship between two quantitative

variables measured on the same individuals variables measured on the same individuals

ScatterplotsScatterplotsExaminingExamining– Look for overall pattern and any deviationsLook for overall pattern and any deviations– Describe pattern with form, strength, and directionDescribe pattern with form, strength, and direction

Drawing Drawing – Uniformly scale the vertical and horizontal axesUniformly scale the vertical and horizontal axes– Label both axesLabel both axes– Adopt a scale that uses the entire available gridAdopt a scale that uses the entire available grid

Categorical Variables Categorical Variables – Add a different color/shape to distinguish between Add a different color/shape to distinguish between

categorical variables categorical variables

Classwork p125 #3.7, 3.10-3.11Classwork p125 #3.7, 3.10-3.11Homework: #3.16, 3.22 and 3.2 BlueprintHomework: #3.16, 3.22 and 3.2 Blueprint

YMS – 3.2YMS – 3.2

CorrelationCorrelation

CorrelationCorrelation

Measures the direction and strength of the Measures the direction and strength of the linear relationship between two linear relationship between two quantitative variables quantitative variables

Facts About CorrelationFacts About Correlation

Makes no distinction between explanatory and Makes no distinction between explanatory and response variablesresponse variables

Requires both variables be quantitativeRequires both variables be quantitative

Does not change units when we change units of Does not change units when we change units of measurementmeasurement

Sign of Sign of rr indicates positive or negative association indicates positive or negative association

rr is inclusive from -1 to 1 is inclusive from -1 to 1

Only measures strength of linear relationshipsOnly measures strength of linear relationships

Is not resistant Is not resistant

In Class ExercisesIn Class Exercises

p146 #3.28, 3.34 and 3.37p146 #3.28, 3.34 and 3.37

Correlation Guessing GameCorrelation Guessing Game

HomeworkHomework

3.3 Blueprint3.3 Blueprint

YMS – 3.3YMS – 3.3

Least-Square RegressionLeast-Square Regression

RegressionRegression

Regression LineRegression Line– Describes how a response variable Describes how a response variable yy changes as an changes as an

explanatory variable explanatory variable xx changes changes

LSRL of LSRL of yy on on xx – Makes the sum of the squares of the vertical Makes the sum of the squares of the vertical

distances of the data points from the line as small as distances of the data points from the line as small as possible possible

Line should be as close as possible to the points Line should be as close as possible to the points in the vertical direction in the vertical direction – Error = Observed (Actual) – Predicted Error = Observed (Actual) – Predicted

LSRLLSRL

Equation of the LSRLEquation of the LSRL

SlopeSlope InterceptIntercept

Coefficient of determination – rCoefficient of determination – r22

The fraction of the variation in the values of The fraction of the variation in the values of yy that is explained by the least-squares regression that is explained by the least-squares regression of of yy on on xx

Measures the contribution of Measures the contribution of xx in predicting in predicting yy

If If xx is a poor predictor of is a poor predictor of yy, then the sum of the , then the sum of the squares of the deviations about the mean squares of the deviations about the mean (SST) and the sum of the squares of deviations (SST) and the sum of the squares of deviations about the regression line (SSE) would be about the regression line (SSE) would be approximately the same. approximately the same.

Understanding r-squared: Understanding r-squared: A single point simplificationA single point simplification

Al CoonsBuckingham Browne & Nichols School

Cambridge, MAal_coons@bbns.org

Error eliminated by y-hat model

yError w.r.t. mean model

Error eliminated by y-hat model

Proportion of error eliminated by Y-hat model

Error w.r.t. mean model=

r2 = proportion of variability accounted for by the given model (w.r.t the mean model).

Error eliminated by y-hat model

yError w.r.t. mean model

Proportion of error eliminated by Y-hat model

Error w.r.t. mean model=

= ~

Error eliminated by y-hat model

Facts about Facts about Least-Squares Regression Least-Squares Regression

Distinction between explanatory and response Distinction between explanatory and response variables is essentialvariables is essentialA change of one standard deviation in A change of one standard deviation in xx corresponds to a change of corresponds to a change of rr standard deviations in standard deviations in yyLSRL always passes through the pointLSRL always passes through the pointThe square of the correlation is the fraction of the The square of the correlation is the fraction of the variation in the values of variation in the values of yy that is explained by the that is explained by the least-squares regression of least-squares regression of yy on on xx

Classwork: Transformations and LSRL WSClasswork: Transformations and LSRL WS

Homework: #3.39 and ABS Matching to Plots Extension Homework: #3.39 and ABS Matching to Plots Extension Question (we’ll finish the others in class)Question (we’ll finish the others in class)

Residuals Residuals

observed y – predicted y orobserved y – predicted y or

Positive values show that data point lies above Positive values show that data point lies above the LSRLthe LSRL

The mean of residuals is always zeroThe mean of residuals is always zero

Residual PlotsResidual Plots

A scatterplot of the regression residuals A scatterplot of the regression residuals against the explanatory variableagainst the explanatory variable

Helps us assess the fit of a regression lineHelps us assess the fit of a regression line

Want a random patternWant a random pattern

Watch for individual points with large Watch for individual points with large residuals or that are extreme in the residuals or that are extreme in the xx direction direction

Outliers vs. Influential Outliers vs. Influential ObservationsObservations

Outlier Outlier – An observation that lies outside the overall An observation that lies outside the overall

pattern of the other observationspattern of the other observations

Influential observation Influential observation – Removing this point would markedly change Removing this point would markedly change

the result of the calculationthe result of the calculation

Classwork: Residual Plots WSClasswork: Residual Plots WS

Homework: p177 #3.52 and 3.61Homework: p177 #3.52 and 3.61

Doctor’s for the PoorDoctor’s for the Poor

This will be graded for accuracy!This will be graded for accuracy!

Ch 3 ReviewCh 3 Review

p176 #3.50-3.51, 3.56, 3.59, 3.69, p176 #3.50-3.51, 3.56, 3.59, 3.69, 3.76-3.773.76-3.77