doane chapter 12a

7/29/2019 Doane Chapter 12a

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Bivariate Regression (Part 1)

C h

a p t er

12

Visual Displays and Correlation Analysis Bivariate Regression

Regression Terminology Ordinary Least Squares Formulas

Tests for Significance


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Visual Displays andCorrelation Analysis

Begin the analysis of bivariate data (i.e., twovariables) with a scatter plot .

A scatter plot- displays each observed data pair ( x i , y i ) as a dot

on an X/Y grid- indicates visually the strength of the relationship

between the two variables

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Visu al Disp lay s


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Visu al Disp lay s


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The sample correlation coefficient (r ) measuresthe degree of linearity in the relationship between

X and Y . -1 < r < +1

r = 0 indicates no linear relationship In Excel, use =CORREL(array1,array2),where array1 is the range for X and array2 is therange for Y .


Correlat ion A nalys is

Strong negative relationship Strong positive relationship


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Strong PositiveCorrelation

Weak PositiveCorrelation


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Weak NegativeCorrelation

Strong NegativeCorrelation


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No Correlation

Nonlinear Relation


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r is an estimate of the population correlationcoefficient r (rho).

To test the hypothesis H 0: r = 0, the test statisticis:

The critical value t a is obtained from Appendix Dusing n = n 2 degrees of freedom for any a .

Find the p -value using Excels function=TDIST(t,deg_freedom,tails) or MINITAB.


Tests for Sign if icance


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Equivalently, you can calculate the critical valuefor the correlation coefficient using

This method gives a benchmark for the

correlation coefficient. However, there is no p -value and is inflexible if you change your mind about a .


Tests for Sign if icance


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Step 1: State the HypothesesDetermine whether you are using a one or two-tailed test and the level of significance ( a ).

H 0: r = 0H 1: r 0

Step 2: Calculate the Critical Value

For degrees of freedom n = n -2, look up thecritical value t a in Appendix D, then calculate


Steps in Test in g i f r = 0


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Step 3: Make the DecisionIf the sample correlation coefficient r exceeds the

critical value r a , then reject H 0.If using the t statistic method, reject H 0 if t > t a or if the p -value < a .


Steps in Test in g i f r = 0


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A quick test for significance of a correlation ata = .05 is | r | > 2/ n


Quick Rule for S igni f icance


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As sample size increases, the critical value of r becomes smaller.

This makes it easier for smaller values of thesample correlation coefficient to be consideredsignificant.

A larger sample does not mean that thecorrelation is stronger nor does its significanceimply importance.


Role of Sam p le Size


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Bivariate Regression

Bivariate Regression analyzes the relationshipbetween two variables.

It specifies one dependent (response ) variableand one independent ( predictor ) variable.

This hypothesized relationship may be linear,quadratic, or whatever.


Wh at is B ivar iate Regress ion ?


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Bivariate Regression


Mod el Form


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Regression Terminology

Unknown parameters areb0 Intercept

b1 Slope The assumed model for a linear relationship is

y i = b0 + b1 x i + ei for all observations ( i = 1, 2, , n)

The error term is not observable, is assumednormally distributed with mean of 0 and standarddeviation s .


Mod els and Parameters


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The fitted model used to predict the expected value of Y for a given value of X is

y i = b 0 + b 1 x i


Mod els and Parameters

The fitted coefficients areb 0 the estimated intercept

b 1 the estimated slope Residual is e i = y i - y i . Residuals may be used to estimate s , the

standard deviation of the errors.

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Step 1:- Highlight the data columns.

- Click on the Chart Wizard and choose Scatter Plot

- In the completed graph, click once on the pointsin the scatter plot to select the data

- Right-click and choose Add Trendline- Choose Options and check Display Equation


Fi t t ing a Regress io n o n a Scat ter Plot in Excel


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Fi t t ing a Regress io n o n a Scat ter Plot in Excel


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Ordinary Least Squares Formulas

The ordinary least squares method ( OLS )estimates the slope and intercept of theregression line so that the residuals are small.

The sum of the residuals = 0

The sum of the squared residuals is SSE


Slope and Intercept


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The OLS estimator for the slope is:

The OLS estimator for the intercept is:


Slope and Intercept

or


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We want to explain the total variation in Y aroundits mean ( SST for Total Sums of Squares )

The regression sum of squares ( SSR ) is theexplained variation in Y

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A ssess ing F it


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The error sum of squares ( SSE ) is theunexplained variation in Y

If the fit is good, SSE will be relatively smallcompared to SST .

A perfect fit is indicated by an SSE = 0. The magnitude of SSE depends on n and on the

units of measurement.McGraw-Hill/Irwin 2007 The McGraw-Hill Companies Inc All rights reserved

A ssess ing F it


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R 2 is a measure of relative fit based on acomparison of SSR and SST .


Coeff ic ient o f Determ inat ion

0 < R 2 < 1

Often expressed as a percent, an R 2 = 1 (i.e.,100%) indicates perfect fit.

In a bivariate regression, R 2 = ( r )2

R 2 is a measure of relative fit based on acomparison of SSR and SST .


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The standard error (s yx ) is an overall measure of model fit.


Stand ard Error o f Regress ion

If the fitted models predictions are perfect(SSE = 0), then s yx = 0. Thus, a small s yx

indicates a better fit. Used to construct confidence intervals. Magnitude of s yx depends on the units of

measurement of Y and on data magnitude.


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Standard error of the slope:


Con fidenc e Intervals fo r Slope and Intercept

Standard error of the intercept:


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Confidence interval for the true slope:

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Con fidenc e Intervals fo r Slope and Intercept

Confidence interval for the true intercept:


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If b1 = 0, then X cannot influence Y and theregression model collapses to a constant b0 plusrandom error.

The hypotheses to be tested are:

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Hyp oth esis Tests


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A t test is used with n = n 2 degrees of freedomThe test statistics for the slope and intercept are:


Hyp oth esis Tests

t n-2 is obtained from Appendix D or Excel for agiven a .

Reject H 0 if t > t a or if p -value < a .

Slope:

Intercept:


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Using Exc el


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Usin g MegaStat


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Us in g MINITA B


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Applied Statistics inBusiness and Economics

End of Chapter 12 Part 1 of 2

doane chapter 12a

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