Chapter Thirteen Copyright © 2006 John Wiley & Sons, Inc. Bivariate Correlation and Regression

Chapter Thirteen Chapter Thirteen

Copyright © 2006John Wiley & Sons, Inc.

Bivariate Correlation

and Regression

John Wiley & Son, Inc 2

1. To comprehend the nature of correlation analysis.

2. To understand bivariate regression analysis.

3. To become aware of the coefficient of determination, R2.

4. To understand Spearman Rank Order correlation.

Learning Objectives


To understand bivariate regression analysis.

Bivariate Analysis of Association

• Bivariate Techniques– Statistical methods of analyzing the relationship

between two variables.• Multivariate Techniques

– When more than two variables are involved• Independent Variable (Predictor)

– Affects the value of the dependent variable• Dependent Variable (Criterion)

– explained or caused by the independent variable


• Types of Bivariate Procedures– Bivariate regression– Pearson product moment correlation– Spearman rank-order correlation– Two group t-tests – chi-square analysis of cross-tabulation or

contingency tables – ANOVA (analysis of variance) for two groups

Bivariate Analysis of Association



Bivariate Regression• Bivariate Regression Defined

– Analyzing the strength of the linear relationship between the dependent variable and the independent variable.

• Nature of the Relationship– Plot in a scatter diagram

• Dependent variable– Y is plotted on the vertical axis

• Independent variable– X is plotted on the horizontal axis

• Linear Relationship

• Nonlinear Relationship



Y

XA - Strong Positive Linear Relationship

Exhibit 13.1Types of Relationships Found in Scatter Diagrams

Bivariate Regression Example

To understand bivariate regression analysis.Bivariate

Regression


Y

X

B - Positive Linear Relationship



Regression


Y

XC - Perfect Negative Linear Relationship



Regression


XD - Perfect Parabolic Relationship


Y


Regression


Y

XE - Negative Curvilinear Relationship



Regression


Y

X

F - No Relationship between X and Y



Regression


where

Y = dependent variable

X = independent variablee = error

b = estimated slope of the regression line

a = estimated Y intercept

Y = a + bX + e

• Least Squares Estimation Procedure– Results in a straight line that fits the actual

observations better than any other line that could be fitted to the observations.


Regression


Values for a and b can be calculated as follows:

XiYi - nXYb =

X2i - n(X)2

n = sample size

a = Y - bX

X = mean of value X

Y = mean of value y


Regression


To become aware of the coefficient of determination, R2.

• The Regression Line– Predicted values for Y, based on calculated values.

• Strength of Association: R2

– Coefficient of Determination, R2 • The measure of the strength of the linear relationship

between X and Y.• Coefficient of determination measures the percentage of

the total variation in Y that is explained by the variation in X

• The R2 statistic ranges from 0 to 1.

Bivariate Regression


R2 =explained variance

total variance

explained variance =

total variance - unexplained variance

R2 =total variance - unexplained variance

total variance

= 1 -unexplained variance

total variance




R2 = 1 -unexplained variance

total variance

= 1 - (Yi - Yi)2n

I = 1

(Yi - Y)2n

I = 1




• Statistical Significance of Regression Results

• The total variation is a measure of variation of the observed Y values around their mean.

• It measures the variation of the Y values without any consideration of the X values.

Total variation =

Explained variation + Unexplained variation




Total variation: Sum of squares (SST)

SST = (Yi - Y)2n

i = 1

Yi 2n

i = 1=

Yi 2n

i = 1

n




Sum of squares due to regression (SSR)

SSR = (Yi - Y)2n

i = 1

Yi

n

i = 1= a

Yi

n

i = 1

nb Xi Yi

n

i = 1+

2




Error sums of squares (SSE)

SSE = (Yi - Y)2n

i = 1

Y2i

n

i = 1= a Yi

n

i = 1b XiYi

n

i = 1




0 XXiX

(X, Y)

a

Y

Total Variation

Explained variation

Y

Unexplained variation

Exhibit 13.7 Measures of Variation in a Regression

Yi =a + bXi


• Hypotheses Concerning the Overall Regression

– Null Hypothesis Ho

• There is no linear relationship between X and Y.

– Alternative Hypothesis Ha:• There is a linear relationship between X and Y.




• Hypotheses about the Regression Coefficient b

– Null Hypothesis Ho

• b = 0

– Alternative Hypothesis Ha:• b 0

– The appropriate test is the t-test.




Correlation Analysis To comprehend the nature of correlation analysis

• Correlation for Metric Data - Pearson’s Product Moment Correlation

– Correlation• The degree to which changes in one variable (the dependent

variable) are associated with the changes in another

– Correlation analysis• Analysis of the degree to which changes in one variable are

associated with changes in another variable.

– Pearson’s product moment correlation• Correlation analysis technique for use with metric data


R = +- R2√

R can be computed directly from the data:

R = n XY - ( X) - ( Y)

[n X2 - ( X) 2] [n Y2 - Y)2]√

Correlation Analysis To comprehend the nature of correlation analysis


To understand Spearman Rank Order correlation.

• Correlation Using Ordinal Data: Spearman’s Rank-Order Correlation– To analyze the degree of association between two ordinal

scaled variables.– Spearman’s Coefficient of Rank-Order Coefficient R—the

appropriate measure for analyzing ordinal data and like the coefficient of correlation R, has a lower limit of -1 and an upper limit of +1.

• Conclusions regarding rankings:– 1. Positively correlated– 2. Negatively correlated– 3. Independent

Correlation Analysis


• Bivariate Analysis of Association

• Bivariate Regression

• Correlation Analysis

SUMMARY


The End

Copyright © 2006 John Wiley & Son, Inc

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Chapter Thirteen Copyright © 2006 John Wiley & Sons, Inc. Bivariate Correlation and Regression