Chapter Thirteen

Bivariate Correlation and Regression

Chapter Thirteen

2

To comprehend the nature of correlation analysis.

To understand bivariate regression analysis.

To become aware of the coefficient of determination of R .

To understand Spearman rank-order correlation.

Chapter Thirteen ObjectivesChapter Thirteen Objectives

Chapter Thirteen

Bivariate Techniques:• Statistical methods of analyzing the relationship between variables.

Independent Variable:• Variable believed to affect the value of the dependent variable.

Bivariate Analysis of Association

Chapter Thirteen

Dependent Variable:• Variable expected to be explained or caused by the independent variable.

Bivariate Regression Analysis:• The analysis of the strength of the linear relationship between

variables when one is considered the independent variable and the other is the dependent variable.

Bivariate Analysis of Association

Chapter Thirteen

No Apparent Relationship Between X and Y

X

Y

Perfect Positive Relationship Between X and Y

X

Y

Y

X

Perfect Negative Relationship Between X and Y Parabolic Relationship Between X and Y

X

Y

Types of RelationshipsAs Found in Scatterplot Diagrams

Chapter Thirteen

General Positive Relationship Between X and Y

X

Y

No Apparent Relationship Between X and Y

X

YY

X

Negative Curvilinear Relationship Between X and Y

General Negative Relationship Between X and Y

X

Y

Types of RelationshipsAs Found in Scatterplot Diagrams

Chapter Thirteen

X

Y

• Used to fit data for X and Y not plotted;• Enables estimation of non-plotted data points;• Results in a straight line that fits the actual observations (plotted dots) better than any other line that could be fit to the observations.

Least-Square Estimation Procedure

Chapter Thirteen

Values for “a” and “b” can be calculated as follows:

X iY i - nXYb =

X2i - n(X)2

a = Y - bXn = sam p le s iz e

X = m e an o f v a lu e X

Y = m e a n o f v a lu e y

W h e re :

W here:Y = dependen t variab le

X = independen t variab le

e = erro r

b = estim ated slope of the regression line

a = es tim ated Y in tercep t

Y = a + bX + e

E stim a ting the best line o f fit:

Least-Square Estimation Procedure

Chapter Thirteen

Coefficient of Determination:• Percentage of the total variation in the dependent variable explained by the manipulation of the independent variable(s).

Pearson Correlation:• Analysis of the degree to which changes in one variable are associated with changes in another for use with metric data.

The Strength of Association - R :• The coefficient of determination: the percentage of the total variation in the dependent variable explained by the independent variable.

R 2 = T o ta l V aria tio n - U nex p la in ed V aria tionT o ta l V aria tion

R = + o r - R 2

R2 = 1 - (Yi - Yi) 2n

I = 1

(Yi - Y) 2n

I = 1

2

Measures of Association

Chapter Thirteen

Total Variation: Sum of Squares (SST)

SST = (Yi - Y)2n

i = 1

Yi 2n

i = 1=

Yi 2n

i = 1

n

Sum of Squares

Chapter Thirteen

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

Sum of Squares

Chapter Thirteen

Error Sums of Squares (SSE)

SSE = (Yi - Y)2n

i = 1

Y2i

n

i = 1= a Yi

n

i = 1 b XiYi

n

i = 1

Sum of Squares

Chapter Thirteen

CorrelationAssessing Measures of Association

Measure of Association using interval or ratio data.

Measure of Association using ordinal or rank order data.

Chapter Thirteen

Measures of Association:

• Do not mean there is a causal relationship between the relevant variables;

• Could simply represent coincidence between the relevant variables;

• Should be taken in context and with the timeliness of both data sets in mind;

• Can be used in conjunction with cross tabulations of the relevant data to add another perspective to the results.

CorrelationAssessing Measures of Association

Chapter Thirteen

Bivariate AnalysisLeast SquaresMeasures of AssociationScatterplotsSum of Squares

Index

Index