Chapter 10

DESCRIBING RELATIONSHIPS USING CORRELATION AND REGRESSION

Going ForwardYour goals in this chapter are to learn:• How to create and interpret a scatterplot• What a regression line is• When and how to compute the Pearson r• How to perform significance testing of the

Pearson r• The logic of predicting scores using linear

regression and 2r

Understanding Correlations

Correlation Coefficient

• A correlation coefficient is a statistic that describes the important characteristics of a relationship

• It simplifies a complex relationship involving many scores into one number that is easily interpreted

Distinguishing Characteristics

• A scatterplot is a graph of the individual data points from a set of X-Y pairs

• When a relationship exists, as the X scores increase, the Y scores change such that different values Y tend to be paired with different values of X

A Scatterplot Showing the Existence of a Relationship Between the Two Variables

Linear Relationships

• A linear relationship forms a pattern following one straight line

• The linear regression line is the straight line that summarizes a relationship by passing through the center of the scatterplot

Positive and Negative Relationships

• In a positive linear relationship, as the X scores increase, the Y scores also tend to increase

• In a negative linear relationship, as the scores on the X variable increase, the Y scores tend to decrease

Scatterplot of a Positive Linear Relationship

Scatterplot of a Negative Linear Relationship

Nonlinear Relationships

In a nonlinear relationship, as the X scores increase, the Y scores do not only increase or only decrease: at some point, the Y scores alter their direction of change.

Scatterplot of a Nonlinear Relationship

Strength of a Relationship

The strength of a relationship is the extent to which one value of Y is consistently paired with one and only one value of X

•The larger the absolute value of the correlation coefficient, the stronger the relationship

•The sign of the correlation coefficient indicates the direction of a linear relationship

Correlation Coefficients

• Correlation coefficients may range between –1 and +1. The closer to ±1 the coefficient is, the stronger the relationship; the closer to 0 the coefficient is, the weaker the relationship.

• As the variability in the Y scores at each X becomes larger, the relationship becomes weaker

Correlation Coefficient

A correlation coefficient tells you• The relative degree of consistency with which

Ys are paired with Xs• The variability in the group of Y scores paired

with each X• How closely the scatterplot fits the regression

line• The relative accuracy of prediction

A Perfect Correlation (±1)

Intermediate Strength Correlation

No Relationship

The Pearson Correlation Coefficient

Pearson Correlation Coefficient

Describes the linear relationship between two interval variables, two ratio variables, or one interval and one ratio variable.

The computing formula is

])()([])()([

))(()(2222 YYNXXN

YXXYNr

Step-by-Step

Step 1. Compute the necessary components:• • • •

X2)( X

2XXY

•

•

•

•

Y2)( Y

2YN

Step-by-Step

• Step 2. Use these values to compute the numerator

• Step 3. Use these values to compute the denominator and then divide to find r

))(()( YXXYN

])()([])()([ 2222 YYNXXN

Significance Testing of the Pearson r

Two-Tailed Test of the Pearson r

• Statistical hypotheses for a two-tailed test

• This H0 indicates the r value we obtained from our sample is because of sampling error

• The sampling distribution of r shows all possible values of r that occur when samples are drawn from a population in which = 0

0:;0:0 aHH

Two-Tailed Test of the Pearson r

Two-Tailed Test of the Pearson r

• Find appropriate rcrit from the table based on– Whether you are using a two-tailed or one-tailed

test– Your chosen – The degrees of freedom (df) where df = N – 2,

where N is the number of X-Y pairs in the data

• If robt is beyond rcrit, reject H0 and accept Ha

• Otherwise, fail to reject H0

One-Tailed Test of the Pearson r

• One-tailed, predicting positive correlation

• One-tailed, predicting negative correlation

0:;0:0 aHH

0:;0:0 aHH

An Introduction to Linear Regression

Linear Regression

Linear regression is the procedure for predicting unknown Y scores based on known correlated X scores.•X is the predictor variable•Y is the criterion variable•The symbol for the predicted Y score is (pronounced Y prime)

Y

Linear Regression

The equation that produces the value of at each X and defines the straight line that summarizes the relationship is called the linear regression equation.

Y

Proportion of VarianceAccounted For

• The proportion of variance accounted for describes the proportion of all differences in Y scores that are associated with changes in the X variable

• The proportion of variance accounted for equals 2r

Example 1

For the following data set of interval/ratio scores, calculate the Pearson correlation coefficient.

X Y

1 8

2 6

3 6

4 5

5 1

6 3

Example 1 Pearson Correlation Coefficient

• Determine N• Calculate

• Insert each value into the following formula

])()([])()([

))(()(2222 YYNXXN

YXXYNr

,X ,)( 2X ,2XXY

,Y ,)( 2Y,2Y and

Example 1 Pearson Correlation Coefficient

X X 2 Y Y 2 XY

1 1 8 64 8

2 4 6 36 12

3 9 6 36 18

4 16 5 25 20

5 25 1 1 5

6 36 3 9 18

X = 21 X 2 = 91 Y = 29 Y 2 = 171 XY = 81

N = 6

Example 1 Pearson Correlation Coefficient

88.0374.139

123

]185][105[

609486

)29()171(6[])21()91(6[

)29)(21()81(6

)()([])()([

))(()(

22

2222

YYNXXN

YXXYNr

Example 2 Significance Test of the Pearson r

Conduct a two-tailed significance test of the Pearson r just calculated. Use = .05.•df = N – 2 = 6 – 2 = 4•rcrit = 0.811

•Since robt of –0.88 falls beyond the critical value of –0.811, reject H0 and accept Ha.

•The correlation in the population is significantly different from 0

Example 3 Proportion of Variance Accounted For

Calculate the proportion of variance accounted for, using the given data.

Proportion of variance accounted for is

7744.088.0 22 r