Chapter 15Correlation and Regression

PowerPoint Lecture Slides

Essentials of Statistics for the Behavioral Sciences Seventh Edition

by Frederick J. Gravetter and Larry B. Wallnau

Chapter 15 Learning Outcomes

Concepts to review

• Sum of squares (SS) (Chapter 4)– Computational formula– Definitional formula

• z-Scores (Chapter 5)

• Hypothesis testing (Chapter 8)

15.1 Introduction to Correlation and Regression

• Measures and describes a relationship between two variables.

• Characteristics of relationships– Direction (negative or positive)– Form (linear is most common)– Strength

Figure 15.1 Scatterplot for correlational data

Figure 15.2 Examples of positive and negative relationships

Figure 15.3 Examples of different values

for linear relationships

15.2 The Pearson Correlation

• Measures the degree and the direction of the linear relationship between two variables

• Perfect linear relationship – Every change in X has a corresponding

change in Y– Correlation will be –1.00 or +1.00

y separatelY and X of variablity

Y and X ofity covariabilr =

Sum of Products (SP)

• Similar to SS (sum of squared deviations)

• Measures the amount of covariability between two variables

∑ −−= ))(( YX MYMXSP

SP – Computational formula

• Definitional formula emphasizes SP as the sum of two difference scores

• Computational formula results in easier calculations

n

YXXYSP ∑ ∑∑ −=

Calculation of the Pearson correlation

• Ratio comparing the covariability of X and Y (numerator) with the variability of X and Y separately (denominator)

YX SSSS

SPr =

Figure 15.4 Example 15.3 Scatterplot

Pearson Correlation and z-scores

• Pearson correlation formula can be expressed as a relationship of z-scores.

1

1

−=

−=

∑

∑

n

zzr :Population

n

zzr :Sample

YX

YX

Learning Check

• A scatterplot shows a set of data points that are clustered loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data?

Learning Check

• A scatterplot shows a set of data points that are clustered loosely around a line that slopes down to the right. Which of the following values would be closest to the correlation for these data?

Learning Check TF

• Decide if each of the following statements is True or False.

Answer TF

20402010

)20)(20(20 −=−=−=SP

15.3 Using and Interpreting the Pearson Correlation

• Correlations used for prediction

• Validity

• Reliability

• Theory verification

Figure 15.5 Number of churches and number of serious crimes

Interpreting correlations

• Correlation does not demonstrate causation

• Value of correlation is affected by the range of scores in the data

• Extreme points – outliers – have an impact

• Correlation cannot be interpreted as a proportion.– To show the shared variability, need to square

the correlation

Figure 15.6 Restricted range and correlation

Figure 15.7 Influence of outlier on correlation

Coefficient of determination

• Coefficient of determination measures the proportion of variability in one variable that can be determined from the relationship with the other variable.

2rionDeterminat of oefficientC =

Figure 15.8 Three degrees of linear relationship

15.4 Hypothesis Testing with the Pearson Correlation

• Pearson correlation is usually computed for sample data, but used to test hypotheses about the relationship in the population.

• Population correlation shown by Greek letter rho (ρ)

• Nondirectional: H0: ρ = 0 and H1: ρ ≠ 0

• Directional: H0: ρ ≤ 0 and H1: ρ > 0

Figure 15.9 Correlation of sample and population

Hypothesis Test for Correlations

• Sample correlation used to test population ρ• Degrees of freedom (df) = n – 2• Hypothesis test can be computed using

either t or F.• Critical Values have been computed

– See Table B.6– A sample correlation beyond ± Critical Value

is very unlikely– A sample correlation beyond ± Critical Value

leads to rejecting the null hypothesis.

Partial correlation

• A partial correlation measures the relationship between two variables while controlling the influence of a third variable by holding it constant

)1)(1(

)(22yzxz

yzxyxyzxy

rr

rrrr

−−

−=⋅

Figure 15.10 Controlling the impact of a third variable

15.5 Alternative to the Pearson Correlation

• Pearson correlation has been developed– for linear relationships– for interval or ratio data

• Other correlations have been developed for– non-linear data– other types of data

Spearman correlation

• Pearson correlation formula is used with data from an ordinal scale (ranks)– Used when both variables are measured on

an ordinal scale– Used when relationship is consistently

directional but may not be linear

Figure 15.11 Consistent nonlinear positive relationship

Figure 15.12 Scatterplot showing scores and ranks

Ranking tied scores

• Tie scores need ranks for Spearman correlation

• Method for assigning rank– List scores in order from smallest to largest– Assign a rank to each position in the list– When two (or more) scores are tied, compute

the mean of their ranked position, and assign this mean value as the final rank for each score.

Special formula for the Spearman correlation

• The ranks for the scores are simply integers

• Calculations can be simplified– Use D as the difference between the X rank

and the Y rank for each individual to compute the rs statistic

)1(

61

2

2

−−= ∑

nn

Drs

Point-Biserial Correlation

• Measures relationship between two variables– One variable has only two values

(dichotomous variable)

• Same situation as the independent samples t-test in Chapter 10– Point-biserial r2 has same value as the r2

computed from t-statistic– t-statistic evaluates the significance – r statistic measures its strength

Phi Coefficient

• Both variables (X and Y) are dichotomous– Both variables are re-coded to values 0 and 1– The regular Pearson formulas is used

Learning Check

• Participants were classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship?

Learning Check - Answer

• Participants were classified as “morning people” or “evening people” then measured on a 50-point conscientiousness scale. Which correlation should be used to measure the relationship?

Learning Check

• Decide if each of the following statements is True or False.

Answer

15.6 Introduction to Linear Equations and Regression

• The Pearson correlation measures a linear relationship between two variables

• The line through the data– Makes the relationship easier to see– Shows the central tendency of the relationship– Can be used for prediction

Figure 15.13 Regression line

Linear equations

• General equation for a line– Equation: Y = bX + a– X and Y are variables– a and b are fixed constant

Figure 15.14 Graph of a linear equation

Regression

• Regression is the method for determining the best-fitting line through a set of data– The line is called the regression line

• Ŷ is the value of Y predicted by the regression equation for each value of X

• (Y- Ŷ) is the distance of each data point from the regression line: the error of prediction

• Regression minimizes total squared error

Figure 15.15 Distance between data point & the predicted point

Regression equations

• Regression line equation: Ŷ = bX + a

• The slope of the line, b, can be calculated

• The line goes through (MX,MY) so

X

Y

X s

sr or

SS

SPb =

XY bMMa −=

Figure 15.16 X and Y points and regression line

Figure 15.17 Perfectly fit regression line and regression line for Example 15.13

Correlation and the standard error

• Predicted variability in Y scores:SSregression = r2 SSY

• Unpredicted variability in Y scores: SSresidual = (1 - r2) SSY

Standard error of estimate

• Regression equation makes prediction

• Precision of the estimate is measured by the standard error of estimate

2

)ˆ( 2

−

−= ∑

n

YY

dfSSresidual

Testing significance of regression

• Analysis of Regression– Similar to Analysis of Variance– Uses an F-ratio of two Mean Square values– Each MS is a SS divided by its df

Mean squares and F-ratio

residual

residualresidual df

SSMS =

regression

regressionregression df

SSMS =

residual

regression

MS

MSF =

Figure 15.18 Partitioning of the SS and df in Analysis of Regression

Figure 15.19 Plot of data in Demonstration 15.1

Learning Check

• A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7?

Learning Check - Answer

• A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7?

Learning Check

• Decide if each of the following statements is True or False.

Answer