Chapter 15 Correlation 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 Explain & compute Pearson correlation to variables’ relationship 2 Test hypothesis about population correlation with sample r 3 Explain & compute Spearman correlation 4 Compute point-biserial correlation and phi-coefficient 5 Explain & compute linear regression equation to predict Y values 6 Evaluate significance of regression equation

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

Sum of Products (SP) Similar to SS (sum of squared deviations) Measures the amount of covariability between two variables

SP – Computational formula Definitional formula emphasizes SP as the sum of two difference scores Computational formula results in easier calculations

Calculation of the Pearson correlation Ratio comparing the covariability of X and Y (numerator) with the variability of X and Y separately (denominator)

Figure 15.4 Example 15.3 Scatterplot

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

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? A 0.75 B 0.35 C -0.75 D -0.35

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? A 0.75 B 0.35 C -0.75 D -0.35

Learning Check TF Decide if each of the following statements is True or False. T/F A set of n = 10 pairs of X and Y scores has ΣX = ΣY = ΣXY = 20. For this set of scores, SP = –20 If the Y variable decreases when the X variable decreases, their correlation is negative

Answer TF True False The variables change in the same direction, a positive correlation

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.

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

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

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 Point-biserial correlation Spearman correlation 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? A Pearson correlation B Spearman correlation C Point-biserial correlation D Phi-coefficient

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? A Pearson correlation B Spearman correlation C Point-biserial correlation D Phi-coefficient

Learning Check Decide if each of the following statements is True or False. T/F The Spearman correlation is used with dichotomous data In a test for significance of a correlation, the null hypothesis states that the population correlation is zero.

Answer The Spearman correlation uses ordinal (ranked) data False The Spearman correlation uses ordinal (ranked) data True Zero indicates no relationship

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

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

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

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? A 14 B 25 C 31 D Cannot be determined

Learning Check - Answer A linear regression has b = 3 and a = 4. What is the predicted Y for X = 7? A 14 B 25 C 31 D Cannot be determined

Learning Check Decide if each of the following statements is True or False. T/F It is possible for the regression equation to have none of the actual data points on the regression line. If r = 0.58, the linear regression equation predicts about one third of the variance in the Y scores.

Answer True The line is an estimator. When r = .58, r2 = .336

Any Questions? Concepts? Equations?