Correlations

Outline What is a correlation? What is a correlation? What is a scatterplot? What is a scatterplot? What type of information is provided by a correlation coefficient What type of information is provided by a correlation coefficient Pearson correlation Pearson correlation How is pearson calculated How is pearson calculated Hypothesis testing with pearson Hypothesis testing with pearson Correlation & causation Correlation & causation Factors affecting correlation coefficient Factors affecting correlation coefficient Coefficient of determination Coefficient of determination Correlation in research articles Correlation in research articles Other types of correlation Other types of correlation

Distinguishing Characteristics of Correlation Correlational procedures involve one sample containing all pairs of X and Y scores Correlational procedures involve one sample containing all pairs of X and Y scores Neither variable is called the IV or DV Neither variable is called the IV or DV Use the individual pair of scores to create a scatterplot Use the individual pair of scores to create a scatterplot

Correlation Coefficient Describes three characteristics of the relationship: Describes three characteristics of the relationship: 1.Direction 2.Form 3.Degree

What Is A Large Correlation? Guidelines: Guidelines: –0.00 to <±.30 – low –±.30 to <±.50 – moderate –>±.50 – high While 0 means no correlation at all, and 1.00 represents a perfect correlation, we cannot say that.5 is half as strong as a correlation of 1.0 While 0 means no correlation at all, and 1.00 represents a perfect correlation, we cannot say that.5 is half as strong as a correlation of 1.0

Pearson Correlation Used to describe the linear relationship between two variables that are both interval or ratio variables Used to describe the linear relationship between two variables that are both interval or ratio variables The symbol for Pearson’s correlation coefficient is r The symbol for Pearson’s correlation coefficient is r The underlying principle of r is that it compares how consistently each Y value is paired with each X value in a linear manner The underlying principle of r is that it compares how consistently each Y value is paired with each X value in a linear manner

Calculating Pearson r The degree to which X & Y vary together (covary) The degree to which X & Y vary apart The degree to which X & Y vary together (covary) Variability due to error and chance

Calculating Pearson r There are 3 main steps to r: There are 3 main steps to r: 1.Calculate the Sum of Products (SP) 2.Calculate the Sum of Squares for X (SS X ) and the Sum of Squares for Y (SS Y ) 3.Divide the Sum of Products by the combination of the Sum of Squares

Pearson Correlation - Formula

1) Sum of Products To determine the degree to which X & Y covary (numerator) To determine the degree to which X & Y covary (numerator) –We want a score that shows all of the deviation X & Y have in common –Sum of Products (also known as the Sum of the Cross-products) –This score reflects the shared variability between X & Y –The degree to which X & Y deviate from the mean together SP = ∑(X – M X )(Y – M Y )

Sums of Product Deviations Computational Formula Computational Formula  n in this formula refers to the number of pairs of scores

2) Sum of Squares X & Y For the denominator, we need to take into account the degree to which X & Y vary separately For the denominator, we need to take into account the degree to which X & Y vary separately –We want to find all the variability that X & Y do not have in common –We calculate sum of squares separately (SS X and SS Y ) –Multiply them and take the square root

2) Sum of Squares X & Y The denominator: The denominator: =

Hypothesis testing with r Step 1) Set up your hypothesis Step 1) Set up your hypothesis –Ho: ρ = 0; There is no correlation in the population between the number of errors and the number of drinks –H1: ρ ≠ 0; There is a correlation in the population between the number of errors and the number of drinks

Hypothesis testing with r Step 2) Find your critical r-score Step 2) Find your critical r-score –Alpha and degrees of freedom α =.05, two-tailed α =.05, two-tailed Degrees of freedom: n – 2 Degrees of freedom: n – 2

Hypothesis testing with r Step 3) Calculate your r-obtained Step 3) Calculate your r-obtained Step 4) Compare the r-obtained to r- critical, and make a conclusion Step 4) Compare the r-obtained to r- critical, and make a conclusion –If r-obtained < r-critical = fail to reject Ho –If r-obtained > r-critical = reject Ho

Correlation and Causality A statistical relationship can exist even though one variable does not cause or influence the other A statistical relationship can exist even though one variable does not cause or influence the other Correlational research CANNOT be used to infer causal relationships between two variables Correlational research CANNOT be used to infer causal relationships between two variables

Correlation and Causality When two variables are correlated, three possible directions of causality When two variables are correlated, three possible directions of causality –1st variable causes 2nd –2nd variable causes 1st –Some 3rd variable causes both the 1st and the 2nd There is inherent ambiguity in correlations There is inherent ambiguity in correlations

Factors Affecting Correlation: Watch out for “outliers” r = -.08 r 2 =.016 r =.85 r 2 =.72

Factors Affecting Correlation: Restriction of Range No relationship here Strong relationship here

Coefficient Of Determination The squared correlation (r 2 The squared correlation (r 2 ) measures the proportion of variability in the data that is explained by the relationship between X and Y Coefficient of Non-Determination (1-r 2 ): percentage of variance not accounted for in Y Coefficient of Non-Determination (1-r 2 ): percentage of variance not accounted for in Y

Correlation in Research Articles Coleman, Casali, & Wampold (2001). Adolescent strategies for coping with cultural diversity. Journal of Counseling and Development, 79,

Other Types of Correlation Spearman’s Rank Correlation Spearman’s Rank Correlation –variable X is ordinal and variable Y is ordinal Point-biserial correlation Point-biserial correlation –variable X is nominal and variable Y is interval Phi-coefficient Phi-coefficient –variable X is nominal and variable Y is also nominal Rank biserial Rank biserial –variable X is nominal and variable Y is ordinal

Example #2 Hours (X) Errors (Y)

Create Scatterplot

Step 1- Set Up Your Hypothesis Ho: Ho: H1: H1:

Step 2 - Find critical r-score Alpha and degrees of freedom Alpha and degrees of freedom α =.05, two-tailed α =.05, two-tailed Degrees of freedom: n – 2 Degrees of freedom: n – 2

Step 3 - Calculate r-obtained

Step 4 - Compare R-obtained To R-critical, & Make A Conclusion Step 4 – Compute r 2