Correlation Examining the relationship between interval-ratio variables

Transcription

1 Correlation Examining the relationship between interval-ratio variables Young Americans Stress Hours of Exercise Y = X + e Older Americans Stress Hours of Exercise Y = X + e

2 PEARSON CORRELATION COEFFICIENT (r) A measure of association reflecting both the strength and the direction of the association between two interval-ratio level variables. Definitional formula: Interpretation: Simultaneously indicates the strength and direction of the relationship and how tightly the points are clustered about the line Ranges from -1.0 to 1.0 Values close to 0 indicate no relationship; values close to -1.0 or 1.0 indicate strong relationships Rule of thumb (in absolute values):.00 to.30 = weak.31 to.60 = moderate.61 to 1.00 = strong

3 Example: relationship between Education and Income for a sample of 12 Education Income Name (X) (Y) Shirley Bob Dwayne Phil Lenora Roberta Hanna Karen Ron May Steve Bo Income by Years of education Income in $1,000's Years of Education

4 Educ. Inc. Name (X) (Y) X - X (X - X ) 2 (Y - Y) (Y - Y ) 2 (X -X )(Y - Y) Shirley Bob Dwayne Phil Lenora Roberta Hanna Karen Ron May Steve Bo SUM Mean of X = S X / N Mean of Y = S Y / N Variation of X = S (X - X ) 2 Variation of Y = S (Y - Y ) 2 Covariation of X and Y = S (X - X )(Y - Y )

5 Educ. Inc. Name (X) (Y) X - X (X - X ) 2 (Y - Y) (Y - Y ) 2 (X -X )(Y - Y) Shirley Bob Dwayne Phil Lenora Roberta Hanna Karen Ron May Steve Bo SUM Mean of X = S X / N = 160 / 12 = Mean of Y = S Y / N = 275 / 12 = Variation of X = S (X - X ) 2 = Variation of Y = S (Y - Y ) 2 = Covariation of X and Y = S (X - X )(Y - Y ) = SO: r = / [(246.68)(848.96)] =.85 = High, positive correlation between education and income

6 Hypothesis testing for Pearson s r We want to know if the relationship revealed by the sample r reflects a. a real relationship between the two variables in the population b. chance sampling error when in reality the two variables are unrelated in the population Follow the same 6 steps as before: 1. Check Assumptions Random sample Variables roughly normally distributed in the population Linear relationship Homoscedasticity Sampling distribution is normally distributed 2. State the research and null hypotheses H 1 : r 0 H 0 : r = 0 3. Identify the sampling distribution and the test statistic assuming that the assumptions above are met, the sampling distribution of r will follow the t-distribution so the test statistic is the t-statistic (t-obtained) 4. Choose an alpha-level and locate the region of rejection Find the critical value of t in Appendix Table B with df = N 2 5. Calculate the test-statistic: 6. Make a decision. Reject the null if t-obtained is larger than the t-critical

7 ELABORATION OF BIVARIATE RELATIONSHIPS: Examining the net relationship between an independent variable (X) and the dependent variable (Y) after controlling for one or more other variables (Z) Goal is to determine whether the raw observed bivariate correlation between X and Y represents a direct relationship, or is more appropriately attributable to the association of the variables with a control variable Does the relationship between X and Y persist when Z is controlled? Type of Relationship Direct Relationship Spurious Relationship Intervening Relationship Possible outcomes after introducing statistical controls: Change when Z is controlled (difference between bivariate and partial correlation) correlation between X and Y does not change or changes little correlation is reduced substantially or completely wiped out correlation is reduced substantially or completely wiped out Theoretical / Causal Implications Support for the idea that X has a direct effect on Y Contradicts the idea that X has a direct effect on Y Supports the idea that X affects Y but does so by affecting Z Suppressing Relationship correlation increases Supports the idea that X affects Y Conditional Relationship / impossible to detect with a single Interaction correlation coefficient Supports the idea that the relationship between X and Y differs under different conditions

8 Partial correlation coefficient: Indicates the strength and direction of the relationship between X and Y when a third variable, Z, is controlled. where: r xy.z represents the correlation between X and Y while controlling for Z r xy represents the bivariate correlation between X and Y r xz represents the bivariate correlation between X and Z r yz represents the bivariate correlation between Y and Z Interpretation: the relationship between X and Y while controlling for the influence of Z strength and direction same as with bivariate correlation