Understanding Correlations: Analyzing Relationships Between Variables in Statistics

1. Intro to Stats Correlations

2. Correlation Coefficients • Captures how the value of one variable changes when the value of the other changes • Use it when: • Test the relationship between variables • Only two variables at a time

3. Correlation coefficient • Ranges from -1 to +1 • A Pearson correlation is based on continuous variables • Important to remember this is a relationship for a group, not each person • Reflects the amount of variability shared by two variables

4. Computations rxy = n ΣXY - ΣX ΣY [n ΣX2 – (ΣX)2][n ΣY2 - (ΣY)2] • rxy = correlation coefficient between x & y • n = size of sample • X = score on X variable • Y = score on Y variable

5. Example 1 • A study of the relationship between years of post-high school education and income (in 1,000s)

6. Example 1 • 1. State hypotheses • Null hypothesis: no relationship between years of education and income • ρeducation*income = 0 • Research hypothesis: relationship between years of education and income • reducation*income ≠ 0

7. Computations rxy = n ΣXY - ΣX ΣY [n ΣX2 – (ΣX)2][n ΣY2 - (ΣY)2] • rxy = correlation coefficient between x & y • n = size of sample • X = score on X variable • Y = score on Y variable

8. Example 1 • 6. Determine whether the statistic exceeds the critical value • .95 > .81 & .92 • So it does exceed the critical value • 7. If over the critical value, reject the null • & conclude that there is a relationship between years of education and income

9. Example 1 • In results • There was a significant positive correlation between years of education and income, such that income increased as years of education increased, r(4) = .95, p < .05 (can say p < .01).

10. Why you can’t say “x caused y” • Even though you may suspect there’s a causal relationship you can only make causal statements if: • X definitely preceded Y • X was manipulated so that it was the only probable factor that could cause changes in Y • When talking correlations, you can use “relationship”, “relate”, “associated”

11. Figure: Scatterplot Income (in 1,000s) Years of Education

12. Table: Correlation Matrix

13. Interpretations .80 to 1.0 Very strong .60 to .80 Strong .40 to .60 Moderate .20 to .40 Weak .00 to .20 Weak/ None

14. Interpretations • Coefficient of Determination • Percentage of variance in one variable that is accounted for by variance in the other • Square the correlation coefficient • If r = .70 r2 = .49 49% of variance is shared (or variance in one is explained by variance in other)

15. Coefficient of Determination • Cases/ people will have different scores on measures • Correlations reflect the extent to which people’s scores tend to “move” together • Variables are correlated if they share variability • So coefficient of determination estimates how much of the differences among people on one measure is associated with differences among people on the other measure