Correlational Research: Objectives, Calculating r, Interpreting r, Corrupting r, Sample Size, Applications of r

1. Slides to accompany Weathington, Cunningham & Pittenger (2010), Chapter 10: Correlational Research

2. Objectives • Correlation • Corrupting r • Sample size and r • Reliability and r • Validity and r • Regression • Regression to the mean

3. Correltion • Correlational Method • Vs. • Correlational Statistic • -what’s the difference?

4. Calculate r • Sum of z score products / N r = ∑ ZxZy/N • NOTE: N is number of Pairs

5. Correlation • It’s about linear relationship • As X increases, so does Y (positive) • As X increases, Y decreases (negative • Relationships vary in terms of their “togetherness” • Figure 10.1

6. Interpreting r • Magnitude • Sign • As an estimate of explained variance • r2 = coefficient of determination • Proportion of variance shared by 2 variables • 1 - r2 = coefficient of nondetermination • Unshared variance • Figure 10.2

7. r = .35

8. r and Causality • Large r do not indicate a causal relationship • Why? • Temporal order • Missing “third variables”

9. Corrupting r: Nonlinearity • Sometimes a straight line does not adequately describe the relationship between two variables

10. Corrupting r: Truncated Range • See Figure 10.4 • Develops when poor sampling biases the results • If sample fails to capture normal range of possible scores, your r will reflect this abnormal variance

11. Corrupting r: Extreme Scores • Extreme/multiple populations • If a subgroup in your sample is dramatically different than the rest of your sample r may be inaccurate • Outliers • If you have a few scores that are very large or small this can affect r

12. Sample Size Matters • Just as M reflects µ, r reflects ρ • Your estimate is more accurate as your confidence interval around it decreases in size • A larger sample size tends to help • See Table 10.1

13. Applications of r: Reliability • Test-retest • Relating test scores from two administrations • Interrater • Correlating ratings from two raters • Internal consistency (Cronbach’s Alpha α) • Relating scores on multiple items in a test with each other (agreement) • Should be strong if measuring the same construct

14. Improving Test Reliability • Include more items in your scale • Same principle as taking more measurements or replicating your study multiple times • Average of 15 measurements more reliable than average of 3 • Can use Spearman-Brown prophecy formula to tell you how many more items you need to add to an existing measure

15. Applications of r: Validity • Construct • Convergent • (think of two that converge) • Discriminant (divergent) • (Think of two that diverge) • Criterion-related • Concurrent • Predictive

16. Figure 10.7

17. Regression • Using r to predict one variable from another • Translating r into an equation: • Y’ = a + b(X) • b = ΔY/ΔX • Y’ = 5 + 3X  As X increases 1, Y increases 3, starting from Y = 5 when X = 0 • (See Fig 10.8 for 4 reg lines)

18. Y = 5 + 3(X)

19. Regression Lines • Line of best fit Σ(Y – Y’) = 0 • Unless r = 1.00, Y’ is best we can do • Standard error of estimate = SD for Y around Y’ • Can build CI around this

20. Mediation & Moderation • Mediation occurs when the relationship between X and Y is partially or fully explained by the presence of a mediator, M • Moderation occurs when the relationship between X and Y is different depending on the level of some third variable, Z • It’s easier to understand with figures…

22. Regression to the Mean (fig 10.11) • A threat to internal validity • Over time, scores will tend toward their M • When rxy < 1.00: |(X – Mx| > |(Y’ – My)| • In sports, the "Sophomore Slump” • May influence your interpretations or conclusions of data gathered over time

23. What is Next? • Multiple Regression • http://home.ubalt.edu/tmitch/632/multiple%20regression%20palgrave.pdf • Demonstration of lab 2 analysis