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

This presentation covers the objectives of correlational research, calculating and interpreting r, factors that can corrupt r, the importance of sample size, and applications of correlation in reliability and validity testing.

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

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  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…

  21. 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

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

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