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

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

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

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

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

Interpreting r Magnitude Sign As an estimate of explained variance –r 2 = coefficient of determination Proportion of variance shared by 2 variables –1 - r 2 = coefficient of nondetermination Unshared variance –Figure

r =.35 XY 7

r and Causality Large r do not indicate a causal relationship Why? 1)Temporal order 2)Missing “third variables” 8

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

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 10

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 11

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

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 13

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 14

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

Figure

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) 17

Y = 5 + 3(X) 18

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 19

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… 20

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

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