Slides to accompany Weathington, Cunningham & Pittenger (2010),

Chapter 10: Correlational Research

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

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Correltion• Correlational Method

– Vs.

• Correlational Statistic

• -what’s the difference?

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Calculate r• Sum of z score products / N

r = ∑ ZxZy/N

• NOTE: N is number of Pairs

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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

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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

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r = .357

r and Causality• Large r do not indicate a causal

relationship• Why?

1) Temporal order

2) Missing “third variables”

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Corrupting r: Nonlinearity• Sometimes a straight line does not

adequately describe the relationship between two variables

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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

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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

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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

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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

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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

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Applications of r: Validity• Construct

– Convergent • (think of two that converge)

– Discriminant (divergent)• (Think of two that diverge)

• Criterion-related– Concurrent– Predictive

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Figure 10.7

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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)

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Y = 5 + 3(X)

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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

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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 rxy < 1.00:

|(X – Mx| > |(Y’ – My)|

• In sports, the "Sophomore Slump”• May influence your interpretations or

conclusions of data gathered over time22

What is Next?• Multiple Regression• http://home.ubalt.edu/tmitch/632/mu

ltiple%20regression%20palgrave.pdf• Demonstration of lab 2 analysis

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