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