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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X
Y
r = .35
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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
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
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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…
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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 time
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What is Next?
• Multiple Regression
• http://home.ubalt.edu/tmitch/632/multip
le%20regression%20palgrave.pdf
• Demonstration of lab 2 analysis
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