Confidence Intervals and Hypothesis Testing with Correlation Coefficients

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Confidence Intervals and Hypothesis Testing with
Correlation Coefficients
Aron, Aron Coups, Chapter 3 (Appendix)
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Inspecting Scatterplots The Bivariate Normal
Distribution
Weight
Height
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Inspecting Scatterplots The Bivariate Normal
Distribution
Weight
Height
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Inspecting Scatterplots The Bivariate Normal
Distribution
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Confidence Intervals for Correlation Coefficients
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Confidence Intervals for Correlation Coefficients

• Biased and Unbiased estimators

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Confidence Intervals for Correlation Coefficients

• r is a sample statistic and r is a population
  parameter that represents the correlation between
  two variables (X and Y) in the population.
• Confidence intervals for correlations must be
  computed differently from confidence intervals
  about means (or mean differences) because the
  sampling distribution of r is skewed,
  particularly as r approaches 1 and -1.

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Confidence Intervals for Correlation Coefficients

• To deal with this problem we use the Fisher
  r-to-Zr transformation.
• Formally Zr 0.5 loge (1 r)/(1 - r)
• or, you can look it up in tables (Handout)
• or, use the function Fisher(r) in Excel to find
  Zr and FisherInv(Zr ) to find r.
• The advantage of Zr is that its sampling
  distribution is normal.

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Confidence Intervals for Correlation Coefficients

• We next ask what is the standard error of Zr?
• SEZr or
• Because Zr is normally distributed and its
  standard error(SEZr) is defined, we can place a
  95 confidence interval around Zr as follows
• CI Zr 1.96 (SEZr )
• The limits of the CI can then be converted back
  to rs using FisherInv (Zr) in Excel.

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Confidence Intervals for Correlation Coefficients
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Testing whether r is different from 0

• When r 0 and the sample size is relatively
  large, the sampling distribution of r will be
  normal with a standard error of
• which can be estimated by
• Therefore, we can calculate a t-statistic for a
  correlation coefficient as

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Testing whether r is different from 0
0.43
r
32
N
0.82
(1-r
2
)
30
N - 2
2.609
t
obs
2.042
0.014
t
p
crit(2-tailed)
1.697
0.007
t
p
crit(1-tailed)
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Testing whether r is different from 0
-0.41
r
29
N
0.83
(1-r
2
)
27
N - 2
-2.336
t
obs
2.052
0.03
t
p
crit(2-tailed)
1.703
0.01
t
p
crit(1-tailed)
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Testing whether r is different from 0
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Testing whether r is different from 0
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Testing whether r is different from a known r

• When r ? 0 the sampling distribution of r will
  not be normal (in general) so the Fisher
  transform is used.
• Non-directional test (that r .5, a .05)

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Testing whether r is different from a known r

• When r ? 0 and the sampling distribution of r
  will not be normal (in general) so the Fisher
  transform is used.
• Directional test (that r gt .5, a .05)

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Testing whether r is different from a known r
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Testing whether r is different from a known r
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Testing whether two independent correlations
differ from each other

• Again, r is converted to Zr and the
  z-distribution is consulted

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Testing whether two independent correlations
differ from each other
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Testing whether two independent correlations
differ from each other