Topics:%20Significance%20Testing%20of%20Correlation%20Coefficients

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Topics Significance Testing of Correlation
Coefficients

• Inference about a population correlation
  coefficient
• Testing H0 ?xy 0 or some specific value
• Testing H0 ? xy 0 for two or more correlations
  based on the same sample
• Inference about a difference between population
  correlation coefficients
• Testing H0 ? xy1 - ? xy2 0 (or ? xy1 ? xy2 )

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Inference about a Correlation Coefficient

• Purpose to determine whether two variables (X
  and Y) are linearly related in the population.
• H0 ?xy 0
• H0 ?xysome specified value

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Test of Correlation Coefficient

• H0 ?xy some specified value
• H1 ?xy not some specified value ( lt or gt than
  some specified value)
• Transform sample and population correlation
  coefficients to Zr and Z?
• Calculate t zobserveddistance of transformed rxy
  from the transformed population ?xy in standard
  error points
• Test against zcritical (determined from table for
  chosen level of significance)

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Sampling Distribution of rxy
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Example

• Study of relationship between achievement
  motivation and performance in school (grade point
  average). Theory and prior research suggests
  that the correlation between these two variables
  is positive and moderately high (.50)
• The observed correlation in this study was .75
  based on N63

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Test of Correlation Coefficient One Sample

• H0 ?xy .50
• H1 ?xy not .50
• Level of Significance .05
• Verify Assumptions
• Independence of score pairs
• Bivariate Normality
• n gt 30

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Assumptions

• Independence where the pair of scores for any
  particular student is independent of the pair of
  scores of every other student.
• Bivariate Normality For each value of X, the
  values of Y are normally distributed for each
  value of Y the values of X are normally
  distributed each variable normally distributed
• Sample Size n gt 30

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Bivariate Normal
For each value of X the Y scores are normally
distributed
For each value of Y the X scores are normally
distributed
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Example Contd

• Find Fisher Z transformation for rxy and ?xy
  (from a Table I)
• r .75 so Zr .973
• ? .50 so Z? .549
• Set up Zrobserved Zr-Z?/sZ to get distance of
  Zr from Z? in standard error points
• Computation formula for Zrobserved
• (Zr-Z?) (sqrt n-3)
• (.973 - .549)/7.75
• (.424)(7.75) 3.29

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Example Contd

• Find zcritical (from table or memory) 1.96
• Decision Rule
• Reject H0 if absolute value of zrobserved gt 1.96
  (3.29 is greater than 1.96)
• Do not reject H0 if absolute value of zrobserved
  lt 1.96
• Conclusion the relationship between achievement
  motivation and school performance (grade point
  average) is greater than the specified value of
  .50

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The Simple Approach When H0 ?xy 0

• H0 ?xy 0 H1 ?xy gt 0
• Sample size 102
• r .24
• Compare robserved with r critical (.05,df100)
  .1638 (from Table G)
• Since .24 gt .1638 can reject the null
  hypothesis and conclude that there is a positive
  correlation in the population--our best estimate
  of that correlation is .24

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Example Testing Two or More Correlation
Coefficients

• Working example Suppose the following measures
  were collected on 82 subjects GPA,.
  Self-concept, and locus of control
  (internal-external)

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Testing H0 ?xy 0 for Two or More Correlations
Based on Same Sample (N82)
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Testing H0 ?xy 0 for Two or More Correlations
Based on Same Sample

• H0
• H1 (non-directional)
• Level of significance ? .01 (level of
  significance)
• Assumptions
• Number of Variables
• Number of dfs
• Critical Value (from Table H)
• Decisions and conclusions

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Inference about Difference between Population
Correlation Coefficients

• To determine whether or not the observed
  difference between two correlation coefficients
  (r1-r2) may be due to chance or represents a
  difference in population coefficients

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Example Testing Difference between Two
Correlation Coefficients

• To determine whether or not the observed
  difference between two correlation coefficients
  (r1-r2) may be due to chance or represents a
  difference in population coefficients
• In the Overachievement Study, the correlation
  between SAT scores and GPA was .0214 for the
  sample of 40 subjects. However the correlation
  between SAT and GPA for men was -.2369 and for
  women was .3760.

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Example Differences (cont)

• H0
• H1 (non directional)
• Significance Level ? .05
• Check assumptions
• Convert sample rs to Zrs (Table I) male Zrmale
  -.239 female Zrfemale .394
• Compute standard error of the difference between
  correlations srmale-rfemale .34 (via
  formula)
• Calulate zrmale- rfemale(observed) Zrmale- -
  Zrfemale/ srmale-rfemale -.633/.34 -1.86
• Find z rmale-rfemale(critical) /-1.96 (.05,
  two-tailed)
• Decision and Conclusion