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Correlation Mechanics

Correlation Mechanics. Covariance. The variance shared by two variables When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg) cov (x,y) = pos. When X and Y move in opposite directions cov (x,y) = neg. When no constant relationship

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Correlation Mechanics

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  1. Correlation Mechanics

  2. Covariance • The variance shared by two variables • When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg) • cov (x,y) = pos. • When X and Y move in opposite directions • cov (x,y) = neg. • When no constant relationship • cov (x,y) = 0

  3. Covariance • Covariance is not easily interpreted on its own and cannot be compared across different scales of measurement • Solution: standardize this measure • Pearson’s r:

  4. Covariance and correlation

  5. Computational formula

  6. Example • IQ and GPA for 12 students • IQ = 110,112,118,119,122,125,127,130,132,134, 136,138 • GPA = 3.0,1.7,2.0,2.5,3.9,3.5,3.7,3.8,2.2,3.7,3.8,4.0 • Sum of IQ = 1503 • Sum of GPA = 37.8 • Sum of IQ*GPA = 4786.1 • Sd IQ = 9.226 • Sd GPA = .833

  7. Calculation

  8. Calculate r • Calculate the Pearson product moment correlation for the following data to see if there is a relationship between how fast one drives on the highway and scores on a measure of type A personality. • Speed Test Score 65 34 75 45 72 40 61 37 68 39 • = 341 195 • ()2 = 23379 7671

  9. Solution • r = .833 • Interpretation? • Strong relationship but… • p = .08 • The observed p-value is really best used as a check on r in terms of a sampling distribution, rather than a determination of significance

  10. Significance test for correlation • A correlation is an effect size – i.e., standardized measure of amount of covariation • R2 = amount of variability seen in y that can be explained by the variability seen in x • A 1- or 2-tailed significance test can be done in an effort to infer to a population • Typically though, correlations are considered descriptive • The sig. test result will depend on the power of study (i.e., higher N, more likely to be sig) • Alternatively look up r tables with df = N - 2

  11. Significance test for correlation • We will use the t-distribution as we have to use our sample data as estimates of the population parameters (pop. sd not known)

  12. Test of the difference between two rs • Since r has limits of +1, the larger the value of r, the more skewed its sampling distribution about the population  (rho)

  13. Transformation of r to r' • Fisher’s transformation will change r into one that is approximately normally distributed • With standard error (sr‘)

  14. Now we can calculate a z value (not t because our standard error is not estimated by the sample statistic)

  15. Test that r equals some specified value • Now we’re talking about an interesting hypothesis test • This is the equivalent to the one-sample z-test for a mean except now we are testing our sample r vs. some specified  • Again we transform our r, and this time we will transform also

  16. Test the difference of two dependent rs • Suppose we wanted to test to see if there is a difference between creativity and a person’s motivation scores (external and internal) • rec = .10 • ric = .59 • rei = .05 • In other words, I’m testing to see if there is a difference between rec and ric

  17. Solution • Yikes! For N = 30, t = 2.21 • tcv (27)= 2.052  Reject H0

  18. Confidence interval on  •  'L < ' < 'U • Then convert upper and lower ' values to 

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