Bivariate Statistics

1. Bivariate Statistics Y Nominal Ordinal Interval Nominal2 Rank-sum t-test Kruskal-Wallis H ANOVA Ordinal Spearman rs (rho) Interval Pearson r Regression X

2. October 31 Sir Francis Galton Karl Pearson http://www.york.ac.uk/depts/maths/histstat/people/

3. Source: Raymond Fancher, Pioneers of Psychology. Norton, 1979.

4. A correlation coefficient is a numerical expression of the degree of relationship between two continuous variables.

5. Pearson’s r -1  r  +1 -1    +1

6. SampleC XC _ sc SampleD XD n _ sd Population n SampleB XB _  µ sb n SampleE XE SampleA XA _ _ se sa n n

7. SampleC SampleD rXY Population rXY SampleB XY rXY SampleE SampleA _ rXY rXY

8. Pearson’s r -1  r  +1 -1    +1 Pearson’s r is a function of the sum of the cross-product of z-scoresfor x and y.

9. Pearson’s r  zxzy r = N

10. SampleC SampleD rXY Population rXY SampleB XY rXY SampleE SampleA _ rXY rXY

11. r N - 2 t = 1 - r2 The familiar t distribution, at N-2 degrees of freedom, can be used to test the probability that the statistic r was drawn from a population with  = 0 H0 :  XY = 0 H1 :  XY  0 where

12. Some uses of r • Association of two variables • Reliability estimates • Validity estimates

13. Factors that affect r • Non-linearity • Restriction of range / variability • Outliers • Reliability of measure / measurement error

15. Spearman’s Rank Order Correlation rs Point Biserial Correlation rpb

16. Pearson’s r -1  r  +1 -1    +1 Pearson’s r can also be interpreted as how far the scores of Y individuals tend to deviate from the mean of X when they are expressed in standard deviation units.

17. Pearson’s r -1  r  +1 -1    +1 Pearson’s r can also be interpreted as the expected value of zYgiven a value of zX. tend to deviate from the mean of X when they are expressed in standard deviation units. Theexpected value of zY is zX*r If you are predicting zY from zX where there is a perfect correlation (r=1.0), then zY=zX.. If the correlation is r=.5, then zY=.5zX.