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Bivariate Statistics Y Nominal X Ordinal Interval Nominal Ordinal Interval 2 Rank-sum Kruskal-Wallis H t-test ANOVA Spearman rs (rho) Pearson r Regression October 31 Sir Francis Galton Karl Pearson http://www.york.ac.uk/depts/maths/histstat/people/ Source: Raymond Fancher, Pioneers of Psychology. Norton, 1979. A correlation coefficient is a numerical expression of the degree of relationship between two continuous variables. 80 RDG 70 60 50 40 30 30 40 50 60 MATH 70 80 Pearson’s r -1 r +1 -1 +1 Sample _ C XC sc n Sample _ D XD sd n Population Sample _ B µ Sample _ E XE se n n XB sb Sample _ A XA sa n SampleC rXY SampleD Population rXY XY _ E Sample rXY SampleB rXY SampleA rXY Pearson’s r -1 r +1 -1 +1 Pearson’s r is a function of the sum of the cross-product of z-scores for x and y. Pearson’s r r= zx zy N SampleC rXY SampleD Population rXY XY _ E Sample rXY SampleB rXY SampleA rXY 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 r N-2 t= 1 - r2 Some uses of r • Association of two variables • Reliability estimates • Validity estimates Factors that affect r Non-linearity Restriction of range / variability Outliers Reliability of measure / measurement error Johnson & Newport, scaled properly, with new ranges age <20 and >20. All Subjects English Proficiency 300 200 r=-.87 r=-.49 10 30 100 0 0 20 Age of Arrival 40 Spearman’s Rank Order Correlation rs Point Biserial Correlation rpb 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. Pearson’s r -1 r +1 -1 +1 Pearson’s r can also be interpreted as the expected value of zY given a value of zX. tend to deviate from the mean of X when they are expressed in standard deviation units. The expected 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.
