Obs Actual salary Predicted ResidualA DATA SET (P. 99 of Cohen & Cohen, 1983)GRAPHIC REPRESENTATION OF BIVARIATE RELATIONS1BIVARIATE CORRELATION ANDREGRESSION2Multiple Correlation/Regression-Reveals associations b/w a DV and multiple continuous/categorical IVs-Does not indicate whether associations are “causal.”-Inferences of causality are limited by method in which data were gathered. In particular, limited to the extent to which method satisfies criteria of an Experimental Design-Manipulate the IV(s)-Randomly assign participants to levels of the IV(s)-Methodologically control extraneous variables3Organization of Lecture-Quick review of standard deviation and Z-scores-Bivariate Correlation/Regression4Standard Deviation-Average amount of variability in a variable-Two formulas, one for population (-) and sample (s)NSSNXX2)( 11)(2nSSnXXsXdenominator of sample formula adjusts for tendency of samples to underestimate variability in a population.-Cohen & Cohen use different notationsd = - and ds~=slower case x = XX e.g., )( XXxDividing Y/X5-Many formulas involve -Y/-X which is SSY/SSX when n is equal-xyxxyyxxyySSSSSSNNSSNSSNSS * -xyxxyyxxyySSSSSSnnSSnSSnSS1*111Z-scoresXz6-Provides a common metric to compare variables measured on different scales…measured in standard deviations from themean.-If entire distribution is transformed into z-scoresz-distribution will have a mean =0standard deviation = 1shape of the transformed distribution will have the same shape as the original distribution. A DATA SET (P. 99 of Cohen & Cohen, 1983)Subject Salary($) PhD Pubs Sex Citations1 18000 1 2 0 12 19961 2 4 0 073 19828 5 5 1 14 17030 7 12 1 05 19925 10 5 0 06 19041 4 9 0 17 27132 3 3 1 08 27268 8 1 0 19 32483 4 8 0 010 27029 16 12 1 411 25362 15 9 0 012 28463 19 4 0 313 32931 8 8 0 514 28270 14 11 0 015 38362 28 21 0 3PhD = years since PhD, Pubs = # of publications, Sex (0 = male, 1 = female), Citations = # of times pubs were cited. Cohen & Cohen (1983). Applied multiple regression/correlation analysis for the behavioral sciences (p. 99). Hillsdale: NJ.GRAPHIC REPRESENTATION OFBIVARIATE RELATIONS8-Can visualize a bivariate relations with a scatter plot-Plot paired values of X,Y variables-E.g, plot values of salary and publications-Obtaining a Scatter Plot in SASproc plot;plot salary*pubs; run;Positive Linear Association Plot of salary*pubs. Legend: A = 1 obs, B = 2 obs, etc. salary ‚ ‚ 40000 ˆ ‚ ‚ A ‚ ‚ ‚ ‚ 35000 ˆ ‚ ‚ ‚ A ‚ A ‚ ‚ 30000 ˆ ‚ ‚ A A ‚ ‚ A A A ‚ ‚ A 25000 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 20000 ˆ A B ‚ A ‚ ‚ A ‚ A ‚ ‚ 15000 ˆ ‚ Šƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 pubs9Negative Linear Association(reverse scored publications) Plot of salary*rpubs. Legend: A = 1 obs, B = 2 obs, etc. salary ‚ ‚ 40000 ˆ ‚ ‚ A ‚ ‚ ‚ ‚ 35000 ˆ ‚ ‚ ‚ A ‚ A ‚ ‚ 30000 ˆ ‚ ‚ A A ‚ ‚ A A A ‚ ‚ A 25000 ˆ ‚ ‚ ‚ ‚ ‚ ‚ 20000 ˆ B A ‚ A ‚ ‚ A ‚ A ‚ ‚ 15000 ˆ ‚ Šƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒƒˆƒƒ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 rpubs10Non-Linear Association Plot of y*x. Legend: A = 1 obs, B = 2 obs, etc. y ‚ ‚10 ˆ ‚ ‚ ‚ ‚ ‚ ‚ ‚ A A ‚ ‚ ‚ ‚ A A ‚ ‚ ‚ ‚ ‚ A A ‚ ‚ ‚ ‚ ‚ ‚ 0 ˆ ‚ Šˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒ 0 10 x11QUANTIFYING LINEAR ASSOCIATIONS:PEARSON CORREALTION COEFFICIENT12-Visualization: deceiving and no metric for expressing strength of the relationship-Pearson correlation coefficient (r)-ranges from –1 to +1-Valence of r indicates direction of relationship-indicates negative association+indicates positive association-Magnitude of r indicates strength of the relationship0 = no linear association-1 = perfect linear associationSalary and Publications13r = .46So, positive and moderately strong association between salaryand publications, in our sample.Equations for r14-Multiple equations:In terms of Z-scoresnZZrYX, if Z is calculated using the population standard deviation (-)
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