LINEARITY PSY 201 suppose you get r 0 Does that mean there is no relation between the data sets r is partly an index of how well a straight line fits the data set r 0 903 Professor Greg Francis many aspects of the data may affect the value of r Lecture 11 correlation Is there a relationship between IQ and problem solving ability Linearity of data Final Examination Score CORRELATION Introduction to Statistics in Psychology Homogeneity of group 60 50 40 400 Size of group 450 500 550 600 650 Quantitative SAT Score Restricted range 2 3 NONLINEARITY NONLINEARITY NONLINEARITY when data points don t fall along a single line nonlinear data r 0 05 there are lots of types of nonlinearities It can get complicated curvilinear relationship r 0 131 70 80 60 60 50 40 30 450 500 550 600 650 700 Y 60 40 20 50 0 0 40 5 10 15 X Quantitative SAT Score 30 400 450 500 550 600 650 r 0 91 700 Anxiety Measure 1000 500 Y 400 Final Examination Score Final Examination Score r 0 20 100 0 500 0 5 10 X 4 5 6 15 700 BOTTOM LINE HOMOGENEITY HOMOGENEITY Pearson r is an index of a linear relationship between variables suppose you get r 0 and you cannot detect any type of nonlinear relationship Does this mean there is no relationship between the variables consider the covariance formula sxy rxy sx sy if another nonlinear relationship exists r might not notice it Pearson r measures only simple relationships between variables if r is small you might want to plot a scattergram to look at the data to notice if other relationships exist Correlation measures how variable X changes with variable Y if one doesn t change much there won t be a strong correlation similarly sy is going to be small so we divide a small number by a small number 7 8 9 HOMOGENEITY HOMOGENEITY HOMOGENEITY intutively the effects of homogeneity can be subtle suppose we looked at the relationship among only the best students those with final exam scores above 44 degree to which X and Y vary together degree to which X and Y vary separately relationship between SAT scores and Final exam grade r 0 92 if one of those variables or both is not varying much at all r will be small 44 44 Final Exam Grade you need enough variability across both sets of scores to adequately measure correlation r 0 62 Final Exam Grade r Not necessarily it may be that the data does not have enough variation in it where covariance is X X Y Y sxy n 1 if there is little change in Y from Y sxy is going to be small because variations in X X will be weighted by small values of Y Y 42 42 40 38 40 400 38 450 500 550 400 450 500 550 600 650 700 Quantitative SAT Scores 10 600 Quantitative SAT Scores 11 12 650 700 HOMOGENEITY SIGNIFICANCE SIZE OF GROUP or worst students those with final exam scores below 40 if you have r 0 it may be because there is not enough variation in your data set suppose you have only two data points r 0 62 e g IQ and problem solving is probably unrelated among a group of geniuses 42 60 40 IQ and problem solving is probably unrelated among a group of idiots 38 Y Final Exam Grade 44 you can always draw a straight line connecting them which implies perfect correlation r 1 0 40 20 400 450 500 550 600 650 700 Quantitative SAT Scores correlation drops IQ and problem solving is probably strongly related among a mix of geniuses idiots and normals 22 24 26 28 30 32 34 X correlation doesn t tell us anything useful 13 14 15 SIZE OF GROUP RESTRICTED RANGE RESTRICTED RANGE if you have enough data points for correlation to be meaningful 2 and you have enough variation in the data then if you sample data from a limited range you may not be able to trust the correlation values in general if you sample from the general population not just college students you would get a larger range of IQs e g suppose you want to study relationships between IQ and creativity you may find a much weaker correlation e g r 0 12 we will see later that it is important in determining the accuracy of the relationship hypothesis testing if you sample college students you will probably get IQ s between 110 and 140 perhaps you find a strong correlation e g r 0 78 40 Creativity measure size of group is not important in determining the value of r 35 30 25 20 80 90 100 Creativity measure 30 25 110 115 120 125 130 135 140 IQ 16 110 IQ 35 17 18 120 130 140 RESTRICTED RANGE INTERPRETATION OF r SCALE OF r of course it could be that you fail to find a large r over a restricted range but a larger range finds a large r this is slightly different from the issue of homogeneity if we calculate a value of r How do we know what it means values of r are ordinal measures of correlation How do we compare r values for different data sets higher r values indicate larger correlation in general Rule of thumb equal spacings of r values may not indicate equal spacings of correlation a correlation measure applies only to the range of values used to compute it you cannot extend the correlation value to other ranges 0 9 0 7 0 5 0 3 0 0 r to 1 0 to 0 9 to 0 7 to 0 5 to 0 3 Interpretation Very high correlation High correlation Moderate correlation Low correlation Little if any correlation 19 20 CONCLUSIONS NEXT TIME Pearson r coefficient of determination size causation interpretation Spearman s rho Does TV make you drink 22 23 thus r 0 90 is not twice as correlated as r 0 45 the difference in correlation between r 0 90 and r 0 75 is not the same as the difference in correlation between r 0 60 and r 0 45 21