Introduction to Statistics in Psychology INTERPRETATION OF r SCALE OF r if we calculate a value of r values of r are ordinal measures of correlation PSY 201 Professor Greg Francis How do we know what it means Lecture 12 How do we compare r values for different data sets higher r values indicate larger correlation correlation Rule of thumb equal spacings of r values may not indicate equal spacings of correlation Does TV make you drink 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 positive correlation Little if any correlation 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 2 3 VARIANCE VARIANCE VARIATION we can interpret r in terms of variance the idea is embedded in mathematical models assume you want to predict the final exam score when you know the SAT score deviation of a final exam score from the mean value can be due to deviation accounted for by SAT scores or due to something else also indicates proportion of individual differences that can be associated with individual differences of another variable line predicts score could go in reverse too Final Exam Grade 44 42 44 Final Exam Grade correlation coefficient indicates relationships between variables 42 40 38 40 400 38 450 500 550 400 450 500 550 600 650 700 Quantitative SAT Scores 4 600 Quantitative SAT Scores 5 6 650 700 VARIATION CORRELATION Spearman it turns out that s2 r2 a2 sy we have studied the correlational coefficient called the Pearson r special case of the Pearson r s2y the total variance in y s2a the variance in Y associated with variance in X thus r2 is the proportion of variance in Y accounted for with variance in X we are skipping the mathematical details thank you formula looks much different 6 d2 1 n n2 1 where one limitation is that it only works for quantitative data interval or ratio sometimes we want to calculate correlations of ordinal ranked data e g does ranking near the top of SAT scores correlate with ranking near the top of Final Exam scores we might not know the actual scores n number of paired ranks d difference between paired ranks called the coefficient of determination 7 8 9 CALCULATION CALCULATION SPEARMAN Ties take average rank important points if we calculated r from the ranked data we would get r 0 93 Y X rank Y rank d 68 4 1 3 55 8 7 1 65 1 2 1 42 14 12 2 64 2 3 1 45 11 10 1 56 6 5 5 5 1 59 5 4 1 56 3 5 5 2 5 42 13 12 1 38 12 14 2 50 9 9 0 37 15 15 0 42 10 12 2 53 6 5 8 1 5 1 Ranking order must be the same for both sets of data Highest to lowest or lowest to highest 2 Tied ranks take the average value of the rank positions 3 If we calculated the r value for the ranks it would equal as long as there are no tied ranks 1 6 32 12 1 2 15 225 1 1 10 6 36 50 0 93 3360 11 when we calculated r from the raw scores we got r 0 90 text is misleading graph of ranked data 0 93 15 FINAL EXAM GRADE RANK X 595 520 715 405 680 490 565 580 615 435 440 515 380 510 565 10 5 2 4 6 8 10 SAT SCORE RANK 12 12 14 CAUSALITY CAUSATION TV Teens Drinking in behavioral sciences we often look for causalities to try to determine how to make decisions e g High school students who watch lots of television and music videos are more likely to start drinking alcohol than other youngsters while those who rent movies are at less risk Associated Press 3 November 1998 SAT scores and Final Exam scores in a statistics class may be highly correlated but no one would claim that doing well on the SAT causes someone to get a good grade in statistics in some places war there is a high correlation between bullets in the brain and being dead that does not mean that being dead causes bullets to form in the brain in fact it is probably the other way around getting a good grade and statistics may be caused by being smart causation cannot be established through quantitative methods getting a high SAT score may be caused by being smart establishment of causation requires understanding of the variables and their roles being smart causes both variables to be highly correlated found correlation between TV and music video watching and drinking among 9th graders implied that glamorization of drinking on TV led to increased drinking in viewers 13 14 15 TV Teens Drinking CORRELATION CONCLUSIONS may be true but parents and peers are known to have a very big influence and this study did not control for those influences Other coefficients of correlation Pearson r correlation does not necessarily imply causation USA Today got it right they quoted ABC s Julie Hoover who likened the conclusions to saying gynecologists make women pregnant because there are so many pregnant women in their offices Notice this does not mean that TV programs do not influence drinking It just means that a single correlation cannot demonstrate causality Some TV programs do influence drinking behavior 16 size Nominal data C coefficient Cramer s V interpretation Ordinal data Rank biserial tetrachoric coefficient of determination Interval ratio and nominal point biserial Nonlinear other uses of correlation Inferential statistics sampling theory Predict scores linear regression 17 18 NEXT TIME probability rules significance Why casinos make money 19