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PSY 307 – Statistics for the Behavioral Sciences Chapter 6 – Correlation Midterm Results Top score = 45 Top score for curve = 45 40-53 36-39 31-35 27-30 0-26 A B C D F 7 4 2 8 3 24 Aleks/Holcomb Hint To Find the Cutoff Scores If you know the mean and standard deviation, you can find what x values cut off certain percentages. Solve for k then multiply the k value by the SD and add/subtract that number from the mean to get the cutoff scores. Does Aleks Quiz 1 Predict Midterm Scores? Adding a Prediction (Regression) Line Provides More Information r = .56 Does Time Spent on Aleks Predict Quiz Grades? r = .16 Sometimes the Relationship is Not Linear r = .16 r = .47 (quadratic) Lying With Statistics This is the graph as published in a Wall Street Journal editorial (7/13), where they claimed that reducing corporate taxes results in greater revenue. Treating Norway as an outlier, the data instead shows that as taxes increase, so do revenues – the opposite conclusion. Which is right? The correct graph is the one with the best fit – where most of the data points are close to the line drawn (right). Describing Relationships Positive relationship – high values tend to go with high values, low with low. Negative relationship – high values tend to go with low values, low with high. No relationship – no regularity appears between pairs of scores in two distributions. Relationship Does Not Imply Causality A relationship can exist without being a CAUSAL relationship. Correlation does not imply causation. Third variable problem -- a third variable is causing both of the variables you are measuring to change – e.g., popsicles & drowning. The direction of causality cannot be determined from the r statistic. Chocolate and Nobel Prizes http://www.nejm.org/doi/full/10.10 56/NEJMon1211064 Scatterplots One variable is measured on the x-axis, the other on the y-axis. Positive relationship – a cluster of dots sloping upward from the lower left to the upper right. Negative relationship – a cluster of dots sloping down from upper left to lower right. No relationship – no apparent slope. Example Positive Correlations r=1.0 r=.39 r=.85 r=.17 Example Negative Correlations r=-.94 r=-.54 r=-.33 Note that the line slopes in the opposite direction, from upper left to lower right. Strength of Relationship The more closely the dots approximate a straight line, the stronger the relationship. A perfect relationship forms a straight line. Dots forming a line reflect a linear relationship. Dots forming a curved or bent line reflect a curvilinear relationship. More Examples http://www.stat.uiuc.edu/courses/stat100/java/GCApplet/GCAppletFrame.html Correlation Coefficient Pearson’s r –a measure of how well a straight line describes the cluster of dots in a plot. Ranges from -1 to 1. The sign indicates a positive or negative relationship. The value of r indicates strength of relationship. Pearson’s r is independent of units of measure. Interpreting Pearson’s r The value of r needed to assert a strong relationship depends on: The size of n What is being measured. Pearson’s r is NOT the percent or proportion of a perfect relationship. Correlation is not causation. Experimentation is used to confirm a suspected causal relationship. Calculating Pearson’s r r= S zxzy _______ n–1 This formula is most useful when the scores are already z-scores. Computational formulas – use whichever is most convenient for the data at hand. Sum of the Products (SP) r SPxy SS x SS y where SPxy ( X X )(Y Y ) XY ( X )( Y ) and SS x ( X X ) 2 X 2 ( X ) 2 n n Computational Formulas Outliers An outlier that is near where the regression line might normally go, increases the r value. r=.457 r=.336 An outlier away from the regression line decreases the r value. Dealing with Outliers Outliers can dramatically change the value of the r correlation coefficient. Always produce a scatterplot and inspect for outliers before calculating r. Sometimes outliers can be omitted. Sometimes r cannot be used. http://www.stat.sc.edu/~west/javahtml/Regression.html Other Correlation Coefficients Spearman’s rho (r) – based on ranks rather than values. Used with ordinal data (qualitative data that can be ordered least to most). Point biserial correlation -correlations between quantitative data and two coded categories. Cramer’s phi – correlation between two ordered qualitative categories.