PSY 307 – Statistics for the Behavioral Sciences Chapter 6 – Correlation

Midterm Results Top score = 45 Top score for curve = 45 40-53 A 7 36-39 B 4 31-35 C 2 27-30 D 8 0-26 F 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

Does Time Spent on Aleks Predict Quiz Grades?

Sometimes the Relationship is Not Linear 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/10.1056/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

Example Negative Correlations Note that the line slopes in the opposite direction, from upper left to lower right. r=-.54

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 S zxzy r = _______ 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)

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.