Correlation is a technique used to determine the co-occurrence of two or more variables. Correlation coefficients range from -1 to 1, with values closer to these extremes indicating a stronger linear relationship between variables. Positive correlations indicate that high values in one variable are associated with high values in the other, while negative correlations show that high values in one variable are associated with low values in the other. Scatter plots can be used to display the relationship between two variables and identify if the correlation is positive, negative, or zero. The size of the correlation coefficient corresponds to the width of an imaginary ellipse that can be drawn around the points in a scatter plot.

1. HPH 7310 Correlations and Covariates Chapter 9/443-456 Correlation is a technique for determining the co-occurrence of two or more variables. Covariates are variables that co-occur with either your IV or your DV or your descriptive and subject variables.

2. Correlations Overview Variables taken 2 at a time, and we ask to what extent variable of the pair is related to the other variable If subject has high score on one variable, does this imply that same subject is likely to score high on the other variable? Or does high on one imply low on other? Or does high on one have no relation to the score on other variable.

3. Correlation coefficient The statistic that allows us to answer those questions is the correlation coefficient Correlation coefficients are stats that characterize nature of the relationship between 2 variables Scatterplots are a useful way to display relationship between 2 variables.

4. Values of r Correlation coefficient is a # between -1 & 1 My ballpark interpretations of size of r. Absolute value of 0 to .3 small .31 to .40 small to moderate .41 to .55 moderate .56 to .75 rather high .76 to 1.0 high

5. Scatter Plots Positive r Negative r Zero r (a “cloud”)

6. Note 2 characteristics of scatter diagrams: First characteristic - the slope Slope determines the sign of r : positive, negative, or 0 Positive r Negative r Zero r (a “cloud”) That line’s angle is the slope

7. 2nd characterstic of scatter diagram: the width of the imaginary ellipse that can be drawn around most of the points in the scatter diagram Positive r Negative r Zero r (a “cloud”)

8. The width of the imaginary ellipse that can be drawn around most of the points of the scatter diagram corresponds to the magnitude of the correlation coefficient. If ellipse is extremely narrow, magnitude is 1. If the ellipse is as wide as it can be, thus making it a circle the magnitude is zero.

9. Positive correlation implies that relatively large values of one variable are associated with relatively large values of other variable, & that small values of one are associated with small values of other Negative correlation implies that relatively large values of one variable are associated with relatively small values of other variable. Measure of Correlation There are several measures of correlation, depending on level of measurement of data. We’ll talk about 1 type of correlation coefficient: Pearson’s product moment correlation coefficient r (Pearson's r), which requires interval/ration level data.

10. Factors Affecting the Size of r 1. restriction of the range. If the total range of one or both of the variables we are measuring is made smaller, the correlation between 2 variables is likely to be reduced. income Price of Car income Price of Car vs

11. Factors Affecting the Size of r 2. Nonlinear relationships. Correlations we’ve discussed thus far involve fitting a straight line What if the data is nonlinear? To correct this problem, we fit a quadratic term (we’ll cover that later) Exam score Anxiety Anxiety Exam Score

12. Factors Affecting the Size of r 3. Outliers. Which scatterplot illustrates a larger correlation?

13. Correlation usually does NOT imply causation spurious correlations exist (e.g., ice cream consumption and violent crime example) direction of causality an important issue (e.g., anxiety and test performance versus test performance and anxiety)

14. Creating Correlations in SPSS “ The correlation between depression and feelings of hopelessness is significantly different from zero. The size and direction of the correlation suggests that people who are depressed tend to have increased feelings of hopelessness (or vice versa)”.

15. Creating Correlations in SPSS “ The correlation between current family income and breakdown of physical health is small, yet significantly different than zero”. The size and direction of the correlation suggests a small tendency for current family income to decrease as the breakdown of physical health increases. (or vice versa)”.