Correlation examining relationships

Five Descriptive Questions What is the middle of the set of scores? What is the middle of the set of scores? How spread out are the scores? How spread out are the scores? Where do specific scores fall in the distribution of scores? Where do specific scores fall in the distribution of scores? What is the shape of the distribution? What is the shape of the distribution? How do different variables relate to each other? How do different variables relate to each other?

Correlation Once you know: Once you know: –Middle –Spread –Shape –Relative position of specific cases It is now useful to know relationships between variables. It is now useful to know relationships between variables.

Correlation Direction of Relationships Direction of Relationships Positive or Negative Positive or Negative Magnitude of Relationships Magnitude of Relationships Weak, Moderate, Strong Weak, Moderate, Strong Scatterplots Scatterplots Outliers Outliers

Correlation Quantitative index of association Quantitative index of association Scaling of Pearson r Scaling of Pearson r –1 = perfect negative relationship –1 = perfect negative relationship 0 = no relationship 0 = no relationship +1 = perfect positive relationship +1 = perfect positive relationship Most common measure of association for interval and ratio variables Most common measure of association for interval and ratio variables

Examples Parent educational level and student academic achievement Parent educational level and student academic achievement Parent income or SES and student academic achievement Parent income or SES and student academic achievement Coping strategies and perceived stress Coping strategies and perceived stress

Correlation For positive correlations between two variables: For positive correlations between two variables: High values on x tend to be associated with high values on y High values on x tend to be associated with high values on y Low values on x tend to be associated with low values on y Low values on x tend to be associated with low values on y

r= NC State System Level Data

Correlation For negative correlations between two variables: For negative correlations between two variables: Low values on x tend to be associated with high values on y Low values on x tend to be associated with high values on y High values on x tend to be associated with low values on y High values on x tend to be associated with low values on y

r=-.613

r= NC State System Level Data

r= NC State System Level Data

Interpretation Guidelines Correlation is not causality. Correlation is not causality. Correlation is necessary for causal inference, but not sufficient. Correlation is necessary for causal inference, but not sufficient. Causal inference requires experimental designs. Causal inference requires experimental designs.

Interpretation Guidelines Rum use and number of people entering the priesthood. Rum use and number of people entering the priesthood. Square footage of home and student academic achievement. Square footage of home and student academic achievement. Percent of women in a state who earn high salaries and percent of public officials who are women. Percent of women in a state who earn high salaries and percent of public officials who are women.

Interpretation Guidelines The third variable problem. The third variable problem. –SES and home size. The risk factor vs. causal agent problem. The risk factor vs. causal agent problem. –Length of time smoking and life expectancy. The direction of causality problem. The direction of causality problem. –Productivity and job satisfaction

Interpretation Guidelines R assumes a linear relationship. R assumes a linear relationship. R will underestimate curvilinear relationships. R will underestimate curvilinear relationships. Restriction of range will lower correlation. Restriction of range will lower correlation. Outliers, gaps in distributions, non-normal distributions can all influence r. Outliers, gaps in distributions, non-normal distributions can all influence r. Be aware of subgroups. Be aware of subgroups.

Interpretation Guidelines Examine the scatterplot. Examine the scatterplot. Examine the distributions of both variables. Examine the distributions of both variables. Be aware of the other descriptive statistics on both variables. Be aware of the other descriptive statistics on both variables.

Interpreting Magnitude

Outliers You can look at outliers in the univariate case (within the distribution of a single variable) and in the bivariate case (within the scatterplot of points representing values on two variables). You can look at outliers in the univariate case (within the distribution of a single variable) and in the bivariate case (within the scatterplot of points representing values on two variables). Examine the scatterplots for values out of the pattern. Examine the scatterplots for values out of the pattern.

What would you expect? Teacher age Teacher age Classroom quality Classroom quality

r=-.279

What would you expect? Perceived stress Perceived stress Depression Depression

r=.582

What would you expect? Depression Depression Self-acceptance Self-acceptance

r=-.596

What would you expect? Emotional Exhaustion Emotional Exhaustion Depersonalization Depersonalization

r=.574