Correlation: Relationship between Variables

Statistical Relationships versus Deterministic Relationships Deterministic: if we know the value of one variable, we can determine the value of the other exactly. e.g. relationship between volume and weight of water. Statistical: natural variability exists in both measurements. Useful for describing what happens to a population or aggregate.

Measuring Strength Through Correlation A Linear Relationship Correlation (or the Pearson product-moment correlation or the correlation coefficient) represented by the letter r: Indicator of how closely the values fall to a straight line. Measures linear relationships only; that is, it measures how close the individual points in a scatterplot are to a straight line.

Features of the Correlation Coefficient It indicates the strength of the relationship and whether there is a positive or negative relationship. The correlation coefficient is a number between -1 and 1. A positive correlation indicates that the variables increase together. A negative correlation indicates that as one variable increases, the other decreases. Correlation of +1/-1 indicates a perfect linear positive/negative relationship between the two variables; as one increases, the other increases/decreases at a constant rate (a deterministic linear relationship).

Features of the Correlation Coefficient Correlation of zero could indicate no linear relationship between the two variables, or that the best straight line through the data on a scatterplot is exactly horizontal. The closer the correlation is to 1 or -1, the stronger the relationship. The closer it is to 0, the weaker the relationship. Crude estimate: > |.5|? most likely a relationship < |.3|? correlation essentially non-existent |.3| < r < |.5|? gray area! Correlations are unaffected if the units of measurement are changed. For example, the correlation between weight and height remains the same regardless of whether height is expressed in inches, feet or millimeters (as long as it isn’t rounded off).

What to discuss? To discuss the relationships between two variables we look at: Statistical Significance Strength of Correlation

Statistical Significance A relationship is statistically significant if that relationship is stronger than 95% of the relationships we would expect to see just by chance.

Statistical Significance What r value is statistically significant? It depends on the size of the sample. The following table indicates the lowest r value that would indicate a statistically significant relationship for a given sample size. A more complete table is in the Excel file. Critical Values for the Correlation Coefficient  Number of Points 95% Confidence 3 0.997 4 0.950 5 0.878 6 0.811 7 0.754 8 0.707 9 0.666 10 0.632 11 0.602 12 0.576 13 0.553 14 0.532 15 0.514 16 0.497 17 0.482 18 0.468 19 0.456 20 0.444

Two Warnings about Statistical Significance Even a minor relationship will achieve “statistical significance” if the sample is very large. A very strong relationship won’t necessarily achieve “statistical significance” if the sample is very small.

Example - Verbal SAT and GPA Scatterplot of GPA and verbal SAT score. The correlation coefficient is .485, indicating a moderate positive relationship. Higher verbal SAT scores tend to indicate higher GPAs as well, but the relationship is nowhere close to being exact.

Example - Husbands’ and Wifes’ Ages and Heights Scatterplot of British husbands’ and wives’ heights (in millimeters); r = .36 Scatterplot of British wives’ and husbands’ and ages; r = .94 Discuss the strength of the relationships. Are they positively or negatively correlated?

Example - Occupational Prestige and Suicide Rates Plot of suicide rate versus occupational prestige for 36 occupations. Correlation of .109. There is a weak positive relationship If outlier removed r drops to .018.

Statistical Relationships What is the difference between correlation and regression? Correlation: measures the strength of a relationship between two measurement variables. Regression: uses the equation of the trendline to predict one measurement variable from another.