Statistics review 7: Correlation and regression

Abstract

The present review introduces methods of analyzing the relationship between two quantitative variables. The calculation and interpretation of the sample product moment correlation coefficient and the linear regression equation are discussed and illustrated. Common misuses of the techniques are considered. Tests and confidence intervals for the population parameters are described, and failures of the underlying assumptions are highlighted.

Figure 1

Scatter diagram for ln urea and age

Figure 2

Correlation coefficient (r) = +0.9. Positive linear relationship.

Figure 3

Correlation coefficient (r) = -0.9. Negative linear relationship.

Figure 4

Correlation coefficient (r) = 0.04. No relationship.

Figure 5

Correlation coefficient (r) = -0.03. Nonlinear relationship.

Figure 6

Subgroups in the data resulting in a misleading correlation. All data: r = 0.57; males: r = -0.41; females: r = -0.26.

Figure 7

Regression line for ln urea and age: ln urea = 0.72 + (0.017 × age).

Figure 8

Regression line obtained by minimizing the sums of squares of all of the deviations.

Figure 9

Total, explained and unexplained deviations for a point.

Figure 10

Regression line, its 95% confidence interval and the 95% prediction interval for individual patients.

Figure 11

(a) Scatter diagram of y against x suggests that the relationship is nonlinear. (b) Plot of residuals against fitted values in panel a; the curvature of the relationship is shown more clearly. (c) Scatter diagram of y against x suggests that the variability in y increases with x. (d) Plot of residuals against fitted values for panel c; the increasing variability in y with x is shown more clearly.

Figure 12

Plot of residuals against fitted values for the accident and emergency unit data.

Figure 13

Normal plot of residuals for the accident and emergency unit data.