Correlation and Simple Linear Regression

Authors: Kelly H. Zou, Kemal Tuncali, Stuart G. Silverman
Department of Radiololgy, Brigham and Women’s Hospital (K.H.Z., K.T., S.G.S.)
Department of Health Care Policy (K.H.Z.), Harvard Medical School, 180 Longwood Ave, Boston, MA 02115
Summary

In this paper the concepts of correlation and simple linear regression are reviewed. Two commonly used correlations Pearson and Spearman ρ are used here. Both correlation coefficients take the values from -1 (negatively correlated) to 0 (uncorrelated) to +1 (positively correlated). Linear correlation can be computed from the data and their means using the formula
 
where subscript i denotes the data point and bar represents the mean. Spearman’s ρ is the correlation of ranks which are the relative orders of the continuous data. Correlation can be interpreted using the following table.
Interpretation of Correlation Coefficient
Correlation Coefficient Value    Direction and Strength
-1.0    Perfectly negative
-0.8    Strongly negative
-0.5    Moderately negative
-0.2    Weakly negative
0.0    No association
+0.2    Weakly positive
+0.5    Moderately positive
+0.8    Strongly positive
+1.0    Perfectly positive

The purpose of regression analysis is to find the impact on predictor variable on the outcome. A simple regression model contains only one dependent and one independent variable. The model is represented by the following mathematical form
 
Where a is the intercept and b is the slope and e is the error term.
Least square method is used to find the straight line through data of X and Y.
The dose response time is shown using these analyses. They are shown below.
 

By: Aamir Alaud-din
aamiralauddin@gist.ac.kr