Introduction to Correlation and Regression Analysis

SAS® has many tools that can be used for data analysis. From Freqs and Means to Tabulates and Univariates, SAS can present a synopsis of data values relatively easily. However, there is a difference between what the data are, and what the data mean. In order to take this next step, I would like to go beyond the basics and introduce correlation and hypothesis testing using regression models. A brief statistical background will be included, along with coding examples for correlation and linear regression. INTRODUCTION When analyzing vast amounts of data, simple statistics can reveal a great deal of information. However, it is often more important to examine relationships within the data, especially in medical and social sciences. Through correlation measures and hypothesis testing, these relationships can be studied in depth, limited only by the data available to the researcher. This paper and corresponding presentation will attempt to explain these powerful tools with a statistical background and programming examples. CORRELATION: STATISTICAL BACKGROUND In layman’s terms, correlation exists when two variables have a linear relationship beyond what is expected by chance alone. When examining data in SAS, correlation reveals itself by the relationship between two variables in a dataset. The most common measure of correlation is called the “Pearson Product-Moment Correlation Coefficient”. It is important to note that while more than two variables can be analyzed when looking for correlation, the correlation measure only applies to two variables at a time, by definition: ρ x,y = /σ x σ y It is apparent when examining the definition of correlation that measures from only two variables are included, namely the covariance between the two variables {cov(x,y)} and the standard deviation of each (σxσy). The result of this calculation is the correlation between the two variables. This measure can range from -1 to 1, inclusive. A value of -1 represents a “perfect negative correlation”, while a value of 1 represents a “perfect positive correlation”. The closer a correlation measure is to these extremes, the “stronger” the correlation between the two variables. A value of zero means that no correlation is observed. It is important to note that a correlation measure of zero does not necessarily mean that there is no relationship between the two variables, just that there is no linear relationship present in the data that is being analyzed. It is also sometimes difficult to judge whether a correlation measure is “high” or “low”. There are certain situations where a correlation measure of 0.3, for example, may be considered negligible. In other circumstances, such as in the social sciences, a 0.3 correlation measure may suggest that further examination is needed. As with all data analysis, the context of the data must be understood in order to evaluate any results. 1 Statistics and Data Analysis SAS Global Forum 2008