Correlation and Regression 9.1 Introduction

Abstract

In many applications, we may want to study the underlying nature of relationships among the variables. Furthermore, we may also want to utilize these relationships for predicting or estimating the values of some variables (outcome variables) on the basis of the given values of other variables (explanatory variables). By exploring the underlying relationships, we can find whether there is any association between variables as well as whether there is dependence of outcome variables on explanatory variables that can provide necessary inputs required for useful decisions. Some examples of these relationships are (i) relationship between height and weight, (ii) relationship between weight and cholesterol level, (iii) relationship between income and expenditure on health care, etc. In these studies, we are interested in answering several important questions, some of which are (i) Is there a relationship between the variables? What is the nature of this rela-tionship? What is the strength of this relationship? (ii) If there is a relationship between the variables, how can we formulate it mathematically? How can we interpret it? What are the policy implications? What is the significance of such finding in explaining the underlying truth in population? Suppose that we are interested in studying the relationship between the height of the person and his weight in a certain population of a community. In this case, we select a random sample of size n from this population, and then we measure the height (X) and the weight (Y) of each individual in the sample. The random sample of pairs of variables is ðX 1 ; Y 1 Þ; ðX 2 ; Y 2 Þ;. . .; ðX n ; Y n Þ: The observations of the sample will be a set of pairs ðx 1 ; y 1 Þ; ðx 2 ; y 2 Þ;. . .; ðx n ; y n Þ