Dr Hidayathulla Shaikh Correlation and Regression

Objectives At the end of the lecture student should be able to – Define correlation and correlation coefficient Discuss each type correlation coefficient Define regression and regression coefficient Explain regression.

Introduction When two continuous characters or two sets of variable are measured in the same person such as height and weight, weight and cholesterol etc, And at other times the same characters or variables are measured in two related groups such as tallness in parents and tallness in children, study of Intelligence Quotient (IQ) in brothers and sisters, and so on… Then this relationship or association between two quantitatively measured or continuous variables is called Correlation.

When the two variables are measurable in the quantitative units such as height and weight, temperature and pulse rate etc. Then it is often necessary and possible to know that not only there is any association and relationship between them or not, but also we can know the degree or extent of such association/relationship. Hence the extent or degree of relationship between two sets of figures/variables is measured in terms of another parameter called correlation co-efficient, and is denoted by letter “r”. The correlation coefficient ranges from minus one (-1) to plus one (+1), that is -1 < r < +1

Lets study with an eg – the study of correlation between two variables (temperature and pulse rate) in 5 persons…. The careful examination of this hypothetical data reveals that, the pulse rate rises by 8 beats with one degree rise in temperature. Correlation determines the relationship between two variables, but this do not prove that one variable is responsible for the change in another variable. The cause of change may be because of other factors also. Sl noTemperature (F)Pulse rate

Types of Correlation There are 5 types of Correlation depending on its extent and direction – 1) Perfect Positive Correlation 2) Perfect Negative Correlation 3) Moderately Positive Correlation 4) Moderately Negative Correlation 5) Absolutely No Correlation Each type is described graphically by scatter diagram. In the scatter diagram one variable is represented on X axis and other variable on Y axis.

1) Perfect Positive Correlation – Here the two variables denoted by X & Y are directly proportional and fully correlated with each other. The correlation coefficient r = +1 that is both variables rise or fall in the same proportion, which means if one variable increases other also increases and if one variable decreases other also decreases. The graph forms a straight line rising from the lower end of both X & Y axis, when scatter diagram is drawn all points fall on straight line.

2) Perfect Negative Correlation - Here X & Y values are inversely proportional to each other, that is when one values increases other decreases in the same proportion. The correlation coefficient r = -1, here also the graph shows no scatter, the graph will contain all observations on a straight line.

3) Moderately Positive Correlation – In this case the values of correlation coefficient r lies between 0 and +1 that is 0 < r < 1, eg age of husband and age of wife. Here the scatter will be there around an imaginary line, rising from lower extreme values of both variables.

4) Moderately Negative Correlation – In this case the values of correlation coefficient r lies between -1 and 0 that is -1 < r < 0, eg – age and vital capacity in adult, income and infant mortality rate. Here also the scatter diagram will be of the same type, but the mean imaginary line will rise from the extreme values of one variable.

5) Absolutely No Correlation – Here the value of correlation coefficient r is zero, indicating that no linear relationship exists between two variables. There is no mean or imaginary line indicating correlation, here X is completely independent of Y for eg – height and pulse rate. The magnitude of correlation coefficients (r), either positive or negative is indicated by the closeness of dots to imaginary line indicating scatter or trend of correlation. When points are so scattered that no imaginary line can be drawn, then the correlation is zero.

Regression Regression is a change in the measurements of a variable, on positive side or a negative side, beyond the mean. In experimental sciences after understanding the correlation between two variable, There are situation when it is necessary to estimate or predict the value of one character (say variable Y) from the knowledge of other character (say variable X) such as to estimate weight when height is known. And this estimate is done by finding a constant called Regression Coefficient and is denoted by “b”

The former variable weight (Y) which is to be estimated is called dependent variable. And the latter variable that is height (X) which is known is called the independent variable. Regression coefficient is a measure of the change in one dependent (Y) character with one unit change in the independent character (y).

Correlation gives the degree and direction of relationship between two variables. Where as regression analysis enables us to predict the values of one variable on the basis of the other variable. Thereby the cause and effect relationship between the two variables is understood precisely. For eg – tobacco usage and stages of oral cancer.

When the corresponding values are plotted on a graph, a straight line called the regression line or the mean correlation line (Y on X) is obtained. The same was referred to as an imaginary line while explaining various types of correlation.