Correlation and regression
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Correlation measures the relationship between two or more variables, showing whether they move in the same or opposite directions. Correlation coefficient r ranges from -1 to 1, with higher positive or negative values indicating stronger linear relationships. Regression finds the functional relationship between a dependent variable Y and independent variable X. It models Y as a linear combination of X and estimates coefficients to best fit the data. Assumptions for correlation and regression include continuous, normally distributed variables with independent pairs of observations.
Correlation and regression
- 2. CORRELATION Correlation is the study of relationship between two or more variables. Suppose we have two continuous variables X and Y and if the change in X affects Y, the variables are said to be correlated. In other words, the systematic relationship between the variables is termed as correlation.
- 3. • When only two variables are involved the correlation is known as simple correlation and when more than two variables are involved the correlation is known as multiple correlation. • When the variables move in the same direction, these variables are said to be correlated positively and if they move in the opposite direction they are said to be negatively correlated. • When there are two related variables their joint distribution is known as bivariate normal distribution and if there are more than two variables their joint distribution is known as multivariate normal distribution.
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- 5. Correlation coefficient: The measures of the degree of relationship between two continuous variables is called correlation coefficient. It is denoted by r. The correlation coefficient r is given as the ratio of covariance of the variables X and Y to the product of the standard deviation of X and Y.
- 6. Assumptions: Correlation coefficient r is used under certain assumptions, they are 1. The variables under study are continuous random variables and they are normally distributed. 2. The relationship between the variables is linear. 3. Each pair of observations is unconnected with other pair (independent)
- 7. Properties: 1. The correlation coefficient value ranges between –1 and +1. 2. The correlation coefficient is not affected by change of origin or scale or both. 3. If r > 0 it denotes positive correlation r< 0 it denotes negative correlation. r = 0 then the two variables x and y are not linearly correlated.(i.e)two variables are independent. r = +1 then the correlation is perfect positive r = -1 then the correlation is perfect negative.
- 8. Regression is the functional relationship between two variables and of the two variables one may represent cause and the other may represent effect. The variable representing cause is known as independent variable and is denoted by X. The variable X is also known as predictor variable or repressor. The variable representing effect is known as dependent variable and is denoted by Y. Y is also known as predicted variable. REGRESSION
- 9. The relationship between the dependent and the independent variable may be expressed as a function and such functional relationship is termed as regression. When there are only two variables the functional relationship is known as simple regression and if the relation between the two variables is a straight line is known a simple linear regression. When there are more than two variables and one of the variables is dependent upon others, the functional relationship is known as multiple regression.
- 10. The regression line is of the form y=a+bx where a : constant or intercept b : regression coefficient / slope
- 11. Assumptions: 1. The x’s are non-random or fixed constants. 2. At each fixed value of X the corresponding values of Y have a normal distribution about a mean. 3. For any given x, the variance of Y is same. 4. The values of y observed at different levels of x are completely independent
- 12. Properties of Regression coefficients: 1. The range of regression coefficient is -∞ to +∞ 2. Regression coefficients are independent of change of origin but not of scale. 3. If r=1 angle between two regression line is “zero degree. If r=0 the regression lines are perpendicular to each other. 4.If variables X and Y are independent then the regression coefficients are Zero. 5. Also if one regression coefficient is positive the other must be positive and if one regression coefficient is negative the other must be negative. ie. if b1>0, then b2>0 and if b1<0, then b2<0. 6.The two regression lines intersect at the point of means of X and Y.