Correlation and Regression.pptx
- 1. Correlation and Regression Dr. Yogesh A. Garde Assistant Professor (Agril. Statistics)
- 2. Introduction • Univariate analysis: The study related to the characteristics of only variable such as height, weight, ages, marks, wages, etc. • Bivariate analysis: The statistical Analysis related to the study of the relationship between two variables. • Univariate population: A population that is characterized by a single variable, e.g. population of height of students • Bivariate population: When two variables are simultaneously studied in a single population, e.g. the height and weight of the students
- 3. Simple Correlation • A measure of degree or extent of linear relationship between two variable X and Y. • Correlation is an analysis of the co-variation between two or more variables • It is a unit free measure • Uses of Correlation: • It is used in physical and social sciences. • It is useful for economists to study the relationship between variables like price, quantity etc. Businessmen estimates costs, sales, price etc. using correlation. • It is helpful in measuring the degree of relationship between the variables like income and expenditure, price and supply, supply and demand etc. • Sampling error can be calculated. • It is the basis for the concept of regression
- 4. Scatter Diagram: • The simplest method of studying the relationship between two variables diagrammatically • we cannot get the exact degree or correlation between the two variables (Range: -1 to +1)
- 6. Types of Correlation: • Positive and negative Correlation: • Linear and non-linear(Curvi-linear) Correlation • Partial and total Correlation • Simple and Multiple Correlation
- 7. • Positive: If the two variables tend to move together in the same direction (increase/decrease) • Negative: If the two variables tend to move together in the opposite direction (one increase other decrease) • Linear: If the ratio of change between the two variables is a constant • Non linear: If the amount of change in the one variable does not shows a constant ratio of the amount of change in the other • Simple: study only two variables • Multiple: study more than two variables simultaneously • Partial: study of two variables excluding some other variable • Total: all variable and facts are taken into account
- 8. Measures of Correlation • Scatter diagram method/Graphic method • Algebraic method: Karl Pearson’s coefficient of correlation
- 9. • Karl pearson, a great biometrician and statistician, suggested a mathematical method for measuring the magnitude of linear relationship between the two variables For grouped data, this formula can write using letter u and v; where u= (x-a)/h And v= (Y-b)/k
- 10. Assumptions for Correlation Coefficient • Assumption of Linearity • Variables being used to know correlation coefficient must be linearly related. You can see the linearity of the variables through scatter diagram. • Assumption of Normality • Both variables under study should follow Normal distribution. They should not be skewed in either the positive or the negative direction. • Assumption of Cause and Effect Relationship • There should be cause and effect relationship between both variables, for example, Heights and Weights of children, Demand and Supply of goods, etc. When there is no cause and effect relationship between variables then correlation coefficient should be zero. If it is non zero then correlation is termed as chance correlation or spurious correlation. For example, correlation coefficient between: 1. Weight and income of a person over periods of time; 2. Rainfall and literacy in a state over periods of time
- 11. Properties of Correlation • Correlation coefficient lies between –1 and +1 • r = +1 perfect positive correlation and r = -1 perfect negative correlation between the variables. • ‘r’ is independent of change of origin and scale. • If X and Y are two independent variables then correlation coefficient between X and Y is zero, i.e. Corr(x, y)=0 • It is a pure number independent of units of measurement. • Correlation coefficient is the geometric mean of two regression coefficients • The correlation coefficient of x and y is symmetric. rxy = ryx
- 12. Example:
- 13. Exercise:
- 14. Spearman’s Rank Correlation: • Edward Spearman in 1904 • No assumption about the parameters of the population is made. This method is based on ranks. • It is useful to study the qualitative measure of attributes like honesty, color, beauty, intelligence, character, morality etc. It is also denoted by ‘ρ’. The value of ‘r’ lies between -1 and +1. If r = +1, there is complete agreement in order of ranks and the direction of ranks is also same. If r = -1, then there is complete disagreement in order of ranks and they are in opposite directions.
- 15. Example: Thus there is a positive association between ranks of Statistics and Mathematics.
- 16. IF Tied/Repeated ranks Corrected factor
- 17. 1. 2.
- 18. Standard error and Probable error of r • Standard error of r or S.E. (r) = 1−𝑟2 𝑛 • Probable error of r or P.E. (r) = 0.6745 x 1−𝑟2 𝑛 • S.E. and P.E. are used as measures the reliability of coefficient correlation (r)
- 19. Coefficient of concurrent deviations • A very simple and casual method of finding correlation when we are not serious about the magnitude of the two variables is the application of concurrent deviations. • This method involves in attaching a positive sign for a x- value (except the first) if this value is more than the previous value and assigning a negative value if this value is less than the previous value. • Coefficient of concurrent deviations also lies between –1 to +1, If (2c–m) > 0, then we take the positive sign both inside and outside the radical sign and if (2c–m) < 0, we are to consider the negative sign both inside and outside the radical sign. m = number of pairs of deviations c = no. of positive signs in the product of deviation column
- 20. Example: m = number of pairs of deviations, m = 7 c = No. of positive signs in the product of deviation column, c = 2 Exercise
- 21. Simple linear regression • Regression: is the measure of the average relationship between two or more variables in terms of the original units of the data. • It is functional relationship between a dependent variable, Y with one or more independent variables, X is called Regression equation. • The term regression is coined by Sir Francis Galton • Types of Regression: a) Simple and Multiple b) Linear and Non –Linear c) Total and Partial
- 22. • Simple: In case of simple relationship only two variables are considered • Multiple: more than two variables are involved. On this while one variable is a dependent variable the remaining variables are independent ones. • Linear: The linear relationships are based on straight-line trend, the equation of which has no-power higher than one. • Non-linear: In the case of non-linear relationship curved trend lines are derived. The equations of these are parabolic. • Total: all the important variables are considered. normally, they take the form of a multiple relationships because most economic and business phenomena are affected by multiplicity of cases. • Partial: In the case of partial relationship one or more variables are considered, but not all, thus excluding the influence of those not found relevant for a given purpose.
- 23. Linear Regression Equation: • If two variables have linear relationship then as the independent variable (X) changes, the dependent variable (Y) also changes. These equations show best estimate of one variable for the known value of the other. The equations are linear. • Linear regression equation of Y on X is Y = a + byxX + e Where: a= intercept; byx =slope of the line/regression coefficient; e= is error distributed as N(0,σ2) Principle of ‘Least Squares’ the constants ‘a’ and ‘b’ can be estimated with by applying the `least squares method’.
- 24. It can be written as: Where, r= coeff. of correlation σ y = SD of y and σ x = SD of x 2 Is minimized by partially differentiating it w.r.t. a and b respectively and equating them to zero.
- 25. Assumptions of Linear Regression • The X’s are non random or fixed constants. For example, it may apply 40, 60, 80 and 100 kg of N2 per ha to 10 plots each. Then it may observe the crop yield (Y) corresponding to each of these selected level (X). • At each fixed value of X, the corresponding values of Y have a Normal distribution about theoretical mean. • For any given x, the variance of Y is the same or homoscedastic. • The y’s observed at different value of X are completely independent.
- 26. Properties of Regression Coefficient:
- 28. Example:
- 29. Exercise:
- 30. Test of significance of coefficient of correlation • Null hypothesis H0: ρ = 0 • Alternative hypothesis H1: ρ ≠ 0 • If n is very large and in fact ρ ≠ 0 then, test a sample correlation coefficient against a population correlation coefficient • 𝑍 = 𝑟−𝜌 1−𝑟2 𝑛 − 1 • The analysis of variance technique can be used; • 𝐹 = 𝑟2 (1−𝑟)2 𝑛 − 2 With (n-2) d.f. With (1, n-2) d.f.
- 31. Test of significance of coefficient of regression • Test of significance of byx 𝑡 = 𝑏𝑦𝑥−𝛽 𝑆.𝐸.(𝑏𝑦𝑥) where 𝛽 is the population regression coefficient 𝑆. 𝐸. 𝑏𝑦𝑥 = 1 𝑛 (𝑦𝑖 − 𝑦𝑖(𝑒𝑠𝑡))2 (𝑛 − 2) (𝑥𝑖− 𝑥)2 By solving, 𝑆. 𝐸. 𝑏𝑦𝑥 = 𝑦𝑖 2−𝑏𝑦𝑥 2 . 𝑥𝑖 2 (𝑛−2) 𝑥𝑖 2 • Que. Differentiate Correlation and regression With (n-2) d.f.
- 32. • Test of significance of byx using ANOVA • F= v1/v2 with d.f. (1, n-2) S.V d.f. S.S. M.S. Regression Error 1 N-2 ( 𝑥𝑦)2 𝑥2 𝑦2 − (𝑥𝑦)2 𝑥2 𝑣1 = ( 𝑥𝑦)2 𝑥2 𝑣2 = 1 𝑛 − 2 𝑦2 − (𝑥𝑦)2 𝑥2 Total N-1 𝑦2
- 33. Thank You