Regression Analysis: “Regression is that technique of statistical analysis by which when one variable is known the other variable can be estimated. In regression analysis the variable, which is known, is called independent variable and which is to be estimated is called dependent variable.”

3 Nondependent and Dependent Relationships Types of Relationship Types of Relationship Nondependent (correlation) -- neither one of variables is target Example: protein and fat intake Nondependent (correlation) -- neither one of variables is target Example: protein and fat intake Dependent (regression) -- value of one variable is used to predict value of another variable. Example: ACT and MCAT scores for medical applicants, MCAT is the dependent and ACT is the independent variable Dependent (regression) -- value of one variable is used to predict value of another variable. Example: ACT and MCAT scores for medical applicants, MCAT is the dependent and ACT is the independent variable Statistical Expressions Statistical Expressions Correlation Coefficient -- index of nondependent relationship Correlation Coefficient -- index of nondependent relationship Regression Coefficient -- index of dependent relationship Regression Coefficient -- index of dependent relationship

Regression: 3 Main Purposes To describe (or model) To describe (or model) To predict (or estimate) To predict (or estimate) To control (or administer) To control (or administer)

Simple Linear Regression Statistical method for finding Statistical method for finding the “line of best fit” the “line of best fit” for one response (dependent) numerical variable for one response (dependent) numerical variable based on one explanatory (independent) variable. based on one explanatory (independent) variable.

Difference between Correlation and Regression: 1.Degree and nature: Correlation studies the relationship between two or more series but regression analysis measures the degree and extent of this relationship thereby providing a base for estimation.

2.Cause and effect relationship: Correlation specifies the relationship between two series and it can specify as to what extent is the cause and what is effect. Whereas in regression the value of which series is known is called independent series and whose value is to be predicted is called dependent series. The independent series is cause and independent series is effect.

3. Limit of co-efficient: The limit of co-efficient of correlation is plus minus 1 but this is not the case with regression co-efficient. But the product of both the regression co- efficient cannot become greater than 1.

Regression Lines: The regression analysis between two related series of data is usually done with the help of diagrams. On the scatter diagram obtained by plotting the various values of related series X and Y, two lines of best fit are drawn through the various points of the diagrams, which are called regression lines.

Why two regression lines?  When there are two series then the lines of regression will also be two.  If the variable values of two series are named as X and Y then one regression line is called X on Y and the other is called Y on X.

Deviations Taken From Arithmetic Mean (i)Regression Equations of X on Y X – X = r  x ( Y – Y)  y Here r  x is known as the regression coefficient of X on Y  y It is also denoted by b xy or b1. It measures the change in X corresponding to a unit Change in Y Also b xy =  xy /  y 2

(ii)Regression Equations of Y on X Y – Y = r  y ( X – X)  x Here r  y is known as the regression coefficient of Y on X  x It is also denoted by b yx or b2. It measures the change in Y corresponding to a unit Change in X Also b yx =  xy /  x 2 Where x = X –( mean of X series) y = Y –( mean of Y series)

Also Regression equation of X on Y (X – X ) = b 1 (Y- Y) Regression equation of Y on X (Y – Y) = b 2 (X – X) Here r is known as the coefficient of correlation between X and Y series. and r =  b 1 x b 2

Least square method: X on Y:Y on X:  X = n.a + b.  Y  Y = n.a + b.  X  XY =  Y a + b  Y  XY =  X a + b  X X = a + byY = a + bx 2 2

Example: Calculate the regression equations taking deviations of items from the mean of X and Y series. XY

Deviations taken from assumed mean (i)Regression Equations of X on Y X – X = r  x ( Y – Y)  y Here r  x = N  dx dy – (  dx  dy) / N  dy 2 (  dy) 2  y (ii)Regression Equation of Y on X Y – Y = r  x ( X – X)  y Here r  x = N  dx dy – (  dx  dy) / N  dx 2 (  dx) 2  y

Example: From the following data of the rainfall and production of rice, find (i) the most likely production corresponding to rainfall 40 cm (ii) the most likely rainfall corresponding to production 45 kgs. Rainfall(cm)Prod (kgs) Mean3550 Std deviation58 Coefficient of correlation between rainfall and production =.8

Example: Obtain the lines of regression: XY

Example: From the following series X and Y, find out the value of: (i)Two regression coefficients. (ii)Two regression equations. (iii) Most likely value of X when Y is 34. (iv) Most likely value of Y when X is 47.

Series X Series Y

Example : The two regression equations are as follows: 20 X - 3Y= 975………………..(i) 4 Y – 15 X + 530= 0………………….(ii) Find out (i) Mean value of X and Y (ii) The coefficient of correlation between X and Y (iii)Estimate the value of Y, when X = 90; and that of X when Y = 130.

Example: For a certain X and Y series, the two lines of regression Are given below 6Y – 5X = 90 15X – 8Y = 130 Variance of X series is Find the mean value of X and Y series 2. Coefficient of correlation between X and Y series. 3. Standard deviation of Y series.

Real Life Applications Estimating Seasonal Sales for Department Stores (Periodic)

Real Life Applications Predicting Student Grades Based on Time Spent Studying

Practice Problems Measure Height vs. Arm Span Find line of best fit for height. Predict height for one student not in data set. Check predictability of model.

Practice Problems Is there any correlation between shoe size and height? Does gender make a difference in this analysis?

Practice Problems Can the number of points scored in a basketball game be predicted by The time a player plays in the game? By the player’s height?

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