7_Regression - Linear Regression Analysis Correlation Simple Linear Regression The Multiple Linear Regression Model Least Squares Estimates R2 and

7_Regression - Linear Regression Analysis Correlation...

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Linear Regression Analysis Correlation Simple Linear Regression The Multiple Linear Regression Model Least Squares Estimates R 2 and Adjusted R 2 Overall Validity of the Model ( F test) Testing for individual regressor ( t test) Problem of Multicollinearity Gaurav Garg (IIM Lucknow)
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Smoking and Lung Capacity Suppose, for example, we want to investigate the relationship between cigarette smoking and lung capacity We might ask a group of people about their smoking habits, and measure their lung capacities Cigarettes ( X ) Lung Capacity ( Y ) 0 45 5 42 10 33 15 31 20 29 Gaurav Garg (IIM Lucknow)
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Scatter plot of the data We can see that as smoking goes up, lung capacity tends to go down. The two variables change the values in opposite directions. 0 5 10 15 20 25 0 10 20 30 40 50 Lung Capacity Gaurav Garg (IIM Lucknow)
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Height and WeightConsider the following data of heights and weights of 5 women swimmers:Height (inch):62 64 65 66 68Weight (pounds):102108115128132We can observe that weight is also increasing with height. 61 62 63 64 65 66 67 68 69 0 20 40 60 80 100 120 140 Gaurav Garg (IIM Lucknow)
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Sometimes two variables are related to each other. The values of both of the variables are paired. Change in the value of one affects the value of other. Usually these two variables are two attributes of each member of the population For Example: Height Weight Advertising Expenditure Sales Volume Unemployment Crime Rate Rainfall Food Production Expenditure Savings Gaurav Garg (IIM Lucknow)
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is ,
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Properties of Covariance: Cov(X+a, Y+b) = Cov(X, Y) [not affected by change in location] Cov(aX, bY) = ab Cov(X, Y) [affected by change in scale] Covariance can take any value from -∞ to +∞ . Cov(X,Y) > 0 means X and Y change in the same direction Cov(X,Y) < 0 means X and Y change in the opposite direction If X and Y are independent, Cov(X,Y) = 0 [other way may not be true] It is not unit free. So it is not a good measure of relationship between two variables. A better measure is correlation coefficient. It is unit free and takes values in [-1,+1]. Gaurav Garg (IIM Lucknow)
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Correlation Karl Pearson’s Correlation coefficient is given by When the joint distribution of X and Y is known When observations on X and Y are available Gaurav Garg (IIM Lucknow) ) ( ) ( ) , ( ) , ( Y Var X Var Y X Cov Y X Corr r XY 2 2 2 2 )] ( [ ) ( ) ( , )] ( [ ) ( ) ( ) ( ) ( ) ( ) , ( Y E Y E Y Var X E X E X Var Y E X E XY E Y X Cov n i i n i i n i i i y y n Y Var x x n X Var y y x x n Y X Cov 1 2 1 2 1 ) ( 1 ) ( , ) ( 1 ) ( ) )( ( 1 ) , (
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Properties of Correlation Coefficient Corr(aX+b, cY+d) = Corr(X, Y), It is unit free. It measures the strength of relationship on a scale of -1 to +1 . So, it can be used to compare the relationships of various pairs of variables. Values close to 0 indicate little or no correlation Values close to +1 indicate very strong positive correlation. Values close to -1 indicate very strong negative correlation. Gaurav Garg (IIM Lucknow)
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