Linear Regression and Correla/on. Correla/on and Regression Analysis. Three Ques/ons 9/14/14. Chapter 13. Dr. Richard Jerz

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1 Linear Regression and Correla/on Chapter 13 Dr. Richard Jerz 1 Correla/on and Regression Analysis Correla/on Analysis is the study of the rela/onship between variables. It is also defined as group of techniques to measure the associa/on between two variables. Regression Analysis is a technique used to express the rela/onship between two variables. If the rela/onship is assumed to be a straight line, this is called linear regression. 2 Three Ques/ons 1. Are two variables related? (correla/on analysis) 2. Is there a linear rela/onship between two variables? (linear regression analysis) 3. How strong are these rela/onships? 3 1

2 9/14/14 Correla/on and Linear Regression 4 Correla/on 5 Correla/on & Regression Example The sales manager of Copier Sales of America, which has a large sales force throughout the United States and Canada, wants to determine whether there is a rela/onship between the number of sales calls made in a month and the number of copiers sold that month. The manager selects a random sample of 10 representa/ves and determines the number of sales calls each representa/ve made last month and the number of copiers sold. 6 2

3 Step 1: Look at the Data (Plot the Data) A ScaXer Diagram is a chart that portrays the rela/onship between the two variables. It is the usual first step in correla/on analysis The Dependent Variable is the variable being predicted or es/mated. The Independent Variable provides the basis for es/ma/on. It is the predictor variable. 7 Step 2: Are they correlated? 8 The Coefficient of Correla/on, r The Coefficient of Correla/on (r) is a measure of the strength of the rela/onship between two variables. It requires interval or ra/o- scaled data. It can range from to Values of or 1.00 indicate perfect and strong correla/on. Values close to 0.0 indicate weak correla/on. Nega/ve values indicate an inverse rela/onship and posi/ve values indicate a direct rela/onship. 9 3

4 Correla/on Coefficient Interpreta/on 10 Correla/on Coefficient Equa/on, r 11 Coefficient of Determina/on The coefficient of determina/on (r 2 ) is the propor/on of the total varia/on in the dependent variable (Y) that is explained or accounted for by the varia/on in the independent variable (X). It is the square of the coefficient of correla/on. It ranges from 0 to 1. It does not give any informa/on on the direc/on of the rela/onship between the variables. 12 4

5 Example: Correla/on Coefficient How do we interpret a correlation of 0.759? First, it is positive, so we see there is a direct relationship between the number of sales calls and the number of copiers sold. The value of is fairly close to 1.00, so we conclude that the association is strong. However, does this mean that more sales calls cause more sales? No, we have not demonstrated cause and effect here, only that the two variables sales calls and copiers sold are related. 13 Coefficient of Determina/on (r 2 ) - Example The coefficient of determina/on, r 2,is 0.576, found by (0.759) 2 This is a propor/on or a percent; we can say that 57.6 percent of the varia/on in the number of copiers sold is explained, or accounted for, by the varia/on in the number of sales calls. 14 Step 3: How strong are they correlated? 15 5

6 Tes/ng the Significance of the Correla/on Coefficient H 0 : ρ = 0 (the correlation in the population is 0) H 1 : ρ 0 (the correlation in the population is not 0) Reject H 0 if: t > t α/2,n-2 or t < -t α/2,n-2 16 Example: Tes/ng the Significance of the Correla/on Coefficient H 0 : ρ = 0 (the correlation in the population is 0) H 1 : ρ 0 (the correlation in the population is not 0) Reject H 0 if: t > t α/2,n-2 or t < -t α/2,n-2 t > t 0.025,8 or t < -t 0.025,8 t > or t < Tes/ng the Significance of the Correla/on Coefficient Equivalently, you can calculate the cri/cal value for the correla/on coefficient using This method gives a benchmark for the correla/on coefficient. However, there is no p- value and is inflexible if you change your mind about α. 18 6

7 Tes/ng Rela/onship with r Step 1: State the Hypotheses Determine whether you are using a one or two- tailed test and the level of significance (a). H0 : ρ = 0 H 1 : ρ 0 Step 2: Calculate the Cri/cal Value For degrees of freedom n = n - 2, determine t α then calculate 19 Tes/ng Rela/onship with r Make the Decision If the sample correla/on coefficient r exceeds the cri/cal value r α, then reject H 0. If using the t sta/s/c method, reject H 0 if t > t α or if the p- value α. 20 Ex: Tes/ng the Significance of the Correla/on Coefficient The computed t (3.297) is within the rejection region, therefore, we will reject H 0. This means the correlation in the population is not zero. From a practical standpoint, it indicates to the sales manager that there is correlation with respect to the number of sales calls made and the number of copiers sold in the population of salespeople. 21 7

8 Step 4: Is there a linear rela/onship? Regression 22 Example: Robot Repeatability Data

9 Bivariate Regression Analysis Bivariate Regression analyzes the rela/onship between two variables. It specifies one dependent (response) variable and one independent (predictor) variable. This hypothesized rela/onship may be linear, quadra/c, or whatever. 25 Linear Regression Unknown parameters are β 0 Intercept β 1 Slope The assumed model for a linear rela/onship is y i = β 0 + β 1 x i + e i, for all observa/ons (i = 1, 2,, n) The error term is not observable, is assumed normally distributed with mean of 0 and standard devia/on σ. 26 Linear Regression Model Y ˆ = a + bx Y Y Hat, is the es/mate of Y given X a is the Y- intercept b is the slope of the line X is any value of the independent variable 27 9

10 Differences between Actual Y Es/mated Y 28 Least Squares Principle Determining a regression equa/on by minimizing the sum of the squares (the variance) of the ver/cal distances between the actual Y values and the predicted values of Y. 29 Calcula/ng a and b (least squares) b = XY 2 X nxy 2 nx a = Y bx 30 10

11 Example: Finding the Regression Equa/on Y = X Y = (20) 31 Y = The regression equation is: Y = a + bx Step 5: Plot the Es/mated and the Actual Y s 32 Compu/ng the Es/mates of Y Step 1 Using the regression equa/on, subs/tute the value of each X to solve for the es/mated sales Tom Keller Y = X Y = (20) Soni Jones Y = X Y = (30) 33 Y = Y =

12 Step 6: Predict other values The regression equation is: Y = a + bx Y = X Y = (20) Y = Assump/ons Underlying Linear Regression For each value of X, there is a group of Y values, and these Y values are normally distributed. The means of these normal distribu/ons of Y values all lie on the straight line of regression. The standard devia/ons of these normal distribu/ons are equal. The Y values are sta/s/cally independent. This means that in the selec/on of a sample, the Y values chosen for a par/cular X value do not depend on the Y values for any other X values. 35 Assump/ons: Graphic 36 12

13 Confidence Intervals for Slope and Intercept Confidence interval for the true slope: Confidence interval for the true intercept: 37 Ex: Confidence Interval Es/mate Step 4 Use the formula above by subs/tu/ng the numbers computed in previous slides Thus, the 95 percent confidence interval for the average sales of all sales representa/ves who make 25 calls is from up to copiers. 38 Ex: Predic/on Interval Es/mate Step 2 Using the informa/on computed earlier in the confidence interval es/ma/on example, use the formula above. If Sheila Baker makes 25 sales calls, the number of copiers she will sell will be between about 24 and 73 copiers 39 13

14 End of Chapter