Multiple Linear Regression estimation, testing and checking assumptions

Transcription

1 Multiple Linear Regression estimation, testing and checking assumptions Lecture No. 07

2 Example 1 The president of a large chain of fast-food restaurants has randomly selected 10 franchises and recorded for each franchise the following information on last year s net profit and sales activity. For these data, there will be one dependent variable (y net profit) and two independent variables (x 1 counter sales; x 2 drive-through sales). The form of the sample regression equation will be y = b 0 + b 1 x 1 + b 2 x 2.

3 The multiple linear regression model

4 The multiple linear regression model The sample regression equation is based on observed values for the dependent and independent variables. I has the form: Determination of the best-fit multiple regression equation is according to the least-squares criterion, in which the sum of the squared deviations between observed and estimated values of y is minimized.

5 Assumptions for multiple linear regression 1. Linear relationship between the independent and dependent variables 2. All variables are normally distributed 3. No multicollinearity independent variables must not be highly correlated 4. Independence of random errors (residuals) 5. Homoscedasticity (homogeneity of variance) 6. Normally distributed errors

6 Example 1 Solution in Excel

7 Example 1 Solution in Statistica

8 Example

9 Example

10 Example 1

11 Example 1 The scatter diagram for the fast-food data can be visualized as a room where each of 10 helium-filled balloons is held to the floor by a string. The length of each string is the observed value of y for that data point. The least-squares regression equation (not shown) passes through the data and takes the form of a two-dimensional surface, or plane.

12 Example 1 - Point Estimation Using the Regression Equation Point estimates are made by substituting a set of x values into the regression equation and calculating the estimated value of y. For example, if a franchise had sold x 1 = $5.0 million over the counter and x 2 = $7.4 million at the drive-through, we would estimate its net profit as $1.05 million:

13 The Multiple Standard Error of Estimate For our example, there are k = 2 independent variables and n = 10 data points. We calculate the error sum of squares

14 Example 1 - Confidence Interval for the Conditional Mean of y For the sample of 10 fastfood franchises, the summary results from these calculations are important in determining both the multiple standard error of estimate and the coefficients of multiple correlation and determination. (Data are in millions of dollars.)

15 Confidence Interval for the Conditional Mean of y

16 Example 1 - Confidence Interval for the Conditional Mean of y

17 Example 1 - Confidence Interval for the Conditional Mean of y Solution in Statistica

18 Example 1 - Confidence Interval for the Conditional Mean of y Solution in Statistica

19 Prediction Interval for an Individual y Observation

20 Example 1 - Prediction Interval for an Individual y Observation

21 Example 1 - Prediction Interval for an Individual y Observation

22 Example 1 - Prediction Interval for an Individual y Observation Solution in Statistica

23 Testing the Partial Regression Coefficients

24 Example 1 - Testing the Partial Regression Coefficients

25 Example 1 - Testing the Partial Regression Coefficients

26 Interval Estimation for the Partial Regression Coefficients

27 Example 1 - Interval Estimation for the Partial Regression Coefficients

28 Checking the assumptions for multiple linear regression 1. Linear relationship between the independent and dependent variables 2. All variables are normally distributed 3. No multicollinearity independent variables must not be highly correlated 4. Independence of random errors (residuals) 5. Homoscedasticity (homogeneity of variance) 6. Normally distributed errors

29 Example 2 - Checking the assumptions for multiple linear regression Solution in Statistica

30 Example 2 - Linear relationship between the independent and dependent variables 1 2 3

31 Example 2 - Linear relationship between the independent and dependent variables 1 2 3

32 Example 2 - Linear relationship between the independent and dependent variables 1 2 Checking the linear relationship between Y and X1 X7

33 Example 2 - Linear relationship between the independent and dependent variables 1

34 Example 2 - All variables are normally distributed

35 Example 2 - All variables are normally distributed 3 1 2

36 Example 2 - All variables are normally distributed p value of Shapiro-Wilk test review all graphs

37 Example 2 - No multicollinearity 1 2 3

38 Example 2 - No multicollinearity 1 2 3

39 Example 2 - No multicollinearity 1

40 Example 2 - Independence of random errors (residuals) 1 Homoscedasticity (homogeneity of variance) Normally distributed errors

41 Example 2 - Independence of random errors (residuals) 1 Homoscedasticity (homogeneity of variance) Normally distributed errors 2 3

42 Example 2 - Independence of random errors (residuals) Homoscedasticity (homogeneity of variance) Normally distributed errors

43 Example 2 - Independence of random errors (residuals) 1 Homoscedasticity (homogeneity of variance) Normally distributed errors 2

44 Example 2 - Independence of random errors (residuals) Homoscedasticity (homogeneity of variance) Normally distributed errors

45 Example 2 - Checking the assumptions for multiple linear regression Solution in Excel

46 Example 2 - Linear relationship between the independent and dependent variables 1 2 3

47 Example 2 - Linear relationship between the independent and dependent variables 1 2 3

48 Example 2 - All variables are normally distributed

49 Example 2 - All variables are normally distributed

50 Example 2 - No multicollinearity 1 2 3

51 Example 2 - No multicollinearity

52 Example 2 - Independence of random errors (residuals) 1 Homoscedasticity (homogeneity of variance) Normally distributed errors

53 Example 2 - Independence of random errors (residuals) Homoscedasticity (homogeneity of variance) Normally distributed errors

54 Example 2 - Independence of random errors (residuals) Homoscedasticity (homogeneity of variance) Normally distributed errors

55 Example 2 - Independence of random errors (residuals) Homoscedasticity (homogeneity of variance) Normally distributed errors