Statistical Reasoning in education- Multiple linear regression

PMC500

1. PMC 500 STATISTICAL REASONING IN EDUCATION Lecture 9: Multiple Linear Regression Dr. Nor Asniza Ishak Dr. AhmadZamri Khairani

2. 1. Multiple Linear Regression Multiple linear regression is used to predict scores on one variable using scores from two or more different variables based on a multiple regression plane.  The variable that is being predicted is known as the dependent or criterion variable (Y )  The variables that are used to predict scores are known as the independent or predictor variables (X1 , X2 , ... Xk )

3. 2. Multiple Regression Plane The multiple regression plane is a plane that best represents the linear relationship between two or more predictor variables (X1 , X2 , ... Xk ) and one criterion variable (Y ) based on the method of least squares

4. 2. Multiple Regression Plane The method of least squares fits the multiple regression plane in such a way that the sum of squares of the difference between the actual value of Y and the predicted value of Y (Y’ ) is a minimum Or in symbol,

5. 2. Multiple Regression Plane To build the multiple regression plane, the equation of the multiple regression plane ( multiple regression equation ) needs to be determined.

6. 3. Multiple Regression Equation

7. 3. Multiple Regression Equation

8. 3. Multiple Regression Equation Example: To determine the multiple regression equation involving the linear relationship between two predictor variables (X1 and X2 ) and one criterion variable (Y )

9. 3. Multiple Regression Equation

10. 3. Multiple Regression Equation

11. Example

12. Obtaining a 3-D Scatterplot using SPSS Create a data file Graphs > Chart Builder ... > Scatter/Dot Move Simple 3-D Scatter to Chart preview box Move the criterion variable, Y to the Y-axis: box Move the predictor variables X1 to the X-axis box and X2 to the Z- axis: box Click OK Double-click on the 3-D scatterplot Click Options > Show Grid Lines > Close Click on any point Click Elements > Show Data Labels > Close Close Chart Editor

13. Example Cost of advertisement (X1) and number of students (X2) are the predictor variables Sale of books (Y) is the criterion variable Plot graph Y against X1 and X2 to produce a 3-D Scatterplot by using SPSS

14. 3-D Scatterplot

15. Obtaining the multiple regression equation

16. 4. Obtaining the Multiple Regression Equation Using SPSS  Create a data file Click Analyze > Regression > Linear When the Linear Regression box is displayed, move the criterion variable (Y ) to the Dependent: box and move the predictor variables (X1 and X2) to the Independent(s): box Click OK

17. 4. Obtaining the Multiple Regression Equation Using SPSS

18. Meaning of a The value of a = 6.40 means that the predicted value of sale of books (Y’) is 6.40 thousand Ringgit Malaysia (or RM 6 400) when the advertising cost of advertisement (X1) is zero and the number of students (X2) is zero

19. Meaning of b 1 The value of b1 = 20.49 means that the predicted value of sale of books (Y’) increases by 20.49 thousand ringgit Malaysia (i.e., RM 20 490) for a 1-unit change (i.e., RM 1 000) in the cost of advertisement (X1) when the number of students (X2) remains constant.

20. Meaning of b 2 The value of b2 = 0.28 means that the predicted value of sale of books (Y’) increases by 0.28 thousand Ringgit Malaysia (i.e., RM 280) for a 1-unit change (i.e., 1000) in the number of students (X2) when the cost of advertisement (X1) remains constant.

21. 5. Errors of Prediction

22. 6. Standard Error of the Estimate

23. 6. Standard Error of the Estimate

24. 6. Standard Error of the Estimate n multiple linear regression, the predictor variables should have a strong linear relationship with the criterion variable, but have a weak linear relationship between themselves The smaller the standard error of the estimate, the better is the prediction of the value of the criterion variable from the values of the predictor variables

25. 7. Obtaining the Standard Error of the Estimate Using SPSS

26. Standard error of the estimate

27. Effect Size R Square (R2 ) is the effect size for multiple linear regression Shows the percentage of variance in the criterion variable (Y) that is explained by the linear combination of the predictor variables.

28. Adjusted R Square

29. 8. Testing the Multiple Correlation Coefficient for Statistical Significance

30. 8. Testing the Multiple Correlation Coefficient for Statistical Significance

31. 8. Testing the Multiple Correlation Coefficient for Statistical Significance Steps for the significance test: State the null and alternative hypotheses Set the significance level for rejecting the null hypothesis Analyse the data using SPSS Make a decision about the null hypothesis Draw a conclusion

32. Example 1 A researcher wanted to predict the sale of wood from the price of wood and thickness of wood. Table 1 shows the sale of wood, price of wood and thickness of wood for 30 pieces of wood that were sold by a hardware shop. Test the hypothesis of whether the price of wood and thickness of wood are significant predictors of the sale of wood at the 0.05 level of significance. Which predictor variable is the better predictor?

33. Step 1: State the null and alternative hypotheses

34. Step 2: Set the significance level for rejecting the null hypothesis Example: Set α = .05, If p ≤ .05, reject the null hypothesis. If p > .05, fail to reject the null hypothesis.

35. Step 3: Analyse the data using SPSS Create a data file Analyze > Regression > Linear Move the criterion variable (Y ) to the Dependent: box Move the predictor variables (X1 and X2) to the Independent(s): box Statistics... > Confidence intervals Level (95 %): > Descriptives > Casewise diagnostics Continue Plots... Move *ZRESID to Y box Move *ZPRED to X box Choose Histogram and Normal probability plot Continue > OK

36. Step 4: Make a decision about the null hypothesis

37. Step 5: Draw a Conclusion

38. Step 5: Draw a Conclusion Which predictor variable is a significant predictor of the sale of wood? Price of wood or thickness of wood or both?

39. Step 5: Draw a Conclusion

40. Step 5: Draw a Conclusion

41. Step 5: Draw a Conclusion Which predictor variable is a better predictor of the sale of wood? The price of wood or the thickness of wood?

42. Step 5: Draw a Conclusion

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