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SW388R6 Data Analysis and Computers I Slide 1 Testing Assumptions of Linear Regression Detecting Outliers Transforming Variables Logic for testing assumptions

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SW388R6 Data Analysis and Computers I Slide 2 Assumptions of regression Based on information from the data set 2001WorldFactbook.sav, is the following statement true, false, or an incorrect application of a statistic? Assume that the assumptions of linear regression are satisfied. Use.05 for alpha in the regression analysis and.01 for the diagnostic tests. A simple linear regression between "population growth rate" [pgrowth] and "birth rate" [birthrat] will satisfy the regression assumptions if we choose to interpret which of the following models. 1 The original variables including all cases 2 The original variables excluding extreme outliers 3 The transformed variables including all cases 4 The transformed variables excluding extreme outliers *** 5 The quadratic model including all cases 6 The quadratic model excluding extreme outliers 7 None of the proposed models satisfies the assumptions The transformed variables excluding extreme outliers is the correct answer. [Feedback: 4743 characters] TESTING MODEL: ORIGINAL VARIABLES, USING ALL CASES The linear regression of "birth rate" [birthrat] by "population growth rate" [pgrowth] satisfied one of the regression assumptions (independence of errors). The Durbin- Watson statistic (1.93) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied. However, three assumptions were violated (linearity, homogeneity of error variance, and normality of the residuals). The lack of fit test (F(157, 59) = 1.78, p =.006) indicated that the assumption of linearity was violated. The Breusch-Pagan test (Breusch-Pagan(1) = , p <.001) indicated that the assumption of homogeneity of error variance was violated. The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(218) = 0.81, p <.001) indicated that the assumption of normality of errors was violated. TESTING MODEL: ORIGINAL VARIABLES, OMITTING EXTREME OUTLIERS One extreme outliers were found in the data. Montserrat was an extreme outlier (the cook's distance ( ) was larger than the cutoff value of ,the leverage ( ) was larger than the cutoff value of and the studentized residual (-9.173) was smaller than the cutoff value of -4.0). The linear regression of birthrat by "population growth rate" [pgrowth] satisfied two of the regression assumptions (linearity and independence of errors). The lack of fit test (F(156, 59) = 0.94, p =.617) indicated that the assumption of linearity was satisfied. The Durbin-Watson statistic (2.01) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied. However, two assumptions were violated (homogeneity of error variance and normality of the residuals). The Breusch-Pagan test (Breusch-Pagan(1) = 29.24, p <.001) indicated that the assumption of homogeneity of error variance was violated. The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(217) = 0.97, p <.001) indicated that the assumption of normality of errors was violated. SELECTING A TRANSFORMATION The logarithm of "birth rate" [LG_birthrat] with a value of for the Shapiro-Wilk statistic was the transformation that was most normal for the dependent variable "birth rate" [birthrat]. The logarithm of "population growth rate" [LG_pgrowth] with a value of for the Shapiro-Wilk statistic was the transformation that best approximated a normal distribution for the independent variable "population growth rate" [pgrowth]. TESTING MODEL: TRANSFORMED VARIABLES, INCLUDING ALL CASES The linear regression of logarithm of "birth rate" [LG_birthrat] by logarithm of "population growth rate" [LG_pgrowth] satisfied two of the regression assumptions (linearity and independence of errors). The lack of fit test (F(157, 59) = 1.38, p =.080) indicated that the assumption of linearity was satisfied. The Durbin-Watson statistic (1.94) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied. However, two assumptions were violated (homogeneity of error variance and normality of the residuals). The Breusch-Pagan test (Breusch-Pagan(1) = 29.02, p <.001) indicated that the assumption of homogeneity of error variance was violated. The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(218) = 0.96, p <.001) indicated that the assumption of normality of errors was violated. TESTING MODEL: TRANSFORMED VARIABLES, EXCLUDING EXTREME OUTLIERS One extreme outliers were found in the data. Montserrat was an extreme outlier (the cook's distance ( ) was larger than the cutoff value of ,the leverage ( ) was larger than the cutoff value of and the studentized residual (-9.173) was smaller than the cutoff value of -4.0). The linear regression of logarithm of "birth rate" [LG_birthrat] by logarithm of "population growth rate" [LG_pgrowth] satisfied all of the regression assumptions (linearity, homogeneity of error variance, normality of the residuals, and independence of errors). The lack of fit test (F(156, 59) = 1.14, p =.288) indicated that the assumption of linearity was satisfied. The Breusch-Pagan test (Breusch-Pagan(1) = 0.82, p =.367) indicated that the assumption of homogeneity of error variance was satisfied. The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(217) = 0.99, p =.357) indicated that the assumption of normality of errors was satisfied. The Durbin-Watson statistic (1.96) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied.

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SW388R6 Data Analysis and Computers I Slide 3 Run the script - 1 Select Run Script from the Utilities menu.

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SW388R6 Data Analysis and Computers I Slide 4 Run the script - 2 Navigate to the folder where you downloaded the script. Click on the Run button to run the script. Highlight the script (.SBS) file to run.

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SW388R6 Data Analysis and Computers I Slide 5 Assumption of linearity - 1 Highlight the dependent variable in the list of variables. Click on the arrow button to move the variable to the text box for the dependent variable.

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SW388R6 Data Analysis and Computers I Slide 6 Assumption of linearity - 1 Highlight the independent variable in the list of variables. Click on the arrow button to move the variable to the list box for the independent variable.

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SW388R6 Data Analysis and Computers I Slide 7 Initial test of conformity to assumptions Run the regression with all cases to test the initial conformity to the assumptions.

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SW388R6 Data Analysis and Computers I Slide 8 The Durbin-Watson statistic (1.93) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied.

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SW388R6 Data Analysis and Computers I Slide 9 The lack of fit test (F(157, 59) = 1.78, p =.006) indicated that the assumption of linearity was violated.

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SW388R6 Data Analysis and Computers I Slide 10 The Breusch-Pagan test (Breusch- Pagan(1) = , p <.001) indicated that the assumption of homogeneity of error variance was violated.

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SW388R6 Data Analysis and Computers I Slide 11 The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(218) = 0.81, p <.001) indicated that the assumption of normality of errors was violated.

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SW388R6 Data Analysis and Computers I Slide 12 One extreme outliers were found in the data. Montserrat was an extreme outlier (the cook's distance ( ) was larger than the cutoff value of ,the leverage ( ) was larger than the cutoff value of and the studentized residual (-9.173) was smaller than the cutoff value of - 4.0).

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SW388R6 Data Analysis and Computers I Slide 13 The script will remove the extreme outliers by clicking on the Exclude extreme outliers button. We could exclude the cases one at a time by selecting the case in the list of cases included and clicking on the arrow button, or we can use the script.

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SW388R6 Data Analysis and Computers I Slide 14 Case number 136, Montserrat, is added to the list of cases to exclude.

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SW388R6 Data Analysis and Computers I Slide 15 To see whether or not removing the outlier resolves the violation of assumptions, run the regression again. Run the regression with all cases to test the initial conformity to the assumptions.

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SW388R6 Data Analysis and Computers I Slide 16 This is an example of a strong linear relationship. The red lowess (loess in SPSS) smoother is almost completely straight throughout the range of the data. The rate of change in the dependent variable is the same for all values of the independent variable.

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SW388R6 Data Analysis and Computers I Slide 17 Removing the one extreme outlier solved the violation of the assumption of linearity. The lack of fit test (F(156, 59) = 0.94, p =.617) indicated that the assumption of linearity was satisfied.

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SW388R6 Data Analysis and Computers I Slide 18 The Durbin-Watson statistic (2.01) fell within the acceptable range from 1.50 to 2.50, indicating that the assumption of independence of errors was satisfied.

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SW388R6 Data Analysis and Computers I Slide 19 The Breusch-Pagan test (Breusch- Pagan(1) = 29.24, p <.001) indicated that the assumption of homogeneity of error variance was violated.

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SW388R6 Data Analysis and Computers I Slide 20 The Shapiro-Wilk test of studentized residuals (Shapiro-Wilk(217) = 0.97, p <.001) indicated that the assumption of normality of errors was violated.

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SW388R6 Data Analysis and Computers I Slide 21 Since removing outliers did not solve all of our violations, we will try transformations of the variables. We restore all of the cases to the analysis by clicking on the Include all cases button.

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SW388R6 Data Analysis and Computers I Slide 22 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 23 There is a statistical procedure named the Box-Cox transformation which SPSS does not compute and which I have not added to the script. However, we can use the test of normality as a surrogate. As the statistical value of the Shapiro-Wilk statistic gets larger, it is associated with a higher probability. We will select the transformation with the largest Shapiro- Wilk statistic as the transformation which best “normalizes” the variable, provided it is at least 0.01 larger than the statistical value for the untransformed variable. For this variable, we would choose the Logarithmic transformation. Choosing one transformation does not mean that is is particularly effective, only that it is better than the others.

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SW388R6 Data Analysis and Computers I Slide 24 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 25 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 26 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 27 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 28 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 29 First, click on the dependent variable to select it. Click on the Test normality button.

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SW388R6 Data Analysis and Computers I Slide 30 Since removing outliers did not solve all of our violations, we will try transformations of the variables. We restore all of the cases to the analysis by clicking on the Include all cases button.

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