Stat 501 Sept. 22 Lab Significance tests that might be used for examining possible difficulties with a model:For the tests listed below, the null hypothesis is the desired result. In each case, the null is that some particular aspect of the regression model is correct. 1. Test the null hypothesis that the errors are normally distributed H0: εi ~ n(0,σ2)- Ryan-Joiner test (the test called the correlation test in chapter 3)- Anderson-Darling test (the default test in Minitab). It’s a bit more powerful than the Ryan-Joiner.- Kolmogorov-Smirnov (an option in Minitab). It’s relatively weak compared to the test just listed.I advise against its use. Minitab: Store the residuals using the Storage button in the regression dialog. Then, use Stat>Basic Stats>Normality Test with the stored column of residuals as the variable. 2. Test the null hypotheses that the errors have constant variance. H0: 22i where 2i represents the variance for unit i.- Brown-Forsythe (also called modified Levene). Described in section 3.6.- Breusch-Pagan – described on pages 118-119. We won’t do these, other than familiarize ourselves with the fact that they could be done. My experience is that in practice these tests aren’t necessary. 3. Test whether the equation used for E(Y) is correct. For a straight-line model the null hypothesis isi1oioX)Y(E:H . Caution: This is not the same as testing01!- Pure Error test (F test for Lack of Fit described in section 3.7). To do this test, it’s necessary that some observations have the same values of x. - Data subsetting approach (available in Minitab). This is based on comparing the MSE from the model to an MSE from a piecewise model where potentially different lines are used for different regions of x. Minitab: In Stat>Regression>Regression, use Options. These two tests are available in the resulting dialog box. Lowess Smoothing (pages 138-139 of the book).This is essentially a weighted moving average of y-values used to identify the pattern of a plot. We average y-values for points with x-values that are close to each other. When smoothed values are plotted, the result is a line that might be helpful for identifying whether the pattern is linear or curvilinear. Minitab: Use Graph>Scatterplot. After selecting the desired type of plot and specifying the variables, select Data View, click on the Smoother tab, and then click on Lowess.Name _____________________________ PSU e-mail ID _______________ Sept. 22Go to the website www.stat.psu.edu/~rho/501data/ Click on the link for the dataset lab-sept22.mtw. The first two columns give Y = the concentration of a chemical solution, X = time (hours) since preparation of the solution. 1. Graph Y versus X. Include a Lowess smoother line as part of the graph. (See bottom of previous page for how to do this.) What are the noteworthy features of this plot?2. Use Stat>Regression>Regression to fit a straight line model to the data. Store the residuals using the Storage button. Also, use the Options button and click on the Pure Error test. Write the null hypothesis for the “pure error” lack of fit F-test.3. In the output, find the F-value and the p-value for the pure error lack of fit F test. F = p-value = 4. What conclusion can be made based on the pure error lack of fit F test?5. Do an Anderson-Darling test of whether the errors have a normal distribution or not. Write the null hypothesis, report the p-value and state a conclusion.6. Refer to the previous part. The output included a normal probability plot. In general, what is the ideal pattern for this plot? 7. Use Calc>Calculator to create a new column in Minitab that contains the log base 10 of Y. Graph this column versus X. Describe the result. 8. Do a regression in which the log base 10 is the response variable (and the original x is still the x). Writethe estimated equation. 9. Give the F-value and p-value for the pure error lack of fit test, and state a conclusion about this situation based on this10. For the log data, use the Option button in Stat>Regression>Regression to get a 95% prediction interval for the log concentration when time x = 2. Write the interval.11. Convert the prediction interval found in the previous part to an interval for the concentration (in the original scale, not the log scale).--------------------Columns 4 and 5 contain data for Time = number of days of growing for a tumor being grown in a lab andSize = size of tumor in cc.11. Graph y = Size versus x = Time. Comment on the pattern.12. Use Stat>Regression>Regression with Y = Size and X = Time. Use the options button to request a Pure Error test. What happens? Why does this happen?13. Return to the regression dialog and select the data subsetting lack of fit test rather than the pure error test. Report the result of this test. 14. What would you try next as a model for this
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