Diagnostics – Part II

1. Diagnostics – Part II Using statistical tests to check to see if the assumptions we made about the model are realistic

2. Diagnostic methods • Some simple (but subjective) plots.(Then) • Some formal statistical tests. (Now)

3. Simple linear regression model The response Yi is a function of a systematic linear component and a random error component: with assumptions that: • Error terms have mean 0, i.e., E(i) = 0. • i and j are uncorrelated (independent). • Error terms have same variance, i.e., Var(i) = 2. • Error terms i are normally distributed.

4. Why should we keep NAGGING ourselves about the model? • All of the estimates, confidence intervals, prediction intervals, hypothesis tests, etc. have been developed assuming that the model is correct. • If the model is incorrect, then the formulas and methods we use are at risk of being incorrect. (Some are more forgiving than others.)

5. Summary of the tests we’ll learn … • Durbin-Watson test for detecting correlated (adjacent) error terms. • Modified Levene test for constant error variance. • (Ryan-Joiner) correlation test for normality of error terms.

6. The Durbin-Watson test for uncorrelated (adjacent) error terms Durbin-Watson test statistic • Compare D to Durbin-Watson test bounds in Table B.7: • If D > upper bound (dU), conclude no correlation. • If D < lower bound (dL), conclude positive correlation. • If D is between the two bounds, the test is inconclusive.

7. Example: Blaisdell Company Seasonally adjusted quarterly data, 1988 to 1992. Reasonable fit, but are the error terms positively auto-correlated?

9. Blaisdell Company Example: Durbin-Watson test • Stat >> Regression >> Regression. Under Options…, select Durbin-Watson statistic. • Durbin-Watson statistic = 0.73 • Table B.7 with level of significance α=0.01, (p-1)=1 predictor variable, and n=20 (5 years, 4 quarters each) gives dL= 0.95 and dU=1.15. • Since D=0.73 < dL=0.95, conclude error terms are positively auto-correlated.

10. For completeness’ sake … one more thing about Durbin-Watson test • If test for negative auto-correlation is desired, use D*=4-D instead. If D* < dL, then conclude error terms are negatively auto-correlated. • If two-sided test is desired (both positive and negative auto-correlation possible), conduct both one-sided tests, D and D*, separately. Level of significance is then 2α.

11. Modified Levene Test for nonconstant error variance • Divide the data set into two roughly equal-sized groups, based on the level of X. • If the error variance is either increasing or decreasing with X, the absolute deviations of the residuals around their group median will be larger for one of the two groups. • Two-sample t* to test whether mean of absolute deviations for one group differs significantly from mean of absolute deviations for second group.

12. Modified Levene Test in Minitab • Use Manip >> Code >> Numeric to numeric … to create a GROUP variable based on the values of X. • Stat >> Regression >> Regression. Under Storage …, select residuals. • Stat >> Basic statistics >> 2 Variances … Specify Samples (RESI1) and Subscripts (GROUP). Select OK. Look in session window for Levene P-value.

13. Example: How is plutonium activity related to alpha particle counts?

14. A residual versus fits plot suggesting non-constant error variance

15. Plutonium Alpha Example: Modified Levene’s Test Levene's Test (any continuous distribution) Test Statistic: 9.452 P-Value : 0.006 It is highly unlikely (P=0.006) that we’d get such an extreme Levene statistic (L=9.452) if the variances of the two groups were equal. Reject the null hypothesis at the 0.01 level, and conclude that the error variances are not constant.

16. (Ryan-Joiner) Correlation test for normality of error terms in Minitab • H0: Error terms are normally distributed vs. HA: Error terms are not normally distributed • Stat >> Regression >> Regression. Under storage…, select residuals. • Stat >> Basic statistics >> Normality Test. Select residuals (RESI1) and request Ryan-Joiner test. Select OK.

17. 100 chi-square (1 df) data values

18. Normal probability plot and test for 100 chi-square (1 df) data values

19. 100 normal(0,1) data values

20. Normal probability plot and test for 100 normal(0,1) data values

21. Normal probability plot for Tree diameter (X) and C-dating Age (Y)

22. Tree diameter and Age Example: Ryan-Joiner Correlation Test

23. Some closing comments • Checking of assumptions is important, but be aware of the “robustness” of your methods, so you don’t get too hung up. • Model checking is an art as well as a science. • Do not think that there is some definitive correct answer “in the back of the book.” • Use your knowledge of the subject matter.