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### Diagnostics – Part II

Using statistical tests to check to see if the assumptions we made about the model are realistic

Diagnostic methods

- Some simple (but subjective) plots.(Then)
- Some formal statistical tests. (Now)

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.

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.)

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.

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.

Example: Blaisdell Company terms

Seasonally adjusted quarterly data, 1988 to 1992.

Reasonable fit, but are the error terms positively auto-correlated?

Blaisdell Company Example: Durbin-Watson test terms

- 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.

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.

Modified Levene Test for nonconstant error variance Durbin-Watson test

- 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.

Modified Levene Test in Minitab Durbin-Watson test

- 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.

Plutonium Alpha Example: varianceModified 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.

(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.

100 chi-square (1 df) data values in Minitab

100 normal(0,1) data values data values

Some closing comments Age (Y)

- 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.

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