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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.

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Diagnostics – Part II

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diagnostics part ii

Diagnostics – Part II

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

diagnostic methods
Diagnostic methods
  • Some simple (but subjective) plots.(Then)
  • Some formal statistical tests. (Now)
simple linear regression model
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
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
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
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
Example: Blaisdell Company

Seasonally adjusted quarterly data, 1988 to 1992.

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

blaisdell company example durbin watson test
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.
for completeness sake one more thing about durbin watson test
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
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.
modified levene test in minitab
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.
plutonium alpha example modified levene s test
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.

ryan joiner correlation test for normality of error terms in minitab
(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.
some closing comments
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.