Tukey's Studentized Range (HSD) Test for logcountBonferroni (Dunn) t Tests for logcountScheffe's Test for logcountTukey's Studentized Range (HSD) Test for logcountBonferroni (Dunn) t Tests for logcountIES 612/STA 4-573/STA 4-576Spring 2005Week 7 – IES612-week07-lecture.docANOVA MODELS – model adequacyNumeric data – samples from “t’ populations obtainedAssume Yij ~ independent N(i, 2) or Yij = i + ij with ij ~ independent N(, 2)i = 1,2, …, t (populations or treatments)j = 1, 2, …, ni (observations)Residual definition? .iijijyye Assumption Checking? Addressing?Constant variance? Plot eij vs. sample means and look for a pattern- Transformation (e.g. log, sqrt)- Weighted Least SquaresNormal responses? - Normal probability plot (normal scores vs. residual quantiles)- Histogram? Boxplot? Stemplot?- 68% of standardized residuals with -1 and +1 (95% within -2 and +2)- Transformation (sqrt – count responses, arcsin-sqrt – proportions, log – right skewed responses)- GLiMsIndependent? Plot residuals vs. order of observations? Often implicit part of the designAnalysis that reflects dependence?Outliers? Large standardized residuals? Check? Run analysis with andwithout points? Rank-based methods?Why not worry about quality of fits? Devote a unique parameter to each group so model specification is not much of an issue.1Example: How about the Meat study – log(bacterial growth) with different conditions Consider the adequacy of the model:Log(Bacterial growth)ij = i + ij for i=1, 2, 3, 4 (packaging condition) and j=1, 2, 3. Plot of resid*condition. Legend: A = 1 obs, B = 2 obs, etc. resid ‚ ‚ 0.4 ˆ ‚ ‚ A A A ‚ ‚ 0.2 ˆ A ‚ A A ‚ ‚ A ‚ 0.0 ˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‚ A ‚ ‚ ‚ -0.2 ˆ A ‚ A ‚ ‚ ‚ -0.4 ˆ ‚ A ‚ A ‚ ‚ -0.6 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ CO2 mixed plastic vacuum Condition2Plot of resid*yhat. Legend: A = 1 obs, B = 2 obs, etc.resid ‚ ‚ 0.4 ˆ ‚ ‚ A A A ‚ ‚ 0.2 ˆ A ‚ A A ‚ ‚ A ‚ 0.0 ˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ ‚ A ‚ ‚ ‚ -0.2 ˆ A ‚ A ‚ ‚ ‚ -0.4 ˆ ‚ A ‚ A ‚ ‚ -0.6 ˆ ‚ Šƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒ 3 4 5 6 7 8 yhat* Variances look constant [know pattern with increasing mean response]* None of the residuals stand out and look “outlying”3Plot of resid*nscore. Legend: A = 1 obs, B = 2 obs, etc. resid ‚ ‚ 0.4 ˆ ‚ ‚ B A ‚ ‚ 0.2 ˆ A ‚ A A ‚ ‚ A ‚ 0.0 ˆ ‚ A ‚ ‚ ‚ -0.2 ˆ A ‚ A ‚ ‚ ‚ -0.4 ˆ ‚ A ‚ A ‚ ‚ -0.6 ˆ ‚ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒˆƒƒ -2 -1 0 1 2 Rank for Variable resid* normal probability plot looks approximately linear4Example: How about the Meat study – (bacterial growth) with different conditions Suppose we analyzed bacterial growth directly (i.e. a log-transformation not used)Consider the adequacy of the model:(Bacterial growth)ij = i + ij for i=1, 2, 3, 4 (packaging condition) and j=1, 2, 3. Plot of resid*condition. Legend: A = 1 obs, B = 2 obs, etc. resid ‚ 1000 ˆ ‚ ‚ ‚ ‚ ‚ ‚ A 500 ˆ ‚ ‚ ‚ ‚ A A ‚ ‚ A A 0 ˆƒƒCƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒAƒƒ ‚ A ‚ ‚ ‚ A ‚ ‚ -500 ˆ ‚ ‚ ‚ ‚ A ‚ ‚ -1000 ˆ Šƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒˆƒƒ CO2 mixed plastic vacuum Condition5Plot of resid*yhat. Legend: A = 1 obs, B = 2 obs, etc. resid ‚ 1000 ˆ ‚ ‚ ‚ ‚ ‚ ‚ A 500 ˆ ‚ ‚ ‚ ‚ A