lect8 - IE 410 Notes for Lecture 8 Model Adequacy The ANOVA...

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IE 410 Notes for Lecture 8 Model Adequacy The ANOVA, and its analysis rely on certain assumptions. These assumptions are implicit in the model:

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Y ij = μ + τ i + ε ij s.t. i n = 1 τ i = 0 ε ij ~ NID(0 , σ 2 )
The basic assumptions are: 1. ε ij are NORMALLY distributed - Bell Shaped - No outliers - Note that Y ij will generally not be normal in total since they may have different means

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2. ε ij are independent - No relevant variables/factor have been omitted - No time-based effect is affecting response 3. The variance of each ε ij is constant (and = σ 2 regardless of treatment level)
Our model adequacy checking procedure is: - Check that the assumptions are reasonable - If they appear to be violated, take remedial actions. No ANOVA is complete without checking the adequacy of the model.

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We need to learn: - Diagnostic tools (today) - Remedial tools (later) The primary diagnostic tool is residual analysis, particularly residual plots.
Residuals e ij = Y ij - Yhat ij = Y ij - (model prediction of Y ij ) = Y ij - ( μ hat + τ hat i ) = Y ij - μ hat i

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Residuals can be considered as "estimates" of the random error terms ε ij That is: ε ij = Y ij - μ i e ij = Y ij - μ hat i
The residuals ε ij are close to ε ij , but are not independent observations on the ε ij For example i n = 1 e ij = 0 Thus there is some (fairly "small") amount of dependency built into the residuals

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Checking the NORMALITY assumption Check if the residuals appear to follow the Normal distribution. 1. Use some formal hypothesis tests 2. Look at the histogram 3. Look at the Normal Probability Plot
1. Goodness of fit tests These are formal statistical tests for testing: H 0 : data are from some specified distributed. These are Non-parametric tests.

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There are many different tests for goodness of fit. We can spend a little time trying to understand them.
Different Goodness of fit tests - Chi-squared goodness of fit test* - Kolmogorov Type goodness of fit tests - Kolmogorov-Smirnov test* - Lillefors test - Cramer-Von-Mises Type tests - Anderson-Darling test - Shapiro-Wilk Test* * In statgraphics

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