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Handling Violations of Assumptions in One-Way ANOVA: Data Transformation and Analysis

In this analysis, we address a One-Way ANOVA problem (3.26) where the assumptions may be violated. We estimate the mean response for Failure Time by calculating predicted values and residuals. When issues arise, we utilize the Box-Cox transformation to adjust the data, enabling us to better fulfill ANOVA assumptions. Following the transformation, we plot residuals vs. predicted values, test for equal variances, and ensure normality through the Shapiro-Wilk test. Proper reporting of results utilizes untransformed data to maintain validity in p-values while ensuring accurate interpretation of treatment group means.

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Handling Violations of Assumptions in One-Way ANOVA: Data Transformation and Analysis

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  1. One-Way ANOVA Problem 3.26, when assumptions are violated

  2. Estimates of terms: We can estimate the mean response for Failure Time for problem 3.26from the data by which is called the Predicted Value and which is called the Residual Value.

  3. OK, what about when things go wrongProblem 3.26 data

  4. Plot that data

  5. Put in means and error bars

  6. Plot Residuals vs. Predicted

  7. A clear problem exists but run the test(s) for equal variances

  8. What to do, what to do….. Use Box-Cox transformation to transform the data!!!!!!!!!!!!!!

  9. Do the transformation and..

  10. Plot again

  11. Residuals vs. Predicted

  12. Testing equal variances

  13. Now Normality Plot of Residuals

  14. Test of Normality Goodness-of-Fit Test Shapiro-Wilk W Test W Prob<W 0.985985 0.9869 Note: Ho = The data is from the Normal distribution. Small p-values reject Ho.

  15. Whew, now do ANOVA….

  16. Compare Trt. groups with Tukey

  17. Reporting results after the analysis of transformed data • The purpose of diagnostics is to make sure that the assumptions are at least approximately correct, in which case the p-values reported for significance tests of means comparisons are valid. • In reporting results, such as means of treatment groups, generally use the untransformed or raw data. • Transformations of the data are used to obtain valid p-values, summaries of the original (untransformed or raw) data are used to report results.

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