1. ANOVA Assumptions Consider the linear model X ij =  +  i +  ij i)  is a fixed constant common to all observations ii)The  ij are independent and normally distributed with E[  ij ]=0 and Var[  ij ]=  2 iii) The deviations from the overall mean for the I treatments are such that   i = 0

2. Under these assumptions: E[X ij ]=  i Var [X ij ]=  2 and X ij is normally distributed which facilitates the use of ANOVA for testing hypothesis about the equality of the means

3. In real world experiments, however, either the normality and/or equal variances assumptions are often violated. How robust is the ANOVA test to these violations?

4. Normality Assumption It was established by Cochran and Hay that the ANOVA test is very robust with respect to non-normality. Regardless, the plausibility of a normal assumption for X ij under a fixed i may be established through Normal Probability Plots (NPP)

5. Normal Probability Plots (4.6) A NPP is a plot of the observed data values against the z-percentiles of the standard normal distribution. If the plotted points do not deviate greatly from a straight 45 o line, then it is plausible to assume that our data is normally distributed

6. If the plotted points fall in an S shape, then it is plausible to assume that our data from a heavy-tailed distribution If the plotted points fall in a backwards S shape, then it is plausible to assume that our data from a light-tailed distribution If the plotted points fall in a middle curved shape, then it is plausible to assume that our data from a positively skewed distribution

7. Equal Variances Assumption It was established by Welsh and Box that the ANOVA procedure is robust to mild departures from the equal variances assumption for equal replications If there is a large departure from the equal variances assumption and/or mild departures with extremely unequal replications, a variance stabilizing transformation should be used if possible

8. Variance Stabilizing Data Transformations If Var [X ij ] = g(  i ) (that is the variance is a known function of the mean) then the transformation h(X ij ) such that Var [X ij ] is approximately the same for each i is given by

9. Common Transformations

10. Further Comments on Data Transformations Does a data transformation destroy the other needed properties such as normality and independence? Answer: Generally No! In fact, the presence of non-normality and unequal variances are often related. It has been shown that transformations to stabilize the variance often helps to correct non-normality in the data

11. Example A small restaurant has 4 different locations in the local area. The management is interested in whether the true average of complaints received per restaurant differs by location. The number of complaints at each restaurant was counted and recorded for 30 consecutive months. Test the appropriate hypothesis at  =.05 los.

12. Location1:{1,1,2,2,2,2,3,3,3,3,3,3,3,4,4, 4,4,4,4,4,5,5,5,6,6,6,6,6,7,8} Location2:{3,3,3,3,4,4,4,4,5,5,5,5,6,6,6, 6,6,6,7,7,7,7,8,8,8,9,10,10,12,13} Location3:{1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,4, 4,4,4,4,4,5,5,5,6,7,7,7,8,9} Location4:{0,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3, 3,3,3,3,3,3,4,4,4,4,4,5,5,5}

13. Plot of the Data Sets

14. Location1:{1,1,1.41421,1.41421,1.41421,1.41 421,1.73205,1.73205,1.73205,1.73205,1.7320 5,1.73205,1.73205,2.,2.,2.,2.,2.,2.,2.,2.23607,2.23607,2.23607,2.44949,2.44949,2.44949,2.44 949,2.44949,2.64575,2.82843} Location2:{1.73205,1.73205,1.73205,1.73205,2.,2.,2.,2.,2.23607,2.23607,2.23607,2.23607,2.449 49,2.44949,2.44949,2.44949,2.44949,2.44949,2. 64575,2.64575,2.64575,2.64575,2.82843,2.8284 3,2.82843,3.,3.16228,3.16228,3.4641,3.60555} Transformed (x 0.5 ) Data Sets

15. Location4:{0,1,1,1,1,1.41421,1.41421,1.41421,1.41421,1.41421,1.73205,1.73205,1.73205,1.7320 5,1.73205,1.73205,1.73205,1.73205,1.73205,1.7 3205,1.73205,1.73205,2.,2.,2.,2.,2.,2.23607,2.23 607,2.23607} Location3:{1,1,1,1.41421,1.41421,1.41421,1.41 421,1.73205,1.73205,1.73205,1.73205,1.73205, 1.73205,1.73205,1.73205,2.,2.,2.,2.,2.,2.,2.2360 7,2.23607,2.23607,2.44949,2.64575,2.64575,2.6 4575,2.82843,3.}