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~~That is, our data are consistent with the idea that the (population) variances are the same among all ﬁve groups.~~

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~~Use a calculator to transform the data and compute new group means and standard deviations.~~

~~One question to ask would be: Which groups are diﬀerent from which other groups?~~

~~This method is the most liberal of the three we will discuss, producing intervals that are often much narrower than those produced by the other two methods and thus is more likely to identify diﬀerences (either real or false).~~

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~~It does so by using a smaller individual error rate for each of the intervals.~~

~~Residuals versus ﬁtted values for cancer survival times What does Levene's test tell us?~~

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~~Test for Homogeneity of Variances Levene's Test for Homogeneity of Variances tests the null hypothesis of equality of variances by computing the absolute deviations between the observations and their group medians, and then applying an ANOVA model to those absolute deviations.~~

~~Thus, Levene's test operates on four groups (cells) of data, corresponding to the four combinations of factor levels.~~

~~The procedure, as described above, sounds somewhat complicated if we need to compute every step, but most software will perform this test for you from the original data, without you having to make the intermediate calculations.~~

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~~The answer is simple: Levene's test is a form of an ANOVA itself, which means that we couldn't introduce it until we had ﬁnished introducing the ANOVA model.~~

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~~Use a calculator or spreadsheet to compute the comparison values.~~

~~In particular, we discussed the fact that all error terms need to come from a distribution with the same variance.~~

~~Then the median value of each group is computed.~~

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~~We ﬁnd our data to be consistent with the condition that the population variances are equal.~~

~~Or, put another way, our data are consistent with the null hypothesis.~~

~~If you ignore the two outliers, the remaining six points lie very close to a line (see the ﬁgure below).~~

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~~Bonferroni Adjustment: A Conservative Approach On the other end of the spectrum is the Bonferroni method.~~

~~Do the points suggest a line?~~

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~~What transformation is suggested?~~

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~~And the more intervals we compute, the more likely it is that we will have at least one interval that doesn't capture its intended parameter.~~

~~But, in its favor, it has a small chance of missing actual diﬀerences that exist.~~

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~~Compute the means and standard deviations for the four groups.~~

~~This exercise was invented to show the method described above at work using simple numbers.~~

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~~Compute the average and standard deviation for each group.~~

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~~We observed in Chapter 5 that the spreads seem to be rather similar in this graph.~~

~~Why Worry about Multiple Tests?~~

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~~We prefer to think of the sections of this chapter as topics rather than subdivisions of a single theme.~~

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~~So our decision to proceed with the ANOVA model in Chapter 5 for these data is justiﬁed.~~

~~Finally, the absolute deviation between each point and the median of its group is calculated.~~

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~~Plot the residuals versus the comparison values.~~

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~~In that case, when we looked at the original data, we had some concerns about the equality of the variances.~~

~~Moreover, the residual plot also carries a lot of subjectivity with it: No two groups will have exactly the same spread.~~

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~~If the null hypothesis is not rejected, then the condition of equality of variances required for the original ANOVA analysis can be considered met.~~

~~What diﬀers between the methods is the critical value.~~

~~Additional Topics in Analysis of Variance In Chapters 5 and 6, we introduced you to what we consider to be the basics of analysis of variance.~~

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~~What transformation, if any, is suggested?~~

~~They also kept track of which month the data came from since there was likely to be a relationship between admissions and the season of the year.~~

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~~We also discuss the relative merits of all three methods.~~

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~~Residuals from Additive Model.~~

~~The researchers collected admissions data for 17.~~

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~~Supplementary Exercises A: Transforming to equalize standard deviations.~~

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~~Neither of these guidelines is very satisfying.~~

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~~ANOVA and introduces you to ideas that, while not strictly necessary for a beginning analysis, will substantially strengthen an analysis of data.~~

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