ANOVA is often used in medical research to simultaneously assess the effectiveness of several treatments for a given condition. One such study looked at different treatments for dandruff. The treatments were 1% pyrithione zinc shampoo (PyrI), the same shampoo but with instructions to shampoo two times (PyrII), 2% ketoconazole shampoo (Keto), and a placebo shampoo (Placebo). Subjects were randomized to each group, and initially each group had 112 volunteers, except the placebo group which was only assigned 28. Each volunteer was examined for dandruff flakes before and after six weeks of treatment. Dandruff flaking was measured on a scale from 0 to 80. Initially, there were no significant differences between the groups. During the clinical trial, 3 dropped out from the PyrII group and 6 from the Keto group. No patients dropped out of the other two groups.
Below is a summary of the final results for each group:
## N Mean SD
## Keto 106 16.02830 0.9305149
## Placebo 28 29.39286 1.5948827
## PyrI 112 17.39286 1.1418110
## PyrII 109 17.20183 1.3524999The data can be found here:
dandruff = read.csv("http://people.hsc.edu/faculty-staff/blins/classes/spring18/math222/data/dandruff.txt")Last time we looked at this data, we make a confidence interval for the contrast between the Ketoconazole shampoo vs. the two Pyrithione Zinc shampoo options. An alternative follow-up would be to make confidence intervals for the pairwise differences between the average flaking levels with different shampoos. If you make more than one 95% confidence interval, then you cannot be 95% certain that all of your confidence intervals will be valid (contain the relevant parameter). To fix this problem, statisticians will often use the Bonferroni correction: To be confident at the \((1-\alpha)\)-level that all of your \(k\) confidence intervals are valid, you should aim for a \((1 - \alpha/k)\) significance level for each individual confidence interval.
Use the formula: \[\bar{x}_i - \bar{x}_j \pm t^{**} \sqrt{MSE \left( \frac{1}{N_i} + \frac{1}{N_j} \right)}\] to make confidence intervals for the three pairwise differences between the 3 shampoo options. Be sure to use the Bonferroni correction to find the correct \(t^*\) value to get 95% confidence.
An alternative to the Bonferroni method is Tukey’s Honest Significant Differences (HSD) which you can get using the R command: TukeyHSD(yourAOV, conf.level=0.95). The Tukey method is a little more complicated then the Bonferroni method, but it has a convenient R command and it is a little less conservative while still being accurate.