In the previous section, we decided to perform this test only after examining the data and noting that Managers One and Two had the highest and lowest mean performances. In a sense, this means that we have implicitly performed \(\binom{5}{2} = 5(5-1)/2 = 10\) hypothesis tests, rather than just one, as discussed in Section 13.3.2. Hence, we use the TukeyHSD() function to apply Tukey’s method in order to adjust for multiple testing. This function takes as input the output of an ANOVA regression model, which is essentially just a linear regression in which all of the predictors are qualitative. In this case, the response consists of the monthly excess returns achieved by each manager, and the predictor indicates the manager to which each return corresponds.

> returns <- as.vector(as.matrix(fund.mini))
> manager <- rep(c("1", "2", "3", "4", "5"), rep(50, 5))
> a1 <- aov(returns ~ manager)
> TukeyHSD(x = a1)
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = returns ~ manager)

$manager
    diff        lwr       upr     p adj
2-1 -3.1 -6.9865435 0.7865435 0.1861585
3-1 -0.2 -4.0865435 3.6865435 0.9999095
4-1 -2.5 -6.3865435 1.3865435 0.3948292
5-1 -2.7 -6.5865435 1.1865435 0.3151702
3-2  2.9 -0.9865435 6.7865435 0.2452611
4-2  0.6 -3.2865435 4.4865435 0.9932010
5-2  0.4 -3.4865435 4.2865435 0.9985924
4-3 -2.3 -6.1865435 1.5865435 0.4819994
5-3 -2.5 -6.3865435 1.3865435 0.3948292
5-4 -0.2 -4.0865435 3.6865435 0.9999095

The TukeyHSD() function provides confidence intervals for the difference between each pair of managers (lwr and upr), as well as a p-value. All of these quantities have been adjusted for multiple testing. We can plot the confidence intervals for the pairwise comparisons using the plot() function.

plot(TukeyHSD(x = a1))

plot

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