Tukey's test

Tukey’s test is good choice for post-hoc comparisons. Specifically, Tukey’s Honestly Significant Difference (HSD) test is implemented using TukeyHSD(). The input should be the output from aov(). In our example, this is our variable out.

     >  TukeyHSD(out, ordered = T)

     Tukey multiple comparisons of means
     95 factor levels have been ordered

    Fit: aov(formula = DIVERSTY ~ ZNGROUP, data = d)

    $ZNGROUP
     diff lwr upr p adj
    3-4 0.44000000 -0.15739837 1.0373984 0.2095597
    1-4 0.51972222 -0.09606192 1.1355064 0.1218677
    2-4 0.75472222 0.13893808 1.3705064 0.0116543
    1-3 0.07972222 -0.53606192 0.6955064 0.9847376
    2-3 0.31472222 -0.30106192 0.9305064 0.5153456
    2-1 0.23500000 -0.39863665 0.8686367 0.7457444

The additional argument ordered = T just ensures the the differences are positive. (Compare this result to TukeyHSD(out)).

In all cases, we see that the data indicate that the difference really lies between groups 2 and 4. To adequately interpret and summarize this information, we have to follow the rules for interpreting post-hoc tests. We found:

$\mu _{4} = \mu _{3} = \mu _{1}$ and $\mu _{3} = \mu _{1} = \mu _{2}$.

We could also summarize this result on the boxplot using letters. Here, I will have to expand the y-axis limits using the ylim argument to give room for the letters. I will also use tapply(d$DIVERSTY, d$ZNGROUP, max) to extract the maximum diversity values in each group. These (plus a little extra space) become the y-coordinates for the new letters.

     >  bp = boxplot(d$DIVERSTY ~ (d$ZNGROUP), ylim = c(0, 3), xlab = "Zn Group",
    + ylab = "Diatom Diversity", names = c("B", "L", "M", "H"))
     >  text(1:4, tapply(d\$DIVERSTY, d\$ZNGROUP, max) + 0.15, c("ab", "a",
    + "ab", "b"))

Figure 8.3: Boxplot of the diatom diversity data. Here, letters represent the results of Tukey’s post-hoc comparisons of group means.