Post Hoc Tests – Tukey HSD


Tukey’s Honest Significant Difference (HSD) test is a post hoc test commonly used to assess the significance of differences between pairs of groups. This applies for example to one-way ANOVA, factorial design ANOVA, …

As usual, the null hypothesis states that the means of the tested groups are equal. The assumptions are:

  • the observations are independent,
  • normality of distribution,
  • equality of variance

NB: these are the same assumptions as for one-way ANOVA or multi-way ANOVA; on other words, if you have been allowed (or if you allowed yourself ) to conduct an ANOVA test, then it is ok to run Tukey’s test.

 

The function is TukeyHSD() and the syntax is TukeyHSD(results.aov, groups, conf.level) where results.aov is the vector that stores the results of the function aov() that you have previously run when performing the ANOVA and conf.level specifies the confidence level (usually fixed at 0.95).

Let’s take the example from one-way ANOVA.  The code for the dataframe and the analysis (with ANOVA results) is:

size <-c (25,22,28,24,26,24,22,21,23,25,26,30,25,24,21,27,28,23,25,24,20,22,24,23,22,24,20,19,21,22)
location <- as.factor(c(rep("ForestA",10), rep("ForestB",10), rep("ForestC",10)))
my.dataframe <- data.frame(size,location)
results <- aov(size~location, data=my.dataframe)
summary(results)

 

Now we just have to transfer the object “results” from the function aov() into TukeyHSD():

results.post.hoc <- TukeyHSD(results, conf.level=0.95)
results.post.hoc

 

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The output displays the results of all pairwise comparisons among the tested groups (here three groups: ForestA, ForestB and ForestC, thus 3 comparisons): you’ll find the actual difference between the means under diff and the adjusted p-value (p adj) for each pairwise comparison. As denoted by the yellow frame, the only significant difference to be reported in this test is between the means of the groups ForestB and ForestC.

Note that there exists a graphical output of the Tukey HSD test and the function plot(TukeyHSD( )) displays it:

par(mar=c(8,8,8,8))
plot(TukeyHSD(results), las=1)

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Shown above are the difference between the group means and the confidence intervals. Thus, you may easily visualize the differences between groups and understand the level of significance (for example: p-value greater than- but close to 0.05 (p adj=0.0619) in the case of ForestC-ForestA correlated with the right whisker very close to the dashed line…)

Alternative function from the package multcomp:

The package multcomp offers various functions and functionalities which are sometimes more practical or adapted to our problems. Among them is the function glht() which can be used to perform Tukey’s test. Note that it must be used on the resulting output of the function lm(). Reusing our previous example, here is the code to run the test:

library(multcomp)
lm.results <- lm(size~location)
Tukey.HSD.results <- glht(lm.results, linfct=mcp(size='Tukey'))
summary(Tukey.HSD.results)

And here is the output with all three comparisons, p-values and corresponding stars. Note that the values of the p-values are very, very close to those printed by TukeyHSD(), but they are not strictly identical…

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