**Tukey’s Honest Significant Difference (HSD)** test is a *post hoc* test commonly used to assess the significance of differences between pairs of group means. Tukey HSD is often a follow up to one-way ANOVA, when the F-test has revealed the existence of a significant difference between some of the tested groups. Running the test in R involves using the function `TukeyHSD()`

which does not require any additional package installation.

The null hypothesis `H0`

for the test states that the means of the tested groups are equal.

We will reuse the example introduced here (one-way ANOVA). To create the corresponding dataframe, use the following code:

```
# response variable
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)
# predictor variable
location <- as.factor(c(rep("ForestA",10), rep("ForestB",10), rep("ForestC",10)))
# dataframe
my.dataframe <- data.frame(size,location)
```

The assumptions are:

- the observations are
**independent**, **normality**of distribution,**homogeneity of variance**.

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

`TukeyHSD()`

The syntax is `TukeyHSD(aov(response ~ predictor), conf.level)`

where `response`

is the response variable, `predictor`

is the predictor variable, and `conf.level`

is the confidence level that you want to define (usually fixed at 0.95). Let’s apply it to our example:

`TukeyHSD(aov(size ~ location), conf.level=0.95)`

```
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = size ~ location)
##
## $location
## diff lwr upr p adj
## ForestB-ForestA 1.3 -1.097245 3.69724537 0.3835917
## ForestC-ForestA -2.3 -4.697245 0.09724537 0.0619108
## ForestC-ForestB -3.6 -5.997245 -1.20275463 0.0025530
```

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. Looking at the last column, the only significant difference to be reported in the present test is between the means of the groups `ForestB`

and `ForestC`

.