### Bartlett’s test

Bartlett’s test allows you to compare the variance of two or more samples to determine whether they are drawn from populations with equal variance. It is suitable for normally distributed data. The test has the null hypothesis that the variances are equal and the alterntive hypothesis that they are not equal.^{1}

This test is useful for checking the assumptions of an analysis of variance.

You can perform Bartlett’s test with the `bartlett.test`

function. If your data is in stacked form (with the values for both samples stored in one variable), use the command:

> bartlett.test(`values`

~`groups`

,`dataset`

)

where `values`

is the name of the variable containing the data values and `groups`

is the name of the variable that specifies which sample each value belongs too.

If your data is in unstacked form (with the samples stored in separate variables) nest the variable names inside the `list`

function as shown below.

> bartlett.test(list(`dataset`

$`sample1`

,`dataset`

$`sample2`

,`dataset`

$`sample3`

))

If you are unsure whether your data is in stacked or unstacked form, see the article Stacking a dataset in R for examples of data in both forms.

## Example 10.10. Bartlettâ€™s test using the `PlantGrowth` data

Consider the `PlantGrowth`

dataset (included with R), which gives the dried weight of three groups of ten batches of plants, where each group of ten batches received a different treatment. The `weight`

variable gives the weight of the batch and the `groups`

variable gives the treatment received (either `ctrl`

, `trt1`

or `trt2`

). To view more information about the dataset, enter `help(PlantGrowth)`

. To view the data, enter the dataset name:

> PlantGrowth

weight group 1 4.17 ctrl 2 5.58 ctrl 3 5.18 ctrl 4 6.11 ctrl 5 4.50 ctrl 6 4.61 ctrl 7 5.17 ctrl 8 4.53 ctrl 9 5.33 ctrl 10 5.14 ctrl 11 4.81 trt1 12 4.17 trt1 13 ... 30 5.26 trt2

Suppose you want to use Bartlettâ€™s test to determine whether the the variance in weight is the same for all treatment groups. A significance level of 0.05 will be used.

To perform the test, use the command:

> bartlett.test(weight~group, PlantGrowth)

This gives the output:

Bartlett test of homogeneity of variances data: weight by group Bartlett's K-squared = 2.8786, df = 2, p-value = 0.2371

From the output we can see that the p-value of 0.2371 is not less than the significance level of 0.05. This means we cannot reject the null hypothesis that the variance is the same for all treatment groups. This means that there is no evidence to suggest that the variance in plant growth is different for the three treatment groups.

[1] Montgomery, D.C. and Runger, G.C., 2007. *Applied Statistics and Probability for Engineers*, 4th ed. John Wiley & Sons.