Multivariate Bartlett Test

TL;DR

Tests whether variances across multiple groups differ significantly.
Applied to multivariate data organized with rows as individuals and columns as variables.
Uses a p-value (commonly compared to α = 0.05) to decide if observed variance differences are unlikely to be due to chance.

Definition

The Multivariate Bartlett test is a statistical test used to determine whether there is significant differences in the variances of several groups. This test is an extension of the standard Bartlett test, which is used to compare the variances of two groups.

Explanation

To perform the Multivariate Bartlett test, data are gathered from multiple groups and organized into a matrix where each row represents an individual and each column represents a variable (for example, height or weight). A statistical software program is used to compute the test, which produces a p-value indicating the likelihood that observed differences in variances between groups are due to chance. If the p-value is less than a pre-determined level of significance (usually 0.05), the differences in variances between the groups are considered statistically significant.

Examples

Example 1 — Heights and weights across three groups

Suppose we have data on the heights and weights of 10 individuals in each of three different groups - Group A, Group B, and Group C. We organize this data into a matrix, with each row representing a different individual and each column representing a different variable (i.e. height or weight).

After performing the Multivariate Bartlett test, we find that the p-value is 0.01. This means that there is a 1% probability that the differences in the variances between the groups are due to chance. Since this probability is less than our pre-determined level of significance (0.05), we can conclude that there is a significant difference in the variances between the groups.

Example 2 — Randomized controlled trial on blood pressure

Consider data from a randomized controlled trial with two groups of patients - Group 1 and Group 2. The data are organized into a matrix where each row is a patient and each column is a variable (i.e. blood pressure before and after treatment).

After performing the Multivariate Bartlett test, we find that the p-value is 0.04. This means there is a 4% probability that the differences in the variances between the groups are due to chance. Because 0.04 is less than the usual significance level 0.05, we conclude there is a significant difference in the variances between the groups, indicating the new drug is likely to have a different effect on blood pressure in the two groups of patients.

Use cases

Analysis of randomized controlled trials to check whether treatment groups have different variances in outcome measures (example: blood pressure before and after treatment in two groups).

Bartlett test