# 100% Solution Of What To Do If Homogeneity Of Variance Is Violated

Before I start telling you the solution of what to do if homogeneity of variance is violated let me tell you about my thoughts. I can still remember, how hard it was to convince my classmates that research was actually an important part of our educational journey.

I mean, sure, we were being taught about how to find and analyze data, but that wasn’t really what we were after in school – right?

Wrong.

``Research is one of the key skills that you need if you want to succeed as a journalist or businessperson. And the same goes for students studying biomedical sciences. Without good research methods under your belt, it will be extremely difficult to make significant advances in knowledge on topics like cancer, neurodegenerative diseases, or infertility.``

So if you’re thinking of embarking on a research project at some point in your career. Or if you’re currently studying one, it’s important to familiarize yourself with the basics of homogeneity of variance (HOV).

In this blog post, we will discuss what homogeneity of variance is and why it’s important. We will also discuss what to do if homogeneity of variance is violated and the consequences of violating this assumption and how to correct it. Finally, we’ll provide a few examples to illustrate the concept.

So whether you’re a researcher conducting statistical analysis or just trying to understand the basics of the process, this blog post is for you.

## Definition of homogeneity of variance

First of all, how many of you don’t know what homogeneity of variance is?

Good. Because we’re going to focus on it a little bit more in this blog post.

Simply put, homogeneity of variance is an assumption that scientists make when they’re trying to study a sample.

In other words, when they’re looking at how different groups of data (e.g., students or patients) are related to one another. Basically, the assumption states that the variances within each group of data are equal.

Still hard to understand? Let me explain it more simply. When scientists collect data from different groups of people, they want to make sure that the variance in each group of data is equal. Researchers ensure the accuracy and validity of their research results by making this assumption.

If one group’s variance is greater than another group’s variance, it could mean two things:

1) Their study may not have accounted for differences between the groups; or

2) The sample size may not have been large enough to accurately measure the variance of each group.

## Importance of homogeneity of variance in statistical analysis

Now, understanding the importance of homogeneity of variance is key to understanding statistical analysis.

Specifically, when scientists collect data, they need to make sure that the variances within their group of data are equal. This is to come up with accurate results. Otherwise, the research could be inaccurate and lead to false conclusions.

Here are some reasons why homogeneity of variance is so important:

1) It ensures accuracy in data analysis.

2) It helps to prevent bias from creeping into research results.

3) It provides an accurate picture of how different groups are related to one another

## Identifying Violation of Homogeneity of Variance

Don’t you think, I should talk about the ways one can identify a violation of homogeneity of variance?

Well, that’s what I’m going to do in this section.

### 1. Levene’s test

Levene’s test is a statistical method for detecting violations of homogeneity of variance, which is an assumption of many parametric tests. The test compares the variances of different groups to determine if they are equal.

The test works by comparing the absolute deviations of each group’s observations from their respective group mean. The researchers obtain the variance measure for each group by squaring and averaging the deviations.

Then, they calculate the test statistic as the ratio of the largest variance to the smallest variance and compare it to a critical value from an F-distribution.

This comparison determines whether the variances are equal.

If the p-value of Levene’s test is less than the significance level (usually 0.05), it means that the variances are not equal. This actually indicates a violation of homogeneity of variance.

In this case, the researcher should use a non-parametric test or use a different method to correct for non-homogeneity of variance.

It should be noted that Levene’s test is considered a robust test and it can be used with different types of data. But it has been shown to be less powerful than other tests for detecting small violations of homogeneity of variance, such as Brown-Forsythe test or Bartlett’s test.

### 2. Bartlett’s test

Now let me explain Bartlett’s test. Bartlett’s test is one of the most commonly used tests for detecting violations of homogeneity of variance and it is a parametric test.

The test works by comparing the variances of different groups to determine if they are equal. It does this by calculating the variance of each group divided by its sample size, then dividing that number by 2 to get an estimate of degrees of freedom (DF).

For example, let’s say there are n observations in Group A and m observations in Group B. Then their DF would be (n-1 )*(m-1), or 3.

Next, Bartlett’s test is used to compare Group A’s DF to Group B’s DF. If they are not equal, this suggests that the variances of the groups are not equal and may be due to heterogeneity of variance.

### 3. Fligner-Killeen test

Our third test is the Fligner-Killeen test. This test was developed to overcome some of the weaknesses of Bartlett’s test.

One weakness of Bartlett’s test is that it can be fooled by outliers. To avoid this, Fligner-Killeen tests take into account the size of each group and how well its variance correlates with its sample size.

So if there are a lot of small groups or groups with an abnormal variance, then that will affect their DF (and potentially lead to a false positive result).

However, if there are a lot of large groups or groups with a normal variance, then their DF will be unaffected. Fligner-Killeen tests are also parametric and can be used to test for homogeneity of variances in any sample size.

## Examples of situations where homogeneity of variance is likely to be violated

I know, things are getting a bit complex now. But hopefully, this will help you to understand when homogeneity of variance is likely to be violated and what tests can be used to assess this. Let’s see a few examples below for a better understanding:

Example no.1: One example of a situation where homogeneity of variance might be violated is if there are different groups with very different sample sizes. This would lead to an estimate of DF that isn’t accurate because it would ignore the size of each group.

Example no.2: Another example is if one group’s variance is significantly higher than the other groups. This could mean that there’s a difference in how well each group’s data corresponds to the sample size.

Example no.3: Finally, if one group has an abnormally high variance, this might be an indication that there is something wrong with the data. In this case, Fligner-Killeen tests would be able to identify this and help you to investigate further.

## Handling Violation of Homogeneity of Variance

After reading all the above, hopefully, you’re more confident when it comes to detecting homogeneity of variance violations. In most cases, an error will be detected by one of the tests mentioned above. And you’ll be able to carry out further analysis without having to worry about a false positive result.

Now the next step is to decide what to do if an error is detected. In most cases, the answer will be clear. You’ll need to carry out a suitable test of variance in order to assess how well the groups’ data correspond to each other. However, there are occasions where this won’t be possible or where it would be too complicated.

In these cases, you may wish to use one of the approaches outlined below to help you make an informed decision about whether homogeneity of variance is likely still be violated.

#### A. Data transformation techniques

One approach is to try and homogenize the data using one of the data transformation techniques mentioned earlier. This could involve doing something like transferring all the values from one group to another, or transforming them so that they’re more homogeneous.

Here are the three most used data transformation techniques to handle the violation of homogeneity of variance:

##### 1. Log transformation

Log transformation is a method for transforming data that is skewed or has a heavy-tailed distribution into a more symmetric and normal-like distribution.

The log transformation is defined as y = log(x), where x is the original variable and y is the transformed variable. The log transformation can be applied to any variable that is greater than zero. And it has the effect of compressing large values while expanding small values.

##### 2. Square root transformation

This is a useful transformation when homogeneity of variance is not violated but there are major differences between groups in terms of size.

If the square root transformation is a method for transforming data that has a heavy-tailed distribution into a more symmetric and normal-like distribution.

The square root transformation is defined as y = √x, where x is the original variable and y is the transformed variable.

The square root transformation is used to reduce the influence of outliers and extreme values on the data. This can help to improve the normality of the distribution and make it more amenable to statistical analysis.

##### 3. Box-Cox transformation

The Box-Cox transformation is a method for transforming non-normal data into a normal or nearly normal distribution. It is a family of power transformations that can be applied to a variable to stabilize its variance and make it more normal in distribution.

The Box-Cox transformation is defined as:

y = (x^λ – 1) / λ if λ ≠ 0

y = log(x) if λ = 0

Here, x is the original variable, y is the transformed variable, and λ is a parameter that is estimated from the data. The value of λ that produces the best approximation to normality is the one that maximizes the log-likelihood of the data.

The Box-Cox transformation can be used to address violations of the assumption of homogeneity of variance, as it can help to stabilize the variance and make it more constant across different groups or samples.

#### B. Non-parametric tests

Let me be honest, I was overwhelmed while writing this part.

Here the non-parametric tests are statistical tests that do not make assumptions about the distribution of the data and are used to handle violations of assumptions.

These tests are often used when the assumptions of parametric tests, such as normality and homogeneity of variance, are not met.

Some examples of non-parametric tests that can be used to handle violation of homogeneity of variance are:

##### 1) The Kruskal-Wallis test:

This is the first and easy of the tests to use. It is a test of homogeneity of variance that is based on the Wilk-Samplestatistic statistic.

The test is performed by first ordering the data set into groups. And then comparing each group’s variance using a one-way analysis of variance (ANOVA).

χ2 tests can also be used to compare variances between groups, but they are less powerful than an ANOVA because they do not allow for multiple comparisons.

##### 2) The Wilcoxon-Mann-Whitney test:

This test is used to compare two independent groups or samples. If the groups or samples are not Normally Distributed, the Kruskal-Wallis test can be used to adjust for the variance of each group.

The Wilcoxon-Mann-Whitney test is based on an unpaired sample t-test and assumes that data from two groups are Normally distributed with the same variance.

##### 3) The Friedman test:

Our final test is the Friedman test. It is used to compare two groups when it is suspected that one of the groups has an abnormally high variance.

The Friedman test uses a chi-square statistic and assumes that data from both groups come from a normal distribution with equal variance.

Also Read: Why Homogeneity Of Variance Is Important

#### C. Robust statistical methods

Consider the test’s robustness when handling homogeneity of variance violations. Various statistical methods have unique strengths and weaknesses to handle non-normality.

##### 1) The Bootstrap:

The bootstrap is a sampling method that allows for the sample size to be increased without increasing the error rate. This makes it an ideal method for testing whether data come from a Normally Distribution.

Bootstrapping works by randomly selecting n samples from data set X, and then calculating the median and variance of each sample. This is repeated n times, and the median and variance of the bootstrap samples are used to estimate the population means and variance.

##### 2) Huber-White standard errors

Huber-White standard errors are an alternative to the error of the mean. They allow for unequal variance in groups and can be used to test for homogeneity of variance.

To use Huber-White standard errors, first, create ANOVA tables using one group as the treatment variable and two Groups as covariates. Then, estimate the variance of each group by taking its sample variances and dividing them by their sample sizes. Finally, use the Huber-White statistic to compare these variances to an accepted value.

## Examples of how violations can lead to incorrect conclusions

Oh boy, I am not done yet! There are a few more cases where incorrect conclusions can be drawn from data if homogeneity of variance is not met.

These include:

Selection bias: If a study only includes a certain subset of individuals that may not be representative of the overall population, the conclusions drawn from the study may not be applicable to the broader population.

For example, if a study on the effectiveness of a medication only includes participants who are over the age of 60, the results may not be generalizable to individuals under the age of 60.

Confirmation bias: If a researcher only looks for evidence that supports their existing beliefs or hypotheses, they may overlook or ignore evidence that contradicts their beliefs. This can lead to incorrect conclusions about the topic being studied.

Observer bias: The researcher may affect the conclusions of the study if their own beliefs or expectations influence how they interpret and report data. For example, if a researcher expects a certain outcome and unconsciously influences the results in order to achieve that outcome, the conclusions of the study may be incorrect.

## Conclusion

So, this is it. You have successfully learned how to identify and correct violations of homogeneity of variance in a study.

What have you learned? Let me summarize it again for you.

Basically, Homogeneity of variance is an assumption that scientists make when studying a sample, which states that the variances within each group of data are equal.

Ensuring accurate research results and valid conclusions, the post explains how to address violations of homogeneity of variance and correct any associated consequences.

The post also provides examples of methods for detecting violation of homogeneity of variances such as Levene’s test and Bartlett’s test.

Knowing what corrective action to take when homogeneity of variance is violated makes it easier to confirm any violation and determine the necessary steps.

Keep in mind that these corrections are only a first step – eventually, researchers will need to replicate their findings with an independent sample in order to be sure they’ve accurately measured the effect of interest.