Equal variance assumption what is it

The null hypothesis states that the population is normal. The alternative hypothesis states that the population is non-normal. The Anderson-Darling statistic is a measure of how far the plot points fall from the fitted line in a probability plot.

The statistic is a weighted squared distance from the plot points to the fitted line with larger weights in the tails of the distribution. For a specified data set and distribution, the better the distribution fits the data, the smaller this statistic will be. When interpreting a graphical summary report for normality:. For some processes, such as time and cycle data, the data will never be normally distributed. Non-normal data are fine for some statistical methods, but make sure your data satisfy the requirements for your particular analysis.

In simple terms, variance refers to the data spread or scatter. Statistical tests, such as analysis of variance ANOVA , assume that although different samples can come from populations with different means, they have the same variance.

Equal variances homoscedasticity is when the variances are approximately the same across the samples. Unequal variances heteroscedasticity can affect the Type I error rate and lead to false positives. If you are comparing two or more sample means, as in the 2-Sample t-test and ANOVA, a significantly different variance could overshadow the differences between means and lead to incorrect conclusions.

Minitab offers several methods to test for equal variances. The most common way to determine if this assumption is met is to created a plot of residuals vs. If the residuals in this plot seem to be scattered randomly around zero, then the assumption of homoscedasticity is likely met. If this assumption is violated, the most common way to deal with it is to transform the response variable using one of the three transformations:.

Log Transformation: Transform the response variable from y to log y. By performing these transformations, the problem of heteroscedasticity typically goes away. Another way to fix heteroscedasticity is to use weighted least squares regression. This type of regression assigns a weight to each data point based on the variance of its fitted value.

Essentially, this gives small weights to data points that have higher variances, which shrinks their squared residuals. When the proper weights are used, this can eliminate the problem of heteroscedasticity. Your email address will not be published. Skip to content Menu. Posted on April 11, by Zach. The most common statistical tests and procedures that make this assumption of equal variance include: 1.

ANOVA 2. Linear Regression This tutorial explains the assumption made for each test, how to determine if this assumption is met, and what to do if it is violated. There are two ways to test if this assumption is met: 1. Results from an experiment to compare yields as measured by dried weight of plants obtained under a control and two different treatment conditions. The red lines are the residuals. Now by squaring and adding the length of those individual lines, you will get a value that tells you how well the mean our model describes the data.

A small number, tells you the mean describes your data points well, a bigger number tells you the mean describes your data not so well. This number is called the Total Sums of Squares :.

Now you do the same thing for the residuals in your treatment Residual Sums of Squares , which is also known as the noise in the treatment levels :. Lastly, we need to determine the signal in the data, which is known as the Model Sums of Squares , which will later be used to calculate whether the treatment means are any different from the grand mean:.

Now the disadvantage with the sums of squares is that they get bigger as the sample size increase. To express those sums of squares relative to the number of observation in the data set, you divide them by their degrees of freedom turning them into variances.

So after squaring and adding your data points you are now averaging them using their degrees of freedom:. This results in the Model Mean Square and the Residual Mean Square both are variances , or the signal to noise ratio, which is known as the F-value:. The F-value describes the signal to noise ratio, or whether the treatment means are any different from the grand mean. The F-value is now used to calculate p-values and those will decide whether at least one of the treatment means will be significantly different from the grand mean or not.

Now I hope you can see that the assumptions are based on calculations with residuals and why they are important. Since we adding , squaring and averaging residuals, we should make sure that before we are doing this, the data in those treatment groups behaves similar , or else the F-value may be biased to some degree and inferences drawn from this F-value may not be valid.

Edit: I added two paragraphs to address the OP's question 2 and 1 more specifically. Normality assumption : The mean or expected value is often used in statistics to describe the center of a distribution, however it is not very robust and easily influenced by outliers. The mean is the simplest model we can fit to the data. Since in ANOVA we are using the mean to calculate the residuals and the sums of squares see formulae above , the data should be roughly normally distributed normality assumption.

Instead once could use the median for example see non parametric testing procedures. Homogeneity of variance assumption : Later when we calculate the mean squares model and residual , we are pooling the individual sums of squares from the treatment levels and averaging them see formulae above.

By pooling and averaging we are losing the information of the individual treatment level variances and their contribution to the mean squares. Therefore, we should have roughly the same variance among all treatment levels so that the contribution to the mean squares is similar.

This is how I see it for myself. You need some assumptions to decide what you want to compare and to calculate the p-values. The most useful distribution is the normal one because of the CLT , that's why it's the most commonly used.

If your data it's not normally distributed you need at least to know what's its distribution in order to calculate something. Homoscedasticity is a common assumption also in regression analysis, it just makes things easier. We need some assumptions to start with. The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams?

Learn more. Ask Question.