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*Assumption of normality ; Homogeneity of variance.*

- Do you know the difference between homoscedasticity and heteroscedasticity ?
- Homoscedasticity
- Methods for Detecting and Resolving Heteroskedasticity

Heteroscedasticity is a hard word to pronounce, but it doesn't need to be a difficult concept to understand. Put simply, heteroscedasticity also spelled heteroskedasticity refers to the circumstance in which the variability of a variable is unequal across the range of values of a second variable that predicts it. A scatterplot of these variables will often create a cone-like shape, as the scatter or variability of the dependent variable DV widens or narrows as the value of the independent variable IV increases. The inverse of heteroscedasticity is homoscedasticity, which indicates that a DV's variability is equal across values of an IV.

## Do you know the difference between homoscedasticity and heteroscedasticity ?

One of the assumptions of an anova and other parametric tests is that the within-group standard deviations of the groups are all the same exhibit homoscedasticity. If the standard deviations are different from each other exhibit heteroscedasticity , the probability of obtaining a false positive result even though the null hypothesis is true may be greater than the desired alpha level.

To illustrate this problem, I did simulations of samples from three populations, all with the same population mean. There have been a number of simulation studies that have tried to determine when heteroscedasticity is a big enough problem that other tests should be used.

Heteroscedasticity is much less of a problem when you have a balanced design equal sample sizes in each group. Early results suggested that heteroscedasticity was not a problem at all with a balanced design Glass et al.

The problem of heteroscedasticity is much worse when the sample sizes are unequal an unbalanced design and the smaller samples are from populations with larger standard deviations; but when the smaller samples are from populations with smaller standard deviations, the false positive rate can actually be much less than 0.

You should always compare the standard deviations of different groups of measurements, to see if they are very different from each other. However, despite all of the simulation studies that have been done, there does not seem to be a consensus about when heteroscedasticity is a big enough problem that you should not use a test that assumes homoscedasticity.

If you see a big difference in standard deviations between groups, the first things you should try are data transformations. A common pattern is that groups with larger means also have larger standard deviations, and a log or square-root transformation will often fix this problem. It's best if you can choose a transformation based on a pilot study, before you do your main experiment; you don't want cynical people to think that you chose a transformation because it gave you a significant result.

If the standard deviations of your groups are very heterogeneous no matter what transformation you apply, there are a large number of alternative tests to choose from Lix et al.

The most commonly used alternative to one-way anova is Welch's anova , sometimes called Welch's t —test when there are two groups. Non-parametric tests, such as the Kruskal—Wallis test instead of a one-way anova, do not assume normality, but they do assume that the shapes of the distributions in different groups are the same.

This means that non-parametric tests are not a good solution to the problem of heteroscedasticity. All of the discussion above has been about one-way anovas. Homoscedasticity is also an assumption of other anovas, such as nested and two-way anovas , and regression and correlation.

Much less work has been done on the effects of heteroscedasticity on these tests; all I can recommend is that you inspect the data for heteroscedasticity and hope that you don't find it, or that a transformation will fix it. There are several statistical tests for homoscedasticity, and the most popular is Bartlett's test.

Use this test when you have one measurement variable , one nominal variable , and you want to test the null hypothesis that the standard deviations of the measurement variable are the same for the different groups. Bartlett's test is not a particularly good one, because it is sensitive to departures from normality as well as heteroscedasticity; you shouldn't panic just because you have a significant Bartlett's test.

An alternative to Bartlett's test that I won't cover here is Levene's test. It is less sensitive to departures from normality, but if the data are approximately normal, it is less powerful than Bartlett's test. While Bartlett's test is usually used when examining data to see if it's appropriate for a parametric test, there are times when testing the equality of standard deviations is the primary goal of an experiment. For example, let's say you want to know whether variation in stride length among runners is related to their level of experience—maybe as people run more, those who started with unusually long or short strides gradually converge on some ideal stride length.

You could measure the stride length of non-runners, beginning runners, experienced amateur runners, and professional runners, with several individuals in each group, then use Bartlett's test to see whether there was significant heterogeneity in the standard deviations. I have put together a spreadsheet that performs Bartlett's test for homogeneity of standard deviations bartletts.

It allows you to see what the log or square-root transformation will do. It also shows a graph of the standard deviations plotted vs. This gives you a quick visual display of the difference in amount of variation among the groups, and it also shows whether the mean and standard deviation are correlated. None of these is close to significance, so there's no real need to worry. The graph of the untransformed data hints at a correlation between the mean and the standard deviation, so it might be a good idea to log-transform the data:.

You have to enter the variances not standard deviations and sample sizes, not the raw data. This modification of the program from the one-way anova page does Bartlett's test.

Glass, G. Peckham, and J. Consequences of failure to meet assumptions underlying fixed effects analyses of variance and covariance.

Review of Educational Research Harwell, M. Rubinstein, W. Hayes, and C. Journal of Educational Statistics Lix, L. Keselman, and H. Consequences of assumption violations revisited: A quantitative review of alternatives to the one-way analysis of variance F test. Skills to Develop Parametric tests assume that data are homoscedastic have the same standard deviation in different groups.

To learn how to check this and what to do if the data are heteroscedastic have different standard deviations in different groups. What to do about heteroscedasticity You should always compare the standard deviations of different groups of measurements, to see if they are very different from each other.

Bartlett's test There are several statistical tests for homoscedasticity, and the most popular is Bartlett's test. How to do Bartlett's test Spreadsheet I have put together a spreadsheet that performs Bartlett's test for homogeneity of standard deviations bartletts. The graph of the untransformed data hints at a correlation between the mean and the standard deviation, so it might be a good idea to log-transform the data: Fig.

References Glass, G. Contributor John H. McDonald University of Delaware.

## Homoscedasticity

This is known as constant variance or homoscedasticity. When this assumption is violated, the problem is known as heteroscedasticity. It is sensitive to departures from normality. The Levene test is an alternative test that is less sensitive to departures from normality. You can perform the test using 2 continuous variables, one continuous and one grouping variable, a formula or a linear model.

Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. It only takes a minute to sign up. In my view, they are both defined by the existance of correlation between the x-variable and the residuals. This is also more or less confirmed by for example wikipedia I cant post more than two links, but if you google endogenity econometrics and heteroscedasticity you can verify this. They explain homoscedasticity by stating that the variances of the n disturbances exist and are all equal and cannot be thus influenced by the x-variables? Furthermore, they state that the condition basically means that the explanatory variables should be asymptotically uncorrelated with the disturbances.

Heteroskedasticity occurs when the variance for all observations in a data set are not the same. In this demonstration, we examine the consequences of heteroskedasticity, find ways to detect it, and see how we can correct for heteroskedasticity using regression with robust standard errors and weighted least squares regression. As mentioned previously, heteroskedasticity occurs when the variance for all observations in a data set are not the same. Conversely, when the variance for all observations are equal, we call that homoskedasticity. Why should we care about heteroskedasticity? In the presence of heteroskedasticity, there are two main consequences on the least squares estimators:. Most real world data will probably be heteroskedastic.

In presence of heteroscedasticity the estimates of regression are nothing but the difference between the is homoscedastic error term.

## Methods for Detecting and Resolving Heteroskedasticity

One of the assumptions of an anova and other parametric tests is that the within-group standard deviations of the groups are all the same exhibit homoscedasticity. If the standard deviations are different from each other exhibit heteroscedasticity , the probability of obtaining a false positive result even though the null hypothesis is true may be greater than the desired alpha level. To illustrate this problem, I did simulations of samples from three populations, all with the same population mean. There have been a number of simulation studies that have tried to determine when heteroscedasticity is a big enough problem that other tests should be used.

Homoscedasticity versus heteroscedasticity. Homoscedasticity can be also called homogeneity of variance, because it is about a situation, when the sequence or vector of rando variable have the same finite variance. And as we probably know already — variance measures how far a set of numbers is spread out.

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#### Bartlett's test

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