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Homoscedasticity and homogeneity of effect-sizes assumptions are popular with regression analysis and Anova respectively.These assumptions create lot of confusion at least in my mind. I am not clear about particularly homoscdasticity and how it is different from homogeneity assumption for Anova ?

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    $\begingroup$ Homoscedasticity means equal variances. I would expect whenever homogeneity is mentioned in a statistical context, it would also imply that something is constant on average, but quite what would depend on context. As you don't explain the doubt ("perhaps"?) and give precisely zero evidence for the claim of "a lot of confusions" I can only match your two sentences by my two sentences. This gives essentially minimal substance to respond to. I'd call that a lack of research effort. $\endgroup$ – Nick Cox May 2 '14 at 14:23
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    $\begingroup$ Subhash, if you could edit your question to explain what you mean by "homogeneity"--which out of context is a vague term--then it would be less problematic to answer. $\endgroup$ – whuber May 2 '14 at 18:50
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    $\begingroup$ It depends on what thing we are considering the homogeneity of. Homogeneity of variance is homoscedasticity. Homogeneity of something that is distinct from variance will be distinct from homoscedasticity. $\endgroup$ – Glen_b -Reinstate Monica Jun 1 '14 at 4:05
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    $\begingroup$ It's really bizarre that you decided to accept a new answer that has by now -4 downvotes instead of gung's answer with +9 upvotes. That's a really strange choice. I downvote your question (-1) to steer other users away from this thread. $\endgroup$ – amoeba says Reinstate Monica Nov 18 '15 at 21:44
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I disagree with every answer here. Homogeneity of variance means similar variance among grouped scatterplots. Homoscadasticity is a normal distribution occurring for each point on the x-axis (predictor variable) thus there must be a similar kurtosis across every point of the predictor variable which may seem like homogeneity of variance, but it is not the same thing.

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    $\begingroup$ Homoscedasticity [not scad] does not imply a normal distribution at all. As its roots imply it is a matter of (approximately) equal scatter, with nothing else implied. Nor does homoscedasticity imply that we have a continuous axis any where, as it could also be defined for qualitatively distinct distributions. Here is a trivial example. I imagine several uniform distributions on the same interval. It follows immediately that they have the same variance and the set-up is homoscedastic. $\endgroup$ – Nick Cox Nov 17 '15 at 18:17
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    $\begingroup$ Similar (even equal) kurtosis is also quite distinct from equal variance. The same kurtosis is consistent with differing variance. More generally, you're announcing dissent here: so, what precisely is wrong with the existing answer (I count only one)? $\endgroup$ – Nick Cox Nov 17 '15 at 18:20
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    $\begingroup$ This characterization of homoscedasticity is so far from the usual meaning that I feel obliged to downvote the answer as a warning to those who might be new to the term. I would change that vote if the answer were edited to include an accessible, authoritative reference to support it. $\endgroup$ – whuber Nov 17 '15 at 18:27
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    $\begingroup$ This answer needs to support its claims $\endgroup$ – Glen_b -Reinstate Monica Nov 19 '15 at 0:53
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    $\begingroup$ I looked at your links but could find nothing in them to support your claims. Both of them illustrate the conventional meaning of heteroscedasticity. Neither invokes normality or kurtosis in the definition. (Kurtosis, by the way, has little to do with the shape of the normal distribution and is not synonymous with it). Thus, they both contradict, rather than support, your answer. I believe the reason @NickCox pointed out the correct spelling was not to be critical, but only to help readers search related material. (The search engine on this site does not do well at identifying misspellings.) $\endgroup$ – whuber Dec 12 '15 at 17:37
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(Note: by "homogeneity", I assume you mean "homogeneity of variance".)

They are, in essence, two different names for the same assumption, which might be called in more colloquial English "constant variance of the errors" (of course, in practice we do not have access to the true errors, only the residuals, which are what we actually check). The term "homogeneity of variance" is traditionally used in the ANOVA context, and "homoscedasticity" is used more commonly in the regression context. But they both mean that the variance of the residuals is the same everywhere.

If you are having trouble understanding homo- / heteroscedasticity, I have several posts about the topic that may be helpful for you:

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    $\begingroup$ Typo here @Gung: it is homosc. that implies that variance is the same. Strictly, homosc. is an assumption about errors, or conditional distributions, not residuals. $\endgroup$ – Nick Cox May 2 '14 at 14:47
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    $\begingroup$ Homegeneity has also a wider meaning of samples being the similar in some sense, i.e. as opposed to heterogeneity. $\endgroup$ – Aksakal May 2 '14 at 15:08
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    $\begingroup$ I'd say it's usually given in full as "homogeneity of variance" - as @Aksakal says, "homogeneity" is broader. [I took the liberty of correcting the typo Nick pointed out.] $\endgroup$ – Scortchi - Reinstate Monica May 2 '14 at 15:29
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    $\begingroup$ This is helpful but I would qualify it a little. For example, I've seen references to homogeneity in relation to possibly mixed distributions for the case in which a distribution is from a single source; and in relation to spatial processes. So, homogeneity need not mean homogeneity of variance. For all I know, this goes beyond what the OP had in mind, but it's a fair comment given the current wording of the question. $\endgroup$ – Nick Cox May 2 '14 at 17:30
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    $\begingroup$ Good point, @NickCox. I added a caveat. $\endgroup$ – gung - Reinstate Monica May 2 '14 at 17:34

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