Since this question has not garnered a response in the first (nearly) two weeks, let me ask this another way: 

Would heteroscedasticity only refer to regression where variance differs for each estimated residual, whereas heterogeneity of variance refers to differences in variance between groups, whether sets of population values, or in regression, sets of estimated residuals? 

Can heterogeneity of variance, if applicable to regression, be used to refer to both homoscedastic and heteroscedastic sets of estimated residuals? 

What about multilevel models? 

Should the same term, 'heterogeneity of variance,' be applied to both (1) population sets/groups of data, and, (2) in regression, to sets/groups of estimated residuals? 

Please feel free to comment on any one, or more, of the above parts to this question.

Thank you. 

Does it matter so much? In both cases, you are conditioning the distribution of rv's with respect either to a class identifier (eg girl vs boy) or to a continuous variable (eg time).

One important issue remains modelling such "heterogeneous" variances: see eg Goldstein's works.

JLF

Well, i think that it can matter. I think that terminology can sometimes be confusing, especially when different people use the same words to mean different things, or different words to mean the same thing. For example, when I say "prediction," I do not mean a forecast. Yes, context can tell you, but being clear from the start is helpful, I think. Plus, I did not know which of the terms in this question was meant on an occasion or two on ResearchGate.

Thank you for your response.

James,

If we think of two-sided t-tests as special cases of one-way anova, and anova as special cases of linear regression, then we may think of homogeneity of group variances as special cases of homoscedasticity and the same for heterogeneity of variances and heteroscedasticity. On the other side, heteroscedasticity means lack of homogeneity in the conditional variances at some points of the predictor variable and this is the same as heterogeneity in the conditional variances. Then, in my humble opinion, there is no problem to use one term or another. I don't know references where authors give different meanings to these terms :(

In one case, as I recall, there was no regression involved, just two data samples, and it was portrayed that if their variances - i.e., unconditional variances - were different, then that was heteroscedasticity. I don't think so.

Thinking basically in terms of finite populations, and considering the choices, as spelled out by Jean-Louis Foulley, I do not think of "heteroscedasticity" in terms of conditioning with respect to class identifiers.  You can have heteroscedasticity caused by model misspecification or a data quality problem, but basic (say, "essential") heteroscedasticity is due to a relative size measure for each member of the population, which was visualized by Ken Brewer, and is described in the paper at the following location:

 https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity

I'm not saying that I know what view Ken would have had of this overall, just that this is the way that I see it. 

James,

Surely I’m interpreting your question and your answer in a wrong way. Please excuse me.

If we have Y1= (Y11, Y12, … Y1,n1) and Y2 = (Y21, Y22, … Y2,n2) and define Y = (Y11, Y12, … Y1,n1, Y21, Y22, … Y2,n2) and X = (1, 1, … , 1, 0, 0, … 0), with 1 repeated n1 times, and 0 repeated n2 times, and if we adjust a linear model Y = Beta0 + Beta1 * X, then the t-statistic used for testing H0: Beta1 = 0 is the same we find in the two-independent-samples T-test (the Student T test assumes equal variances in the two populations). This allows us to see two-independent-samples Student’s T-test as a special case of linear regression.

The equality of variances in the two populations becomes the same as the constant residual variance condition on Y along the X values. Based on this, in my humble opinion, it doesn’t matter so much to use one term or another. However, it is possible that in some special scenarios, the distinction of these terms is very important.

best regards.

I'm not certain, but I guess your point is the same as that of Jean-Louis: that conditioning on a class identifier could be equated to conditioning on continuous (or other numerical) data.  Here you are using x equal to only 1 or 0.  But if x is a size measure, as found in "Essential Heteroscedasticity," noted previously, then this seems not very useful as a size measure, especially if one of only two possible measures is zero. 

I'd use the term "heteroscedasticity" when referring to weighted least squares, where each member of the population has a measure of its relative 'size.'  Perhaps "heterogeneity of variance" is a good term for use with regression with 'class identifiers.'  I don't know what would be good terminology when someone compares unconditional variances between data samples.  I don't think a term is really required for that.  I certainly would not call that "heteroscedasticity," as I noted earlier that I recall having seen done recently. 

However, if considering class identifiers, I would generally expect heteroscedasticity in each case.  Say you had gender as an identifier.  The subpopulation in each case would generally be modeled such that each member of that population has a different measure of 'size.'  See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity.

I have thought of these as synonyms used according to the background of the writer. I have failed to see how a "change in variability as x increases" is different from "change in variance in different treatments."

I run an experiment growing plants with different levels of nitrogen fertilizer. I will use 25, 50, 100, and 200 kilograms nitrogen per hectare.

What I am saying is that I don't see a difference if I find that the variance in yield increases with increasing nitrogen versus I find that variance in yield is different between 25 versus 200 kg/ha. Why should I call one heteroscedastic and the other heterogeneity of variance? It is all changes in variability correlated with different states of the independent variable(s).

Is there is some key difference that is important in a mathematical proof wherein the distinction matters?

Mathematically, see https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, the size measure determines the degree of heteroscedasticity: the coefficient of heteroscedasticity.  This is not only key to 'how much' heteroscedasticity we have, it describes the essential mathematical nature of heteroscedasticity.  (However, there can be other influences, such as what is described in https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity.) 

Non-mathematically, the argument you stated well Timothy, has appeal.  When the class identifier works in as a measure, as in your example, that might be used, but depending upon the other types of identifiers people might use, and other independent variables, and the sigmas, the term "heterogeneity of variance" starts to have more appeal for that. 

Heteroscedasticity may refer either to 1) a statistically significant increase or decrease in the variance of the effect by a particular predictor across the predictor values or 2) statistically significant increases or decreases in each of the variances in the effects by more than one particular predictor across the values of the predictors. The predictor (or predictors) may be specified in the case of "pure" heteroscedasticity or unspecified in the case of "impure" heteroscedasticity. In the case of impure heteroscedasticity with unspecified predictor(s), it seems by definition that there still remains the assumption of increasing or decreasing variance(s) in the effect(s) of the individual unspecified predictor(s). The unspecified predictor(s) in impure heteroscedasticity are likely to be causally important missing variables that result in a misspecified regression model.

Like heteroscedasticity, heterogeneity may involve a specified predictor or predictors, or an unspecified predictor or predictors. However, in heterogeneity the focus on the increase or decrease in the variance of an effect is not necessarily tied to particular specified or unspecified predictor(s). For instance, the statistically significant increase or decrease in the variance of the effect may occur across a subset, or across the full set, of specified and unspecified predictors, and this may occur despite a lack of evidence for pure or impure heteroscedasticity of any particular predictor. This means that the set of myriad control variables (specified or not) that mop up noise due to sampling and context-based differences contribute (along with the primary and secondary predictors) to heterogeneity even if there is no evidence of heteroscedasticity within any particular control variable.

Correction for heteroscedasticity is likely to reduce heterogeneity. However, heterogeneity may or may not include pure and/or impure heteroscedasticity. Heteroscedasticity would seem to be subsumed within the wider concept of heterogeneity.

Richard, Can you provide a simple example using data that you are familiar with? Not actual data, but just illustrate the ideas. I have 50 patients and I want to know if drug BX2 is effective in reducing depression. "This" is heteroscedasticity #1, "this is heterscedasticity #2, and "this" is heterogeneity.

Another question: Say I have an equation y=mx+b+e. It shows that the error is an increasing function of y. I now go back and run the model with a new variable z such that y=mx+nz+b+e. I find that in this equation e is not related to y. Do I still have heterscedasticity (pure heterscedasticity), and is it still a problem?

"Essential" (or "pure") heteroscedasticity is an increase in conditional variance of predicted y, with increasing predicted y, say y*. In the two cases above, y* = mx+b, and then y* = mx+nz+b.

See my "Essential Heteroscedasticity" paper noted further above please.

Isn't most heteroscedasticity a reflection of the problem that we cannot design experiments large enough to accommodate all of the important variables? We always miss a few. So essential/nonessential (pure/impure) heteroscedasticity is largely irrelevant. The phenomena may be all pure heteroscedasticity because there is no possible way to run an experiment that controls/manipulates all of the relevant variables.

1) Essential heteroscedasticity is very important, and is expected. 

2) Please review your definitions, and reread the question. 

I keep running across abstracts for papers talking about various samples with different variances, and calling that "heteroscedasticity," when they mention nothing at all about regression. I don't see that as valid. Heteroscedasticity has to do with the changing variance of y conditional upon x or predicted y. I do not see it as different unconditional variances of y in different samples (from different populations I presume). That is a different kind of application.

What do you, my fellow travelers, think?