I'd like you to participate in a simple experiment. I'd like to limit this to simple linear and multiple linear regression with continuous data. Do you have such a regression application with real data? Some cases with several independent variables might be helpful here. If you have done a statistical test for heteroscedasticity, please ignore that, regardless of result. We are looking at a more measureable degree of heteroscedasticity for this experiment.
To be sure we have reasonably linear regression, the usual graphical residual analysis would show no pattern, except that heteroscedasticity may already show with e on the y-axis, and the fitted value on the x-axis, in a scatterplot.
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Here is the experiment:
1) Please make a scatterplot with the absolute values of the estimated residuals, the |e|, on the y-axis, and the corresponding fitted value (that is, the predicted y value, say y*), in each case, on the x-axis. Then please run an OLS regression through those points. (In excel, you could use a "trend line.") Is the slope positive?
A zero slope indicates homoscedasticity for the original regression, but for one example, this would not really tell us anything. If there are many examples, results would be more meaningful.
2) If you did a second scatterplot, and in each case put the absolute value of the estimated residual, divided by the square root of the fitted value, that is |e|/sqrt(y*), on the y-axis, and still have the fitted value, y*, on the x-axis, then a trend line through those points with a positive slope would indicate a coefficient of heteroscedasticity, gamma, of more than 0.5, where we have y = y* + e, and e = (random factor of estimated residual)(y*^gamma). Is the slope of this trend line positive in your case?
If so then we have estimated gamma > 0.5. (Note, as a point of reference, that we have gamma = 0.5 in the case of the classical ratio estimator.)
I'd also like to know your original equation please, what the dependent and independent variables represent, the sample size, whether or not there were substantial data quality issues, and though it is a very tenuous measure, the r- or adjusted R-square values for the original linear or multiple linear regression equation, respectively. I want to see whether or not a pattern I expect will emerge.
If some people will participate in this experiment, then we can discuss the results here.
Real data only please.
Thank you!
I have developed methods of measure co-dependent variables using what I call implicit regression ... I could suggestion a structure to model if I could see a scatter plot of the data - please review my publications and let me know if you are interested in applying my methods.
Let's all focus on the question at hand.
Attached are Figures 3c and 3d from a paper in 1993, found at the following URL: https://www.researchgate.net/publication/263809034_Alternative_to_the_Iterated_Reweighted_Least_Squares_Method_-_Apparent_Heteroscedasticity_and_Linear_Regression_Model_Sampling
From Figure 3c we might think that this is homoscedastic, so gamma = 0. (For the classical ratio estimator, CRE, gamma = 0.5.) But here I took the process I asked for in the question above further, and Figure 3d shows the results of those further steps, where the graph there indicates gamma > 0.7.
I am asking for various other examples. For those who do not want to do the two regression-analysis-type graphs requested, could you please just send me an excel file with the dependent and independent variables data? Any real data examples are welcomed, and appreciated, but especially multiple regression examples.
Thank you.
PS - Also attached here is a scatterplot for hydroelectric data using multiple regression where I plotted predicted-y, i.e. y*, on the x-axis. The odd shape is because two geographic regions were combined which should not have been combined. When data were grouped properly, that made a big difference. I need to know such background about any examples. Thank you!
Attached is the published version of just such an analysis. Hope you find it helpful. David
Thanks @David -
I take it that it is OK for me to use the data in Appendix 1? Would you happen to have an excel file of those data?
Thanks again - Jim
@Jim sure, the data is all in the public domain. I am sorry but I no longer have this data in computer readable form. Best wishes, David
Could be conditionally linear ... do you have a time measurement?
Sorry it took so long to get back with you.
For some reason, Research Gate is no longer sending me notification. Checking into it now.
Rebecca -
No, sorry, that graph was not meant for you. This question is about how many practical examples of regression in 'real life' are heteroscedastic, and to a degree that falls in a range that Ken Brewer described in his book, discussed in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity. That scatterplot was about a problem when data were mixed that should have been modeled separately.
In the two experimental graphs described in the question, if the second one actually used the predicted y values from the corresponding weighted least squares regression, it would be more exact. Nevertheless, it is intended to see for which applications that level of heteroscedasticity or greater shows up.
Perhaps you could start a different thread to discuss what you are doing. It does not help here. I was just looking for a variety of real world linear and multiple linear regression applications beyond ones I had experienced.
Thanks - Jim
Here are some results:
https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity
and a spreadsheet tool for estimating or considering a default value for the coefficient of heteroscedasticity:
https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx
both found in
https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression
I developed these methods to handle conditional relationships and therefore should capture such effects - let me run a few simulations and I will get back with you shortly - from the graph, my concern is the sample size
I just read your response.
I am curious to why you state so emphatically that my methods are NOT relevant.
I find your comments rude and unnecessary.
I was answering a question Research Gate indicated I might be able to help with.
I will not offer my insights again.
Thank you to all those who tried to answer the question, and for your attention. I do appreciate it. I have also picked up examples elsewhere, and so far, under project https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, I have used these examples in https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity, and https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx. I still have examples left over, including one that is a good one for showing that when regressors are picked which do not capture an important aspect, in that example, estimated residuals are found to be homoscedastic. Ideally, this would not be the case. I plan one more paper using these examples.
Finally, in sections 2 and 3 of "Estimating the Coefficient of Heteroscedasticity," different ways of determining the degree of heteroscedasticity are described, and you can see why I picked the one I did. It is the one to use with Ken Brewer's explanation as to why the degree of heteroscedasticity should fall in the range he describes, as discussed in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity. He uses the measure I have also used here: the coefficient of heteroscedasticity.
Again, thank you.