I am running a series of Generalized Additive Models (GAMs) and the response variable and the predictors are highly skewed. Because of this, I square-root transformed the response, and transformed the other predictors as needed. One of the predictors also needed to be rescaled because its values occupied a scale that was too different from the other predictors (ran into convergence issues). In my linear model, the relationship looked something like this:
y0.5 = B0 + B1(x10.25) + B2(x20.25) + B3(log(x3)) + B4(scaled(x4)). My question is this: when I plot my results (i.e. x vs. y), is it better to plot the transformed data or the untransformed data? Untransformed data would be more intuitive for readers looking at the plots, but they wouldn't directly represent our statistical results since we ran GAMs with transformed data. I'm curious about this because the shapes of the GAM curves change depending on whether the data are transformed or not and my primary interest is in the shape of the curve. Does anyone have thoughts about this?
Glen -
You said that "...the response variable and the predictors are highly skewed. Because of this, I square-root transformed the response, and transformed the other predictors as needed." However, skewed data does not really call for transformations. If you have the best predicted values, y*, such that y = y* + e, then if data are highly skewed, this may emphasize the fact that y* varies substantially in size, which may emphasize the need to account for heteroscedasticity, if that is the issue. Is heteroscedasticity of estimated residuals the reason you felt the need to transform data?
I dealt with much simpler modeling for official energy statistics, but the format was still
y = y* + e, where
ei = (e_0i)(yi*^gamma).
Gamma is the coefficient of heteroscedasticity.
Model and/or data quality issues can impact heteroscedasticity as well, but modeling all heteroscedasticity as above, one can find regression weights as
wi = yi*^-2gamma.
Using that in a weighted least squares regression will handle heteroscedasticity.
Note that the best predicted-yi, yi*, is the best size measure to be used here. When not used because it has not been discovered yet, while in the process of doing so, one may be forced to use a substitute, zi. This could be the homoscedastic predicted-y. But if you don't have the best set of predictors, not too many, nor too few, and just the right ones used the right way, this would impact heteroscedasticity.
Please see the following:
Nature and magnitude of heteroscedasticity of estimated residuals in regressions of form
y = y* + (e_0)(z^gamma),
for finite populations:
https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as determined by Ken Brewer,
and
https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity, which contains some examples where an estimate fir the coefficient of heteroscedasticity is determined.
The spreadsheet tool
https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx
can be used to determine a good coefficient of heteroscedasticity for your data and model. (Note that it considers, on a separate spreadsheet page, when a default may be needed for small sample sizes.)
The following might be helpful:
https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and various updates.
Note that because you have to use the homoscedastic estimate of y*, often called y-hat, where I have noted that GS Maddala used y* for the heteroscedastic case, when entering a regression weight into software to obtain the weighted least squares regression predicted values, you actually have to use
wi = zi^-2gamma instead of
wi = yi*^-2gamma, where
zi = y-hati.
Your model can be examined for fit, and perhaps compared to an alternative, by putting results for both on the same scatterplot, using a "graphical residual analysis," with y-hat for each, on the x-axis, and (raw) estimated residuals on the y-axis. To avoid overfitting to a particular sample, please consider "cross-validation." If you have enough data you could compare models on one scatterplot for one sample, and then on another scatterplot for another sample.
It would be interesting and perhaps helpful for you to show a graphical residual analysis scatterplot using your data, without transformations, and before weighting is used in your model, to see both the general fit and the heteroscedasticity. I would very much like to see that scatterplot.
Cheers - Jim