I am still learning a lot about nonlinear regression and I have some questions about residual normality and Homoscedasticity:
1) From what I could find here (https://stats.stackexchange.com/questions/146542/consequences-of-violating-assumptions-of-nonlinear-regression-when-comparing-mod) One user states that normality of residuals is not a necessary assumption for nonlinear regression, is this correct and, if so, can you explain why and provide some literature on it?
2) I have been using graphpad prism as my statistcs tool and it has multiple possible tests for residual normality (D'Agostino-Pearson, Shapiro-Wilk, Anderson Darling). For some of my datasets, different tests give different results (one tests says residuals are normally distributed while the other says no). Prism reccomends D'Agostino-Pearson. Are you on board with this recomendation and could you try and explain to me (a non-mathematician) why the different tests would wield different results?
3) Does non-normality of residuals mean the model selected for nonlinear regression is incorrect?
4) Similarly to above, is Homoscedasticity essential for a a good nonlinear fit? If there is no Homoscedasticity does that mean the chosen model is incorrect and a diffent one should be used?
5) At the moment, in my correlations, I have a couple of replicates for some values of X. Does this affect the Homoscedasticity and/or normality of residuals calculation in a way that I need to account for?
To add a bit more information about this, because it was apparently not clear. Essentially, I tried fitting my data using multiple nonlinear models (polynomial, power, exponential, etc...) and used the AICc to determine the best fit model. However, I am trying to understand if the model with the lowest AICc is, in fact a good model, and I was wondering if failure to comply with non-normality of residuals and/or Homoscedasticity would disqualify the chosen model
I am sure there will be more questions, but for now, I really appreciate the help.
Joao -
I think you might want to try "graphical residual analysis." You could plot the predicted y values on the x-axis, and the estimated residuals on the y-axis. You might place results for two or more models on the same scatterplot for comparison. This is to study model fit, but to avoid overfitting to a particular sample, you might want to look into "cross-validation."
The graphical residual analyses could be used to see if the estimated residual patterns look better in some cases than others, in addition to which model has a tighter fit. (By the way, a polynomial regression is technically a linear regression.) Normality is not a very strict concern.
Heteroscedasticity is to be expected unless there are model or data quality issues, some of which could increase heteroscedasticity, but some may reduce it. For more on heteroscedasticity - which includes extended use of graphical residual analysis - please see my response to
https://www.researchgate.net/post/If_homoscedasticity_criteria_is_not_being_met_in_regression_analysis_what_tools_can_used_to_make_it_homosedastic.
Cheers - Jim
To be clear, regarding #4 in the question, where it says "If there is no Homoscedasticity does that mean the chosen model is incorrect and a diffent one should be used?" I believe the answer is quite the opposite. There is good reason to believe that heteroscedasticity should be expected if the model is very good, data quality is high, and strata or subpopulations are well defined. See https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and updates, for regressions of form y = y* + e, which would include nonlinear regression, and note the idea and work by Ken Brewer discussed in https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity.
The prediction (conditional mean or expectation - the actual curve) will be correct, but the standard errors as well as the confidence and prediction bands will be not very accurate.
If you don't have any theoretical reason to fit a particular model you might just use a spline or a scatterplot smoother (loess) curve.
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