# Is assumption of residual normality and Homoscedasticity in nonlinear regression

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 (Consequences of violating assumptions of nonlinear regression when comparing models and/or datasets) 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 statistics 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 say residuals are normally distributed while the other says no). Prism recommends D'Agostino-Pearson. Are you on board with this recommendation and could you try and explain to me (a non-mathematician) why the different tests would yield different results?

3) Does the non-normality of residuals mean the model selected for nonlinear regression is incorrect?

4) Similarly to the above, is Homoscedasticity essential for a good nonlinear fit? If there is no Homoscedasticity does that mean the chosen model is incorrect and a different 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?

I am sure there will be more questions, but for now, I really appreciate the help.

• It is not even a requirement of linear regression! That assumption has to do with model inference. If you just want to predict and are getting good results, have at it!
– Dave
Jun 1, 2020 at 12:16
• all the questions you made have the same answer if you consider linear regression instead Jun 1, 2020 at 12:17
• 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. Hope this helps Jun 1, 2020 at 12:45
• @JoãoDuarte please do not use "answers" for commenting. As about your comment, I don't know what you mean by "multiple nonlinear models (polynomial, power, exponential, etc...)"? If you used polynomial features for linear regression, this is still a linear regression.
– Tim
Jun 1, 2020 at 12:47
• Esentially, i fit my data on graphpad prism, using the different nonlinear models included in the package, which include, but are not limited to, the ones I have just mentioned. Jun 1, 2020 at 12:49

$$y = f(X) + \varepsilon$$
with $$\varepsilon \sim \mathcal{N}(0, \sigma^2)$$, then when $$f$$ is a linear function, it is a linear regression, but if you replaced $$f$$ with a non-linear function, it'd be a non-linear regression. If you used a non-linear $$f$$ but didn't change anything about your assumptions about the noise $$\varepsilon$$, then the above formulation still assumes that noise is normally distributed, with constant variance $$\sigma^2$$. However you could define a different regression model, where $$f$$ is non-linear and make different assumptions about the noise, where that still would be a non-linear regression model.