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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.

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  • $\begingroup$ 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! $\endgroup$ – Dave Jun 1 '20 at 12:16
  • $\begingroup$ all the questions you made have the same answer if you consider linear regression instead $\endgroup$ – carlo Jun 1 '20 at 12:17
  • $\begingroup$ 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 $\endgroup$ – João Duarte Jun 1 '20 at 12:45
  • $\begingroup$ @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. $\endgroup$ – Tim Jun 1 '20 at 12:47
  • $\begingroup$ 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. $\endgroup$ – João Duarte Jun 1 '20 at 12:49
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There is a single model that we know under the name linear regression and many different non-linear regression models, so it depends what is your model. If you have a regression model defined as

$$ y = f(X) + \varepsilon $$

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.

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  • $\begingroup$ 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 $\endgroup$ – João Duarte Jun 1 '20 at 12:47
  • $\begingroup$ @JoãoDuarte I answered your question. If you have some other questions, please open new question to ask them, but in such case, you'd need to tell us what exactly were the models you fitted, what exactly were the results etc. $\endgroup$ – Tim Jun 1 '20 at 13:00
  • $\begingroup$ i am sorry, but I don't think my question was answered, I did not think that I needed to provide exact data as this was more of a general question, but if that helps, I can, what data would you like me to provide? $\endgroup$ – João Duarte Jun 1 '20 at 13:03

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