What are the most reliable test of goodness of fit in Mathematica

I realise this is possibly a VERY open ended question, and potentially has answers dotted around e.g. here .

What are the most robust methods of testing the result of NonlinearModelFit[...] in Mathematica?

I know already that there are some methods available in Mathematica such as AdjustedRSquared, RSquared, AIC, AICc, and BIC.

The most familiar of these to me personally are AdjustedRSquared and RSquared, but I know many methods of testing exist.

A sub-question to this is, when are these tests (and others, whatever they may be), most appropriate? Does this depend on the model fitted, the number of parameters, the amount of data, and so on?

I strongly appreciate that some of these questions can only be answered subjectively, and I even anticipate some potential off-topic or close requests; after all if there was a catch-all test -- I wouldn't need to ask this question! But I feel having a question where goodness of fit specifically in Mathematica is discussed (especially where some of the more experienced members of the community can contribute) might be beneficial.

• I think I will post a hefty bounty on this when allowed. – Q.P. Aug 22 at 22:32
• Much of what one usually needs to do with NonlinearModelFit output is not "testing" but rather "displaying" features of the fit such a qq-plots of the residuals (QuantilePlot), historgram of residuals (Histogram), plots of residuals vs predicted (ListPlot), plots of residuals vs predictor values. Generally one is looking for gross deviations from the assumptions. Then one wants measures of fit. AdjustedRSquared and RSquared are relative measures of fit (relative to the variability in the response). AIC and AICc provide a relative ranking of models. – JimB Aug 22 at 23:19
• And the square root of "EstimatedVariance" will give you an absolute measure of fit in terms of the units of the response variable. – JimB Aug 22 at 23:21
• Thanks for the input JimB. I generally look at how pattern-less residuals look, and look for a Gaussian distribution, but say I have two models or, models which are functionally the same but have different scaling, and I want to compare them a statistic is the best way to go, no? In which case, which one? – Q.P. Aug 23 at 0:18
• – xzczd Aug 29 at 11:32