NonlinearModelFit ParameterTable interpretation

Question

I'm not quite sure, whether this question is one for this forum. I have problems to interpret the standard errors and p-values for my fit parameters. They are so small, that I don't know, whether I did use the fit in a proper way or not. The parameter values itself are good, they reproduce results from a colleague...
...but let's start from the beginning:

I have a set of real measurement data

data = {{2.15838, 4.77993*10^-7}, {2.18534, 4.19629*10^-7}, {2.21001, 
4.09502*10^-7}, {2.25854, 3.99853*10^-7}, {2.2836, 
3.84017*10^-7}, {2.30747, 3.70902*10^-7}, {2.33081, 
3.59127*10^-7}, {2.35137, 3.50775*10^-7}, {2.37643, 
3.3436*10^-7}, {2.39882, 3.2283*10^-7}, {2.42257, 
3.01332*10^-7}, {2.4461, 2.98024*10^-7}, {2.4732, 
2.89137*10^-7}, {2.49911, 2.82813*10^-7}, {2.52499, 
2.68812*10^-7}, {2.55184, 2.61058*10^-7}, {2.58006, 
2.2438*10^-7}, {2.64673, 2.30085*10^-7}, {2.71751, 
2.05682*10^-7}, {2.78639, 1.92553*10^-7}, {2.88777, 
1.75161*10^-7}, {2.96864, 1.61607*10^-7}, {3.051, 
1.49176*10^-7}, {3.16198, 1.25115*10^-7}, {3.33296, 
1.20133*10^-7}, {3.43768, 1.01685*10^-7}, {3.56674, 
9.66192*10^-8}, {3.68825, 7.75714*10^-8}, {3.79472, 
6.92602*10^-8}, {3.87449, 6.89086*10^-8}, {3.95223, 
6.43365*10^-8}, {4.08792, 5.56352*10^-8}, {4.14613, 
5.2924*10^-8}, {4.21338, 4.94536*10^-8}, {4.24873, 
4.87433*10^-8}, {4.25848, 4.78836*10^-8}, {4.26615, 
4.7465*10^-8}, {4.31084, 4.81383*10^-8}, {4.44111, 
4.64897*10^-8}, {4.52387, 4.56303*10^-8}, {4.56865, 
4.51747*10^-8}, {4.6603, 4.43627*10^-8}, {4.74556, 
4.33014*10^-8}, {4.82304, 4.2891*10^-8}, {4.92704, 
4.18981*10^-8}, {5.0293, 4.11988*10^-8}, {5.09724, 
4.08886*10^-8}, {5.20107, 4.02819*10^-8}, {5.27865, 
4.11364*10^-8}, {5.35455, 4.10741*10^-8}, {5.49134, 
4.0462*10^-8}, {5.62088, 3.70902*10^-8}, {5.68181, 
3.67886*10^-8}, {5.77189, 3.65903*10^-8}, {5.8566, 
3.6443*10^-8}, {5.97267, 3.61518*10^-8}, {6.02845, 
3.63941*10^-8}, {6.08799, 3.6008*10^-8}, {6.24226, 
3.58653*10^-8}, {6.35647, 3.56767*10^-8}, {6.43191, 
3.56767*10^-8}, {6.51399, 3.55366*10^-8}, {6.5876, 
3.54902*10^-8}};

with estimated systematic uncertainties of 10%

yerrors = {4.77993*10^-8, 4.19629*10^-8, 4.09502*10^-8, 3.99853*10^-8,
3.84017*10^-8, 3.70902*10^-8, 3.59127*10^-8, 3.50775*10^-8, 
3.3436*10^-8, 3.2283*10^-8, 3.01332*10^-8, 2.98024*10^-8, 
2.89137*10^-8, 2.82813*10^-8, 2.68812*10^-8, 2.61058*10^-8, 
2.2438*10^-8, 2.30085*10^-8, 2.05682*10^-8, 1.92553*10^-8, 
1.75161*10^-8, 1.61607*10^-8, 1.49176*10^-8, 1.25115*10^-8, 
1.20133*10^-8, 1.01685*10^-8, 9.66192*10^-9, 7.75714*10^-9, 
6.92602*10^-9, 6.89086*10^-9, 6.43365*10^-9, 5.56352*10^-9, 
5.2924*10^-9, 4.94536*10^-9, 4.87433*10^-9, 4.78836*10^-9, 
4.7465*10^-9, 4.81383*10^-9, 4.64897*10^-9, 4.56303*10^-9, 
4.51747*10^-9, 4.43627*10^-9, 4.33014*10^-9, 4.2891*10^-9, 
4.18981*10^-9, 4.11988*10^-9, 4.08886*10^-9, 4.02819*10^-9, 
4.11364*10^-9, 4.10741*10^-9, 4.0462*10^-9, 3.70902*10^-9, 
3.67886*10^-9, 3.65903*10^-9, 3.6443*10^-9, 3.61518*10^-9, 
3.63941*10^-9, 3.6008*10^-9, 3.58653*10^-9, 3.56767*10^-9, 
3.56767*10^-9, 3.55366*10^-9, 3.54902*10^-9};

And I use a NonlinearModelFit with the following function:

c1 = 1.3*10^9; c2 = 9.2;
fitfunc = ((p1*c1^2)/c2)*x*Exp[-p2*x] + p3

while the constants and parameters are physical values. Then I fit the data with weights and do get sensible values for the parameters

nlmFIT = NonlinearModelFit[data, fitfunc, {p1, p2, p3}, x, 
Weights -> 1/yerrors^2, VarianceEstimatorFunction -> (1 &)]

But the interpretation of the goodness of the fit gives me a headache, since the errors and p-values given below are really, really small:

I read here in the forum and elsewhere, but I didn't find a good explanation (which I understood ;-)).
Any ideas and hints are welcome. Is it a result of the small numbers of my parameters? Or should I use other values for the interpretation of my results?

Answer 1

You are right to be suspicious of the standard errors of the parameters. If you would examine nlmFIT["CorrelationMatrix"] you'll find all of the correlations are estimated to be either 1 or -1 which also spells trouble. While the cause of such high magnitude correlations can be an overparameterized model, in this case the problem is that the parameters are on such wildly different scales.

First, consider fitting the model without weights but with multiplicative factors to get all of the parameters roughly on the same scale:

c1 = 1.3*10^9; c2 = 9.2;
fitfunc = ((10^(-23) p1*c1^2)/c2)*x*Exp[-p2*x] + 10^(-8) p3;
nlmFIT = NonlinearModelFit[data, fitfunc, {p1, p2, p3}, x];
nlmFIT["ParameterTable"]

Show the data and the resulting fit:

Show[Plot[nlmFIT[x], {x, 2, 7}, PlotRange -> All, AxesOrigin -> {2, 0},
  Frame -> True, FrameLabel -> {"x", "y"}],
 ListPlot[data, PlotStyle -> Black]]

So on a quick examination the fit is not outrageous. But do you need to add weights? Your estimated weights are by definition exactly proportional to the observed values. To see if that weighting might be reasonable we look at the residuals from the unweighted model fit vs the predicted values:

ListPlot[Transpose[{nlmFIT["PredictedResponse"], nlmFIT["FitResiduals"]}],
 Frame -> True, FrameLabel -> {"Predicted response", "Residual"}]

If the variability increases with increasing predicted values, we would see a "fan shape" for the scatter of points with low variability on the left of the graph and larger and larger amounts of variability as we move to the right of the graph. But we don't see that.

We also don't see a "nice" uniform scatter of points across the range of the predicted values. This can occur for a variety of reasons. One possibility is that there are other terms that should be in the model but aren't. Another (and this would be my guess as to what's going on) would be that the data points are taken across time (essentially the x variable) and there is serial correlation among the observations. Here's a plot of the residuals vs the x variable:

ListPlot[nlmFIT["FitResiduals"], Frame -> True, FrameLabel -> {"x", "Residual"}]

So in summary I don't think your proposed weights are appropriate for this data and model. However, there are still some "lack-of-fit" features that probably should be addressed. What to do about those would depend on how the data is generated (serial correlation over time?, instrument flucation over time? multiple instruments? observations taken in separate groups? etc.).