How can I compute the standard error for coefficients returned from curve fitting functions in the Curve Fitting Toolbox?

I fit my data to a polynomial model using the FIT function from the Curve Fitting Toolbox as follows:

load census
[fitresult,gof1,out1] = fit(cdate,pop,'poly2');

The FIT function returns the following coefficients, along with confidence intervals:

 fitresult =

      Linear model Poly2:
        fit1(x) = p1*x^2 + p2*x + p3
      Coefficients (with 95% confidence bounds):
        p1 =    0.006541  (0.006124, 0.006958)
        p2 =      -23.51  (-25.09, -21.93)
        p3 =  2.113e+004  (1.964e+004, 2.262e+004)

I am interested in finding the standard error of the coefficients rather than the confidence interval.

Answer 1

[MathWorks Support Team]

The FIT function in the Curve Fitting Toolbox does not currently return the standard error or variance information on the estimated coefficients.

You can compute the standard errors from the confidence interval in the following manner.

Let "fitresult" be the result of calling "fit", and "df" be the degrees of freedom:

>> alpha = 0.95
>> ci = confint(fitresult, alpha) 
>> t = tinv((1+alpha)/2, df); 
>> se = (ci(2,:)-ci(1,:)) ./ (2*t) % Standard Error

Alternatively, models fitted using the Statistics Toolbox methods will have their coefficient-covariance matrices calculated and stored automatically as a model property. For example, you can access the coefficient-covariance matrix of a model "mdl" generated as output of the command "fitlm" by the command "mdl.Coefficient." See the following documentation link for more information:

https://www.mathworks.com/help/stats/linearmodel.html#bsz4dm2-1_sep_shared-Coefficients

You can also refer to the following discussion on MATLAB Answers:

https://www.mathworks.com/matlabcentral/answers/34234-how-to-obtain-std-of-coefficients-from-curve-fitting

[Adam Danz]

It would be helpful if the df were defined in the answer. Otherwise the answer is incomplete. For example, is the df the number of observations?

df = out1.numobs; 

Or is the df subtracting the number of coefficients to fit?

df = gof1.dfe; 

[John D'Errico]

df should be the number of observations minus the number of estimated parameters.

[Adam Danz]

So gof1.dfe would be the way to go, then.