MATLAB: How to interpret nlparci

nlinfitnlparciode

Hi,
I am just curious to know, most of the peoples use "nlparci" for estimating confidence intervals.
Suppose, we have data points:
x = [0.5 0.387 0.24 0.136 0.04 0.011];
y = [1.255 1.25 1.189 1.124 0.783 0.402];
& the function is as follows:
f = @(pars,x) pars(1)*x./(pars(2)+x);
With initial guess of parameters:
parguess = [ 1.3 0.03];
For fitting this function is used:
[pars, resid, J] = nlinfit(x,y,f,parguess)
After running the above program, we get:
pars =
      1.3275    0.0265
resid =
     -0.0058    0.0074   -0.0067    0.0127   -0.0160    0.0122
J =
      0.9497   -2.3949
      0.9360   -3.0053
      0.9007   -4.4873
      0.8371   -6.8404
      0.6019  -12.0216
      0.2936  -10.4056
Now here we go for "nlparci":
alpha = 0.05; % this is for 95% confidence intervals
pars_ci = nlparci(pars,resid,'jacobian',J,'alpha',alpha)
We get:
pars_ci =
      1.3005    1.3545
      0.0236    0.0293
This means:
pars(1) is in the range of [1.3005 1.3545] at the 95% confidence level, &
pars(2) is in the range of [0.0236 0.0293] at the 95% confidence level.
^^ BUT WHAT ACTUALLY DOES IT MEANS; I MEAN HOW CAN YOU INTERPRET IT; WHAT ARE THESE RANGES FOR?
Thanks in advance.

Best Answer

‘^^ BUT WHAT ACTUALLY DOES IT MEANS; I MEAN HOW CAN YOU INTERPRET IT; WHAT ARE THESE RANGES FOR?’
In formal terms, the 95% confidence intervals test the hypothesis that the individual parameters are equal to zero, and so not needed in the model. So for instance, in a linear model (not yours here), y=m*x+b, it would test to see if the intercept b was different from zero, so tests the hypothesis that the data could be explained by a line that went through the origin. It would separately test to see if the slope m was different from zero, so tests whether the slope could be explained by a line parallel to the x-axis at the mean of the y-values. If both those tests fail (both parameters are significantly different from zero), the linear model (in this illustration) is a valid explanation of the data.
In your particular situation, the 95% confidence intervals do not include zero, so: (1) the parameters are needed in the model, and (2) your model explains your data as well as it is able.
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