How to interpret nlparci?

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.

Answer 1

[Star Strider]

‘^^ 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.

[Learner]

Thanks for such a clear explanation.

I just want to know, suppose in my above problem the values of "pars_ci" is:

pars_ci =
      -10     100
      -0.02   0.5

1. Then this means both parameters are not needed in the model because limits have zero in between?

2. Do huge difference (from -10 to 100) has any significance over smaller difference (from -0.02 to 0.5)?

[Star Strider]

My pleasure!

1. Correct. They would not be significantly different from zero.
2. If you’re discussing parameter confidence intervals, no. The huge difference simply means the degrees-of-freedom are greater than in the example with the narrower confidence intervals. (The possibilities are that there are either fewer data pairs or more parameters — or both — in the second situation than the first. This has to do with ‘degrees-of-freedom’, calculated by the number of data pairs minus the number of parameters being estimated, i.e. n-p.)

Answer 2

[Fritz Ulysse]

how to test if confidence interval covers ground truth parameters