can you use joint confidence region to form confidence intervals on parameters?In JMP Pro 17, I have generated LR confidence bounds and likelihood contour plots for simulated Weibull data. Under the Life Distribution platform, I am able to get confidence bounds on the Weibull parameters along with contour plots for the joint confidence region of the parameters.
I was wondering if you drop lines tangent to the 0.95 contour and perpendicular to the axes, should these form the 95% confidence bounds for the parameters? From the graph below, it does not appear to be the case. However, I wanted to know if it should be the case. Is there a bug in the software or is this result is expected?
Thanks!
Re: can you use joint confidence region to form confidence intervals on parameters?The following pages may be helpful. Please see "Example 1: Likelihood Ratio Bounds on Parameters".
http://reliawiki.org/index.php/Confidence_Bounds#Likelihood_Ratio_Confidence_Bounds
What the JMP displays are contour plots of the log likelihood. As described on the above page, the contour plots are done with degrees of freedom k=2, so that both parameters can be compared simultaneously.
If you want to calculate a confidence interval based on the plot, you need to generate with degrees of freedom k=1, as we are only determining bounds for one parameter.
Please have a look at the following figure (weibull α and β). I think this is what you want. I think you need to use Nonliner Platform to get this figure.
Please note that above example is confidence interval based on Likelihood.
Yuichi
Re: can you use joint confidence region to form confidence intervals on parameters?Thanks to @yuichi_katsumur for the refence and script. Very useful to understand how the likelihood ratio-based confidence intervals for the individual parameters can match the bounds on a contour plot with an appropriate DF in JMP.
Here is my explanation to @charlie_whitman's question "I was wondering if you drop lines tangent to the 0.95 contour and perpendicular to the axes, should these form the 95% confidence bounds for the parameters?"
No, they should/do not. It is joint confidence bounds vs. individual confidence intervals. Let me explain using the data from @yuichi_katsumur.
In the screenshot below I fit a Weibull to Time, choose the likelihood ratio-based CI, and draw a vertical line that’s tangent to the 0.95 contour and perpendicular to the X-axes on the Location parameter. The value of Location obtained is ~4.258, and the corresponding value on the y-axis for Scale is ~0.79. This pair, (4.258,0.79), represents the largest value of Location and associated value of Scale that satisfies -2 LL Ratio =Chi Sq (0.05, 2), the 95% joint confidence bounds of the Loglikelihood Contour plot). As you can see the individual UCL for Location is 4.02. The value of 0.79 for Scale falls within the 95 %CI, (0.23,1.03).
Re: can you use joint confidence region to form confidence intervals on parameters?Thanks Yuichi! This makes sense. I remember seeing somewhere that you can use the contour plot of the joint distribution to get individual confidence bounds. I had forgotten that the contours need to be using only 1 degree of freedom. This answers the question. Thanks!
Charlie
