topic Crow-AMSAA Confidence Intervals - Delta Method in Discussions

Crow-AMSAA Confidence Intervals - Delta Method

Reliability_JPL — Mon, 25 Mar 2024 14:02:59 GMT

Hello,

I am using JMP 16.2.0 to calculate Reliability Growth and am trying to understand the math behind the profiler confidence intervals on the Crow AMSAA models. The help article states that: "The delta method is applied to this expression to obtain an estimate of its variance. This estimate is used to construct a 95% Wald-based confidence interval. The resulting confidence limits are then transformed using the exponential function to give confidence limits for the estimated cumulative number of events at time t." I have not been able to find an example of utilizing the delta method that I can follow. Is there a way to on JMP to look at the formula/math that goes into these calculations?

Thank you!

-John

Re: Crow-AMSAA Confidence Intervals - Delta Method

peng_liu — Mon, 01 Apr 2024 01:43:47 GMT

The formula/math of the delta method can be found here: https://en.wikipedia.org/wiki/Delta_method

The method can be applied well if the result of g(.) function is approximately normal.

For any g(.) function in hands, the above may not be true. Sometimes a transformation on g(.) function is necessary, e.g. h(.).

For such a situation, apply the delta method on h(g(.)) first, then use the inverse of h(.) to get desired results of g(.).

In other words, get intervals of h(g(.)), say h_Low and h_High, using delta method on h(g(.)), then use h^(-1)(h_Low) and h^(-1)(h_High) to get the intervals for g(.), where h^(-1) is the inverse of h(.).

In above JMP documentation, log function is the h(.).

Re: Crow-AMSAA Confidence Intervals - Delta Method

Reliability_JPL — Tue, 02 Apr 2024 11:50:38 GMT

Thank you for this information but I admit I am still lost. I am uncertain what the resulting formula of the delta method applied to the h(.) should give me. I am attempting to recreate confidence intervals for numerous time inputs simultaneously, so I would love to be able to input my parameters and times of interest in a table and have a JMP formula to create these confidence bands.

Re: Crow-AMSAA Confidence Intervals - Delta Method

peng_liu — Tue, 02 Apr 2024 12:47:24 GMT

At the moment, JMP does not produce a formula in JSL that a user can directly reproduces the results.

The mathematical details are available in statistical analysis books. My reference is Statistical Methods for Reliability Data . Section C.7.4 and C.7.7 in that book are what you need.

The needed results, however, are available in the report, especially the profilers in the report; see this screenshot:

Based on your description, seems that you would like to get a table which lists desired confidence intervals. I suggest looking into the "Remember Settings" feature in the profiler: https://www.jmp.com/support/help/en/17.1/#page/jmp/prediction-profiler-options.shtml

Following is a screenshot of remembered settings, which record confidence intervals at three different time points.