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Dear JMP,
I found a discrepancy between a JMP result and the result in the textbook you cite as a reference. Is this a bug or does JMP use a different calculation than the textbook?
JMP Pro 17.2.0 > Reliability and Survival > Life Distribution platform, fit to a Weibull, Wald confidence intervals > in the Parametric Estimate – Weibull section, all the numbers agree except the scale and Weibull ß confidence intervals (screenshot below).
Reference book is Meeker, Escobar, and Pascual, second edition, 2022. (In JMP documentation, an earlier version of this book is cited as the source for these calculations.) Section 8.3.2 Wald Confidence Intervals for Model Parameters, Example 8.13 Wald Confidence Interval for the Shock Absorber Weibull Shape Parameter, data set attached in JMP format. Book gets for the scale factor [0.201, 0.498] (compare [0.173, 0.460] below) and for the Weibull beta parameter [2.01, 4.97] (compare 2.175, 5.780] below).
All the other parameters agree to at least 3 significant figures, including the location and Weibull alpha confidence intervals, all the standard errors, and the covariance matrix; it is only the scale and Weibull beta that disagrees. The book says (equation 8.7) that the scale LCL, for example, is calculated as (scale best estimate) / w, where w = exp[ (z score of 95% confidence) * (standard error) / (scale best estimate) ]. All those numbers are given by both the book and JMP and all agree, and the calculation gives the result in the book, not the JMP result. So, either JMP has a bug, or it is using a different calculation, in which case, what is JMP’s calculation?
Thank you,
Scott
Accepted Solutions
For the LCLs on the scale and the Weibull beta specifically JMP is building Wald CIs on the extreme‑value parameters on the so called "identity scale," and then transforming endpoints for them. Whereas, Meeker–Escobar–Pascual (2022) seem to construct Wald CIs for strictly positive parameters on the log scale directly and then back‑transform, which yields a different multiplicative form (albeit arguably more intuitive based on the information/ data table shared).
So, JMP fits these:
μ^=location=10.230, σ=scale=0.316, SE(μ)=0.1099, SE(σ)=0.0732
Then directly computes:
Lower and Upper CI on location:
CI 0.95(µ)=µ^ ± zSE(µ^) = 10.230 ± 1.96(0.1099) = (10.015,10.445)
Lower and Upper CI on scale:
CI 0.95(σ)=σ^ ± zSE(σ^) = 0.316 ± 1.96(0.0732) = (0.173, 0.460)
from the reported Estimates and Std Errors that it has fitted as shown above and in the attached summary table appended to the data shared:
The Lower and Upper CI on Weibull α are computed this way which gives the same result as in Meeker and Escobar:
CI 0.95(α) = exp(LCL 0.95(µ)), exp(UCL 0.95(µ)) = (exp(10.015), exp(10.445)) = (22347.770, 34380.492)
But the Lower and Upper CI on Weibull β, computed by 'inversion' of the sigma confidence limit estimates, differs from what you showed per Meeker and Escobar:
CI 0.95(β)=(1/UCL 0.95(σ^), ((1/LCL 0.95(σ^)) = (1/0.460, 1/0.173)=(2.175, 5.780).
The attached modified version of this data with the summary table from Life Distribution appended to it shows the calculations in column formulas next to JMP's reported output, and next to the results calculated using the formulas shared here:
For the LCLs on the scale and the Weibull beta specifically JMP is building Wald CIs on the extreme‑value parameters on the so called "identity scale," and then transforming endpoints for them. Whereas, Meeker–Escobar–Pascual (2022) seem to construct Wald CIs for strictly positive parameters on the log scale directly and then back‑transform, which yields a different multiplicative form (albeit arguably more intuitive based on the information/ data table shared).
So, JMP fits these:
μ^=location=10.230, σ=scale=0.316, SE(μ)=0.1099, SE(σ)=0.0732
Then directly computes:
Lower and Upper CI on location:
CI 0.95(µ)=µ^ ± zSE(µ^) = 10.230 ± 1.96(0.1099) = (10.015,10.445)
Lower and Upper CI on scale:
CI 0.95(σ)=σ^ ± zSE(σ^) = 0.316 ± 1.96(0.0732) = (0.173, 0.460)
from the reported Estimates and Std Errors that it has fitted as shown above and in the attached summary table appended to the data shared:
The Lower and Upper CI on Weibull α are computed this way which gives the same result as in Meeker and Escobar:
CI 0.95(α) = exp(LCL 0.95(µ)), exp(UCL 0.95(µ)) = (exp(10.015), exp(10.445)) = (22347.770, 34380.492)
But the Lower and Upper CI on Weibull β, computed by 'inversion' of the sigma confidence limit estimates, differs from what you showed per Meeker and Escobar:
CI 0.95(β)=(1/UCL 0.95(σ^), ((1/LCL 0.95(σ^)) = (1/0.460, 1/0.173)=(2.175, 5.780).
The attached modified version of this data with the summary table from Life Distribution appended to it shows the calculations in column formulas next to JMP's reported output, and next to the results calculated using the formulas shared here:
