Hello Dr. Huang

I find your research very interesting and with great impact. And in fact unaware of this behavior. Have you performed another experiment other than Ballico's data?. I mainly ask why we take the use of the Welch-Satterthwaite formula for granted and do not even question it when developing an uncertainty budget. I would be grateful if you would support me to carry out what you set in your discussion with a more practical and simple example.

regards

Alejandro Oharriz Calderon

Thank you for your comments. As I mentioned in the discussion, the Welch-Satterthwaite formula is valid for estimating the effective DOF; the Ballico paradox is due to the use of the t-interval in uncertainty estimation. That is, it is the t-interval method that has a serious problem: i.e. producing paradoxical results when the sample size is small (say, 2, 3, 4, …). It is true that the t-distribution (or t-interval) has been considered as the standard way to deal with small samples for over 100 years. The GUM and many statistics textbooks take the t-based uncertainty as the Type A expanded uncertainty. However, after having been working on the subject for over 12 years, I figured out that all t-based methods for the Type A evaluation of uncertainty are flawed. This might be an astonishing statement. But, believe it or not, the t-distribution (or t-interval) actually is a barrier to realistic evaluation of uncertainty. All t-based methods for the Type A evaluation of uncertainty cause unrealistic or paradoxical results. I have discovered the root cause of the problem is the so-called “t-transformation distortion”. The t-distribution (or t-interval) barrier must be removed. That is, all t-based methods for the Type A evaluation of uncertainty must be abandoned. Otherwise, the Type A evaluation of uncertainty would never be right, regardless of frequentist and Bayesian approaches.

By the way, if you are interested in some of my papers on this subject but do not have access to them, please send requests through ResearchGate and I will send you copies privately (I cannot post some of the papers publicly because of copyrights).

The table below shows a comparison of the results obtained with six methods for Ballico’s data (unit: mK).

Methods 1 mK range 10 mK range

GUM’s WS-t (Ballico 2000) 37.39 35.07

Kacker’s Bayesian (Huang 2018) 41.06 43.89

t-based MCM (Huang 2019) 38.53 41.06

WS-z (Huang 2016) 25.75 29.34

Unified theory (Huang 2018) 25.95 29.95

z-based MCM (Huang 2019) 25.99 29.94

Note from the table that the first three methods: GUM’s WS-t, Kacker’s Bayesian, and t-based MCM, all produce unrealistic estimates of uncertainty, or all artificially dilate uncertainty. It is important to note that these three methods all have one thing in common: their formulations are all based on the t-distribution or t statistic. In GUM’s WS-t approach, the t score is applied to the combined standard uncertainty. In Kacker’s Bayesian approach, the standard deviation of the t-distribution, is applied to each of the Type A components. In the t-based MCM, the scaled and shifted t-distribution is assigned to each of the input quantities having Type A uncertainty. The results indicate that these three methods that are based on t-distribution are invalid.

Also note from the table that the other three methods: WS-z, unified theory, and z-based MCM, all produce realistic estimates of uncertainty. These three methods also all have one thing in common: their formulations are all based on the bias correction factor c4 for the sample standard deviation. In the WS-z approach, c4 is applied to the combined standard uncertainty. In the unified theory, c4 is applied to each of the Type A components. In the z-based MCM, c4 is applied to the PDF for each of the input quantities having Type A uncertainty.