True Standard & Some Silver References

  Cupellation appears to be the most accurate assay method for silver but there are some losses, on the order of 0.1%, in the process which requires an adjustment to determine the fineness on the scale of a true standard. This points out the difficulty in specifying a definitive test for the fineness of silver and the care that needs to be taken.

Here are some books and links concerning silver and its history, alloys, assay and metallugy:

aes - Wiktionary  Harper's Dictionary of Classical Literature and Antiquities

Pliny - The Natural History of Metals c. 79 AD

Arbuthnot - Tables of Ancient Coins, Weights and Measures 1727, proportion of gold & silver in coins

Phillips - A Manual of Metallurgy 2nd Ed 1859, assay of the alloys & ores of silver

Phillips - The Mining and Metallurgy of Gold and Silver 1867, concentration of precious metals in lead, smelting

Percy - The Metallurgy of Lead 1870, lead-smelting

Hill - A Handbook of Greek & Roman Coins 1899, quality of metals used

Del Mar - A History of the Precious Metals 1902

Scientific American Cyclopedia of Formulas 1915, silver and copper alloys  dwt

Phase diagram - Wikipedia

Provisional and Definitive Tests

  I've been trying to come up with a good example from the History of Science to illustrate the comparison of test procedures and one is the assessment of the purity of metals. One test might be the use of Archimedes' principle to test the specific gravity of a given sample. On the other hand one might take cupellation as the definitive test for purity but he disadvantage of cupellation is that it is destructive so one might prefer a provisional test like the use of Archimedes' principle.

Cupellation is used in the Trial of the Pyx for the assay of coinage. This method dates from ancient times. Modern silver standards are well regulated. For example sterling silver is required to have a fineness of 925.

A modern example of the purification of metals similar to cupellation is the Czochralski process. X-rays can also be used for assaying metals.

Using Concurrence Counts for Comparisons

  In the last couple of blogs I tried to show how the expected number of good items in a sample can be estimated if the agreement and disagreement of two testers were known for correct assessments on the same set of items. One just needs the counts of the concurrence since from them one can determine what the testers observations were as indicated in this diagram. The counts in the last column are just the sum of the counts for the two path leading to the combined counts.

If one tries to make an estimate N_G using the actual observations one ends up with an estimate of N₀, the number of items in the sample instead.

So the testers' assessments themselves don't help us very much. In general we need a table of the concurrence of the actual number of good and bad items. We can alter the problem by asking how well a pass/fail function test will predict whether an item will function for a specified length of time with those that do being the number of "good" items. So we need to consider diagrams like these for the two testers.

After the true counts of concurrence have been determined on can use them to get the conditional probabilities for the testers which in turn allow us to evaluate the testers and their tests.

Supplemental (Oct 4): The reference to a concurrence matrix above would be more properly be called a concurrence table. The conditional probabilities are part of a matrix since they allow one set to be used to compute the other if one set of probabilities is known. The reason for the failure of the testers' concurrence estimate is due to the presence of false positives in the counts used.

A Commentary on Bayesian Games etc.

  The last few blogs dealt with problems related to Bayesian inference. The chief obstacle to such an analysis is getting a good estimate of the conditional probabilities when the reliability of the observers are unknown. We found that using two observers can give an improved estimate of the expected values of the likely unknown counts which are needed to determine the conditional probabilities.

The formulas for the hidden probabilities require counts of the number of events that the observers agree on. Rutherford gives the formula for computing the probability of two simultaneous events P and Q occurring but his papers don't indicate that he used these formulas to get improved estimates of rates based on the observations of two observers. Fuller's proofreading problem doesn't indicate a method of solution either. I have a copy of Fuller book (3rd Ed. see p. 170) and derived the formulas for the solution to the problem on my own. The proofreading problem and formulas can be found in Ross, Introduction to Probability and Statistics for Engineers and Scientists, p. 234f.

A related problem in Bayesian inference would be the determination of the likely number of false positives and false negatives for the observers using a particular method for assessments. If the conditional probabilities are known one can do so. A problem with statistical analysis is that rare events tend to be excluded from observations. Another problem is extending restricted studies to more general cases where the rates are likely not to be the same.

Observer Bias Affects the Corrections to the Faulty Observations

  Observer bias can affect the correction to the estimates of the hidden probabilities for the occurrence of good and bad items. In the last blog the observer assessments were unbiased for both examples. In the first example below the first observer is less likely to make an error in identifying good items while the second observer makes fewer mistakes on bad items. The estimated probabilities ends up slightly biased in favor of good items since there are more good items than bad in the sample. In the second example below both observers more accurately identify bad items and the estimated probabilities have a slight shift towards the occurrence of bad items.

Since there were 100 items in each sample one would expect the rms error in the estimated mean probabilities to be about 1/√100=0.1 times the rms error in the 100 items in the sample so the mean probability estimates should be accurate to about 3 digits.

Correcting Faulty Observations

  One can use the results of two independent quality tests to improve the estimate of the probability of a finding a good item. The counts for Tester 1 of good and bad items are sesignated N₁ and N₂ and those of Tester 2 are N₃ and N₄ and their probabilities are p and q and p' and q' respectively. Based on the two observers' assessments of the items how can we determine N_G, N_B, p_G and p_B? The answer is to keep track of the number of times, N₁₃, when both N₁ and N₃ are good and the number of times, N₂₄, when both N₂ and N₄ are bad.

Then we can borrow a trick used by Rutherford to improve on the scintillation rates determined by two different observers.

Even with relatively large rates for the counting errors one still get a good estimate of the actual rates.

The mean values indicated were found by averaging the counts for N_G and N_B using N_G+N_B=N₀.

Supplemental (Sep 26): The estimated probabilities for G and B are again the averages for the 10 sets of assessments each involving 100 items. The set of stochastic variables used to generate the data were for p_G, a, d, a' and d' and were randomly set to 1 or 0 based on the rates as was done previously.

Supplemental (Sep 26): See Feller, An Introduction to Probability Theory and Its Applications, Vol I, 2nd ed., p. 160, prob. 23 which cites Rutherford. See also Rutherford &al., Probability Variations in the Distribution of α Particles cited in Rutherford's book linked above.

Uncertainty in the Cross Terms for the Comparison of the Two Quality Assessments

  When comparing the two processes for determining quality of items using the more accurate estimates good (G) and bad (B) and the less accurate estimates pass (P) and fail (F) the uncertainty  in the cross terms can be quite large. The conditional probabilities were determined as follows.

We can recalculate the Excel worksheet that generated the 10 sets of 100 random estimates of the quality of G, B, P, and F items to get a new set of averages and save the numerical values. With 100 of these trials the average of b and c were determined and as well as the root mean square deviations from this average.

One can see that there is quite a bit of variation in the cross terms even though the average turns out to be fairly accurate. A large number of tests of a given set of items is needed to get good estimates of the cross terms b and c in order to check theoretical results.