National Science and Engineering Research Council of Canada had funded David’s 1988 work with NSERCC Grant 7035. National Research Council of Canada had earlier funded David’s 1977 work with NRC Grant 7035. Professor Dr Michel David’s 1977 work did not respect the requirement of functional independence and ignored the concept of degrees of freedom. His 1988 work was just as flawed but a bit more slipshod. My case against geostatistics has been brought to the attention of several NSERCC officials. One of those thought my message should have been sent to Natural Resources Canada. I had done so long ago but to no avail.

Dr Frederik P Agterberg Ex Emeritus Scientist Natural Resources Canada

The text that had been transmitted on December 14, 2011 reads as follows:

Access to Information Request

Professor Dr Michel David’s work is based on research funded by the Natural Science and Engineering Council of Canada (Grant 7035). The author mentioned this grant in his 1977 Geostatistical Ore Reserve Estimation (364 pages) and in his 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation (216 pages). David shows on page 286 of his 1977 work how to derive a set of sixteen (16) distance-weighted averages from a set of nine (9) holes. What he did not derive was the variance of each distance-weighted average. On the contrary, he postulated, “Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics!” As a matter of fact, counting degrees of freedom for that system of equations is a good test to find out whether one really grasps applied statistics.

Geostatistics is an invalid variant of applied statistics. As such it is a scientific fraud. Lord Kelvin (William Thomson 1824-1907) once said, “…when you can measure what you are speaking about, and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge is of the meagre and unsatisfactory kind…” Lord Kelvin knew more about degrees Kelvin and degrees Celsius than about degrees of freedom and the study of climate change. Lord Kelvin and Sir Ronald A Fisher (1890-1960) were marginal contemporaries. Lord Kelvin would have wondered about the wisdom behind assumed spatial dependence between measured values in ordered sets. Sir Ronald A Fisher could have verified spatial dependence by applying his F-test to the variance of a set of measured values and the first variance term of the ordered set.

What I want to know is whether or not any mutation of Matheron’s new science of geostatistics is applied to study climate dynamics or to monitor pollution of lakes and waterways.

Yours truly, J W Merks, President Matrix Consultants Limited 1357 Napier Place Coquitlam, BC Canada V3B 7A3 Phone: 604-941-1213

The National Research Council did so in the 1970s. Professor Dr Michel David was awarded Grant NRC7035 to advance geostatistics. So he plodded away and got his work printed in 1977. But he came up with a peculiar caution. He pointed out that “professional statisticians would find unqualified statements”. The author mentioned it on page VII of what he had come to call Geostatistical Ore Reserve Estimation. But why didn’t he ask a genuine statistician to read his draft? I found out that the author was right after having read my own copy of his book.

David was as smitten with geostatistics in the 1970s as young Matheron was with applied statistics in the 1950s. Alas, Matheron’s pursuit of applied statistics was not to last. On the contrary, Professor Dr Georges Matheron in the 1970s praised geostatistics as a new science. He did so because he had failed to grasp that functions do have variances. Thus, each and every distance-weighted average has its own variance in applied statistics. Just the same, Matheron got hooked on his variance-deprived distance-weighted average. So much so that he got into calling it a kriged estimate. He did it to honor D Krige who worked with distance-weighted averages at Witwatersrand gold mines in South Africa. What went wrong with Matheron’s new science is that the variance of the kriged estimate has gone missing!

A strong case can be made that eulogies be written long before one’s time on this planet comes to an end. One might ask a lawyer to assist in assuring the veracity of one’s credentials. A case in point is Professor Dr George Matheron’s 2000 eulogy. Dr F P Agterberg was his eulogist. He remembered him as the founder of spatial statistics. As luck would have it, Matheron never tested for spatial dependence by applying Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. Given that Matheron’s magnum opus is posted on the web, one might spend a life time to study his work.

Professor Dr Michel David passed away on May 10, 2000. His obituary was put together by Professor Dr Roussos Dimitrakopoulos and Michel Dagbert. The International Association for Mathematical Geology awarded David the W C Krumbein medal in 1988. David became a Fellow of the Royal Society of Canada in the same year. Moreover, CIMMP recognized David’s worldwide achievements in 1989 with the award of the Selwyn G Blaylock medal.

My son and I knew precisely what had gone wrong with geostatistics when I was face-to-face with David on Saturday, March 23, 1991. It was at a seminar on Sampling and Ore Reserves at the Royal York Hotel, Toronto, Ontario. CIM Bulletin in 1990 rejected Precision Estimates for Ore Reserves. We had shown how to test for spatial dependence and how to derive unbiased confidence limits for gold content and grade. David saw fit to nitpick that twenty years of geostatistical literature went missing. He did not ask me a single question. Take a look at what is wrong in Matheron’s new science of geostatistics.

David wrote: Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics! Merks and Merks claim: Deriving the variance of each distance-weighted average AKA kriged estimate, and counting degrees of freedom for that his system of equations might be a good test to find out whether one really grasps applied statistics.

The National Research Council is the Government of Canada’s institute for research and development. As such it has been active since 1916. NRC’s task is to stand on guard for ethics and integrity. It did not know in the 1970s that Matheron’s new science of geostatistics is an invalid variant of applied statistics. Neither did Dr Roger A Blais, a Professor in Economic Geology and a Fellow of the Royal Society of Canada. He made it possible for David to write so much about so little. David also confessed to be indebted to Matheron. David’s writing added up to a batch of bogus statistics. NRC7035 was in place not only for his 1977 Geostatistical Ore Reserve Estimation but also for his 1988 Handbook of Applied Advanced Geostatistical Ore Reserve Estimation.

Dr Isobel Clark in her 1979 Practical Geostatistics derived the variance of the distance-weighted average AKA kriged estimate. Professor Dr Michel David wrote about the famous Central Limit Theorem but did not apply it. Professor Dr Roussos Dimitrakopoulos wrote a touching farewell to Michel David (1945-2000). David is no longer listed under Obituaries of Deceased Fellows. Sic Transit Gloria Mundi!

It all depends on who applies what! Statisticians apply true variances but geostatisticians work with false variances.The problem is that geostatistocrats call theirs kriging variances. The matter of true variances versus kriging variances came up at a seminar sponsored by the PDAC (Prospectors and Developers Association of Canada). The PDAC had set the stage at the Royal York Hotel in Toronto, Ontario, on Saturday, March 23, 1991. It was organized by H E (Buzz) Neal, PEng, William A Roscoe, PhD, PEng, Henrik Thalenhorst, PhD, and Lorne A Wrigglesworth. I had called my talk Sampling in Exploration, Theory and Practice. I was slated first to speak. As luck would have it, I would give the same talk at Mount Isa, Queensland, Australia, on November 3-7, 1992. That’s where I also presented the Conference Dinner address. But that’s one more part of my story!

During my talk Professor Dr Michel David was sitting sort of face to face with me on the first row. A few of his buddies were close by. David himself had put on paper the very first work on Matheron’s new science of geostatistics. He had simply called it Geostatistical Ore Reserve Estimation. Elsevier Scientific Publishing Company had printed in 1977. David himself had predicted in this book that it was not for professional statisticians. He also predicted that statisticians would find many unqualified statements. And he did get that right too! What David did not predict is that he would blow a fuse if and when he was to review a paper that was short on references to geostatistical literature. But that’s exactly what he did as a reviewer for CIM Bulletin. David did so when he reviewed in September 1989 our paper on Precision Estimates for Ore Reserves.

We had decided not to point out what was wrong with geostatistics but to show what made sense in applied statistics. We had tested for spatial dependence between gold grades of bulk samples taken from a set of ordered rounds in a drift. We had done so by applying Fisher’s F-test to the variance of the set and the first variance of the ordered set. We pointed out that each function does have its own variance in applied statistics and that variances of gold contents are additive. What we didn’t do was estimate the intrinsic variance of gold. It would have required that a pair of interleaved primary samples be taken from every crushed round. We mentioned that extraneous variances such as those for dividing whole core sections into halves, and for selecting and assaying test portions of test samples may be subtracted before deriving unbiased confidence limits for contained gold. We were tickled pink that Precision Estimates for Ore Reserves was praised by and published in Erzmetall, October 1991.

David has made peer review at CIM Bulletin a shameful sham. Read what he wrote about our paper: “The authors present their own method for calculating precision estimates for ore reserves without a single reference to 20 years of work in geostatistical ore reserve estimation (see attached references)”. What he had missed were references to Dagbert & Myers, to himself, and to Journel & Huijbregts. In his 1977 Geostatistical Ore Reserve Estimation he did praise “the famous Central Limit Theorem”. What he didn’t show was how to test for spatial dependence between measured values in ordered sets. Neither did he show how to derive unbiased confidence limits for masses of contained metals.

David may have reviewed A study on Kriging Small Blocks. Its authors called attention to the fact that mine planners are often tempted to over-smooth small blocks. Armstrong and Champigny failed to show how to smooth both small and large blocks to perfection. Good grief! That sort of bogus science was approved by and published in CIM Bulletin of March 1988. Nowadays, mineral analysts are blamed when geostatistically predicted grades mess up metallurgical balances in mineral processing plants. It’s all a huge game of chance for mining investors!

Marechal and Serra showed in 1974 how to derive a set of sixteen (16) distance-weighted averages from a set of nine (9) boreholes. David shows the same set on page 286 of his book. Each distance-weighted average is a function of the same set of nine (9) holes. As such, each is blessed with its own variance in applied statistics. Here’s where statistics went missing in geostatistics. The variance-deprived distance-weighted average morphed into a kriged estimate. What’s more, geostatisticians never took to counting degrees of freedom.

Infinite set of kriged estimates within B

David got into calling a kriged estimate a simulated value. Here’s literally what he wrote on page 324 of his 1977 work, “The criticism to this model is obvious. The simulation is not reality. There is only one answer: The proof of the pudding is …! So far the few simulations made which it has been possible to check have a posteriori proved to be adequate”. Nobody knows all of the nonsense I've had to put up with!

The Bre-X fraud made it clear that mining investors ought to be protected! Mining investors in Canada may well be the first in the world to be so protected. The National Securities Regulator takes on this task once the Supreme Court approves it for all of Canada. Now let’s take a quick look at a scenario. A mining investor may have thought that a mineral resource in an annual report looked like a good bet. But what went wrong if its mined grade is significantly lower than predicted? Here’s a cute catch-22! Confidence limits for metal grades and contents of mineral resources need not be disclosed. Yet public opinion polls are reported with 95% confidence limits. Why does the mining industry not do likewise? I did so in 1997 for Barrick Gold. The mining industry ought to revisit what was once hailed as Matheron’s new science of geostatistics. It made landfall on this continent in 1970. It is simple to prove that geostatistics is an invalid variant of applied statistics. Surely, mining investors in Canada would want a National Securities Regulator to investigate the validity of geostatistics.

Our National Securities Regulator launched its Transition Office in June 2009. Mr Douglas M Hyndman was appointed the NSR Chair. The Supreme Court of Canada has not yet ruled on the constitutional validity of a national securities regulator. It seems to make sense at a glance but is fraught with practical pitfalls. Alberta and Quebec prefer provincial fiefdoms. Here’s what I find funny. David’s 1977 Geostatistical Ore Reserve Estimation was put on paper in La Belle Province. Alberta’s oil patch has taken to geostatistics with reckless abandon. That’s why I am pleased that NSR’s Chair is bringing 25 years of experience to this position.

Mining investors do remember the Bre-X fraud but too few grasp how geostatistics converted bogus grades and barren rock into a massive gold resource. I’m not one to search for moral integrity. Searching for scientific integrity is good enough for me. I am pleased that the BCSC Chair has been appointed to chair the NSR Transition Office. He does have what it takes to unravel a scientific fraud. I do so wish the Supreme Court of Canada to rule in favor of a National Securities Regulator.

I drew the attention to the BCSC Chair in my letter of March 24, 2006 to the fact that I had called on the Canadian Council of Professional Engineers and the Canadian Council of Professional Geoscientists to examine whether geostatistics is a scientific fraud or sound science. Neither CCPE nor CCPG took to the task. I also pointed out to have met in Vancouver, BC on January 22, 2006 with Ms Deborah McCombe, PGeo and Dr Greg Gossan, PGeo. At that time both were on staff with securities commissions in Ontario and in British Columbia.

I pointed out that Dr Isobel Clark derived in her 1979 Practical Geostatistics the variance of the distance-weighted average AKA kriged estimate. She was the first and only scholar who derived the distance-weighted average. I pointed out that the author didn’t test for spatial dependence within her sample space by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. I made it clear that all distance-weighted averages converge on the arithmetic mean as the distance between Clark’s sample space and a selected position converges on infinity. That’s why testing for spatial dependence in sample spaces and sampling units is so critical in applied statistics. All I want to know is why professional engineers and professional scientists accept that spatial dependence between measured values may be assumed simply because Stanford’s Professor Dr Andre Journel has said so.

The Supreme Court of Canada is to decide whether or not a National Securities Regulator is in the best interest of Canadian investors. Meanwhile the Chair of NSR’s Transition Office in June 2009. He no longer has to rely on Dr Greg Gossan, his former Chief Mining Advisor.

Mr Douglas M Hyndman, Chair, NSR Transition Office, ought to ask the Canadian Council of Professional Engineers and the Canadian Council of Professional Geoscientists whether or not a statistical fraud does violate any Code of Ethics.

Bre-X’s bogus grades made Busang’s barren rock look like a genuine gold resource. Yet, Professor Dr Michel David didn't know what was wrong with his 1977 Geostatistical Ore Reserve Estimation. He derived a set of sixteen (16) functionally dependent values in a sample space. Each and every one of them is a function of the same set of nine (9) boreholes. That’s why each of them would have been blessed with its own variance in applied statistics. David was inspired by a tale on Random Kriging by A Marechal and J Serra. Matheron himself put on paper Random Functions and Their Application in Geology. He invoked Brownian motion along a straight line to ensure continuity of random functions. This three-some had been brainstorming at the Centre de Morphology Mathematique, Fontainebleau, France. Next, brought Matheron's new science of geostatistics to this continent in June 1970.

Figure 10. –Grades of n samples belonging to nine rectangles P of pattern surrounding x Fig. 203. Pattern showing all the points within B, which are estimated from the same nine holes.

David’s students seem to have thought some 25 years later that he deserved praise for thinking up the first textbook on geostatistics. David’s bash was called Geostatistics for the Next Century. All I wanted David and his buddies to grasp before this century came along were The Properties of Variances. When praise was heaped on David, Bre-X’s rigs were drilling barren rock in the Kalimantan jungle. Good news for Bre-X’s investors kept coming. Soon it was to be bad news!

Applied statistics proved early in 1997 that Bre-X’s crushed core samples were salted with placer gold. That’s why the Mining Standards Task Force (MSTF) was set up. Morley P Carscallen, Vice-Chair, Ontario Securities Commission, and John W Carson, Senior Vice-President, Market Regulation, Toronto Stock Exchange, were MSTF’s Co-Chairs. Here’s what they had pointed out in MSTF’s Interim Report , ”A number of incidents served as an impetus for the formation of a joint task force between the Ontario Securities Commission and the Toronto Stock Exchange. Public confidence in mining stocks was shaken and the industry as a whole suffered a major setback. The most talked about incident was, of course, Bre-X.”

In spite of so much soul searching the Bre-X fraud ended up being the least acted upon incident. The OSC could have acted several years before I proved that Bre-X was a salting scam. I had pointed to the fact that unbiased confidence limits for metal grades and contents can be derived not only for mined ores and mineral concentrates but also for in-situ reserves and resources. I had done so in my letter of November 30, 1994 to John J Drury, PEng and Chairman CIM Ad Hoc Reserve Definition Committee. He responded on October 23, 1995. Who could possibly be against tried, tested and true ISO Standard Methods after the Bre-X salting scam? And why would the world’s mining industry not want unbiased confidence limits for metal contents and metal grades of mineral inventories? Why not turned out to be one very long story!

The Mining Standards Task Force came up with National Instrument 43-101 to define requirements for disclosure of results. It claims to have done so to increase investor confidence. Yet it still doesn’t show how to derive unbiased confidence limits for metal contents and grades of mineral inventories. It’s good news for mining companies but bad news for mining investors. Appendix A in MSTF’s Final Report points to presentations made to the Task Force. I pity those who had to put up with so much mind numbing geostat drivel. Appendix B points to written submissions to the Task Force. AMEC, Geostat Systems International and SNC-Lavalin have not made submissions in writing.

Post-Bre-X standards protect mines much more than investors. Both the OSC and the BCSC knew I had unscrambled the Bre-X fraud. So, I got to meet Ms Deborah McCombe, OSC’s Chief Mining Consultant, and Dr Greg Gosson, BCSC’s Chief Mining Advisor. We met on January 22, 2006 at the BCSC Office in Vancouver, BC. The objective was “to discuss the use of geostatistics in mineral resource and mineral reserve estimates by mining companies.” I handed out copies of my work. I had posted on my website Clark’s hypothetical uranium data in a two-dimensional sample space. She had done what no geostatistician ever got around to doing. She derived the variance of the distance-weighted average aka kriged estimate. Extrapolation shows that the distance-weighted average tends to converge on the arithmetic mean. What she didn’t do was test for spatial dependence. But then, Stanford’s Journel taught her that spatial dependence between measured values in ordered sets may be assumed.

OSC’s Chief Mining Consultant wrote she would appreciate "being apprised of my progress on the development of industry best practices in the application of mathematical statistics to assess the reliability of data." Her letter was copied to OSC’s Chair.

Praise for ASTM is also due for correcting the first name of Dr Jan Visman. I had reported on May 22, 2011 to ASTM’s President that it was misspelled. Catharine Allan, Administrative Assistant, Technical Committee Operations, made the correction and kept me posted. Now that’s the iconic society I got to know so well. ASTM Committee D05 on Coal and Coke has been working with applied statistics ever since Greg Gould became its driving force. One reference in Visman’s 1947 PhD thesis reads, Gould, G B, How to use laboratory tests in judging coal values, Combustion 11, 31-37, 1939-1940. How about that? Visman was already aware of Gould’s work! Gy’s 1967 and 1971 works do refer to Visman but his 1979 Sampling of Particulate Matter no longer does. On the contrary, Gy praised in his Introduction not only Matheron’s theories but also David’s 1977 textbook. Gy’s praise may well be the reason why David blew a fuse when Merks & Merks showed how to test for spatial dependence between gold assays of ordered round in a drift.

Greg Gould’s friends called him GG. When I met Greg for the first time he told me that his number one textbook was Volk’s Applied Statistics for Engineers. I still have my first copy. Volk explores all properties of variances in a chapter called Analysis of Variance. Visman applied the additive property of variances to partition his sampling variance into its composition and distribution components. I divided the set of primary increments into interleaved pairs so as to get a single degree of freedom. Gy would put his set of primary increments in a single basket. Clearly, the French sampling school didn’t respect the concept of degrees of freedom nearly as much as do statisticians. That’s why I put together a paper called The Properties of Variances. I wanted to prove that kriging variances and classical variances are as different as night and day.

GG’s grasp of sampling and statistics made him a valuable member of ASTM Committee E11 on Quality and Statistics. He didn’t attend the meeting of ISO/TC27 at Vancouver, BC in 2009. Here’s where and when SGS’s Charles D Rose talked about multivariate analysis when testing for bias between paired data. He didn’t talk about trace elements in rocks or soils but about bias test data for total moisture and dry ash in coal. The power of Student’s t-test to detect a bias is best defined in terms of Bias Detection Limits and Probable Bias Ranges. C D Rose and R M Srivastava seemed to have solved in 1993 some sort of sampling problem. They did it with a fractal correlation function but without degrees of freedom. Scores of degrees of freedom deprived papers were presented in 1993 at David’s bash. Thus it came to pass that David’s 1977 Geostatistical Ore Reserve Estimation was praised for no reason whatsoever.

British Columbia got stuck with environmental guidelines cooked up by R Mohan Srivastava and his FSI pals. Mo ticked me off by dismissing degrees of freedom. He put up a squabble when Sandra Rubin’s “Whistleblower Raises Doubts over Ore Bodies” was published in the National Post of September 30, 2002. Mo beats around the bush with the best when talking about degrees of freedom. All the same, a gifted geostatistocrat thought Mo was witty.

Geovariances’ Dr M Armstrong Past Editor:De Geostatisticis Coauthor: A Study on Kriging Small Blocks

Both Armstrong and her coauthor failed to grasp why kriging variances rise and fall. They thought mine planners were over-smoothing small blocks. So they cautioned against over-smoothing. As luck would have it, testing for spatial dependence didn’t play much of a role in Matheronian geostatistics.

Stanford’s Journel made geostatistics a piece of cake by assuming spatial dependence between measured values in ordered sets. So, he didn’t even teach his students how to apply Fisher’s F-test.

Stanford’s Professor Dr A G Journel Unencumbered with Fisher’s F-test

Mineral analysts, too, are blamed when metal grades of mined ores are lower than geostatistically “engineered” grades. Mineral analysts do know that interpolation without justification makes no sense in any science. I have talked about sampling and statistics at several of their annual meetings. I even put together a paper on Self-defense for Mineral Analysts.

ASTM awarded me in 1995 for 25 years of continuous membership. Peter S Unger, Vice Chairman, Committee E-11, had written on March 4, 1995, that Ricardo Stone, 1st Vice Chair, would review my notes and contact me. Hennie and I enjoyed luncheon with Carol and Ricardo. I explained that measured gold grades of mined ore were lower than geostatistically engineered gold grades at Hecla’s Grouse Creek mine. Hecla's chief geologist and his dad were both into geostatistics. That’s why the assay laboratory was blamed for low gold grades. I had met its chief assayer at some other mine in the USA. He asked me to visit the Grouse Creek mine and figure out what was wrong. All I did was apply Fisher’s forbidden F-test to gold grades of ordered blast holes. Spatial dependence dissipated into randomness between 20 and 30 m. Yet, the geostatistical model was based on assumed spatial dependence at 100 m. Hecla’s Grouse Creek never made the predicted grade. As fate would have it, geostatistical software was already converting Bre-X’s bogus grade and Busang’s barren rock into a massive phantom gold resource. ASTM Committee D-18 on Soil and Rock is still kriging and smoothing as much as does the world’s mining industry.

ASTM Committee D18 on Soil and Rock had arranged an International Symposium on Geostatistics for Environmental and Geotechnical Applications. The stage was set at the Hyatt Regency, Phoenix, Arizona on January 24-25, 1995. One of its co-chairs was R Mohan Srivastava. His point of view on geostatistics was already an integral part of BC Environment Guidelines. Environment Canada has a handbook called The Inspector’s Field Sampling Manual. I read it when EC took a client of mine to court. Here’s what EC’s inspectors are taught, “Systematic samples taken at regular intervals can be used for geostatistical data analysis, to produce site maps showing analyte locations and concentrations. Geostatistical data analysis is a repetitive process, showing how patterns of analytes change or remain stable over distance or time spans”. It refers to shellfish samples taken at 1-km intervals along a shore, and to water samples taken from varying depths in a water column. It’s just as short of primary data sets and derived statistics as is Matheron’s whole magnum opus. The question is then why geostatistical data analysis underpins the joint study of the Great Lakes by Canadian and US governments. It boils down to blind ambition and blatant contempt for the properties of variances.

Founder of Geostatistics Founder of Spatial Statistics

ASTM Committee D18 was set up after Geostatistics for the Next Century came about somewhat early in 1993. That’s when geostatisticians from far and wide had flocked together at Montreal, Canada. They had come to praise David’s 1977 Geostatistical Ore Reserve Estimation. Nobody asked David why he had not derived the variance of the distance-weighted average. The more so since Dr Isobel Clark did derive this variance in her 1979 Practical Geostatistics. Sadly, she wasn’t present at McGill. All I had wanted to do was point out that functions do have variances. Alas, my view was as unpopular at Geostatistics for the Next Century as it was when Professor Dr Michel David was the chief enforcer of geostatistics with CIM Bulletin.

David’s peers had come not only to praise his 1977 textbook but also to peddle their own geostat stuff. For example, Journel peddled Modeling Uncertainty: Some Conceptual Thoughts. What David had done in this textbook was derive sixteen (16) distance-weighted averages from nine (9) boreholes. He didn’t derive the variance of each distance-weighted average. He didn’t test for spatial dependence between ordered boreholes. Neither did he count degrees of freedom. David did come up an infinite set of what he then called “simulated values”. Journel derived the zero kriging variance of David’s infinite set of simulated values AKA kriged estimates. Here’s in a nutshell what Professor Dr Georges Matheron has taught all of his disciples. Assume spatial dependence between measured values in ordered sets, interpolate between measured values, smooth the least biased subset of some infinite set of kriged estimates, and rig the rules of applied statistics with impunity. He did all of his thinking about Brownian motion along a straight line in this gloomy edifice.

Centre de Géostatistique Centre de Morphology Mathématique

Matheron invoked it on this continent in June 1970. Brownian motion set the stage to assume spatial dependence between measured values in ordered sets rather than test for it by applying Fisher’s F-test. One of Matheron’s most gifted disciples was Stanford’s Journel. Not surprisingly, Journel never did what Matheron had not taught him to do. But surely, Journel knew a bit of spatial stuff!

Merks and MerksPrecision Estimates for Ore Reserves was praised by and published in Erzmetall. All we did was test for spatial dependence between gold grades of an ordered set of twelve (12) rounds in a drift. Here’s what Stanford’s Journel wrote on October 15, 1992 to Professor Dr Robert Ehrlich, JMG’s Editor, “…Mr. Merks’ anger arises fro [sic!] a misreading of geostatistical theory, or a reading too encumbered by classical “Fischerian” [sic!] statistics.” Journel beat a bit more around the bush when he wrote, “The very reason for geostatistics or spatial statistics in general is the acceptance (a decision rather) that spatially distributed data should be considered a priori as dependent one to another, unless proven otherwise.” He ponders on page 6 of his clarification, “I’ll leave it to you to decide whether this letter should be sent to J. W. Merks; however, I strongly feel that Math Geology has had more than its share of detracting invectives.” Professor Dr Robert Ehrlich, JMG’s Editor, wrote on October 26, 1992, “Your feeling that geostatistics is invalid might be correct.” JMG's Editor did attach a copy of Journel’s 6-page letter. Distracting invectives? I’ll say!

How many Stanford students did he teach about Brownian motion along a straight line? What a silly notion when measuring mineral inventories. What’s more, Fisher’s F-test is forbidden where spatial dependence is deemed to exist a priori. That’s why ASTM ought to shred all standard methods cooked up by ASTM Committee D18 on Soil and Rock. These letters and many others are or will soon be posted on my website under Correspondence. Of course, ASTM Committee D18 ought to be dismissed. Soon I’ll post where we were before we met Dr Ricardo R Stone and his partner. He passed away shortly after we met. Ricardo was an IBM Fellow and a Member of ASTM E11 on Quality and Statistics.

The American Society for Testing and Materials fell for geostatistics in the 1990s. It came about when statistically challenged soil and rock experts had cooked up ASTM D5549-Standard Guide for Reporting Geostatistical Site Investigations. I had brought my case against what Professor Dr Georges Matheron himself had come to call a new science to the attention of Mr James A Thomas, President of the American Society for Testing and Materials. I had done so by snail mail on April 19, 1994.

ASTM President

ASTM has yet to sort out when and why geostatistocrats have made such a mess of applied statistics. Matheron and his disciples brought geostatistics all the way to North America in the 1970s. The problem is not that distance-weighted average point grades morphed into kriged estimates to honor D G Krige. The real problem is that variances of distance-weighted average point grades AKA kriged estimates didn’t morph along but got lost. Kriged estimates and kriging variances are the heart and soul of geostatistics. David’s 1977 Geostatistical Ore Reserve Estimation and Journel’s 1978 Mining Geostatistics reject the fact that every kriged estimate has its own variance. Clark’s 1979 Practical Geostatistics, unlike David in 1977 and Journel in 1978, derived not only the distance-weighted average point grade but also its true variance. But here’s the cinch!

Having a PhD in geostatistics seems a must when assuming spatial dependence between measured values in ordered sets. Yet, it’s so simple to apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. It puts on view whether or not orderliness in sampling units or sample spaces dissipates into randomness. Who wouldn’t want to know? Geostatisticians would have known if they ever got around to counting degrees of freedom. Those who scored a passing grade on Statistics 101 are bound to grasp the properties of variances. Some may even know that one-to-one correspondence between functions and variances is sine qua non in applied statistics.

ASTM itself got on track so to speak in 1898. At that time engineers and chemists of the Pennsylvania Railroad needed standard methods. It made me nostalgic to read such an account of courage and vision. At that time there were no degrees of freedom to count. Geostatistocrats have never stooped to count stuff what can neither be seen nor touched. Once upon a time I was in charge with sampling shipments of Pennsylvania anthracite in the Port of Rotterdam. ASTM Standard Methods for coal were specified in contracts between trading partners. It would be a long while before ISO Standards for coal caught up with ASTM Standard Methods. Now I wish ISO will never catch up. Greg Gould asked me a long while ago what I thought about interpolation between bias test data. Good grief! That looked too much like real kriging! That’s why I put together a false test for bias in a previous blog. I wouldn’t want ASTM Committee D05 to dictate how ISO TC27 is to test for bias. But it may well do so!

Those who had master minded ASTM Standard Methods for Soil and Rock must have been smitten silly with geostatistics. My son and I had found out in 1989 why geostatistics is an invalid variant of applied statistics. My work for Barrick Gold in 1997 proved that geostatistical software converted Bre-X’s bogus grades and Busang’s barren rock into a massive phantom gold resource. Student’s t-test proved that crushed core samples had been salted with placer gold. Bre-X’s duplicates for every tenth test sample proved the intrinsic variance of gold at Busang to be statistically identical to zero. That’s the very reason why I’ll always work with applied statistics.

Here’s what I did point out in April 1994. Either fundamental requirements of probability theory and applied statistics are no longer valid or geostatistical theory and practice are fatally flawed. ASTM’s President wrote on May 13, 1994 that he had asked Mr Bob Morgan, Director of Technical Committee Operations, about the role of Committee E11 Quality and Statistics. What I wanted to study but never got was a copy of D5549-94e1 Standard Guide for Reporting Geostatistical Site Investigations. Robert J Morgan, ASTM’s Director Technical Operations, asked me in February 1995 to direct my input to R Mohan Srivastava.

Geological statistician AKA geostatistician

Teaching Mo all I know about sampling and statistics tops my list of things to never do. It would take more than a few blogs to show what Mohan could have done had he grasped in June 1993 what was wrong with David’s 1977 textbook. That’s when Mo and his coauthor went to McGill University. As a matter of fact, that’s where the united geostatocracy went to praise David’s 1977 Geostatistical Ore Reserve Estimation. For crying out loud!

Dr Isobel Clark is the author of Practical Geostatistics. What she did in this 1979 textbook took me by surprise. It would have baffled many a thinker who has never bothered to read works of others. She deserved praise because she had derived the variance of the distance-weighted average. Agterberg in 1974, David in 1977, and Journel in 1978 never took the trouble to derive this variance. It’s a bad omen that the distance-weighted average morphed into a kriged estimate on Matheron’s watch. It was Matheron who had failed to derive the variance of this kriged estimate long before geostatistics was hailed far and wide as a new science.

Dr IC pointed out on the jacket of her textbook, “Geostatistics is the popular name for the application of statistical methods to problems in mining and geology”. She confessed in her Preface that Journel and others at Fontainebleau, France had taught her all she knows about the Theory of Regionalized Variables. Now that’s what got me worried. Matheron’s way of drawing strings of symbols on a blackboard may well have driven his most gifted disciple to routinely assume spatial dependence. All it took is to assume, krige, smooth, a leap of faith, and a lot of luck!

Author 1974 Geomathematics ex NRCan Emeritus Scientist

Dr Frederik P Agterberg, too, took a leap from applied statistics to geostatistics. He went from serial correlations in 1958 to functions without variances in 1974. In contrast, Dr Isobel Clark derived not only the distance-weighted average grade of a set of measured values but also the variance of the set and the variance of its central value. Much of it is detailed in Figure 4.1 and Table 4.1 of her textbook and a few pages about her data set. She fretted a bit on page 72 whether or not the Central Limit Theorem would hold. But it always does! Surely, David himself in 1977 would not have written about if it were false.

Dr IC’s set of hypothetical uranium concentrations seem to have been measured in hypothetical samples selected at positions with real coordinates in a two-dimensional sample space. I wish she had given the mass of each of her hypothetical samples. She reports that her 95% confidence range for a distance-weighted average hypothetical uranium grade of 400 ppm has a lower limit of 95% CRL=350 ppm and an upper limit of 95% CRU=450 ppm. It doesn’t look too bad, does it? Of course, t0.95;4=2.776 from the t-distribution rather than z0.95=1.96 from the normal distribution should have been used to derive the 95% confidence interval for xbar=400 ppm. But who would split hairs at this stage? Sir R A Fisher was already counting degrees of freedom for small data sets when Dr IC was a cute tot.

Assumes with poise!

The question is not so much whether this distance-weighted average hypothetical grade of 400 ppm is an unbiased estimate for the unknown true grade at 1,244 m Easting and 713 m Northing. Of course, I wish I knew her sampling protocol. The question is whether it is an unbiased estimate for the sample space defined by this set of five (5) hypothetical uranium concentrations. The most effective method to test for spatial dependence is to do what door-to-door sales people have always done. Travel from point to point such that each point is called on but once and the shortest distance is traveled. The next step is to derive the variance of the set and the first variance term of the ordered set. Finally, Fisher’s F-test is applied to the variance of the set and this first variance term. The observed F-value of 2.07 between var(x)=4,480 for the set and var1(x)=2,161 for the ordered set is below the tabulated F-value of 4.53 at 95% probability. Hence, her set of measured values does not display a significant degree of spatial dependence. By implication, the distance-weighted average grade of 400 ppm is not an unbiased estimate for this sample space. Dr IC did what Journel had taught her to do. All she did was assume spatial dependence between measured values in the ordered set. Here’s what happens when spatial dependence is assumed and functionally dependent kriged estimates are added to measured values.

Assume, krige, smooth, and be happy!

Dr Isobel Clark is due to teach a 3-day course at Global InfoMine in Vancouver, BC on May 3-5, 2011. She will take students from "no knowledge of statistics or geostatistics to understanding the mysteries of ordinary kriging and its variants in 30 hours (or less)". I wish I were there to shed light on that shameless practice of converting bogus grades and barren rock into a phantom gold resource. But then, why rush the good stuff!

ISO has set the stage to tackle trueness. It did so by issuing an NWIP ( New Work In Progress). I am pleased that ISO/TC69/SC6/WG1 has been entrusted with the task. Accuracy (trueness and precision) is way wide of the mark. Trueness (accuracy and precision) reads but a bit better. I would rather work with either Precision and accuracy or Precision and bias. Trueness makes more sense in a court of law. I have worked on a number of ISO Committees since 1974. I would find it a senseless task to keep track of trueness. Testing for bias always makes sense. Student’s t-test not only shows what’s biased and what’s not but also gives intuitive measures for statistical risks. I am pleased that ISO never took to kriging and smoothing. Unlike ASTM which went along with geostatistical thinking in 1994. ASTM was set up in 1898 by chemists and engineers. A precursor of CIMMP was set up in 1898. CIMMP's geostatistical peer review has been biased since the 1990s.

ISO/TC183 has set up a standard method to derive metal contents and grades of mineral concentrates and ores. It would have been just as simple to set up an ISO Technical Committee to derive metal contents and grades of mineral reserves and resources. But the COSMO brains behind the mining industry were too keen to assume, krige, smooth, and rig the rules of applied statistics. That’s why Bre-X’s bogus grades and Busang’s barren rock morphed so smoothly into a massive phantom gold resource. And that’s when Barrick Gold asked me to assist in testing for bias between paired gold assays determined by cyanide leaching and by fire assaying.

Student's t-test for bias

The observed t-value of 11.258 is statistically significant at 99.9% probability. It’s impossible to salt drill core with placer gold. In contrast, salting crushed core with placer gold is a cinch. Early in 1997 I derived confidence limits for the mass of gold for one of Barrick’s many deposits. My report was deemed worth its weight in gold. Here’s what Richard Rohmer wrote in Golden Phoenix, “For Munk and others affected by Bre-X and Busang, the strange news about de Guzman was the first hint that the find might well not be what Walsh and Felderhof were claiming”. He also pointed out, “Peter Munk was appalled when he read the Strathcona report. The damage inflicted on Canada’s national and international mining industry was beyond what anyone would have thought possible a few short weeks before”. Now that’s what crying out late is all about! I took a look in my crystal ball. It seems as if a Munk Debates for and against geostatistics has been put on ice!

Here's what Peter Munk himself pontificated on the jacket of his Golden Phoenix: "You have to be courageous; you have to learn to take advantages of change. Be non-conventional; don't fritter your energies - be focused; remember to share. Most important, use the biggest weapon of all weapons, the least appreciated yet the most important tool for success, and this is moral integrity; and don't be afraid to dream and don't be afraid to dream big."

Never mind Munk's moral integrity! I dream about scientific integrity in mineral exploration and mining. It’s a piece of cake to derive borehole statistics with spreadsheet software. I like to call such stats the fingerprint of a borehole. Those who have tried like it a lot! Following is a synopsis derived of the fingerprint of one mind-boggling borehole.

Fingerprint of monster borehole

It shows some 5.5 million ounces of gold in 78 million metric tons of ore. SME published in Transactions 2000 a paper onBorehole Statistics with Spreadsheet Software. One reviewer called it “an excellent paper” and the other thought “it would stir up a hornet’s nest”. But the hornets stayed put!

It makes no sense to assume spatial dependence in sample spaces. Fisher’s F-test ought to be applied to the variance of the set of measured values and the first variance term of the ordered set. Interpolation between measured values gives a false positive for spatial dependence. And that’s where geostatistocrats are taking not only mineral reserve and resource estimation but also the study of climate change. Or climate dynamics as I like to call it! I found Loehle’s 2000-year global temperature reconstruction by accident. Loehle and McCulloch pointed out in 2008 that data sets were smoothed with a 30-year running mean. It does create a pretty smooth picture. But here’s what happens with variances when measured values are enriched with variance-deprived kriged estimates.

Test for homogeneity of variances

I partitioned Loehle’s 2000-year set into twenty (20) sets of no more than 100-year each, derived the first variance term of each set, and applied the chi-square test to this set of twenty (20) variances. These variances constitute too homogeneous a set. The probability that this inference is true exceeds 99.9%. The probability that it is false is less than 0.1%.

I trust that ISO/TC69/SC6/WG1 will never assume spatial dependence between measured values in ordered sets. ISO/TC17 on Steel and ISO/TC34 on Food Products are interested in this New Work in Progress. I shall continue to write whatever needs to be put in writing to ensure that applied statistics prevails not only in ISO standards but also in mineral reserve and resource estimation.

A smart student of sorts taught at TU Delft in 1958 a seminar on the skew frequency distribution of ore assay values. It was none other than young Agterberg. And did he teach real statistics in those days! Scores of students at the University of Utrecht grew up with real statistics! Agterberg was no exception. I only found out when I read his 2000 eulogy for Professor Dr Georges Matheron. He brought up that Professor H J de Wijs thought the ratio of element concentration values to be constant regardless of the volume of the block. Here’s ad verbatim Agterberg’s criticism of what Professor H J de Wijs taught at TU Delft in 1958, “…it would be better to apply the conventional method of serial correlation to series of assay values.” Now that was in 1958 Agterberg’s point of view on spatial dependence between measured values in ordered sets. Why then has he swallowed Matheron’s spatial statistics with hook, line, and sinker?

Neither Agterberg nor de Wijs knew in 1958 that Dr Jan Visman’s 1947 PhD thesis on coal sampling was on file at TU Delft. I, too, was unaware of Visman’s work when I was a teaching assistant and a mature student in the early 1960s. I had been chief chemist with Dr Verwey but wanted to know more about sampling and statistics. The exchange of test samples and test results between trading partners was a thankless task to say the least. I knew all about the analytical variance but didn’t know how to estimate the variance of the primary sampling stage. I thought TU Delft would teach me what I wanted to know about sampling and statistics. One professor was H J de Wijs and the other a coal mining engineer. Neither knew of Visman’s work or of the properties of variances. In fact, H J de Wijs was Rector Magnificus when a student of his defended in 1965 a thesis in which transformation matrices played a key role. So, I left TU Delft, went to work for SGS in the Port of Rotterdam, and found out in 1967 about Visman’s 1947 sampling theory. So, Agterberg and I knew that mathematical statistics was shortchanged at TU Delft in those days. I don’t know why matrix and vector analysis were taught but sampling and statistics were ignored.

Dr Frederik P Agterberg had all but forgotten in this century what he had taught in 1958 at TU Delft. Here’s ad verbatim the very first paragraph of his eulogy, “Professor Georges Matheron (1930-2000) made fundamental contributions to science by establishing new theoretical frameworks in spatial statistics, random sets, mathematical morphology and the physics of random media”. Matheron was a French geologist who was called creator of geostatistics and founder of spatial statistics. I would never have praised Matheron’s surreal geostatistics let alone Journel's assumed spatial dependence. Whatever small minute contribution Matheron has made to science fell far short of real statistics. Surely, it did add up to surreal geostatistics. How ironic that he never got around to testing for spatial dependence between measured values in ordered sets. Why then did Dr Frederick P Agterberg see fit to praise Matheron’s work? He is Emeritus Scientist with the Geological Survey of Canada. Most of his 1974 textbook on Geomathematics has passed the test of time. And most of it will last much longer than Matheron’s magnum opus. What a shame that real statistics behind his 1970 and 1974 figures crumbles under scrutiny.

Figure 1 - Geologic prediction problem in 1970 Figure 64 - Typical kriging problem in 1974

Agterberg solved his geologic prediction problem by linear prediction in time series and assuming a two-dimensional autocorrelation function between his set of five (5) points. He has to explain how it came to pass that a geologic prediction problem in 1970 turned into a typical kriging problem by 1974. That was the very year that Elsevier published Agterberg’s Geomathematics. Cramped between its covers are some 600 pages of mostly real statistics, a lot of sound mathematics and a dash of Matheron’s surreal geostatistics. But why did Agterberg add Stationary random variables and kriging to an otherwise readable textbook?

What I see in each figure is a set of five (5) measured values in a two-dimensional sample space. If each of Agterberg’s points were equidistant to Po, then the central value of his set would be its arithmetic mean. David's famous Central Limit Theorem defines the functional relationship between a set of measured values with identical weights and its central value. Agterberg refers to the Central Limit Theorem on pages 166, 206, 207 and 231. The number of degrees of freedom is a positive integer for a set of measured values with identical weights but a positive irrational for a set of measured values with variable weights. Agterberg refers to degrees of freedom on pages 174, 190 and 254. The Central Limit Theorem and the concept of degrees of freedom do not play a role in Chapter 10 - Stationary random variables and kriging. Dr Frederik P Agterberg, the author of Geomathematics and Emeritus Scientist with Natural Resources Canada, ought to explain why his Central Limit Theorem is not blessed with a variance and why his set of measured values is not blessed with degrees of freedom.

When ISO was set up in April 1947 at Paris, France, it was all about nuts and bolts. As a matter of fact, ISO/TC1 Screw heads came first and ISO/TC2 Fasteners was second. Ever since has ISO been setting up a broad range of standards while the world is putting its standards to the test. But I wonder why ISO did err on trueness. Here’s what ISO announced in its Technical Corrigendum 1 on 2005-08-15.

Accuracy (trueness and precision) of measurement methods and results Part 5: Alternative methods for the determination of the precision of a standard measurement method

ISO/TC69, Applications of Statistical Methods, Subcommittee SC 6, Measurement methods and results published the above Technical Corrigendum 1. So what error was SC6 to correct? Of course, trueness and precision should never have been between brackets! What ought to be between brackets are precision and accuracy! A true test for bias would need first of all an unbiased variance estimate. Those who have kriged and smoothed cannot possibly test for bias or estimate precision. Neither can they test for spatial dependence by applying Fisher’s F-test to the variance of bogus data and the first variance term of the bogus data set. So much for kriging and smoothing when we study climate change on our planet!

What I would want between brackets is precision and bias. Derive the variance and then test for bias if enough degrees of freedom are available. Bias detection limits (BDLs) and Probable bias ranges (PBRs) for Type I risks and Type I&II risks are intuitive and powerful measures for the observed bias. Ignorance of precision and bias has irked me as long as have central values without variances. Surely, Matheron and his disciples have brought a big catch of bad science to our world.

I have juxtaposed precision and bias since 1974. That’s when I became a member of CAC/ISO/TC102-Iron ore. I am also a Member of ISO/TC27-Solid mineral fuels, of ISO/TC69-Applications of statistical methods, and of ISO/TC183-Copper, lead, zinc and nickel ores and concentrates. Much of what I have written on sampling and weighing of bulk solids became part of ISO/TC183. My son and I have written a software module onPrecision and Bias for Mass Measurement Techniques. ISO has published it as an ISO Standard. I was told Canadian Copyright was not violated. Merks and Merks found it easy to work with precision and bias. What’s more, we are pleased to be encumbered with Fischerian (sic!) statistics.

The International Organization for Standardization was much on my mind when I posted false and true tests for bias. ISO comes from the Greek word isos which means “equal”. Scores of countries have set up national institutions to interface with ISO. The Standards Council of Canada Act received Royal Ascent in 1970. That’s when the CAC prefix was placed before ISO. I have nothing but praise for Standards Council of Canada. CAC/ISO/TC69 Applications of statistical methods has played a key role in my work. I have always juxtaposed Precision and Bias. But it’s a long a story. And it’s bound to get longer while I’m trying to keep it short. I do want to kill two nutty practices with the same stone. The first is to assume spatial dependence between measured values in an ordered set. The second is to not apply Fisher’s F-test to the variance of a set of measured values and the first variance term of the ordered set. Geostatistocrats assume, krige, smooth and select the least biased subset of any infinite set of kriged estimates. It may well have dazzled those who have never scored a passing grade on Statistics 101. I still find it funny how so few could write so much about so little.

Biased but high degree of precision

Here’s where ISO has created the error to be corrected. Aretrueand false antonyms or not? Wouldn’t the antonym of trueness be falseness, or perhaps falsehood? Of course, I would call this ISO document Trueness (precision and bias) of measurement methods and results. Surely, a significant degree of spatial dependence between measured values in an ordered set does impact precision. But I upped the odds of finding a false positive. I did so by inserting David’s “famous Central Limit Theorem” between each pair of measured values. Pop in more kriged estimates between measured values and bogus spatial dependence may make the odd mind spin. Is it a minor miracle or Matheronian madness?

Spatial dependence between measured values in an ordered set ought to be verified by applying Fisher’s F-test to the variance of the set and the first variance term of the ordered set. When applied to sets of test results for single boreholes I came to call it fingerprinting boreholes. SME’s reviewers liked it a lot. And so will members of ISO/TC69/SC6 once the upshot of spatial dependence on confidence limits for central values is clear. Assuming spatial dependence between measured values and interpolation by inserting functionally dependent values between measured values has made a mess of the study of climate change. Surely, CAC/ISO/TC69/SC6 has a role to play in selecting the most fitting statistical methods.

Every scientist and engineer ought to grasp the properties of variances. Those who don’t should not even try to apply a true t-test for bias. And nobody could do it without counting degrees of freedom. It is an irrefutable fact that a true t-test for bias cannot be applied without counting degrees of freedom. That’s why geologists ought to question the validity of geostatistics. The more so since the author of the very first textbook predicted that "…statisticians will find many unqualified statements…” What David didn’t predict was that he would blow a fuse if somebody did. By the way, keep your copy in a safe place. It may well become a collector’s treasure before this millennium is history. Counting degrees of freedom comes to mind as a topic that does not get the respect it so richly deserves. But I’m getting off my train of thought! Here’s a simple but true test for bias applied to an ancient set of paired data.

Observed t-value significant at 99.9% probability

Scientists and engineers ought to apply Student’s t-test in the same way as W S Gosset himself did. All textbooks on applied statistics teach the t-test. It was Volk’s Applied Statistics for Engineers that taught me all about the t-test. Those who want to apply Student’s t-test for bias the proper way should study Chapter Six The t Test. Study not only Section 6.1.4 The Null Hypothesis but even more so Section 6.1.3 Degrees of Freedom. Here’s what Volk wrote, “…we accept a risk of a 5 per cent chance of being in error”. Next, he pointed out, “This error, of falsely rejecting the null hypothesis, is called an error of Type 1”. What I have done in my work is avoid the term “error” without some appropriate adjective. Risk analysis and loss control have played a key role in my work. That’s why I report Type I risk and combined Type I&II risks as intuitive measures for the power of the t-test.

Bias Detection Limits and Probable Bias Ranges

A simple analogy exists between those types of risks and the role of a fire alarm. The Type I statistical risk refers to the event that a fire occurs but the alarm does not sound. The Type II statistical risk refers to the event that the alarm sounds but no fire occurs. The combined Type I and Type II statistical risks refer to the event that a fire occurs and the alarm sounds. Simple comme bonjour! The next step was to define Probable Bias Ranges. It is true that PBRs may sound counterintuitive to those who are not used to working with applied statistics. But surely, PBRs fit the observed bias bon d’un t!

Dr Pierre Gy’s view on accuracy is spelled out on page 17 of his 1979 Sampling of Particulate Materials. Here’s his take, “Accurate: when the absolute value of the bias is not larger than a certain standard of accuracy”. He could have but didn’t mention Standard Reference Materials. SRMs play an important role in calibrating analytical methods and systems. Analytical chemists need SRM’s with confidence intervals for one or more constituents. Gy’s so-called sampling constant is, in fact, a function of a set of four (4) variables. As such, it does have its own variance. What’s more, Gy’s own Student t-Fisher test has raised more eyebrows than interest.

Posted on my website are scores of statistical tests and techniques. I used to attached it as an Appendix to my reports long before Professor Dr Michel David took such a dim view of applied statistics. My son and I in 1992 had to put together Precision and Bias for Mass Measurement Techniques. It was Part 1 of a series on Metrology in Mining and Metallurgy. I hold Canadian copyrights for Metrology in Mining and Metallurgy and for Behind Bre-X, the Whistleblower’s Story. I have scores of blogs to write before I make up my mind what to complete first.

Testing for bias plays a key role in science and engineering. Student’s t-test is the par excellence test for bias. The t-test for paired data has always played a role in my work. A bias between test results determined at loading and discharge is a cause of conflict between trading partners. The question is always whose test results are biased. A matter of concern in 1967 was dry ash contents of anthracite shipments from the mines in Pennsylvania to the port of Rotterdam. I went to the USA and determined that loss of dust during sample preparation was the most probable cause of bias between dry ash contents. I had done time at TUDelft. So, I knew that carbonaceous shale is softer than anthracite, and that hammer mills tend to crush and grind autogenously. That’s why I thought that loss of fine dust during preparation of primary samples at loading would cause test samples to show lower dry ash contents at discharge in the Port of Rotterdam.

Holmes hammer mill

SGS’s coal testing laboratory in Rotterdam, too, had a Holmes hammer mill. It was similar to the one at loading but ours worked with its spring-loaded container closed. Settlements between buyer and seller were based on test results determined at discharge. So, we couldn’t afford to mess up primary samples by running our hammer mill ajar. What we did do was prepare test samples for analysis in the usual manner. We would then pass the reject of each primary sample through the hammer mill with its container left slightly ajar. We collected dust on sheets of paper placed at 0.5 m and 1.5 m from the hammer mill. Dust that had settled at 0.5 m weighed 26.2 grams and contained 14.6% dry ash. Dust that had settled at 1.5 m weighed 12.1 g and contained 16.5% dry ash. The settlement sample showed 10.40% ash on dry basis whereas our messed-up sample showed 10.26% ash on dry basis. With but one degree of freedom our experiment was not much of a true test for bias. But it proved that the integrity of our settlement samples passed scrutiny. We didn’t determine dry ash in dust collected on our coveralls and face masks. I had a fine team to work with. But I wanted more than a team! I wanted SGS to build a new laboratory as far away as possible from where we were. But SGS was not ready yet. What SGS did do was ask me to set up a laboratory in Vancouver. Now guess what?

When I met Greg Gould for the first time at Rotterdam in 1967 he did already chair ASTM Committee D05 on coal and coke. Greg praised Volk’s Applied Statistics for Engineers so I bought my first copy. He told me about Dr Jan Visman, his work at the Dutch State Mines during the war, his 1947 PhD thesis, and his input in ASTM. I was pleased to meet him in person after we had moved to Canada in October 1969. Dr Jan Visman was an independent thinker. He was as a true a scientist as Greg Gould was a professional engineer. And he was a true PEng! I shall always treasure my copy of Visman’s PhD thesis and our correspondence. I remember our talks with fondness. We talked about the composition and segregation components of his sampling variance. I pointed out the term “segregation” suggests a sampling unit may have been more homogeneous in the past. My friend agreed. That's the reason why the distribution variance and the composition variance add up to the sampling variance. It happened when two Dutchmen talked about sampling in a foreign language. But unlike French sampling experts we did grasp the properties of variances.

The odd reader of my blogs may think I’m a pack rat. I do plead guilty! I want to get back to testing for bias with Student’s t-test. But I need to tell one more tale before talking about false bias testing. Once upon a time Matheron’s new science of geostatistics somehow slipped into bias testing. It came about after ASTM awarded me in 1996 a plaque for 25 years of services. Greg Gould had asked ASTM’s Board of Directors to recognize Dr Jan Visman and his work. ASTM did so but misspelled Jan's first name as Jane! It was the same year that Barrick Gold signed me on to figure out what kind of gold resource Bre-X Minerals had cooked up in the Kalimantan jungle. And it was the time when Greg Gould sent me bias test data that Charles Rose had enhanced by kriging. He had taken to liking to geostatistics. So much so that of one of his papers was approved for David’s 1993 bash at McGill University. Rose talked about A Fractal Correlation Function for Sampling Problems. But one of his many problems was that Mohan Srivastava had lent him a helping hand. I met Rose for the first time in Columbia many years ago. He joined SGS some time after I had left in 1979. ASTM awarded Rose in 2004 the R A Glen Award. He represents the USA on ISO/TC27 on coal. He talked about his take on bias testing during the meeting at Vancouver in 2009. What a waste of my time! SGS announced on April 24, 2008 the strategic acquisition of Geostat Systems International, Montreal, Canada. For crying out loud!

False test for bias

That’s why I decided to show how to apply a false bias test. Firstly, I got the set of paired dry ash contents determined in eleven (11) shipments of Pennsylvanian anthracite at loading and at discharge. Next, I played the kriging game by inserting a kriged estimate between each pair of measured values. Take a look at what I cooked up! The variance of differences between paired data dropped from var(Δx)=0.1078 for a set of eleven (11) measured values to var(Δx)=0.0396 for the embellished set. That's what happens when a set of measured values is enriched with a set of kriged estimates. So much for kriging when testing for bias. Stay tuned for a true test for bias.