False test for bias

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.

How to fingerprint boreholes

I know how to fingerprint boreholes. What’s more, Merks and Merks not only know how to fingerprint boreholes but how to derive unbiased confidence limits for metal contents and grades. We have known all of that since Professor Dr Michel David in February 1990 rejected Precision Estimates for Ore Reserves. He was but one of many similarly gifted geostatistical reviewers with CIM Bulletin. The same paper was praised by and published in Erzmetall, October 1991. What’s more, Borehole Statistics with Spreadsheet Software was approved by and published in SME Transactions 2000. I applied this technique to a large set of test results for a gold deposit in Kazakhstan. When I asked my contact at Barrick Gold what he thought of my report, his response was: “It’s worth its weight in gold”. I didn’t charge quite that much on February 9, 1998.

Here’s what Merks and Merks have decided to propose. We give a short course on how to derive unbiased confidence limits for grades and contents of mineral inventories. Professor Dr Roussos Dimitrakopoulos and all of his staff and students should participate. Of course, Dr Frederik P Agterberg, Emeritus Scientist with Natural Resources Canada, should be invited. Those who are Members of the International Industry Advisory Board ought to attend. McGill staff and students should be given free access. Our fee for this crash course in statistical thinking is C$100,000.
Here's why! It took a long while to unscramble the French sampling school. Here are but a few of the most salient facts. Dr Pierre Gy would put a set of primary increments in a single basket in a manner of speaking. That’s why he didn’t get a single degree of freedom. It may well be why Gy engineered his sampling constant. What he didn’t know was that his sampling constant does have its own variance. Professor Dr Georges Matheron never put a set of core samples from the same borehole in but one basket. So, he was blessed with almost as many degrees of freedom as there were core sections in a borehole. Not quite as many because a set of n boreholes gives df=n-1 degrees of freedom. But Matheron got stuck in a rut. He may have taught his disciples to assume spatial dependence between measured values in ordered sets, and to strip the variance off the distance-weighted average AKA kriged estimate. It would explain why Stanford's Professor Dr Andre G Journel is so unfamiliar with Fisher’s F-test. He never knew that ordered sets of n measured values give dfo=2(n-1) degrees of freedom. Neither did he know that the number of degrees of freedom for a set of measured values with variable weights is a positive irrational. It took a lot of independent statistical thinking to unscramble what French sampling school had cooked up between 1954 and 1974.

Doing more with less at McGill

The First Coming of Professor Dr Roussos Dimitrakopoulos to McGill University came to pass in June 1993. He had come all the way from Down Under to chair Geostatistics for the Next Century. It was a bash where geostatistical thinkers of the world had flocked together. They had come to praise Professor Dr Michel David and his 1977 Geostatistical Ore Reserve Estimation. David wrote in his Introduction on page VII that “…statisticians will find many unqualified statements here”. Not a single thinker asked why David had done so. My son and I found some odd gaffes when we studied David’s stuff. That’s the very reason why we wrote Precision Estimates for Ore Reserves. My abstract for The Properties of Variances raised more eyebrows than interest. I have posted on my website both the paper and its review by CIM Bulletin.

When all praise was said and done and duly recorded Dr RD went back to the University of Queensland, Brisbane, Australia. That’s where he honed his skill to do more with less. So what did McGill University think it got when Dr RD came to stay? Here’s how McGill’s University Relations Office on July 13, 2005 saw his Second Coming: “…It’s no surprise, then, that a world leader in the field of mining engineering has chosen Canada as a place to pursue his scholarship.” What world leader would be driven to merge stochastic mine planning and holistic mining? And Dr RD did all that while the properties of variances stayed as far beyond his grasp as they were in June 1993. Those were the very properties I had wanted to talk about at David’s shindig.

Professor Dr Roussos Dimitrakopoulos
Spanning disciplines, spanning the globe

Doing more with less sounds so forward looking and snobbish. But Stanford had already bested McGill at doing more with less. It did so after Matheron and a few of his disciples took the new science of geostatistics all the way to North America in 1974. Some novel science it turned out to be! The distance-weighted average lost its variance and morphed into a kriged estimate. Spatial dependence between measured values in ordered sets was to be assumed. Degrees of freedom went the way of the dodo. That’s how Professor Dr Georges Matheron and his mates have managed to make a mockery of applied statistics!

Professor Dr Roussos Dimitrakopoulos had come to chair Geostatistics for the Next Century. But he was also keen to speak about Spatiotemporal Modelling: Covariances and Ordinary Kriging Systems. It was a thoughtful touch to talk about David’s work at his bash. The more so since David had written much about covariances and kriged estimates in his 1977 Geostatistical Ore Reserve Estimation. We knew in the late 1990s that David was wrong on page 286 of Chapter 10 The Practice of Kriging. Here’s where the author of the very first textbook on geostatistics had strayed away from applied statistics. Figure 10 in Maréchal and Serra’s 1974 Random Kriging made a comeback in 1977 as David’s Figure 203.

Fig. 203. Pattern showing all the points within B,
which are estimated from the same nine holes.

Here’s what the author wrote in Section 10.2.3.3 Combination of Point and Random Kriging on page 286, “Writing all the necessary covariances for that system of equations might be a good test to find out whether one really understands geostatistics!” David’s exclamation mark may well have been inserted to imply the veracity of the whole quote. The problem is not so much with this statement but with the caption under Figure 203. All the points within B are not estimated but derived from the same nine holes. As such, each and every point is a function of the same set of nine (9) holes. And each and every one of them does have its own variance in applied statistics. No ifs or buts! Counting degrees of freedom would have been a good test to find out whether one really understands applied statistics. If all holes in a set of nine (9) were equidistant, then the number of degrees of freedom would be df=n-1 for the set, and dfo=2(n-1) for the first term of the ordered set. If distances between holes are variable, then the numbers of degrees of freedom are no longer positive integers but positive irrationals. Degrees of freedom do not disappear because Professor Dr Roussos Dimitrakopoulos says so.

I have told my story on the power of applied statistics and the flaws of geostatistics to the Members of CIM’s Vancouver Branch on Friday, January 29, 1993. CIM Bulletin and its peer review process have played a key role in the proliferation of geostatistics. It’s about time to tell my story about sound sampling practices and proven statistical tests.