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Home > Author > Alan Foos
Alan Foos

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  Alan Foos

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BA Botany, MS Soil Fertility, mcl, chemistry minor, Montana State University, 1979

Background Information

In 1979 I ran low on funds as the GI Bill ran out and was concerned about properly supporting my young family, so skipped the PhD that would have been earned in the coming year and took a government job. That was a big mistake, and 30 years later, after retiring to Thailand, it seemed worthwhile to scan my notes and MS Thesis for academic nuggets. My MS thesis was a large randomized block plot at the agricultural experiment station in Bozeman, Montana, Montana State University. There were 25 treatments with four replications for each of sainfoin and alfalfa. Treatments were several NxPxK factorials and paired comparisons for micronutrients. 

It so happened that during the last year of my thesis I had discovered a proof that could augment the design and interpretation of ANOVAs in all scientific areas. I was driven at the time over questions of how extraneous variables may have biased the data in my own experiment.  The proof was too complex to work with in pencil and there were no GUIs back then, only the mainframe and Hewlett Packard RPN calculators. In 1992 I used MathCad to advance the proof, but had little time and no encouragement. In 2007, almost 30 years later, in Thailand, I found that Ventura Publisher's equation editor made it easy to bring the proof to a useful conclusion, a new parameter that should have always been a staple in quality DOE and analysis.

I'd volunteered for Vietnam infantry duty so I could afford a quality education, risking death, with no family support of any kind. On discharge from Dugway Proving Ground in 1969, worse obstacles arose as my checks were stolen and I ended up essentially homeless for the next two years. Getting through undergraduate and graduate school involved hardships too severe to describe, and my career was cut short as I was unable to tolerate the alcohol culture in government work or find resources to finish a PhD. It was with pleasure, and no doubt God's mercy, that I was able to write this theorem after retirement.

Hundreds of universities have viewed the Internet version of the theorem, but have offered little feedback, so it has been taken offline and offered now only in eBook format. I was a top student and a very serious student, but having no formal university ties, the theorem has been difficult to promote and get accepted. It is as good as I say, however, and the eBook gives a detailed derivation and proof with sample uses.  No, I am not a crackpot, and the eBook and theorem are very much worth having for anyone involved in research or mathematics instruction.

Birth Place
Missoula, MT Missoula

MS Soils, Montana State University, 1979, mcl

Additional Information

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Randomized Block Theorem by Alan Foos
Derives and proves the Foos Coefficient of Covariance which is the ratio of covariance to variance among all combinations of elements in a set of data from one replication to another, usually in an ANOVA. For a set of treatments in a randomized block design, the FCC for blocks is the percentage of covariance among treatment pairs across blocks, while the FCC for treatments is the percent covariance, i.e. consistency of change, among block pairs across treatments. The FCC for blocks if 1 means that 100% of the block effects are uniformly changed from block to block, while the FCC for treatments gives the value of uniformity (0 to 1) for treatment effects within blocks. The Foos Coefficient of Covariance is a useful parameter for proper orientation of blocks or for assessing the level of bias for any replicated set of data. Two sections address reputed errors in Einstein's relativity and a more meaningful alternative to calculating clock speed as a function of gravitational field strength. Best and cheapest PDF version from here, direct from author, pay on this site, contact by email at

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