>> take a stab at applying the analysis in the Lee paper to a sample of 
>> 300 bees, and a finding of 0 mites, 1 mite, and 3 mites.

> I've covered this... [ABJ article discussion of binomial distribution, plus 
> extensive attention to comb selection to get "a good sample"]

(These comments are over and above my other post on this same issue in  the "Risk Calculations" thread, but the observations made in that post also apply here.)

The article cited attempted to reduce one of the errors inherent in "sampling" by making some assumptions about which ages of bee are found on which combs at any one moment.   But "good guesses" will not eliminate the uncertainty.  

So, when the error rate is higher than the mite count that is the treatment threshold, for example:

>>>> In May, my threshold for treating is 1 mite in an alcohol wash of 300 bees.

...the honest man is forced to admit that the entire method is a "doomed hive detector" in fall, and not useful in other seasons.

So, two words - "Sticky Boards"...  the less hives you have, the better they should look to you.  Simple, repeatable inherently accurate results with no fiddling about required.  Dump all the scrapings into denatured alcohol, count the mites that float.  Mites counts from this method have known accuracy, as they have been verified by comparing mite drops to full-colony necroscopy.  

To speak simply, and not  introduce any more math to the discussion, it does no good to take a flawed model like a Binomial distribution, and try to overcome its inherent limitations by adding a complex and nearly irreproducible "sampling technique" that would be hard to document or verify as properly implemented, moreso when  a novice is being asked to classify combs by the activity on  those combs.

To advise a novice to attempt to employ such techniques does a disservice to the novice - it gives one a false sense of security similar to the false sense given by a Burgess Fogger loaded with FMGO - yes, it looks impressive, but it is not going to make things any better, and it distracts time and attention away from doing something more productive.

Why is "Binomial" flawed?  To start, the name does what it says  - its binomial.  It works fine for coin flips, but varroa and  bees don't act like coin flips at all.  So what’s in a "sample of bees"?

a)  Bees with no mites
b)  Bees with mites that you detect
c)  Bees with mites that you don't detect 

Right off the bat, the astute observer notices 3 conditions, which binomial distributions (and Poisson distributions) don't address.  By ignoring the real-world conditions, the model has a significant inherent error rate, and while this provokes only a shrug from someone with 100+ hives who loses 2 colonies to "error", the error rate has the most impact on those who can least afford to lose 2 colonies - those with ONLY 2 colonies.

The bulk of the article attempted to sample "the right bees" in the sample of 300 bees, and this brings up another inherent source of error - 300 bees isn't a good enough  sample unless one is lucky enough to cherry-pick the "right" bees.  So, we can add to "what's in a sample of bees":

d) Bees that are very very unlikely to ever have a mite on them in the first place

So, we end  up not ever really knowing the error rate for any one or two colonies.   But novices are still taught to do a "300 bee sample" by people who do not themselves grasp the inherently sloppy nature of the sample or the terrible fit between the model and reality.

This misuse of a binomial distribution to try to model a situation that is anything but a "coin flip" is akin to my detailed statistics, showing my wife that the odds of winning the lottery are so small, that actually buying a ticket does not really make one's chance of winning any better. But her rebuttal is succinct - "You have to be in it to win it"  (so, she buys a weekly ticket for La Mega at the Bodega).   Even the most detailed model is still just a model, and few of them have the level of reliability required by the 2-hive beekeeper.

With enough  300-bee samples, one can transcend all the errors and artifacts, and get an accurate and reliable estimate, but by the time you've done that, the hobbyist might as well go back to using sticky boards, and counting mites in denatured alcohol, as this approach at least has a rigorous basis, and a verified ability to detect the low levels of varroa at which one wants to treat.   For a hobbyist with a statistically  insignificant number of hives, it is a cruel disservice to convince them to use a sampling method  that only "works" when averaged over a statistically significant number of samples.

This seems a case  where the large operation calculates the odds  of everything, but knows the value of nothing, while the backyard beekeeper needs values he can rely upon, every time.

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