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  #21  
Old 14th July 2017, 06:09 PM
Frederator Frederator is offline
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. . . at least a ~67% chance . . .
I say "at least" because I only used the basic probability curve for a single observation. I intuit that the probability curve for 3 successive observations must be considerably steeper as each successive observation is likely to bring their collective average closer to the population mean. But I'm still debating the methodologically proper way to derive those figures.

I'm working with the binomial distribution function in Excel right now, and my first naive attempts return figures like a ~96% probability of an NPE.
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  #22  
Old 14th July 2017, 08:52 PM
Frederator Frederator is offline
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I say "at least" because I only used the basic probability curve for a single observation. I intuit that the probability curve for 3 successive observations must be considerably steeper as each successive observation is likely to bring their collective average closer to the population mean. But I'm still debating the methodologically proper way to derive those figures.

I'm working with the binomial distribution function in Excel right now, and my first naive attempts return figures like a ~96% probability of an NPE.
Oops! Had a logic error. There are 4, not 3 grandparent relationships in this chain of descent. Anyhow, correcting that only returns a slightly higher probability of NPE.

I think this may be the correct methodology.

1. Identify the minimum point in the curve where 4 successive observations will still return a final matching probability of 100%.

2. Calculate the non-cumulative binomial distribution in Excel for 4 successes in 4 trials and where the individual probability being the entire area in the curve under my minimum point.

This represents the chance that my calculated probability of two 4th cousins who connect to MRCAs through their respective direct paternal lines is less than 100%.

Subtracting that figure from 100% would yield a figure representing the one-sided confidence level that the probability of an NPE in this specific scenario is no less than 100%.

Which coincidentally is just about ~96%.
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  #23  
Old 14th July 2017, 08:58 PM
Frederator Frederator is offline
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. . . Calculate the non-cumulative binomial distribution in Excel for 4 successes in 4 trials and where the individual probability being the entire area in the curve under my minimum point. . . .
I guess this is no big deal. It's just X^4 where X=the area of the curve under my minimum point.
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  #24  
Old 14th July 2017, 10:13 PM
Frederator Frederator is offline
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I guess attempting to verbalize my logic might be helpful. If anyone can point out an obvious flaw it would be appreciated.

The final stages of my matching probability calculation are based on the expected volume of ancestral DNA held by each donor. Therefore, I'm only concerned with the variability in grandparental contribution on a net basis. That there are alternative scenarios of fluctuation in contribution percentage at individual points in the chain of descent doesn't matter provided they all result in the same volume of ancestral DNA held by the donor.

So the simplest way of calculating a top-sided, as opposed to iterative adjustment, to my basic calculation is to multiply the preliminary expected volume of ancestral DNA by X^Y, where:

X=the typical grandparental contribution % in this chain of descent/the population average contribution %

Y=the number of grandparental relationships in the chain of descent

In the confidence calculation I use a binomial distribution where the # of successes equals the # of trials, and the number of trials equals the # of grandparental relationships in the line of descent and the iterative probability equals that for the same value (i.e., typical grandparental contribution %) on a single-trial distribution.

I think this is justified because the single-trial distribution is simply a specific instance of the overall binomial distribution. The only differences are the number of trials.

I think it would be a mistake to use the single-trial distribution and assume one trial to achieve the net grandparental contribution % because we know that the average of sample values moves closer to the population average when we increase the sample size. There is simply too much variation among individual observations to reflect any actual multi-generation descent scenario accurately.

I wish there were less technical language to describe this logic. Maybe there is, but I have yet to discover it. In any case, I think this logic is pretty solid.

Last edited by Frederator; 14th July 2017 at 10:28 PM.
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