Thinking clearly about likelihood ratios and siblingship

Charles Brenner, Ph.D.
Forensic mathematics
August 2011
Expected LR distributions, various relationships versus unrelated
LR. At 50% prior, LR=1, 100, 10000 is 50%, 99%, 99.99%

The graphs are simulations based on U.S. Caucasian population and the 15 Identifiler markers. For each relationship, 500 simulated pairs (in particular this is a two-person parentage analysis) with that relationship were generated by the DNA•VIEW simulation module and analyzed. The graphs show the distribution of likelihood ratios (LRs) obtained. Each LR compares the hypothesis that the pair is related as constructed compared to unrelated.

Assuming a prior probability of 50% and assuming the tested relationship is really true,

relationshipmedian LRmean posterior probability (at 50% prior) Remark: median LR and mean posterior are different, not equivalent, ways to summarize the distribution curves.

Why are the results for sibling and half-sibling tests poorer than those for parentage?

The relatively poor performance in establishing half-sibship is simply because it is a relatively remote relationship and so not really all that different from unrelated. Thus, it is only the minority of half- siblings pairs who share enough DNA traits or rare DNA traits that their test result (LR) will put them in under the blue curve and to the right of LR=100.

Parentage and siblingship are in a sense equally close relationships, but in different ways such that parentage turns out to be a bit easier to prove. However, the biggest difference between the sib and parentage graphs isn't the position of their centers, but that the sib graph is broader. That means the possibility of weak evidence — the area under the red curve left of LR=100 for example — is significantly more for a sibling test.

What likelihood ratio indicates a half-sibling relationship?

Same as for any other relationship. People often assume that this is a relative question, but it is not. They imagine that the blue curve (half-sibship) lying left of the other curves somehow implies that the evidential standard for half-sibship could be lower. But that doesn't make sense at all. The strength of evidence is the number on horizontal scale. If some rule such as LR>10 or LR>100 is considered appropriate for accepting a claim of parentage there is no logical reason to use a different rule for some other relationship. Changing the standard for half-siblingship simply because it is harder to prove is comparable to lowering the vision standard for old people to drive a car. What makes a person ok to drive is their absolute vision, not whether they see ok for their age.

Then what number is significant for any relationship?

There is no simple answer. LR=1 is neutral so any LR>1 is evidence of the relationship vs. unrelated. If there is other evidence (i.e. documents, testimony) that is nearly satisfactory in establishing the relationship, then even a small favorable LR such as LR=5 may be enough to make the case convincing. Alternatively, if the LR is large, LR=100 for example, then it may be reasonable to accept the relationship almost regardless of any other evidence. (That is a common practice in civil paternity cases where mere money, child support, is the issue.)

And of course what constitutes a satisfactory amount evidence — what is the standard of proof — is a completely arbitrary policy question. If the desire is to re-unite families whenever possible even at the cost of accepting some imposters, the standard will be low. Some countries take the attitude of admitting immigration applicants unless the DNA tends to disprove their claim. On the other hand if the policy desire is to avoid admitting people who don't qualify and to maintain a low immigration rate, then the standard will be high.

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