Non-identifiable Pedigrees and a Bayesian Solution

Abstract

Some methods aim to correct or test for relationships or to reconstruct the pedigree, or family tree. We show that these methods cannot resolve ties for correct relationships due to identifiability of the pedigree likelihood which is the probability of inheriting the data under the pedigree model. This means that no likelihood-based method can produce a correct pedigree inference with high probability. This lack of reliability is critical both for health and forensics applications.Pedigree inference methods use a structured machine learning approach where the objective is to find the pedigree graph that maximizes the likelihood. Known pedigrees are useful for both association and linkage analysis which aim to find the regions of the genome that are associated with the presence and absence of a particular disease. This means that errors in pedigree prediction have dramatic effects on downstream analysis.In this paper we present the first discussion of multiple typed individuals in non-isomorphic pedigrees, P and Q, where the likelihoods are non-identifiable, P r[G | P, θ] = P r[G | Q, θ], for all input data G and all recombination rate parameters θ. While there were previously known non-identifiable pairs, we give an example having data for multiple individuals.Additionally, deeper understanding of the general discrete structures driving these non-identifiability examples has been provided, as well as results to guide algorithms that wish to examine only identifiable pedigrees. This paper introduces a general criteria for establishing whether a pair of pedigrees is non-identifiable and two easy-to-compute criteria guaranteeing identifiability. Finally, we suggest a method for dealing with non-identifiable likelihoods: use Bayes rule to obtain the posterior from the likelihood and prior. We propose a prior guaranteeing that the posterior distinguishes all pairs of pedigrees.Shortened version published as: B. Kirkpatrick. Non-identifiable pedigrees and a Bayesian solution. Int. Symp. on Bioinformatics Res. and Appl. (ISBRA), 7292:139-152 2012.

Development and testing of a genetic marker‐based pedigree reconstruction system ‘PR‐genie’ incorporating size‐class data

“…(), true pedigrees were not necessarily uniquely maximal in likelihood on a finite suite of loci, that is, the true pedigree may not be the most likely given the data available, or there may be several pedigrees with similar likelihoods. When it is not possible to choose between pedigrees equal in likelihood, these pedigrees are described as nonidentifiable (Kirkpatrick ), and care must be taken when making inferences from these pedigrees. This is the primary factor for low‐heterozygosity simulations having poor accuracy as in these cases sibs or grandparent–grandchild pairs share sufficiently many alleles to be assigned as parent–offspring pairs.…”

Correcting for Cryptic Relatedness in Genome-Wide Association Studies

“…The estimated identity coefficients are then transformed into estimates of the kinship using equations developed by Kirkpatrick (2012Kirkpatrick ( , 2016, which we repeat for convenience. For an identity state, let t ∈ {aa, ab, bb} denote an edge type, for example, ab indicates any edge between the alleles in two different individuals a and b (i.e., and edge between one of the nodes {a 1 , a 2 } and one of the nodes that fall in the same condensed identity states), but the presentation is simpler in the non-condensed setting.…”

Efficient computation of the kinship coefficients

“…Since there is a deterministic mapping between the condensed identity states and the identity states, the expectation can be written in terms of the condensed identity states. The proof for this approach takes as a first step the expression of the kinship as an expectation over inheritance paths, [6] which is easily converted into an expectation over identity states. We will define this expectation, discuss how to compute it, and leave the proof as an exercise.…”