Sampling correction in pedigree analysis

Usually, a pedigree is sampled and included in the sample that is analyzed after following a predefined non-random sampling design comprising several specific procedures. To obtain a pedigree analysis result free from the bias caused by the sampling procedures, a correction is applied to the pedigree likelihood. The sampling procedures usually considered are: the pedigree ascertainment, determining whether a population unit is to be sampled; the intrafamilial pedigree extension, determining what part of the pedigree is to be sampled; and selective censoring of the sampled pedigree, determining whether it should be included in the sample to be analyzed. The probability of pedigree ascertainment is determined by the total set of potential probands in the true pedigree from which the sampled pedigree is obtained and we indicate how the necessary information on this set can be collected. If insufficient information on this set is observed, it is impossible to correct the pedigree likelihood adequately. Here we show that, if only the structure of this set is known, then an ascertainment-model-based pedigree likelihood can be obtained by conditioning on this structure. An ascertainment-model-free (AMF) pedigree likelihood can be correctly constructed by conditioning on all the data in this set, i.e. on both its structure and its phenotypic content. However, if this set has missing data, the AMF likelihood becomes undefined, which limits the utility of this AMF approach originally proposed by Ewens and Shute (1986). We also consider the sampling correction necessary when the pedigrees included in the sample analyzed have been subjected to censoring. The forms of likelihood correction developed here provide asymptotically unbiased estimators of the genetic model only if the formulated model is correct, which means that it must correctly allow for the most important features of the true inheritance of the trait studied. Otherwise, if no special case of the formulated general model is close to the true inheritance model, then the forms of likelihood correction proposed here result in biases, the magnitude and direction of which depend on both the true model and the general analysis model that should subsume it.