Conditional likelihood inference under complex ascertainment using data augmentation

In many applications, particularly in genetics, samples are drawn under complex ascertainment rules.For example, families may only be selected for study if two or more siblings have trait values exceeding some threshold. The correct likelihood for inference in such situations involves the probabilities of ascertainment, and these are frequently intractable. A consistent, but not fully efficient, method of analysis of such studies is proposed. The main idea is to augment the data with additional pseudo-observations simulated under the ascertainment scheme, and to analyse using a conditional likelihood for discrimination between true observations and pseudo-observations. Ascertainment probabilities cancel in this likelihood. The method is illustrated with a simple example involving left-truncated failure times.