Lecture 7: Linkage Analysis I

1. Lecture 7: Linkage Analysis I Date: 9/16/02 Clean up segregation analysis

2. Steps of Segregation Analysis • Identify mating type(s) where the trait is expected to segregate in the offspring. • Sample families with the given mating type from the population. • Sample and score the children of sampled families. • Test H0: “expected segregation ratio” or estimate segregation ratio.

3. But, you knew it wouldn’t always be that easy… • Appropriate mating types may not be identifiable. • Offspring of Dr x Dr cross segregate, but this mating type is indistinguishable from DD x DD and others. • An incompletely penetrant trait. Some appropriate mating types fail to be detected because the trait is invisible. • The trait is rare and you need to enrich your sample for affecteds. You collect a nonrandom sample.

4. Segregation Analysis for Autosomal Recessive Genes

5. Avoid Contaminating Mating Types • Controlled crosses. • Select only the appropriate mating types by ascertaining them through their offspring.

6. Ascertainment Procedure • Definition: An ascertainment procedure is the way in which families come to be included in a study. • The ascertainment procedure may result in incomplete selection or over-selection. • Exclude those families that happen to have all normal offspring. • Include those families which don’t segregate.

7. Ascertainment Terminology I • Definition: Affected individuals that are identified independently of all other individuals are called probands. • Definition: Other affected individuals in a family with a proband are called secondary cases. • Definition: The ascertainment probability () is the probability that an affected individual in the population is identified as a proband.

8. Ascertainment Terminology II • Definition: Complete ascertainment or more appropriately, truncated ascertainment, is the case where =1, so all families with affected children are identified and all their children are probands. Note that this does not imply complete selection. • Definition:Single ascertainment is the situation where all ascertained families will have just one proband.

9. Ascertainment Bias: Sex-Based • Dizygotic twins: Each twin is independently assigned sex with equal probability of being male or female.

10. Ascertainment Bias: Sex-Based (contd) • By randomly sampling males we only identify only ½ of the cases where a male twin has a sister.

11. Ascertainment Bias: Size-Based • Suppose a proportion kx of families have x siblings. • A random individual derives from a sibship of size x with probability:

12. Ascertainment Bias: Size-Based (contd) • The expected number of sibs of a random individual is: • Thus, by sampling random individuals, we tend to sample more from large families and we increase the average number of sibs.

13. Ascertainment Bias: Genetic Samples • In genetic samples, the ascertainment bias is trait-based. • Enrich for a rare trait by sampling an individual who is affected and those who are related to him/her. • Obviously, among those who are related, the probability of being affected is higher than in an unbiased sample from the population.

14. Family Ascertainment Probability

15. Truncated Ascertainment(p =1) • Consider only families of size s. • Let random variable Xi be the number of affected offspring in the ith family. • Xi~Bin(s, 0.25) • In complete ascertainment, all families with Xi>0 are included in the study. • We seek the distribution of Xi | Xi>0.

16. Distribution of Xi | Xi > 0 (Truncated Binomial) • Call this random variable on the new sample space {1,2,…,s}Yi.

17. Expected Segregation Ratio with Truncated Ascertainment • Let pt be the segregation ratio given the truncated ascertainment procedure. biased!

18. Example: Truncated Ascertainment pt = 0.33

19. Truncated Ascertainment: Estimating p • The expression for pt gives us a means of estimating p. We observe pt, assume truncated ascertainment and use p to estimate p. • Indeed, since p is a function of pt, we can use previous variance formula results to get an approximate variance for p. • Unfortunately, the equation is not analytically solvable.

20. Estimating p: EM Algorithm (E) • Incomplete data: the number of unascertained families with 0 affected offspring. Call this Ui at iteration i. • Expectation Step: Assume pi and calculate:

21. Estimating p: EM Algorithm (M) • Compute the maximum likelihood estimate pi+1:

22. Incomplete Ascertainment: The Norm • Any time there are affecteds in the study that are NOT probands, the assumption of truncated ascertainment does not apply. • Instead we have incomplete ascertainment.

23. The Proband Method • Use the proband to identify the mating type, but then leave it out of subsequent calculations.

24. The Proband Method: Estimating p • Again, use the proband only to identify the mating type. Count only the other siblings.

25. Example: Proband Method

26. Singles Method • A single is a proband who is the only proband in a family. Singles are not considered effective observations because they are observed only through their affected status. • Let d be the number of singles in a sample of n families.

27. Singles Method Estimates

28. Singles Method Example • There are 259 singles.

29. Variance For Proband and Singles Method • The proband and singles method both give estimators that are the quotient of two random variables. Approximate equations fr variance exist:

30. Likelihood Method • P(offspring is proband) = pp. • P(family ascertained) = 1-(1- pp)s. • P(r affected and b probands | ascertained) then is:

31. Likelihood Method (contd) • Each family is an independent observation (assuming they are not related). • Newton-Raphson multiple parameter update:

32. Likelihood Method: Testing Hypotheses • Nested models can be tested using the log likelihood ratio. • Interesting hypotheses include: • Complete ascertainment: p=1 • Recessive inheritance: p=0.25 • Complete ascertainment and recessive inheritance: p=1 and p=0.25

33. Likelihood Method: Sample Results Cannot rejective the recessive disease hypothesis. Can reject complete ascertainment hypothesis.

34. Rejecting a Null Hypothesis • Not a single locus. • Ascertainment procedure. • Selection • Environmental effects mimicing phenotype. • Incomplete penetrance.

35. More Complex Ascertainment Models • We have considered ascertainment procedures where the probability of ascertainment was of the form: • Allowing p to vary covers a wide number of cases, but not all. Still imposes a funcional relationship.

36. Summary • Ascertainment procedure and the impact of sampling. • Segregation analysis when the ascertainment procedure is nonrandom. Specifically, recessive trait. • Truncated ascertainment vs. incomplete ascertainment. • Proband method; singles method; likelihood method.