Guang Guo Department of Sociology University of North Carolina at Chapel Hill

1. Mixed or Multilevel Models for Linear Longitudinal Twin/Sib Data&Frailty Survival Model for Survival Twin/Sib Data Guang Guo Department of Sociology University of North Carolina at Chapel Hill

2. Objective • Understand how environmental and genetic factors interact to influence behaviors at different ages (longitudinal questions) • Understand how environmental and genetic factors interact to influence the occurrence an event over time

3. DNA measures would be ideal; but for complex traits, twin/sib analysis is still useful. Because 1, a large number of genes (e.g.,more than 100 for obesity) 2, Twin studies consider the effects of the entire genome. 3,

4. The general mixed model(Searle 1971; Searle et al. 1992) Y: outcome variable X: Observed Predictors u: Unobserved Random Effects Z: Design matrix for u --Natural choice for biometric data --infinite number of models --Software: SAS, SPSS, … --Cross-Sectional Sibling Data: Guo & Wang (2002; Behavior Genetics)

5. Unlikely the usual case, I am primarily interested in the random parameters.

6. -- i: person and j: pair or cluster -- Individuals in same pair subject to the same random effect -- the random effect of age take account of the larger variance at older age -- Age effect is based on a synthetic cohort

7. In scalar form: j: pair or cluster; i: person; t: time period

8. In matrix form—three measures per person over time

12. The Variance-Covariance Matrix of the outcome variable Intra-sib correlation for the same age and same time period

13. Example • The National Longitudinal Study of Adolescent Health (Add Health) • MZ and DZ twins, and full siblings (3,000-4,500) • Three waves: 1994-5, 1995-6, and 2002 • Obesity (BMI): 1994-5 (reported); 1995-6 and 2002 (measured)

14. --- 8 pairs of siblings --- Each person measured three times (Wave 1, Wave 2, & Wave 3)

17. Age: 12 - 26

18. Compound Symmetry Models

19. Calendar Time Effect on Heritability Contrasting intra-pair correlation at Wave I with that at Wave III

20. Gender Effects on Heritability

23. AR(1) models • 33 proc mixed data=temp1 asycov covtest • 33 ! method=ml; • 34 class race order; • 35 model bmi=sex order age race/s; • 36 random intercept nage/subject=pairid; • 37 repeated/subject=id2 type=sp(exp)(calender) rcorr; • 38 title "Mixed model: MZ, AR(time), negative age"; • run; • Unstructured Models • 40 • 41 proc mixed data=temp1 asycov covtest • 41 ! method=ml; • 42 class race calender order; • 43 model bmi=sex order age race/s; • 44 random intercept nage/subject=pairid; • 45 repeated calender/subject=id2 type=un rcorr; • 46 title "Mixed model: DZ, Unknown, negative age"; • 47 run;

24. ----Two ways: 1. Bootstrapping 2. -----

25. Conclusion (1) • The mixed model generates a large number of models taking advantage of longitudinal sibling data • Can estimate heritability as a function of age, time period, age*time period, usual G*E interaction, … • Results can be from a real cohort or pieces of short cohorts. • Can use compound symmetry+random coefficients, ar(1), and unstructured models. • Estimation is well-established and the codes in SAS are simple. • But need considerable computing time and good data. • Challenges: Tests of the random parameters

26. Environmental and Genetic Contributions to Occurrence of an event over time

27. The Method • Standard event history model (i: persons) • Frailty models, i: persons and j: pairs

28. The shared frailty effect w has a gamma distribution with density

29. The heritability can be estimated by Clayton’s characterization of the random effect bivariate hazard model as follows The ratio of the hazard of T1 at any duration t1 given T2=t2, to the hazard of T1 given T2>t2, is a positive constant equal to one plus the gamma variance of the twin-pair specific frailty. An estimated 1/phi of 0.20 means that if one twin has had the event, the risk of having event for the index twin would be 20% higher than that for the same index twin if the other twin had not had the event.

30. Let We need both MZ and DZ twins to have an estimate on the effect of Genetic factors since the theta includes both shared environmental effects and Genetic effects. Can test if the two is significantly different.

31. Example: first sex • The sample consists of US middle and high school students (7-12 graders) • The wave I sample has 20,745 • Whether ever had sexual intercourse • Month and year had sex first time • Self-administered audio-CASI (computer-assisted self interview) • Three waves of data available (1994, 1995, and 2002)

32. Why timing of first sex? • Sexually transmitted infections • Unintended pregnancy • Adolescent child bearing • Highly salient event for adolescents; marker in the transition to adulthood.

33. Add Health Data • Sib data: mz twins and dz twins • Same-sex pairs of DZ twins • The zygosity is determined by DNA markers

34. Self reports of first sex • Reports can’t be easily checked against objective data • Can only assess consistency of self-reports.

35. Impact of inconsistent reports • Do not substantially impact the substantive conclusions regarding age at first sex or on the effects of SES on risk of first sex • But estimated heritability depends heavily on the accuracy of the individually reported dates.

36. Survival function: males & females

37. Survival function by ethnicity

38. Factors influencing age at first sex • SES, Peers, Education, Religiosity, Personality, Ethnicity, Gender and Physical maturity • Early rise of pubertal androgen levels. Particularly testosterone correlates with early onset of sexual ideation and masturbation in males (Udry et al.)

39. Table 1. Hazard Ratios of Piece-Wise Exponential Survival Model with Gamma Shared Frailty on First Sexual Intercourse

40. Table 2. Hazard Ratios of Piece-Wise Exponential Survival Model with Gamma Shared Frailty on First Sexual Intercourse

41. Table 3. Hazard Ratios of Piece-Wise Exponential Survival Model with Gamma Shared Frailty on First Sexual Intercourse

42. Tentative conclusions • The random parameter ( ) is consistently larger for MZ twins than for DZ twins. • Similar results are found for the entire sample, for the male sample, the female sample, the white male sample, and the white female sample. • A theta of 0.30 implies that the risk of a twin having first sex is 30% higher if

43. Tentative conclusions the other twin has already has first sex. • These results suggest a moderate role of genetic factors in the risk of first sex. Should consider possible dominant effect. • Challenges: -- Tests for random parameters. -- Estimate multiple random parameters and test for them (environmental factors (e.g., religiosity) might influence the expression of the genetic factors).