1. A General Modeling Framework for Studying Candidate Genes Copy files from f:\edwin\example

2. Why general modeling framework?
Candidate genes for quantitative traits usually “main effect” on mean.
Genetic advantage more extensive modeling framework
Some candidate genes may be more likely to be detected
One reason is power e.g. (pleiotropic) easier to detect in multivariate study
Some genes may not work in a simple “main effect” fashion e.g. exert their effects in severely deprived environments only, or influence the sensitivity to environmental fluctuations (variance)
Correct tests? e.g. different genotypic variances in selected samples

3. More extensive picture genetic effects
Substantive advantage general modeling framework
More extensive picture genetic effects
Shed new light on traditional research questions
Continuity, change, and heterotypy Comorbidity/pleiotropy Complex traits: Causal mechanisms involving multiple factors
New issues: The interplay between genotypes and environment.
Vulnerability, resilience, and protective factors Risk behavior and the construction of favorable environments Sensitivity to environmental fluctuations
Instrumental function due to unique properties

4. Requirements modeling framework
Genetic effects on the means, variances, and relations between variables
Stratification effects on all these components
Nuclear families of various sizes
Interpretable parameterization
Di- and multi-allelic loci, marker haplotypes, multiple loci simultaneously, and parental genotypes
Easy to fit in existing (Mx) software

5. LISREL based model h(s) = ajk(s) + Bjk(s)h(s) + Gjk(s) + zjk(s)
y(s) = nyjk(s) + Lyjk(s)h(s) + eyjk(s)
x = nxk + Lxk + exk
y subject variables
x family variables

9. x-variables is independent subject plus family variables
Alternative Models
Conditional model
h(s) = ajk(s) + Bjk(s)h(s) + Gjk(s)xs + zjk(s)
y(s) = njk(s) + Ljk(s)h(s) + Kjk(s)xs + ejk(s)
x-variables is independent subject plus family variables
relax assumption full multivariate normality
curvi or non-linear effects x-variables
Disadvantage:
- Optimization,
- Measurement model x-variables
Other modeling frameworks

10. Partitioning parameter matrices
Most matrices:
a) general matrices that are not subscripted represent overall model in all genotype groups and population strata
b) genetic matrices j represent deviations from the general model caused by locus effects
c) matrices that are subscripted k and represent deviations from the general model caused by population stratification

11. How?
Example matrix Beta: Causal effects of subject variables on each other
Bjk(s) = B + Bj(gsI) + Bk(fI)
Main effects are in B that has dimension nh  nh,

12. Genetic effects in term Bj(gsI)
The ng  1 vector gs contains ng dummy variables coding the genotype (haplotype) of subject s
deviations from B thus maximum = #genotypes - 1
sets of dummy variables to study multiple loci simultaneously or effects of parental genotypes
- Bj = [ B1 | B2 |… | Bng]
dimension is nh  (ng  nh),
where B1 is the nh  nh submatrix containing the effects of the first dummy variable, …etc.

13. Example

14. A1A1 subjects

15. A1A2 subjects

16. A2A2 subjects

17. Stratification effects in term Bk(fI)
The nf  1 vector f contains the nf dummy variables used to code family types
deviations thus maximum = #family types - 1
Bk = [ B1 | B2 |… | Bnf]
dimension is nh  (nf  nh),
where B1 is the nh  nh submatrix containing the effects of the first dummy variable, …etc.
 and I select proper matrix for dummy variable

18. Sibling pairs A B F1 F2 F3 F4 F5 Subject Not informative 2 1F1 F2 F3 F4 F5
Subject A B 
Not informative 2 1
of stratification
Informative

19. Two Parents, one “child”
Parent A B
Subject F1 F2 F3 F4 F5
Not informative 2 1
of stratification
Informative

21. Other matrices are partitioned in the same way

24. General interpretation
Genetic effects on:
means are “main” effects
relations between variables are interaction effects
residuals are variance effects

25. Simple example

26. y1 y2 z1 z2 y1 y2
a1(1) a1(2)
a2(1) a2(2) a1 a2
0 b12
b21 0

32. Interactions

33. b21(1) > 0 and b21(2) = 0

34. b21(1) and b21(2) >0

35. Estimation and specification in Mx

36. Expected means and covariances single subject

37. Expected means and covariances whole family

38. Maximize log-likelihood function given the observed data by Raw Maximum likelihood
where the individual log-likelihoods equal
Minus two times the difference between the log likelihoods of two nested models is chi-square distributed with the difference in estimated parameters as the degrees of freedom.

39. Specification Therefore simple program
Most instances selection of matrices
Dimension matrices > boring, errors
Get started
Therefore simple program
Batch or questions

40. MxScript Data structure Matrices to be used File names
Number of (latent) subject variables?
Number of subjects in largest family?
Number of dummy variables for genotypes?
Matrices to be used
Do the subject variables have causal effects on each other? BETA?
GENETIC: causal relations between subject variables? BETA?
STRATIFICATION: means of subject variables? ALPHA?
File names
Name of file with your data? (DOS name)?
Name of the file for the Mx script? (DOS name)

41. Structure Mx script Most instances four groups
Group Function Free parameters Starting values
1 General part yes yes
2 Genetic effects yes
3 Stratification effects yes
4 Fit model to data
Type from DOS-prompt: MxScript <ENTER>
Type from DOS-prompt: MxScript input.dat <ENTER>

42. Example Name data file: example.dat Sibling pairs, no parents
Three genotype groups
Family variables in data file (indicate that you want specify admixture effects)
Starting values: sample drawn from multivariate distribution with means 0 and variances 1.5

43. General part
exercise
BMD
Intensity
Arm
Spine
Duration
Hip

44. Identification measurement model:

45. exercise
BMD
Common pathway?
Intensity
Arm
Independent pathway?
Genetic +
Stratification effects
Spine
Duration
Hip

46. Tests
Common pathway-Estimate model with genetic and stratification effects on means of second latent variable and test for significance of:
Genetic effects
Stratification effects
Genetic + stratification effect
Independent pathway- Estimate model with genetic and stratification effects on means of the indicators of the second latent variable and test for significance of:

47. Free elements
a Full 2 1 Free [Matrices-End matrices section]
Free a 1 1 a 2 1 [After End matrices - free elements]
Free a 1 1 to a 2 1 [After End matrices - free range]

50. Copy files from f:\edwin\solution