1. Gene-Environment Case-Control Studies
Raymond J. Carroll
Department of Statistics
Center for Statistical Bioinformatics
Institute for Applied Mathematics and Computational Science
Texas A&M University
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2. Note the Maroon color scheme! And the green MSU flag.
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3. Apologies to Dr. Seuss
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4. Michigan State Grads at TAMU
Mohsen Pourahmadi Soumen Lahiri
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5. Other Michigan State Contacts
David Ruppert Anton Schick
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6. Outline
Problem: Case-Control Studies with Gene-Environment relationships
Theme I: Logistic regression is lousy for understanding interactions. We make assumptions that can double or triple the effective sample size

7. Outline
Problem: Case-Control Studies with Gene-Environment relationships
Theme II: There is a lousy estimator, and a good one that makes more assumptions. How do you protect yourself if the assumptions fail, and you want to analyze 500,00 SNP?

8. Outline
Problem: Case-Control Studies with Gene-Environment relationships
Theme III: How does all this work with actual data, as opposed to simulated data?

9. Software
SAS and Matlab Programs Available at my web site under the software button
R programs available from the NCI
New Statistical Science paper 2009, volume 24,

10. Basic Problem Formalized
Gene and Environment
Question: For women who carry the BRCA1/2 mutation, does oral contraceptive use provide any protection against ovarian cancer?

11. Basic Problem Formalized
Gene and Environment
Question: For people carrying a particular haplotype in the VDR pathway, does higher levels of serum Vitamin D protect against prostate cancer?

12. Basic Problem Formalized
Gene and Environment
Question: If you are a current smoker, are you protected against colorectal adenoma if you carry a particular haplotype in the NAT2 smoking metabolism region?

13. Retrospective Studies
D = disease status (binary)
X = environmental variables
Smoking status
Vitamin D
Oral contraceptive use
G = gene status
Mutation or not
Multiple or single SNP
Haplotypes

14. Prospective and Retrospective Studies
Retrospective Studies: Usually called case-control studies
Find a population of cases, i.e., people with a disease, and sample from it.
Find a population of controls, i.e., people without the disease, and sample from it.

15. Prospective and Retrospective Studies
Retrospective Studies: Because the gene G and the environment X are sample after disease status is ascertained

16. Basic Problem Formalized
Case control sample: D = disease
Gene expression: G
Environment, can include strata: X
We are interested in main effects for G and X along with their interaction as they affect development of disease

17. Logistic Regression Logistic Function:
The approximation works for rare diseases

18. Prospective Models Simplest logistic model without an interaction
The effect of having a mutation (G=1) versus not (G=0) is

19. Prospective Models Simplest logistic model with an interaction
The effect of having a mutation (G=1) versus not (G=0) is

20. Empirical Observations
Statistical Theory: There is a lovely statistical theory available
It says: ignore the fact that you have a case-control sample, and pretend you have a prospective study

21. When G is observed
Logistic regression is robust to any modeling assumptions about the covariates in the population
Unfortunately it is not very efficient for understanding interactions
Much larger sample sizes are required for interactions that for just gene effects

22. Gene-Environment Independence
In many situations, it may be reasonable to assume G and X are independently distributed in the underlying population, possibly after conditioning on strata
This assumption is often used in gene-environment interaction studies

23. G-E Independence Does not always hold!
Example: polymorphisms in the smoking metabolism pathway may affect the degree of addiction

24. Gene-Environment Independence
If you are willing to make assumptions about the distributions of the covariates in the population, more efficiency can be obtained.
This is NOT TRUE for prospective studies, only true for retrospective studies.

25. Gene-Environment Independence
The reason is that you are putting a constraint on the retrospective likelihood

26. Gene-Environment Independence
Our Methodology: Is far more general than assuming that genetic status and environment are independent
We have developed capacity for modeling the distribution of genetic status given strata and environmental factors
I will skip this and just pretend G-E independence here

27. More Efficiency, G Observed
Our model: G-E independence and a genetic model, e.g., Hardy-Weinberg Equilibrium

28. The Formulation Any logistic model works
Question: What methods do we have to construct estimators?

29. Methodology
I won’t give you the full methodology, but it works as follows.
Case-control studies are very close to a prospective (random sampling) study, with the exception that sometimes you do not observe people

30. Methodology N Total Population Np1 Np0 Controls in the Population
Cases in the
Population
Cases in the
Sample n1 n0
Controls in the
Sample
Missing Cases
Np1-n1
Np0-n0
Missing Controls
% of Controls observed
% of Cases observed

31. Pretend Missing Data Formulation
This means that there is a missing data problem.
The selection into the case control study is biased: cases are vastly over-represented
Ordinary logistic regression computes the probability of disease given the environment, given the gene, and given that the person was selected into the case control study

32. Pretend Missing Data Formulation
This means that there is a missing data problem.
Our method computes the probability of disease and the probability of gene given the environment and given that the person was selected into the case control study
The selection into the case control study is biased: cases are vastly over-represented

33. Methodology
Our method has an explicit form, i.e., no integrals or anything nasty
It is easy to program the method to estimate the logistic model
It is likelihood based. Technically, a semiparametric profile likelihood

34. Methodology We can handle missing gene data
We can handle error in genotyping
We can handle measurement errors in environmental variables, e.g., diet

35. Methodology
Our method results in much more efficient statistical inference

36. More Data What does More efficient statistical inference mean?
It means, effectively, that you have more data
In cases that G is a simple mutation, our method is typically equivalent to having 3 times more data

37. How much more data: Typical Simulation Example
The increase in effective sample size when using our methodology

38. Real Data Complexities
The Israeli Ovarian Cancer Study
G = BRCA1/2 mutation (very deadly)
X includes
age,
ethnic status (below),
parity,
oral contraceptive use
Family history
Smoking
Etc.

39. Real Data Complexities
In the Israeli Study, G is missing in 50% of the controls, and 10% of the cases
Also, among Jewish citizens, Israel has two dominant ethnic types
Ashkenazi (European)
Shephardic (North African)

40. Real Data Complexities
The gene mutation BRCA1/2 if frequent among the Ashkenazi, but rare among the Shephardic
Thus, if one component of X is ethnic status, then pr(G=1 | X) depends on X
Gene-Environment independence fails here
What can be done? Model pr(G=1 | X) as binary with different probabilities!

41. Israeli Ovarian Cancer Study
Question: Can carriers of the BRCA1/2 mutation be protected via OC-use?

42. Typical Empirical Example

43. Israeli Ovarian Cancer Study
Main Effect of BRCA1/2:

44. Israeli Ovarian Cancer Study

45. Haplotypes
Haplotypes consist of what we get from our mother and father at more than one site
Mother gives us the haplotype hm = (Am,Bm)
Father gives us the haplotype hf = (af,bf)
Our diplotype is Hdip = {(Am,Bm), (af,bf)}

46. Haplotypes
Unfortunately, we cannot presently observe the two haplotypes
We can only observe genotypes
Thus, if we were really Hdip = {(Am,Bm), (af,bf)}, then the data we would see would simply be the unordered set (A,a,B,b)

47. Missing Haplotypes
Thus, if we were really Hdip = {(Am,Bm), (af,bf)}, then the data we would see would simply be the unordered set (A,a,B,b)
However, this is also consistent with a different diplotype, namely Hdip = {(am,Bm), (Af,bf)}
Note that the number of copies of the (a,b) haplotype differs in these two cases
The true diploid = haplotype pair is missing

48. Missing Haplotypes
Our methods handle unphased diplotyes (missing haplotypes) with no problem.
Standard EM-algorithm calculations can be used
We assume that the haplotypes are in HWE, and have extended to cases of non-HWE

49. Robustness
Robustness: We are making assumptions to gain efficiency = “get more data”
What happens if the assumptions are wrong?
Biases, incorrect conclusions, etc.
How can we gain efficiency when it is warranted, and yet have valid inferences?

50. Two Likelihoods The two likelihoods lead to two estimators
The former is robust but not efficient
The latter is efficient but not robust
What to do?

51. Empirical Bayes
The idea is to take a weighted average of the model free and model based estimators
The weight depends on how different the estimators are
Relative to how variable the difference is

52. Empirical Bayes
You can actually formally test the hypothesis of whether the model fits the data
It is just a t-test on the difference between the two estimators

53. Empirical Bayes
If the difference is small relative to the variability, then this argues in favor of the model based approach

54. Empirical Bayes
We chose an Empirical Bayes type-approach
Let and
Then

55. Comments on Empirical Bayes
If the model fails, then the estimator converges to the model-free estimator
If the model holds, the estimator estimates the right thing, but is much more efficient than the model-free estimator

56. Example 1: Prostate Cancer
G = SNPs in the Vitamin D Pathway
X = Serum-level biomarker of vitamin D (diet and sun)
The VDR gene is downstream in the pathway, hence unlikely to influence the level of X
Gene-environment independence likely

57. Example 1: Prostate Cancer
3 age groups
9 centers
Two haplotype-serum Vitamin D interactions
Three haplotype main effects

58. Example 2: Colorectal Adenoma
G = SNPs in the NAT2 gene, which is important in the metabolism of
X =Various measures of smoking history
The NAT2 gene may make smokers more addicted
Gene-environment independence unlikely

59. Example 2: Colorectal Adenoma
Two genders
4 age groups
7 common haplotypes as main effects
One haplotype known to affect metabolism
Current and former smoking interactions

60. The NAT2 Example
Current smoking and haplotype interaction coefficient
Current smokers with this haplotype are 50% less likely to develop a colorectal adenoma
Method
Estimate
s.e.
p-value
Model Free
-0.63
0.17
0.014
Independence
-0.33
0.16
0.048
Consistent EB1
-0.59
0.25
0.017

61. The VDR Example
Serum Vitamin D and 000 haplotype interaction coefficient
Men with 1 sd greater Serum vitamin D then the norm are 70% less likely to develop prostate cancer
Method
Estimate
s.e.
p-value
Model Free
-0.21
0.12
0.093
Independence
-0.18
0.08
0.019
Consistent EB1
-0.19
0.021

62. Genome-Wide Association Studies
These methods are routinely applied to GWAS
My last two examples were actually from the PLCO GWAS
Also, can call the environment = other SNP

63. Summary
Case-control studies are the backbone of epidemiology in general, and genetic epidemiology in particular
Their retrospective nature distinguishes them from random samples = prospective studies

64. Summary
We start by assuming relationships between the genes and the “environment” in the population, e.g., independence
This model can be fully flexible
We also, where necessary, specify distributions for genes

65. Summary
We calculated a new likelihood function, leading to more much more precise inferences
The method can handle missing genes, genotyping errors, measurement errors in the environment
Calculations are straightforward via the EM algorithm

66. Summary Forced to face the dilemma
Lousy but robust method
Great but not robust method
We developed a fast, data adaptive, novel way of addressing this issue
In cases where one can predict the outcome, the EB method works as desired

67. Acknowledgments
This work is joint with Nilanjan Chatterjee (NCI) and Yi-Hau Chen (Academia Sinica)

68. Acknowledgments This work is supposed by NCI-R27-CA057030
NHLBI RO1-HL (P.I., N. Chatterjee)
Texas A&M Institute of Applied Mathematics and Computational Science through KAUST (King Abdullah University of Science and Technology)