American Journal of Epidemiology Advance Access originally published online on November 2, 2006
American Journal of Epidemiology 2007 165(3):334-343; doi:10.1093/aje/kwk006
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
PRACTICE OF EPIDEMIOLOGY |
Logic Regression for Analysis of the Association between Genetic Variation in the Renin-Angiotensin System and Myocardial Infarction or Stroke
1 Division of Public Health Sciences, Fred Hutchinson Cancer Research Center, Seattle, WA
2 Department of Biostatistics, School of Public Health and Community Medicine, University of Washington, Seattle, WA
3 Cardiovascular Health Research Unit, University of Washington, Seattle, WA
4 Department of Epidemiology, School of Public Health and Community Medicine, University of Washington, Seattle, WA
5 Department of Health Services, School of Public Health and Community Medicine, University of Washington, Seattle, WA
Correspondence to Charles Kooperberg, Fred Hutchinson Cancer Research Center, 1100 Fairview Avenue North, M3-A410, Seattle, WA 98109-1024 (e-mail: clk{at}fhcrc.org).
Received for publication October 17, 2005. Accepted for publication June 13, 2006.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
Recent developments in genetic sequencing technology now make it possible to genotype large numbers of single nucleotide polymorphisms (SNPs) in large samples. Many association studies using SNP data are now being carried out. Typically, these observational studies establish whether certain haplotypes or individual SNPs are associated with a health outcome. Few methods exist for finding interaction effects among multiple SNPs or between SNPs and environmental factors. In this paper, the authors describe logic regression, an exploratory method with which to identify interactions for further research. They illustrate this method using data from a US case-control study of myocardial infarction and stroke (1995–1999) carried out among 1,614 persons in Washington State who were genotyped for 32 SNPs on five genes in the renin-angiotensin system.
epidemiologic methods; epistasis, genetic; models, statistical; polymorphism, single nucleotide; regression analysis
Abbreviations: ACE, angiotensin-converting enzyme; AGT, angiotensinogen; AGTR, angiotensin II receptor; RAS, renin-angiotensin system; REN, renin; SNP, single nucleotide polymorphism
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
Over the last few years, the number of studies characterizing associations between single nucleotide polymorphisms (SNPs) and disease outcomes has increased dramatically. With the recent developments in genetic sequencing technology, the number and size of these studies will increase further. While single SNP association analyses are straightforward to carry out, they do not make efficient use of the genomic structure.
For many SNP association studies, interest will not be limited to identifying individual SNPs or haplotypes associated with a disease outcome but, equally important, will also involve the identification of interactions between SNPs within a gene (as in a haplotype effect), between genes, or between genes and environmental factors such as drugs, smoking, and alcohol consumption. Few methods exist for finding interaction effects of multiple SNPs or between SNPs and environmental factors (1, 2). Reasons for this shortage of methods may be the large number of potential interactions, which makes it practically impossible to examine all interaction models, and the requirement of multiple-comparisons correction for all models that are examined.
In judging methods for identifying interactions, it is useful to differentiate between three types of interactions that can be of interest.
Within-gene interactions.
Identification of interactions between SNPs within the same gene, or within the same region of a chromosome, is an alternative to using reconstructed haplotypes for association studies. An advantage of using haplotypes is that there are relatively few common ones within a block, and it is thus an easy way to model interactions. However, haplotypes depend on which tagSNPs, which "tag" the desired haplotype block, are genotyped, and there is no guarantee that another study will identify the same haplotypes. If several haplotypes all have increased or decreased risk, it may not be straightforward to recognize which SNPs are associated with the disease. Moreover, it is conceivable that no individual haplotype is significantly associated with the outcome while a simple combination of fewer SNPs is significant, since haplotypes may include SNPs that are not relevant to the disease risk. A within-gene interaction of the type "a variant at SNP 1 and a variant at SNP 3" could potentially identify groups of haplotypes that together produce increased risk.
Between-gene interactions.
The term "epistasis" was coined to describe the effect of masking of the phenotypic effects of one gene by a second gene. More generally, epistasis is considered an interaction between different genes. Epistasis is thought to play a significant role in complex diseases (3, 4). The identification of epistasis can be problematic, since if the effect of one locus is masked by the effect of another locus, the power to detect the first locus is probably reduced.
Gene-environment interactions.
The computational complexity of gene-environment interactions, with respect to the difficulty of identifying such interactions, falls somewhere between the complexity of "within-gene" interactions and "between-gene" interactions. For a study with 25,000 SNPs, there may be approximately 25,0002/2 possible two-SNP interactions, but there are only 25,000q SNP-environment interactions, where q is the (usually small) number of environmental factors under consideration.
Logic regression (5) is an exploratory regression method that is designed for analysis with binary predictors when interest is in possible interactions between predictors. As such, it is well suited for SNP association studies. In this paper, we will briefly describe the logic regression method and apply it to data on 349 myocardial infarction patients, 202 stroke patients, and 1,063 controls for whom 32 SNPs on five genes in the renin-angiotensin system (RAS) were genotyped.
|
MATERIALS AND METHODS |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
RAS data
The RAS plays a central role in maintenance of vascular tone and in salt and water homeostasis. Renin (REN) cleaves angiotensinogen (AGT) to produce angiotensin I, which is converted by angiotensin-converting enzyme (ACE) to angiotensin II, a potent constrictor. Evidence suggests that the 235T allele of the AGT gene is associated with increased risk of hypertension and elevated AGT levels; other related genotypes, such as angiotensin II receptor types 1 and 2 (AGTR1 and AGTR2), have been associated with hypertension and cardiovascular complications.
We carried out this study to investigate whether the effect of ACE inhibitors on risk of incident nonfatal myocardial infarction or stroke differs by ACE or AGT genotype. Our analysis was conducted within a population-based case-control study of members of the Group Health Cooperative of Puget Sound (western Washington State) aged 30–79 years with pharmacologically treated hypertension. The study design has been described elsewhere (6). Cases were persons who survived an incident myocardial infarction or stroke during the period 1995–1999, and controls were eligible if they did not have a history of myocardial infarction or stroke. Controls were randomly sampled from the Group Health Cooperative enrollment files and were frequency-matched to the myocardial infarction cases on age decade, sex, and calendar year of identification.
Current use of antihypertensive medications was determined using computerized pharmacy records. Individual drugs were grouped into major classes: diuretics, ß-blockers, ACE inhibitors, calcium-channel blockers, and vasodilators. Diuretics included both loop and thiazide diuretics.
The SNPs for the RAS data were identified in the SeattleSNPs Variation Discovery Resource (http://pga.gs.washington.edu) by genomic resequencing. Patterns of linkage disequilibrium were used to select a subset of SNPs that tagged major common patterns of variation. Using this method, three tagSNPs for ACE, eight tagSNPs for AGT, 12 tagSNPs for AGTR1, three tagSNPs for AGTR2, and six tagSNPs for REN were genotyped. AGTR2 is located on the X chromosome; thus, analyses were conducted separately for men and women. More details can be found in the paper by Marciante et al. (7).
The current analysis using logic regression tries to identify combinations of SNPs and drug classes that are associated with increased or decreased disease risk. These are slightly different interactions than the ones for which the study was designed.
Logic regression
Assuming a link function h of interest, a traditional interaction model is
 |
The logic regression model is
 |
) or X5), where 1 equals "true," 0 equals "false," and the superscript c refers to the complement. The Zi are additional confounders. Logic regression is an adaptive algorithm which for a given model selects those Li that minimize the residual sum of squares or the deviance. Typically the number of logic trees m selected is small (between 1 and 3), and the Li can be interpreted as "risk factors." We say that the logic tree in the equation above has three leaves. The method is described in detail by Ruczinski et al. (5), and software is available from the R Foundation for Statistical Computing (http://cran.r-project.org) as an R/CRAN package (8).
Optimization of the logic regression model is carried out using a (stochastic) simulated annealing algorithm employing an irreducible Markov chain. At any stage of this algorithm, a logic tree gets modified by replacing predictors (like "X7") or operators (like "and") or by changing the form of the tree (such as adding or deleting another "or Xi"). Modifications are proposed at random; if the proposed model is an improvement over the current model, then it is accepted (it replaces the current model), while if the proposed model is worse than the current model, it is accepted with a probability that depends on the stage of the algorithm and how much worse the proposed model is.
Logic regression for SNP data
Each SNP is coded into two binary covariates: X
= 1 if a person has at least one variant allele and X
= 1 if the person has two variant alleles; both are 0 otherwise. One can see that X and X code the dominant and recessive effects of SNP i, respectively. Logic regression has been applied successfully to the simulated SNP data of the 12th Genetic Analysis Workshop and to a study of heart disease (9, 10).
Model selection
For adaptive regression methods like logic regression, model selection is needed, since more complicated models typically fit data better, even if there is no signal. Model selection can make use of a simple penalty on the model complexity (size), such as Akaike's Information Criterion (11), or it can involve the use of a separate test set, cross-validation, or permutation tests. Within the logic regression software, three model selection tools are available.
- Permutation tests can be used to globally assess whether any combination of SNPs is associated with the response: The outcome (e.g., case-control status) is permuted at random, and the quality of the fit (e.g., deviance) on the real data is compared with the quality of the fit on the permuted data.
- Conditional permutation tests can be used to assess whether combinations of SNPs that are more complicated (because they involve either more complicated expressions or more expressions) than a particular model are more strongly associated with response than the current model. For these tests, the permutations are carried out such that we are guaranteed that even on the permuted data, a particular model can be fitted. Any improvement of the quality of the fit beyond that model is noise and can be compared with the quality of the fit on the real data.
- Cross-validation can be used to assess which complexity of the model has the best predictive performance. For cross-validation, the data are divided into 10 equal parts. Then, 10 times, one partition is left as a test set, and for each possible model complexity the best model is selected using nine out of 10 parts, after which the predicted deviance on the remaining test set part is computed. For each level of complexity, the 10 predicted deviances are added, and the complexity with the smallest overall predicted deviance is selected.
Typically, the conditional permutation approach will suggest larger models than cross-validation, since the conditional permutation approach assesses an association between the predictors and the response, while cross-validation requires that such an association be strong enough to reduce the prediction error.
Missing data
In our RAS data, only 3 percent of the genotypes were missing. Forty-four percent of the cases and controls had at least one missing genotype, and each SNP had at least one missing genotype. Because any combination of SNPs could be involved in an interaction, a complete case analysis would require elimination of 44 percent of the cases and controls. Since we had several SNPs in each of the genes, a haplotype reconstruction was possible. Haplotypes were inferred using a Bayesian population genetic model that uses coalescent theory (PHASE (12)). A reconstructed haplotype implied a value for a missing genotype. When the imputed genotypes were ambiguous, we used the estimated haplotype probabilities as case weights. In some situations, the haplotype reconstruction was ambiguous, but each of these haplotypes implied the same genotype, so the missing genotypes could be imputed unambiguously.
Logic regression for haplotype data
Logic regression selects Boolean combinations of SNPs, thereby implicitly grouping haplotypes. In some situations, the expressions generated by logic regression may be hard to interpret, since even some of the most elementary expressions may involve unrelated SNPs. An alternative is then to require the most elementary expressions, such as "X1 or X2" or "X3 and X4," to involve SNPs within the same gene. Here we ran the publicly available version of logic regression software on haplotypes. For each haplotype, we created two binary predictors based on whether the subject had at least one copy of the haplotype or at least two copies of the haplotype, as for the SNP data.
Processing of the data
Each of the 32 SNPs was replaced by two binary covariates, using the coding described above in the section "Logic regression for SNP data." Missing genotypes were inferred using the PHASE haplotype reconstruction. For analyses of single genes, we used case weights when genotypes were ambiguous, ignoring probabilities that were smaller than 0.05; the total probability covered by these cases was smaller than 0.5 percent of the data for every analysis. For the analysis using all genes, we imputed the most likely genotype, since using case weights for each SNP in each gene would lead to a much larger data set when there were multiple genes. An alternative is to use multiple imputations for missing genotypes. Because, in the current data, the percentage of missing genotypes was small, there was no practical difference in performance between these approaches, as we confirmed for selected analyses; when the percentage of missing genotypes is larger, a multiple-imputations approach leads to the least biased results (13). For the analysis using haplotypes, we used those reconstructed by PHASE and otherwise coded them in the same way as we code SNPs. We used 29 haplotypes that were estimated to occur at a frequency of more than 5 percent; thus, the number of haplotypes was very similar to the number of SNPs in this study. For the drug-genotype interactions, each of the drug classes was coded as a binary variable. In our current analysis, we did not control for other covariates, although they could be added in the logic regression model in a straightforward fashion.
The simulated annealing chains had 1,000,000 model evaluations for the actual model fitting runs and 500,000 model evaluations for the permutation tests and cross-validation runs. Other options used the defaults, unless indicated otherwise.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
Below we describe our analysis of the RAS data using logic regression. In a haplotype analysis, Marciante et al. (7) did find an association between AGT haplotypes and myocardial infarction in these data.
Global permutation tests
Table 1 shows the results of global permutation tests. The tests were carried out with a maximum model size of one logic tree with four leaves. A small model size allows the algorithm to do an almost exhaustive search of all models, yielding permutation p values with little noise.
|
Most genes and gene-drug interactions appeared not to be associated with the outcomes. For most permutation tests, well over 5 percent of the permutations yielded scores (deviances) that were smaller than the score of the best model. The AGT SNPs, and to a lesser extent the AGT haplotypes, showed some association with myocardial infarction. The AGTR2 SNPs in combination with drugs suggested a possible association with myocardial infarction in women. The REN haplotypes, and to a lesser extent the REN SNPs, showed some association with stroke. The AGTR1 SNPs and haplotypes together with the drugs showed a possible association with stroke. Interestingly, all SNPs or haplotypes combined showed a stronger association with stroke than any of the associations with individual genes.
In the remaining analysis, we concentrated on the possible associations of drugs and AGTR2 SNPs with myocardial infarction in women, of AGT SNPs with myocardial infarction, of drugs and AGTR1 haplotypes with stroke, and of a combination of all SNPs with stroke.
Associations of hypertensive drugs and AGTR2 SNPs with myocardial infarction in women
In figure 1, the cross-validation results of the analysis of the AGTR2 SNPs and drugs for myocardial infarction in women suggest that a model with one or two logic trees with two leaves in total has the best predictive performance among the models examined. This agrees with the results of the conditional permutation tests shown in table 2, since that table indicates that the best model would be larger than a model with one logic tree with one leaf but would not be larger than a model with one logic tree with two leaves or a model with two logic trees with two leaves. For table 2 and all further conditional permutation tests, the largest model size used was two logic trees with eight leaves each. If the maximum model size was the same as for table 1, the entry with zero trees and zero leaves in table 2 would be the same as the entry for drugs and AGTR2 SNPs with myocardial infarction in women in table 1. For conditional permutation tests, where we try to determine the size of a model, we typically use a larger model than that used for the unconditional permutation tests. The best model with one logic tree with two leaves is
 |
|
|
AGT SNP associations with myocardial infarction
Table 3 shows the results of the permutation tests for associations of AGT SNPs with myocardial infarction. The effect of the increased maximum model size compared with table 1 is that there does not seem to be much association between AGT SNPs and myocardial infarction. Typically, the magnitude of the association that is identified is smaller when the maximum model size is larger, since even in data where there is no signal, models that are large enough will show some association. The cross-validation analysis (figure 2) suggests that a model with three SNPs has some predictive power. This model, a model with one tree and three leaves, is
 |
|
|
Drug-haplotype interactions with stroke for the AGTR1 gene
The cross-validation results of the analysis of the AGTR1 haplotypes and drugs for stroke (figure 3) suggested that a model with a single logic tree with one or three leaves had the best predictive performance among the models examined. This agrees with the results of the conditional permutation tests shown in table 4, since this table indicates that the best model would be larger than one logic tree with one leaf and maybe even larger than a model with two leaves, but would not be larger than a model with three leaves.
|
|
The model with three leaves for these data is
 |
Effect of all SNPs on stroke
Table 5 summarizes the results of the conditional permutation tests for the relation between all SNPs and stroke. These results suggest that the best model may be of size zero. This again seems to contradict the global permutation tests somewhat. Figure 4, showing the cross-validation results of the analysis of the relation between all SNPs and stroke, suggests that perhaps models with two or three SNPs in two logic trees are slightly better than other model sizes; the figure suggests that these models may have some predictive power. The best model with three SNPs is of some interest, as it combines effects of SNPs in three different genes:
 |
|
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
It is widely acknowledged that the analysis of epistasis is challenging because of the large number of statistical models to be evaluated and the limited sample sizes if all combinations of (tag)SNPs are considered. Although there are situations where there may be sufficient power to identify large gene-gene interactions (15), generally speaking a search for all combinations in large association studies will have limited power because of the number of tests being carried out.
In the next few years, the size of genome association studies is going to increase dramatically, both in the number of samples and (especially) in the number of SNPs genotyped. Researchers will want to find potential gene-gene or gene-environment interactions, and there is a pressing need for methods that can potentially identify such interactions. Besides logic regression, very few such methods currently exist (e.g., see the papers by Ritchie et al. (1) and Foulkes et al. (2)). Methods that employ adaptive selection of models and use a limited number of degrees of freedom (parameters) to model interactions are more likely to successfully identify interactions than methods that are less prudent with study resources.
In this paper, we have illustrated the use of logic regression, an adaptive regression method that uses a Boolean model structure well suited for SNP data and adaptive model selection, on data from a cardiovascular disease case-control study with 32 SNPs. Since logic regression is a well-defined procedure, model selection and multiple-comparisons corrections for the significance level are implicit and do not require further resampling or bootstrapping. Unless a test data set is available to verify selected logic regression models immediately, the adaptive model selection makes logic regression particularly appropriate as an exploratory method for identifying interactions for further research. Logic regression is intended for binary predictors, such as SNPs. If one wished to enter continuous predictors in the interactions, they would have to be dichotomized.
The types of interactions identified by logic regression are not "traditional" interactions, where one predictor modifies the effect of another predictor, but rather combinations of predictors that are associated with increased or decreased disease risk. With a single logic tree, such logic regression identifies a single group of persons at increased (decreased) risk; when the underlying risk profile is more complicated, additional logic trees may be needed.
As was the case for many of the SNP association studies that have been carried out, the association between the SNPs and the RAS data was not very strong. Marciante et al. (7) reached the same conclusion using a haplotype analysis. Nevertheless, we feel that these data illustrate how logic regression can be used to identify local interactions, drug-gene interactions, or gene-gene interactions in an automated fashion. In addition to case-control studies, logic regression can be used for cohort studies, survival analysis, and any other study design for which the model can be formulated as a generalized linear model.
Recently, we developed Monte Carlo logic regression (16). In this variation of logic regression, rather than identify interactions that have a significant association with a clinical outcome, the investigators identify larger numbers of potential interactions using a Markov chain Monte Carlo algorithm. An illustration of this approach on a cardiovascular disease data set with 779 participants and 89 SNPs can be found in the paper by Kooperberg and Ruczinski (16).
|
ACKNOWLEDGMENTS |
|---|
This research was supported in part by grants CA53996 and CA74841 from the National Cancer Institute; by grants HL43201, HL60739, HL68639, HL68986, and HL74745 from the National Heart, Lung, and Blood Institute; and by grants 9970178N and 0270054N from the American Heart Association.
Conflict of interest: none declared.
|
References |
|---|
|
TOPABSTRACTINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONReferences |
|---|
- Ritchie MD, Hahn LW, Roodi N, et al. (2001) Multifactor-dimensionality reduction reveals high-order interactions among estrogen-metabolism genes in sporadic breast cancer. Am J Hum Genet 69:138–47.[CrossRef][Web of Science][Medline]
- Foulkes AS, DeGrutola V, Hertogs K. (2004) Combining genotype groups and recursive partitioning: an application to HIV-1 genetics data. Appl Stat 53:311–23.
- Fijneman RJ, de Vries SS, Jansen RC, et al. (1996) Complex interactions of new quantitative trait loci, Sluc1, Sluc2, Sluc3, and Sluc4, that influence the susceptibility to lung cancer in the mouse. Nat Genet 14:465–7.[CrossRef][Web of Science][Medline]
- Frankel WM and Schork NJ. (1996) Who's afraid of epistasis? Nat Genet 16:371–3.
- Ruczinski I, Kooperberg C, LeBlanc M. (2003) Logic regression. J Comput Graph Stat 12:475–511.[CrossRef]
- Psaty BM, Smith NL, Heckbert SR, et al. (2002) Diuretic therapy, the
-adducin gene variant, and the risk of myocardial infarction or stroke in persons with treated hypertension. JAMA 287:1680–9.[Abstract/Free Full Text] - Marciante KD, Bis JC, Rieder M, et al. (2005) Renin-angiotensin system gene haplotypes and the risk of myocardial infarction and stroke in pharmacologically treated hypertensive patients. (Abstract). Circulation 111:suppl, e323.
- R Development Core Team. (2005) R: a language and environment for statistical computing. (R Foundation for Statistical Computing, Vienna, Austria).
- Kooperberg C, Ruczinski I, LeBlanc ML, et al. (2001) Sequence analysis using logic regression. Genet Epidemiol 21:suppl 1, S626–31.
- Ruczinski I, Kooperberg C, LeBlanc M. (2004) Exploring interactions in high-dimensional genomic data—an overview of logic regression, with applications. J Mult Anal 90:178–95.[CrossRef]
- Akaike H. (1973) Information theory and an extension of the maximum likelihood principle. Second International Symposium on Information Theory (Akademiai KiadoIn Petrov BN and Csaki F (Eds.). , Budapest, Hungary)267–81.
- Stephens M, Smith M, Donnelly P. (2001) A new statistical method for haplotype reconstruction from population data. Am J Hum Gen 68:978–89.[CrossRef][Web of Science][Medline]
- Dai JY, Ruczinski I, LeBlanc M, et al. Comparison of haplotype-based and tree-based SNP imputation in association studies. Genet Epidemiol (in press).
- Agerholm-Larsen B, Nordestgaard BG, Tybjaerg-Hansen A. (2000) ACE gene polymorphism in cardiovascular disease: meta-analysis of small and large studies in whites. Arterioscler Thromb Vasc Biol 20:484–92.
[Abstract/Free Full Text] - Marchini J, Donnelly P, Cardon LR. (2005) Genome-wide strategies for detecting multiple loci that influence complex diseases. Nat Gen 37:413–17.[CrossRef][Web of Science][Medline]
- Kooperberg C and Ruczinski I. (2005) Identifying interacting SNPs using Monte Carlo logic regression. Genet Epidemiol 28:157–70.[CrossRef][Web of Science][Medline]
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  D. J. Miller, Y. Zhang, G. Yu, Y. Liu, L. Chen, C. D. Langefeld, D. Herrington, and Y. Wang An algorithm for learning maximum entropy probability models of disease risk that efficiently searches and sparingly encodes multilocus genomic interactions Bioinformatics, October 1, 2009; 25(19): 2478 - 2485. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||