Volume 1 Supplement 1
Genetic Analysis Workshop 15: Gene Expression Analysis and Approaches to Detecting Multiple Functional Loci
Model selection based on logistic regression in a highly correlated candidate gene region
- Hae-Won Uh^{1}Email author,
- Bart JA Mertens^{1},
- Henk Jan van der Wijk^{1},
- Hein Putter^{1},
- Hans C van Houwelingen^{1} and
- Jeanine J Houwing-Duistermaat^{1}
DOI: 10.1186/1753-6561-1-S1-S114
© Uh et al; licensee BioMed Central Ltd. 2007
Published: 18 December 2007
Abstract
Our aim is to develop methods for identifying a (causal) variant or variants from a dense panel of single-nucleotide polymorphisms (SNPs) that are genotyped on the evidence of previous studies. Because a large number of SNPs are in close proximity to each other, the magnitude of linkage disequilibrium (LD) plays an important role. Namely, highly correlated SNPs may hamper standard methods such as multivariate logistic regression due to multicolinearity between the covariates. Sequences of models with high dimension naturally raise questions about model selection strategies. We investigate three variable selection methods based on logistic regression. The penalties on stepwise selection were imposed using the Akaike's Information Criterion (AIC), and using the lasso penalty. Finally, a Bayesian variable-selection logistic regression model was implemented. The methods are illustrated using the simulated dense SNPs including the causal DR/C locus on chromosome 6. We also evaluate model selection in terms of average prediction error across nine replicates. We conclude that for the Genetic Analysis Workshop 15 (GAW15) data, the newly developed Bayesian selection method performs well.
Background
When a large number of potentially causative sites have been determined, the next question is how to distinguish the sites that have a causal role from the ones that show disease association because of linkage disequilibrium (LD). Stepwise logistic regression has been suggested to identify the relative importance of variants at different sites [1]. In order to deal with high correlation between the single-nucleotide polymorphisms (SNPs), the lasso penalty was applied for model selection, which shrinks some coefficients to zero for sufficiently large penalty [2]. Subsequently, we contrast results with an explicit variable selection implemented within fully Bayesian framework.
Methods
Model selection using penalties
Here β = {β_{0}, β_{ j }} and the vector x_{ i }includes the constant term 1.
In case of a large number of predictors, it is often desirable to determine a smaller subset with the strongest effects. Our first strategy is to consider stepwise selection with Akaike's Information Criterion (AIC) [2], as defined by AIC = -2l(β) + 2k. Here, k is the number of parameters included in the model, and AIC penalizes for the addition of parameters.
where s = 1 and ||β||_{1} = ∑_{ j }|β_{ j }|, the L_{1} norm of the parameter vector β. This lasso-type penalization can be useful for variable selection, because it shrinks some coefficients to zero. Note that only the β_{ j }values are subject to penalization, not the offset β_{0}. An optimal λ can be determined by AIC and 10-fold cross-validation (CV).
From a Bayesian point of view, Eq. (1.2) can be seen as the posterior mode for combining a flat prior β_{0}, and independently normally distributed β_{ j }values. Therefore, lasso parameter λ_{1} can be seen as the inverse of the variance of the prior. From this perspective, it becomes clear why Markov Chain Monte Carlo (MCMC) techniques can be applied for better handling of model uncertainty.
Bayesian model selection
We implemented a fully Bayesian approach to variable selection for the logistic regression model, with hierarchical specification on the regression parameter vector and logit link on the class probabilities. A flat prior was assumed for the intercept term β_{0}, and the β_{ j }values were independently modeled as $N(0,\frac{1}{\zeta}{\nu}^{2}{I}_{k})$-distributed random variables, where ν is a known scale and 1/ζ a rescaling factor, such that ζ has a Gamma(a, b) distribution with a and b positive real numbers. We assumed a uniform prior on SNP location choices (i.e., from a variable selection point of view, all SNPs have equal prior probability). An auxiliary variable approach was implemented to generate the logistic model via mixture modeling within an ordinary normal regression model [5, 6]. A hybrid MCMC sampler was applied, based on random choices between three steps to add (birth), remove (death), or move a SNP to the model. For more details we refer to Mertens [7], Green [8], and Holmes and Held [6]. We performed a simulation of 100,000 iterations and discarded the first 50,000 as burn-in. The classification performance from the model was investigated based on the marginal mean posterior class probabilities. Sensitivities and specificities were presented along with the receiver operating characteristic (ROC) curve. The Bayesian logistic regression variable selection model was implemented in MATLAB.
Evaluation of the model selection regarding prediction performance
Because it is difficult to defend a model that predicts poorly, we also examined prediction performance. The first simulated data set, Replicate 1, was used as a training set to determine the optimal set of variables, and Replicates 2 to 10 were used for validation by calculating average prediction error. This was defined to be the average of the classification errors from the predicted values using selected SNPs for each of nine data sets. The computations were performed with the programming language R modifying various existing packages.
Materials
We used ten replicates of simulated data, a dense map of 17,820 SNPs on chromosome 6, modeled after the rheumatoid arthritis (RA) data. We selected a high LD region of approximately 5 Mb including the trait loci (DR/C locus) with a large trait effect; we knew the "answers". Setting a threshold of p-value < 0.001 using the Cochran-Armitage trend test, we obtained 73 SNPs. Further, we chose 200 cases and 200 controls by the RA status.
Results
Selection of SNPs
Selection methods | No. selected SNPs | Selected SNPs |
---|---|---|
Stepwise/AIC | 9 | 2823 3301 3379 3384 3394 3437^{a} 3439 3474 3477 |
Lasso/AIC | 29 | 2823 2826 2827 2848 2859 3286 3301 3310 3352 3366 3379 3384 3387 3394 3396 3426 3429 3430 3437 3439 3440 3447 3459 3474 3478 3481 3580 3581 3599 |
Lasso/CV | 17 | 2823 2826 2827 2848 3301 3310 3379 3387 3394 3426 3429 3430 3437 3439 3440 3474 3599 |
Bayesian | 2 | 3437 3439 |
Trait locus DR/C* | 3437 |
Evaluation of model selection
Average Prediction Error (SE)^{a} | ||||
---|---|---|---|---|
Selection methods | No. selected SNPs | Without regularization | Ridge penalty | Random forests |
Stepwise/AIC | 9 | 0.1536 (0.0049) | 0.1506 (0.0044) | 0.1586 (0.0068) |
Lasso/AIC | 29 | 0.1572 (0.0051) | 0.1450 (0.0046) | 0.1469 (0.0059) |
Lasso/CV | 17 | 0.1461 (0.0052) | 0.1428 (0.0060) | 0.1558 (0.0052) |
Bayesian | 2 | 0.1306 (0.0052) ^{b} | 0.1306 (0.0052) | 0.1336 (0.0043) |
Trait locus DR/C* | 0.1572 (0.0058) |
Discussion
When we dropped the causal SNP and analyzed data again, by stepwise selection with the AIC and by the lasso penalty method, the same remaining SNPs were selected. The corresponding average prediction error still remained at the same level. Meanwhile, the Bayesian method selected two SNPs, 3436 (with probability 87%) and 3439 (with probability 97%), between which the causal SNP 3437 was located. Because the penalty methods selected several (possibly correlated) SNPs, we applied ridge penalty [9] and random forests [10] to stabilize the system. We found that prediction performance generally improved slightly with ridge penalty regularization (Table 2). We also compared the above findings with situations in which only the causal SNP was included in the model. In all situations, prediction performance based on nine replicates remained almost at the same level for each selection method.
Additionally, we analyzed another candidate region of 5 Mb around the D locus with a small effect and moderate LD, where the causal SNP was not included. Using these data sets, which can be considered as the opposite of those used in the main analyses, none of the methods performed well in terms of prediction performance.
Our Bayesian method can (theoretically) deal with a great number of SNPs, provided that time and facilities are available. The computational efficiency is mainly achieved by integrating out regression coefficients within the ratio of marginal likelihoods. However, the usefulness for genome-wide scan remains to be evaluated.
Conclusion
All methods identified the causal SNP together with other variants. In terms of parsimony of the model and prediction performance, Bayesian method outperformed other methods. When high correlations between the SNPs are characteristics in some candidate region such as in the specific data presented in this paper, and the focus of investigation is to find a causal gene, we conclude that a Bayesian method might perform well to disentangle the structure. Figures 1 and 2 summarize our evaluation results succinctly. The plot of mean posterior class probability indicates relative importance of the selected SNPs: i.e., in average how many times these SNPs were included across all the models simulated. Additionally, the ROC curve indicates how well these models were classified.
Declarations
Acknowledgements
H.W. Uh was funded by the GENOMEUTWIN project which is supported by the European Union Contract No. QLG2-CT-2002-01254.
This article has been published as part of BMC Proceedings Volume 1 Supplement 1, 2007: Genetic Analysis Workshop 15: Gene Expression Analysis and Approaches to Detecting Multiple Functional Loci. The full contents of the supplement are available online at http://www.biomedcentral.com/1753-6561/1?issue=S1.
Authors’ Affiliations
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