## INTRODUCTION

Regression methods are commonly used in statistical analysis, and the recent move to single marker testing in genetic association studies [WTCCC, 2007] has been out of necessity on account of the large number of predictor variables (upwards of 500,000 genetic markers) to be examined. Analyzing markers together in a regression model allows one to consider the impact of markers on other markers; a weak effect may become more visible when other causal effects are already accounted for, and a false signal maybe removed by the inclusion of a stronger signal from a true causal association. However, when the number of markers is larger than the number of test subjects or when variables are highly correlated, standard regression methods become overwhelmed. Penalized regression methods offer an attractive alternative. These methods operate by shrinking the size of the coefficients, pushing the coefficients of markers with little or no apparent effect on a trait down toward zero, reducing the effective degrees of freedom and in many cases performing model selection. Some penalization methods simply reduce the magnitude of the regression coefficients, while others coerce them to be zero. In genetic association analysis, we expect only a few markers to have a real effect on our trait (i.e. to be genuinely causal, or in linkage disequilibrium (LD) with a causal variant). Thus, through use of penalization, we can find the subset of markers most associated with the disease. One potential problem with penalization approaches is that a variable typically enters the model only if it significantly improves prediction. Thus, a variable with a strong marginal effect can be overlooked if other variables explain the effect. However, arguably, one would hope that the selection procedure would select the variables that do indeed best explain the data. Ideally, one strives to mimic the true underlying model, penalizing and thus eliminating non-causal loci, while leaving true causal loci unpenalized. A good penalty should result in minimally biased estimators, a sparse model and continuity to avoid instability in model prediction.

As an illustrative example, see Figure 1. This figure shows association test results for simulated data in a region of high LD containing multiple causal loci. The trend test gives small *P*-values at many loci correlated with a causal locus, making it difficult to localize the causal locus. Penalized methods are more particular, selecting only one or several variables per causal locus.

In the genetics literature, use of these kinds of approach is just starting to emerge. Ridge regression has been used for distinguishing between causative and noncausative variants for quantitative traits [Malo et al., 2008]. For binary traits, the normal exponential-γ (NEG) distribution [Hoggart et al., 2008], elastic net [Cho et al., 2009] and group lasso [Croiseau and Cordell, 2009] have been applied in order to identify important individual single nucleotide polymorphisms (SNPs), while penalized logistic [Park and Casella, 2008] and least angle [Zhang et al., 2008] regression have been used for identifying gene-gene interactions.

Unfortunately, all penalization methods require specification of a penalization parameter (often referred to as the tuning or regularization parameter), and the parameter value yielding the optimal model is data driven. The choice of penalty parameter controls the number of variables selected: the larger the penalty, the smaller the selected subset. The value of the penalization parameter must be chosen, e.g. through cross validation, to avoid selection of a sub-optimal value. An additional problem with penalization approaches is that there are no efficient ways of obtaining a confidence interval or *P*-value for a coefficient. This must be done through a procedure such as bootstrapping, and still does not reflect a true *P*-value in the usual sense due to the complex selection procedure for the reduced model. Wu et al. [2009] have suggested calculating a leave-one-out index for each coefficient (based on likelihood ratio tests, leaving one variable out of the final model at a time), as a useful tool for comparing variables in the model. Another downfall of penalization is that many penalties not only shrink small coefficients but tend to overshrink what should be large coefficients, adversely affecting the value of the model in prediction. However, if the primary interest lies in locating possible disease genes, prediction per se is not of great concern.

Given the current interest in these types of method, the objective of this study is to evaluate the performance of several software packages for performing penalized logistic regression, using computer simulations. We focus on two different problems: first the issue of *detection* of a relevant SNP or region, and second the issue of *differentiation/**localization* i.e. fine mapping within a region and distinguishing those variants that best explain the association and are thus most likely to be causal, or be in strongest LD with the causal variants. These two separate questions have not always been clearly distinguished in the literature—for example, Malo et al. [2008] compared ridge regression with simple and multiple linear regression using receiver operating characteristic (ROC) curves, without acknowledging that the hypotheses being tested by the different methods differed. (The ROC curves for simple linear regression counted detection of a locus in LD with the true functional variant as a type 1 error—as would be correct if one were using linear regression to try to differentiate between causal and noncausal variants—but in fact simple linear regression would generally be used for *detection* of effects, not for *differentiation* of effects, and so such an observation should actually have counted towards the power.)

We simulate data under a variety of disease locus effect size and linkage disequilibrium patterns. We compare several penalties, including the elastic net, the Lasso, a pseudo-ridge, the minimax concave penalty (MCP) and the NEG shrinkage prior to standard single locus analysis (the Armitage Trend Test) and simple forward stepwise regression. Software packages were selected for ease of use and to cover a wide range of penalty functions. We explore how markers enter a model as penalties or *P*-values are varied, and report the sensitivity and specificity for each of the methods.