Logo of nihpaAbout Author manuscriptsSubmit a manuscriptNIH Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
Ann Stat. Author manuscript; available in PMC 2009 Sep 25.
Published in final edited form as:
Ann Stat. 2009 Jan 1; 37(5A): 2178–2201.
PMCID: PMC2752029
NIHMSID: NIHMS109249

HIGH DIMENSIONAL VARIABLE SELECTION

Abstract

This paper explores the following question: what kind of statistical guarantees can be given when doing variable selection in high dimensional models? In particular, we look at the error rates and power of some multi-stage regression methods. In the first stage we fit a set of candidate models. In the second stage we select one model by cross-validation. In the third stage we use hypothesis testing to eliminate some variables. We refer to the first two stages as “screening” and the last stage as “cleaning.” We consider three screening methods: the lasso, marginal regression, and forward stepwise regression. Our method gives consistent variable selection under certain conditions.

Keywords: Lasso, Stepwise Regression, Sparsity

1. Introduction

Several methods have been developed lately for high dimensional linear regression such as the lasso (Tibshirani 1996), Lars (Efron et al. 2004) and boosting (Bühlmann 2006). There are at least two different goals when using these methods. The first is to find models with good prediction error. The second is to estimate the true “sparsity pattern,” that is, the set of covariates with nonzero regression coefficients. These goals are quite different and this paper will deal with the second goal. (Some discussion of prediction is in the appendix.) Other papers on this topic include Meinshausen and Bühlmann (2006), Candes and Tao (2007), Wainwright (2006), Zhao and Yu (2006), Zou (2006), Fan and Lv (2008), Meinshausen and Yu (2008), Tropp (2004, 2006), Donoho (2006) and Zhang and Huang (2006). In particular, the current paper builds on ideas in Meinshausen and Yu (2008) and Meinshausen (2007).

Let (X1, Y1),…,(Xn, Yn) be iid observations from the regression model

Yi=XiTβ+εi
(1)

where ε ~ N(0, σ2), Xi = (Xi1,…, Xip)T [set membership] Rp and p = pn > n. Let X be the n × p design matrix with jth column Xj = (X1j,…, Xnj)T and let Y = (Y1,…, Yn)T. Let

D={j:βj0}

be the set of covariates with nonzero regression coefficients. Without loss of generality, assume that D = {1,…, s}for some s. A variable selection procedure D n maps the data into subsets of S = {1,…, p}.

The main goal of this paper is to derive a procedure D n such that

limsupn(D^nD)1α,
(2)

that is, the asymptotic type I error is no more than α. Note that throughout the paper we use [subset or is implied by] to denote non-strict set-inclusion. Moreover, we want D n to have nontrivial power. Meinshausen and Bühlmann (2006) control a different error measure. Their method guarantees lim supn→∞ P(D nV[empty]) ≤ α where V is the set of variables not connected to Y by any path in an undirected graph.

Our procedure involves three stages. In stage I we fit a suite of candidate models, each model depending on a tuning parameter λ,

S={S^n(λ):λΛ}.

In stage II we select one of those models Ŝ n using cross-validation to select [lambda with circumflex] . In stage III we eliminate some variables by hypothesis testing. Schematically:

datastageISstageIIscreenS^nstageIIID^nclean

Genetic epidemiology provides a natural setting for applying screen and clean. Typically the number of subjects, n, is in the thousands, while p ranges from tens of thousands to hundereds of thousands of genetic features. The number of genes exhibiting a detectable association with a trait is extremely small. Indeed, for Type I diabetes only ten genes have exhibited a reproducible signal (Wellcome Trust 2007). Hence it is natural to assume that the true model is sparse. A common experimental design involves a 2-stage sampling of data, with stages 1 and 2 corresponding to the screening and cleaning processes, respectively.

In stage 1 of a genetic association study, n1 subjects are sampled and one or more traits such as bone mineral density are recorded. Each subject is also measured at p locations on the chromosomes. These genetic covariates usually have two forms in the population due to variability at a single nucleotide and hence are called single nucleotide polymorphisms (SNPs). The distinct forms are called alleles. Each covariate takes on a value (0, 1 or 2) indicating the number of copies of the less common allele observed. For a well designed genetic study, individual SNPs are nearly uncorrelated unless they are physically located in very close proximity. This feature makes it much easier to draw causal inferences about the relationship between SNPs and quantitative traits. It is standard in the field to infer that an association discovered between a SNP and a quantitative trait implies a causal genetic variant is physically located near the one exhibiting association. In stage 2, n2 subjects are sampled at a subset of the SNPs assessed in stage 1. SNPs measured in stage 2 are often those that achieved a test statistic that exceeded a predetermined threshold of significance in stage 1. In essence, the two stage design pairs naturally with a screen and clean procedure.

For the screen and clean procedure it is essential that Ŝ n has two properties as n → ∞

P(DS^n)1
(3)

and

S^n=oP(n)
(4)

where |M| denotes the number of elements in a set M. Condition (3) ensures the validity of the test in stage III while condition (4) ensures that the power of the test is not too small. Without condition (3), the hypothesis test in stage III would be biased. We will see that the power goes to 1, so taking α= αn → 0 implies consistency: P(D n = D) → 1. For fixed α, the method also produces a confidence sandwich for D, namely,

liminfnP(D^nDS^n)1α.

To fit the suite of candidate models, we consider three methods. In Method 1,

S^n(λ)={j:βj(λ)0}

where betaj(λ) is the lasso estimator, the value of β that minimizes

i=1n(YiXiTβ)2+λj=1pβj.

In Method 2, take Ŝ n(λ) to be the set of variables chosen by forward stepwise regression after λ steps. In Method 3, marginal regression, we take

S^n={j:μ^j>λ}

where [mu] j is the marginal regression coefficient from regressing Y on Xj. (This is equivalent to ordering by the absolute t-statistics since we will assume that the covariates are standardized.) These three methods are very similar to basis pursuit, orthogonal matching pursuit and thresholding; see, for example, Tropp (2004, 2006) and Donoho (2006).

Notation

Let ψ = minj[set membership]Dj|. Define the loss of any estimator [beta] by

L(β^)=1n(β^β)TXTX(β^β)=(β^β)T^n(β^β)
(5)

where [Sigma] n = n−1XT X. For convenience, when [beta] [equivalent] [beta] (λ) depends on λ we write L(λ) instead of L([beta] (λ)). If M [subset or is implied by] S, let XM be the design matrix with columns (Xj: j [set membership] M) and let β^M=(XMTXM)1XMTY denote the least squares estimator, assuming it is well-defined. Note that our use of Xj differs from standard ANOVA notation. Write Xλ instead of XM when M = Ŝ n(λ). When convenient, we extend [beta] M to length p by setting [beta] M (j) = 0 for j [negated set membership] M. We use the norms:

v=jvj2,v1=jvj,v=maxjvj.

If C is any square matrix, let [var phi](C) and Φ(C) denote the smallest and largest eigenvalues of C. Also, if k is an integer define

φn(k)=minM:M=kφ(1nXMTXM),andΦn(k)=maxM:M=kΦ(1nXMTXM).

We will write zu for the upper quantile of a standard Normal, so that P(Z > zu) = u where Z ~ N (0, 1).

Our method will involve splitting the data randomly into three groups An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg , An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms109249ig3.jpg . For ease of notation, assume the total sample size is 3n and that the sample size of each group is n.

Summary of Assumptions

We will use the following assumptions throughout except in Section 8.

  • (A1) Yi=XiTβ+εi where εi ~ N (0, σ2), for i = 1, …, n.
    (A2) The dimension pn of X satisfies pn → ∞ and pnc1enc2 for some c1 > 0 and 0 ≤ c2 < 1.
    (A3) s [equivalent] |{j: βj ≠ 0}| = O(1) and ψ = min{|βj|: βj ≠ 0} > 0.
    (A4) There exist positive constants C0, C1 and κ such that P (lim supn→ ∞ Φn(n) ≤ C0) = 1 and P(lim infn→ ∞ [var phi]n(C1 log n) ≥ κ) = 1. Also, P([var phi]n(n) > 0) = 1 for all n.
  • (A5) The covariates are standardized: An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg (Xij) = 0 and E(Xij2)=1. Also, there exists 0 < B < ∞ such that P(|Xjk|B) = 1.

For simplicity, we include no intercepts in the regressions. The assumptions can be weakened at the expense of more complicated proofs. In particular, we can let s increase with n and ψ decrease with n. Similarly, the Normality and constant variance assumptions can be relaxed.

2. Error Control

Define the type I error rate q(D n) = P(D nDc[empty]) and the asymptotic error rate lim supn→ ∞ q(D n). We define the power π(D n) = P (D [subset or is implied by] D n) and the average power

πav=1sjDP(jD^n).

It is well known that controlling the error rate is difficult for at least three reasons: correlation of covariates, high dimensionality of the covariate and unfaithfulness (cancellations of correlations due to confounding). Let us briefly review these issues.

It is easy to construct examples where, q(D n) ≤ α implies that π(D n) ≈ α. Consider two models for random variables Z = (Y, X1, X2):

Model 1Model 2
X1 ~ N (0, 1)X2 ~ N (0, 1)
Y = ψX1 + N (0, 1)Y= ψX2 + N (0, 1)
X2= ρX1 + N (0, τ2)X1= ρX2 + N (0, τ2).

Under models 1 and 2, the marginal distribution of Z is P1 = N (0, Σ1) and P2 = N (0, Σ2) where

1=(ψ2+1ψρψψ1ρρψρρ2+τ2),2=(ψ2+1ρψψρψρ2+τ2ρψρ1).

Given any ε > 0 we can choose ρ sufficiently close to 1 and τ sufficiently close to 0 such that Σ1 and Σ2 are as close as we like and hence d(P1n,P2n)<ε where d is total variation distance. It follows that

P2(2D^)P1(2D^)ε1αε.

Thus, if qα then the power is less than α + ε.

Dimensionality is less of an issue thanks to recent methods. Most methods, including those in this paper, allow pn to grow exponentially. But all the methods require some restrictions on the number s of nonzero βj’s. In other words, some sparsity assumption is required. In this paper we take s fixed and allow pn to grow.

False negatives can occur during screening due to cancellations of correlations. For example, the correlation between Y and X1 can be 0 even when β1 is huge. This problem is called unfaithfulness in the causality literature; see Spirtes, Glymour and Scheines (2001) and Robins, Spirtes, Scheines and Wasserman (2003). False negatives during screening can lead to false positives during the second stage.

Let [mu] j denote the regression coefficient from regressing Y on Xj. Fix js and note that

μjE(μ^j)=βj+1kskjβkρkj

where ρkj = corr(Xk, Xj). If

1kskjβkρkjβj

then μj ≈ 0 no matter how large βj is. This problem can occur even when n is large and p is small.

For example, suppose that β = (10, −10, 0, 0) and that ρ(Xi, Xj) = 0 except that ρ(X1, X2) =ρ(X1, X3) = ρ(X2, X4) = 1 − ε where ε > 0 is small. Then

β=(10,10,0,0)butμ(0,0,10,10).

Marginal regression is extremely susceptible to unfaithfulness. The lasso and forward stepwise, less so. However, unobserved covariates can induce unfaithfulness in all the methods.

3. Loss and Cross-validation

Let Xλ = (Xj: j [set membership] Ŝ n(λ)) denote the design matrix corresponding to the covariates in Ŝ n(λ) and let [beta] (λ) be the least squares estimator for the regression restricted to Ŝ n(λ), assuming the estimator is well defined. Hence, β^(λ)=(XλTXλ)1XλTY. More generally, [beta] M is the least squares estimator for any subset of variables M. When convenient, we extend [beta] (λ) to length p by setting [beta] j(λ) = 0 for j [negated set membership] Ŝ n(λ).

3.1. Loss

Now we record some properties of the loss function. The first part of the following lemma is essentially Lemma 3 of Meinshausen and Yu (2008).

Lemma 3.1

Let Mm+={MS:Mm,DM}. Then,

P(supMMm+L(β^M)4mlogpnφn(m))1.
(6)

Let Mm={MS:Mm,DM}. Then,

P(infMMmL(β^M)ψ2φn(m+s))1.
(7)

3.2. Cross-validation

Recall that the data have been split into groups An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg , An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg , and An external file that holds a picture, illustration, etc.
Object name is nihms109249ig3.jpg each of size n. Construct [beta] (λ) from An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg and let

L^(λ)=1nXiD2(YiXiTβ^(λ))2.
(8)

We would like L (λ) to order the models the same way as the true loss L(λ) (defined after equation (5)). This requires that, asymptotically, L (λ) − L(λ) ≈ δn where δn does not involve λ. The following bounds will be useful. Note that L(λ) and L (λ) are both step functions that only change value when a variable enters or leaves the model.

Theorem 3.2

Suppose that maxλ[set membership]Λnn(λ)| ≤ kn. Then there exists a sequence of random variables δn = OP (1) that do not depend on λ or X, such that, with probability tending to 1,

supλΛnL(λ)L^(λ)δn=OP(knn1c2)+OP(knn).
(9)

4. Multi-Stage Methods

The multi-stage methods use the following steps. As mentioned earlier, we randomly split the data into three parts An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg , An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms109249ig3.jpg which we take to be of equal size.

  1. Stage I. Use An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg to find Ŝ n (λ) for each λ.
  2. Stage II. Use An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg to find [lambda with circumflex] by cross-validation and let Ŝ n = Ŝ n ([lambda with circumflex] )
  3. Stage III. Use An external file that holds a picture, illustration, etc.
Object name is nihms109249ig3.jpg to find the least squares estimate [beta] for the model Ŝ n. Let

D^n={jS^n:Tj>cn}

where Tj is the usual t-statistic, cn = zα/2m and m = |Ŝ n|

4.1. The Lasso

The lasso estimator (Tibshirani 1996) beta(λ) minimizes

Mλ(λ)=i=1n(YiXiTβ)2+λj=1pβj

and let Ŝ n(λ) = {j: betaj (λ) ≠ 0}. Recall that [beta] (λ) is the least squares estimator using the covariates in Ŝ n (λ).

Let kn = A log n where A > 0 is a positive constant.

Theorem 4.1

Assume that (A1)–(A5) hold. Let Λn = {λ: |Ŝ n(λ)| ≤ kn}. Then:

  1. The true loss overfits: P(D [subset or is implied by] Ŝ n*)) → 1 where λ* = argminλ[set membership]Λn L(λ).
  2. Cross-validation also overfits: P(D [subset or is implied by] Ŝ n([lambda with circumflex] )) →1 where [lambda with circumflex] = argminλ[set membership]ΛnL (λ).
  3. Type I error is controlled: lim supn→ ∞ P(DcD n[empty]) ≤ α

If we let α = αn → 0 then D n is consistent for variable selection.

Theorem 4.2

Assume that (A1)–(A5) hold. Let αn → 0 and nαn. Then, the multi-stage lasso is consistent,

P(D^n=D)1.
(10)

The next result follows directly. The proof is thus omitted.

Theorem 4.3

Assume that (A1)–(A5) hold. Let α be fixed. Then (D n; Ŝ n) forms a confidence sandwich:

liminfnP(D^nDS^n)1α.
(11)

Remark 4.4

This confidence sandwich is expected to be conservative in the sense that the coverage can be much larger than 1 − α.

4.2. Stepwise Regression

The version of stepwise regression we consider is as follows. Let kn = A log n for some A > 0.

  1. Initialize: Res = Y, λ = 0, Ŷ = 0, and Ŝ n (λ) = [empty].
  2. Let λλ+ 1. Compute [mu] j = n−1left angle bracketXj, Resright angle bracket for j = 1, …, p.
  3. Let J= argmaxj |[mu] j|. Set Ŝ n(λ) = {Ŝ n(λ −1), J}. Set Ŷ = Xλ[beta] (λ) where β^λ=(XλTXλ)1XλTY and let Res = YŶ .
  4. If λ = kn stop. Otherwise, go to step 2.

For technical reasons, we assume that the final estimator xT[beta] is truncated to be no larger than B. Note that λ is discrete and Λn = {0, 1, …, kn}.

Theorem 4.5

With Ŝ n(λ) defined as above, the statements of Theorems 4.1, 4.2 and 4.3 hold.

4.3. Marginal Regression

This is probably the oldest, simplest and most common method. It is quite popular in gene expression analysis. It is used to be regarded with some derision but has enjoyed a revival. A version appears in a recent paper by Fan and Lv (2008). Let Ŝ n(λ) = {j: |[mu] j| ≥ λ} where [mu] j =n−1 left angle bracketY, Xjright angle bracket.

Let μj = An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg ([mu] j) and let μ(j) denote the value of μ ordered by their absolute values:

μ(1)μ(2)

Theorem 4.6

Let kn → ∞ with kn=o(n). Let Λn = {λ: |Ŝ n(λ)| ≤ kn}. Assume that

minjDμj>μ(kn).
(12)

Then, the statements of Theorems 4.1, 4.2 and 4.3 hold.

The assumption (12) limits the degree of unfaithfulness (small partial correlations induced by cancellation of parameters). Large values of kn weaken assumption (12) thus making the method more robust to unfaithfulness, but at the expense of lower power. Fan and Lv (2008) make similar assumptions. They assume that there is a C > 0 such that |μj| ≥ C|βj| for all j which rules out unfaithfulness. However, they do not explicitly related the values of μj for j [set membership] D to the values outside D as we have done. On the other hand, they assume that Z = Σ−1/2 X has a spherically symmetric distribution. Under this assumption and their faithfulness assumption, they deduce that the μj’s outside D cannot strongly dominate the μj’s within D. We prefer to simply make this an explicit assumption without placing distributional assumptions on X. At any rate, any method that uses marginal regressions as a starting point must make some sort of faithfulness assumptions to succeed.

4.4. Modifications

Let us now discuss a few modifications of the basic method. First, consider splitting the data only into two groups An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg . Then do these steps:

  1. Stage I. Find Ŝ n(λ) for λ [set membership] Λn where |Ŝ n(λ)| ≤ kn for each λ [set membership] Λn using An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg .
  2. Stage II. Find [lambda with circumflex] by cross-validation and let Ŝ n = Ŝ n([lambda with circumflex] ) using An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg .
  3. Stage III. Find the least squares estimate [beta] Ŝ n using An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg . Let D n = {j [set membership] Ŝ n: |Tj| > cn} where Tj is the usual t-statistic.

Theorem 4.7

Choosing

cn=loglogn2knlog(2pn)α
(13)

controls asymptotic type I error.

The critical value in (13) is hopelessly large and it does not appear it can be substantially reduced. We present this mainly to show the value of the extra data-splitting step. It is tempting to use the same critical value as in the tri-split case, namely, cn = zα/2m where m = |Ŝ n| but we suspect this will not work in general. However, it may work under extra conditions.

5. Application

As an example we illustrate an analysis based on part of the Osteoporotic Fractures in Men Study (MrOS, Orwoll et al. 2005). A sample of 860 men were measured at a large number of genes and outcome measures. We consider only 296 SNPs which span 30 candidate genes for bone mineral density. An aim of the study was to identify genes associated with bone mineral density that could help in understanding the genetic basis of osteoporosis in men. Initial analyses of this subset of the data revealed no SNPs with a clear pattern of association with the phenotype; however, three SNPs, numbered (67, 277, 289) exhibited some association in the screening of the data. To further explore the effacacy of the lasso screen and clean procedure we modified the phenotype to enhance this weak signal and then reanalyzed the data to see if we could detect this planted signal.

We were interested in testing for main effects and pairwise interactions in these data; however, including all interactions results in a model with 43,660 additional terms, which is not practical for this sample size. As a compromise we selected 2 SNPs per gene to model potential interaction effects. This resulted in a model with a total of 2066 potential coefficients, including 296 main effects and 1770 interaction terms. With this model our initial screen detected 10 terms, including the three enhanced signals, 2 other main effects and 5 interactions. After cleaning, the final model detected the 3 enhanced signals, and no other terms.

6. Simulations

To further explore the screen and clean procedures, we conducted simulation experiments with four models. For each model Yi=XiTβ+εi where the measurement errors, εi and εij, are iid Normal(0, 1) and the covariates Xij’s are Normal(0, 1) (except for model D). Models differ in how Yi is linked to Xi and the dependence structure of the Xi’s. Models A, B and C explore scenarios with moderate and large p, while Model D focuses on confounding and unfaithfullness.

  1. Null model: β = (0,…,0) and the Xij’s are iid.
  2. Triangle model: βj = δ(10 − j), j = 1,…, 10, βj = 0, j > 10 and Xij’s are iid.
  3. Correlated Triangle model: as B, but with Xij(+1)=ρXij+(1ρ2)1/2εij for j > 1, and ρ = 0.5.
  4. Unfaithful model: Yi = β1Xi1 + β2Xi2 + εi, for β1 = − β2 = 10, where the Xij’s are iid for j = {1, 5, 6, 7, 8, 9, 10}, but Xi2=ρXi1+τεi2,Xi3=ρXi1+τεi10, and Xi4=ρXi2+τεi11, for τ = 0.01 and ρ = 0.95.

We used a maximum model size of kn = n1/2 which technically goes beyond the theory but works well in practice. Prior to analysis the covariates are scaled so that each has mean 0 and variance 1. The tests were initially performed using a third of the data for each of the three stages of the procedure (Table 1, top half, 3 splits). For models A, B and C each approach has Type I error less than ρ, except the stepwise procedure which has trouble with model C when n = p = 100. We also calculated the false positive rate and found it to be very low (about 10−4 when p = 100 and 10−5 when p = 1000) indicating that even when a Type I error occurs, only a very small number of terms are included erroneously. The lasso screening procedure exhibited a slight power advantage over the stepwise procedure. Both methods dominated the marginal approach. The Markov dependence structure in model C clearly challenged the marginal approach. For Model D none of the approaches controlled the Type I error.

Table 1
Size and Power of Screen and Clean Procedures using Lasso, Stepwise and Marginal regression for the screening step. For all procedures α = 0.05. For p = 100, δ = 0.5 and for p = 1000, δ = 1.5. Reported power is πav. The ...

To determine the sensitivity of the approach to using distinct data for each stage of the analysis, simulations were conducted screening on the first half of the data and cleaning on the second half (2 splits). The tuning parameter was selected using leave-one-out cross validation (Table 1, bottom half). As expected this approach lead to a dramatic increase in the power of all the procedures. More surprising is the fact that the Type I error was near α or below for models A, B and C. Clearly this approach has advantages over data splitting and merits further investigation.

A natural competitor to screen and clean procedure is a two-stage adaptive lasso (Zou, 2006). In our implementation we split the data and used one half for each stage of the analysis. At stage one, leave-one-out cross validation lasso screens the data. In stage two, the adaptive lasso, with weights wj = |[beta] j|−1, cleans the data. The tuning parameter for the lasso was again chosen using leave-one-out cross validation. Table 2 provides the size, power and false positive rate (FPR) for this procedure. Naturally, the adaptive lasso does not control the size of the test, but the FPR is small. The power of the test is greater than we found for our lasso screen and clean procedure, but this extra power comes at the cost of a much higher Type I error rate.

Table 2
Size, Power and False Positive Rate (FPR) of Two-stage Adaptive Lasso Procedure

7. Proofs

Recall that if A is a square matrix then [var phi](A) and Φ(A) denote the smallest and largest eigenvalues of A. Throughout the proofs we make use of the following fact. If v is a vector and A is a square matrix then

φ(A)v2vTAvΦ(A)v2.
(14)

We use the following standard tail bound: if Z ~ N(0, 1) then P(|Z| > t) ≤ t−1e−t2/2. We will also use the following results about the lasso from Meinshausen and Yu (2008). Their results are stated and proved for fixed X but, under the conditions (A1)–(A5), it is easy to see that their conditions hold with probability tending to one and so their results hold for random X as well.

Theorem 7.1 (Meinshausen and Yu, 2008)

Let beta(λ) be the lasso estimator.

  1. The squared error satisfies:
    P(β(λ)β222λ2sn2κ2+cmlogpnnφn2(m))1
    (15)
    where m = |Ŝ n(λ)| and c > 0 is a constant.
  2. The size of Ŝ n(λ) satisfies
    P(S^n(λ)τ2Cn2λ2)1
    (16)
    where τ2=E(Yi2).

Proof of Lemma 3.1

Let D [subset or is implied by] M and φ=φ(n1XMTXM). Then

L(β^M)=1nεTXM(XMTXM)1XMTε1n2φXMTε2=1nφjMZj2

where Zj=n1/2XjTε. Conditional on X, ZiN(0,aj2) where aj2=n1i=1nXij2. Let An2=max1jpnaj2. By Hoeffding’s inequality, (A2) and (A5), P(En) → 1 where En={An2}. So

P(max1jpnZj>4logpn)=P(max1jpnZj>4logpn,En)+P(max1jpnZj>4logpn,Enc)P(max1jpnZj>4logpn,En)+P(Enc)P(Anmax1jpnZjaj>4logpn,En)+o(1)P(max1jpnZjaj>2logpn)+o(1)=E(P(max1jpnZjaj>2logpn)X)+o(1)O(12logpn)+o(1)=o(1).

But jMZj2mmax1jpnZj2 and (6) follows.

Now we lower bound L([beta] M). Let M be such that D [not subset] M. Let A = {j: [beta] (j) ≠ 0} [union or logical sum] D. Then |A| ≤ m + s. Therefore, with probability tending to 1,

L(β^M)=1n(β^Mβ)TXTX(β^Mβ)=1n(β^Mβ)TXATXA(β^Mβ)φn(m+s)β^Mβ2=φn(m+s)jA(β^M(j)β(j))2φn(m+s)jDMc(0β(j))2φn(m+s)ψ2.

Proof of Theorem 3.2

Let denote the responses, and X the design matrix, for the second half of the data. Then = Xβ + [epsilon with tilde]. Now

L(λ)=1n(β^(λ)β)TXTX(β^(λ)β)=(β^(λ)β)T^n(β^(λ)β)

and

L^(λ)=n1YXβ^(λ)2=(β^(λ)β)Tn(β^(λ)β)+δn+2nε,X(β^(λ)β)

where δn = ||[epsilon with tilde]||2/n, and ^n=n11XTX and [Sigma]n = n−1 XT X. By Hoeffding’s inequality

P(^n(j,k)n(j,k)>ε)encε2

for some c > 0 and so

P(maxjk^n(j,k)n(j,k)>ε)pn2encε2.

Choose εn = 4/(cn1−c2). It follows that

P(maxjk^n(j,k)n(j,k)>4cn1c2)e2nc20.

Note that

{j:β^j(λ)0}{j:βj0}kn+s.

Hence, with probability tending to 1,

L(λ)L^(λ)δn4cn1c2β^(λ)β12+2ξn(λ)

for all λ [set membership] Λn, where

ξn(λ)=1niI2εiμi(λ)

and μi(λ)=XiT(β^(λ)β). Now β^(λ)β)12=OP((kn+s)2) since ||[beta] (λ)||2 = OP (kn/[var phi](kn)). Thus, ||[beta] (λ) − β||1C(kn + s) with probability tending to 1, for some C > 0. Also, |μi(λ)| ≤ B||[beta] (λ) − β||1BC(kn + s) with probability tending to 1. Let W ~ N (0, 1). Conditional on An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg ,

ξn(λ)=dσni=1nμi2(λ)WσnBC(kn+s)W

so supλΛnξn(λ)=OP(kn/n).

Proof of Theorem 4.1

(1) Let λn=τnC/kn, M = Ŝ n(λn) and m = |M |. Then, P(mkn) → 1 due to (16). Hence, P(λn [set membership] Λn) → 1. From (15),

β(λn)β22O(1kn)+OP(knlogpnn)=oP(1).

Hence, β(λn)β2=oP(1). So, for each j [set membership] D,

βj(λn)βjβj(λn)βjψ+oP(1)

and hence P(minj[set membership]D|betaj(λn)| > 0) → 1. Therefore, Γn = {λ [set membership] Λn: D [subset or is implied by] Ŝ n(λ)} is nonempty. By Lemma 3.1,

L(λn)cmlogpn/(nφ(m))=OP(knlogpn/n).
(17)

On the other hand, from Lemma 3.1,

P(infλΛnΓncL(β^λ)>ψ2φ(kn))1.
(18)

Now, n[var phi]n(kn)/(kn log pn) → ∞ and so, (17) and (18) imply that

P(infλΛnΓncL(β^λ)>L(λn))1.

Thus, if λ* denotes the minimizer of L(λ) over Λn, we conclude that P(λ* [set membership] Γn) → 1 and hence, P(D [subset or is implied by]Ŝ n(λ*)) → 1.

(2) This follows from part (1) and Theorem 3.2.

(3) Let A = Ŝ nDc. We want to show that

P(maxjATj>cn)α+o(1).

Now,

P(maxjATj>cn)=P(maxjATj>cn,DS^n)+P(maxjATj>cn,DS^n)P(maxjATj>cn,DS^n)+P(DS^n)P(maxjATj>cn,DS^n)+o(1).

Conditional on (An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg , An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg ), [beta] A is Normally distributed with mean 0 and variance matrix σ2(XATXA)1 when D [subset or is implied by] Ŝ n. Recall that

Tj(M)=ejT(XMTXM)1XMTYσ^ejT(XMTXM)1ej=β^M,jsj

where M = Ŝ n, sj2=σ^2ejT(XMTXM)1ej and ej= (0, …, 0, 1, 0, …, 0)T where the 1 is in the jth coordinate. When D [subset or is implied by] Ŝ n, each Tj, for j [set membership] A, has a t-distribution with nm degrees of freedom where m = |Ŝ n|. Also, cn/tα/2m → 1 where tu denotes the upper tail critical value for the t-distribution. Hence,

P(maxjATj>cn,DS^nD1,D2)=P(maxjATj>tα2m,DS^nD1,D2)+anα+an

where an = o(1), since |A| ≤ m. It follows that

P(maxjATj>cn,DS^n)α+o(1).

Proof of Theorem 4.2

From Theorem 4.1, P(D nDc[empty]) ≤ αn and so P(D nDc[empty]) → 0. Hence, P(D n [subset or is implied by] D) → 1. It remains to be shown that

P(DD^n)1.
(19)

The test statistic for testing βj = 0 when Ŝ n = M is

Tj(M)=ejT(XMTXM)1XMTYσ^ejT(XMTXM)1ej.

For simplicity in the proof, let us take [sigma with hat] = σ, the extension to unknown σ being straightforward. Let j [set membership] D, x2133 = {M: |M| ≤ kn, D [subset or is implied by] M}. Then,

P(jD^n)=P(jD^n,DS^n)+P(jD^n,DS^n)P(jD^n,DS^n)+P(DS^n)=P(jD^n,DS^n)+o(1)=MMP(jD^n,S^n=M)+o(1)MMP(Tj(M)<cn,S^n=M)+o(1)MMP(Tj(M)<cn)+o(1).

Conditional on An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg [union or logical sum] An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg , for each M [set membership] x2133, Tj(M) = (βj/sj) + Z where Z ~ N (0, 1). Without loss of generality assume that βj > 0. Hence,

P(Tj(M)<cnD1D2)=P(cnβjsj<Z<cnβjsj).

Fix a small ε > 0. Note that sj2σ2/(nκ). It follows that, for all large n, cnβj/sj<εn. So,

P(Tj(M)<cnD1D2)P(Z<εn)enε2/2.

The number of models in x2133 is

j=0kn(pnsjs)kn(pnskns)kn((pns)ekns)knsknpnkn

where we used the inequality

(nk)(nek)k.

So,

MMP(Tj(M)<cnD1D2)knpnknenε20

by (A2). We have thus shown that P(j [negated set membership] D n) → 0 for each j [set membership] D. Since |D| is finite, it follows that P(j [negated set membership] D n for some j [set membership] D) → 0 and hence (19).

Proof of Theorem 4.5

A simple modification of Theorem 3.1 of Barron, Cohen, Dahmen and DeVore (2008) shows that

L(kn)=1nY^knXβ2=oP(1).

(The modification is needed because Barron, Cohen, Dahmen and DeVore (2008) require Y to be bounded while we have assumed that Y is Normal. By a truncation argument, we can still derive the bound on L(kn).) So

β^knβ2L(kn)φn(kn+s)L(kn)κ=oP(1).

Hence, for any ε > 0, with probability tending to 1, ||[beta] (kn) − β||2 < ε so that |[beta] j| > ψ/2 > 0 for all j [set membership] D. Thus, P(D [subset or is implied by] Ŝ n(kn)) → 1. The remainder of the proof of part 1 is the same as in Theorem 4.1. Part 2 follows from the previous result together with Theorem 3.2. The proof of Part 3 is the same as for Theorem 4.1.

Proof of Theorem 4.6

Note that μj^μj=n1i=1nXijεi. Hence, μj^μjN(0,1/n). So, for any δ > 0,

P(maxjμ^jμj>δ)j=1pnP(μ^jμj>δ)pnδnenδ2/2c1enc2δnenδ2/20.

By (12), conclude that D [subset or is implied by] Ŝ n(λ) when λ = [mu] (kn). The remainder of the proof is the same as the proof of Theorem 4.5.

Proof of Theorem 4.7

Let A = Ŝ nDc. We want to show that

P(maxjATj>cn)α+o(1).

For fixed A, [beta] A is Normal with mean 0 but this is not true for random A. Instead we need to bound Tj. Recall that

Tj(M)=ejT(XMTXM)1XMTYσ^ejT(XMTXM)1ej=β^M,jsj

where M = Ŝn, sj2=σ^2ejT(XMTXM)1ej and ej = (0, …, 0, 1, 0, …, 0)T where the 1 is in the jth coordinate. The probabilities that follow are conditional on An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg but this is supressed for notational convenience. First, write

P(maxjATj>cn)=P(maxjATj>cn,DS^n)+P(maxjATj>cn,DS^n)P(maxjATj>cn,DS^n)+P(DS^n)P(maxjATj>cn,DS^n)+o(1).

When D [subset or is implied by] Ŝ n,

β^S^n=βS^n+(1nXS^nTXS^n)11nXS^nTε=βS^n+QS^nγS^n

where QS^n=((1/n)XS^nTXS^n)1,γS^n=n1XS^nTε, and βŜ n (j) = 0 for j [set membership]A. Now, sj2σ^2/(nC) so that

TjnCβ^S^n,jσ^nloglognβ^S^n,jσ^

for j [set membership] Ŝ n. Therefore,

P(maxjATj>cn,DS^n)P(maxjAβ^S^n,j>σ^cnnC,DS^n).

Let γ = n−1XTε. Then,

β^A2γS^nTQS^n2γS^nγS^n2κ2knmax1jpnγj2κ2.

It follows that

maxjAβ^S^n,jknmax1jpnγjκknloglognmax1jpnγj

since κ > 0. So,

P(maxjAβ^S^n,j>σ^cnnloglogn,DS^n)P(max1jpnγj>σ^cnloglognnkn).

Note that γj ~ N (0, σ2/n) and hence

E(maxjγj)2σ2log(2pn)n.

There exists εn → 0 such that P(Bn) → 1 where Bn = {(1 − εn) ≤ [sigma with hat] /σ ≤ (1 + ε)}. So,

P(max1jpnγj>σ^cnloglognnkn)P(max1jpnγj>σ^cn(1εn)loglognnkn,Bn)nknσ(1εn)cnloglognE(maxjγj)α+o(1).

8. Discussion

The multi-stage method presented in this paper successfully controls type I error while giving reasonable power. The lasso and stepwise have similar performance. Although theoretical results assume independent data for each of the three stages, simulations suggest that leave-one-out cross-validation leads to valid Type I error rates and greater power. Screening the data in one phase of the experiment and cleaning in a followup phase leads to an efficient experimental design. Certainly this approach deserves further theoretical investigation. In particular, the question of optimality is an open question.

The literature on high dimensional variable selection is growing quickly. The most important deficiency in much of this work, including this paper, is the assumption that the model Y = XTβ + ε is correct. In reality, the model is at best an approximation. It is possible to study linear procedures when the linear model is not assumed to hold as in Greenshtein and Ritov (2004). We discuss this point in the appendix. Nevertheless, it seems useful to study the problem under the assumption of linearity to gain insight into these methods. Future work should be directed at exploring the robustness of the results when the model is wrong.

Other possible extensions include: dropping the Normality of the errors, permitting non-constant variance, investigating the optimal sample sizes for each stage, and considering other screening methods besides cross-validation.

Finally let us note that the example involving unfaithfulness, that is, cancellations of parameters to make the marginal correlation much different than the regression coefficient, pose a challenge for all the methods and deserve more attention even in cases of small p.

Acknowledgments

The authors are grateful for the use of a portion of the sample from the Osteo-porotic Fractures in Men (MrOS) Study to illustrate their methodology. MrOs is supported by the National Institute of Arthritis and Musculoskeletal and Skin Diseases (NIAMS), the National Institute on Aging (NIA), and the National Cancer Institute (NCI) through grants U01 AR45580, U01 AR45614, U01 AR45632, U01 AR45647, U01 AR45654, U01 AR45583, U01 AG18197, and M01 RR000334. Genetic analyses in MrOS were supported by R01-AR051124. This work was supported by NIH grant MH057881. We also thank two referees and an AE for helpful suggestions.

Appendix

Prediction

Realistically, there is little reason to believe that the linear model is correct. Even if we drop the assumption that the linear model is correct, sparse methods like the lasso can still have good properties as shown in Greenshtein and Ritov (2004). In particular, they showed that the lasso satisfies a risk consistency property. In this appendix we show that this property continues to hold if λ is chosen by cross-validation.

The lasso estimator is the minimizer of i=1n(YiXiTβ)2+λβ1. This is equivalent to minimizing i=1n(YiXiTβ)2 subject to ||β||1 ≤ Ω, for some Ω. (More precisely, the set of estimators as λ varies is the same as the set of estimators as Ω varies.) We use this second version throughout this section.

The predictive risk of a linear predictor [ell](x) = xTβ is R(β) = An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg (Y[ell](x))2 where (X, Y) denotes a new observation. Let γ = γ(β) = (−1, β1, …, βp)T and let Γ = An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg (ZZT) where Z = (Y, X1, …, Xp). Then we can write R(β) = γTΓγ. The lasso estimator can now be written as [beta] n) = argminβ[set membership]Bn) R (β) where R (β) = γT[Gamma] γ and Γ^=n1i=1nZiZiT.

Define

β=argminβB(Ωn)R(β)

where

B(Ωn)={β:β1Ωn}.

Thus, [ell]*(x) = xT β* is the best linear predictor in the set Bn). The best linear predictor is well defined even though An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg (Y | X) is no longer assumed to be linear. Greenshtein and Ritov (2004) call an estimator [beta] n persistent, or predictive risk consistent, if

R(β^n)R(β)P0

as n → ∞.

The assumptions we make in this section are:

  • (B1) pnenξ for some 0 ≤ ξ < 1 and
  • (B2) The elements of [Gamma] satisfy an exponential inequality:
    P(Γ^jkΓjk>ε)c3enc4ε2
    for some c3, c4 > 0 and
  • (B3) There exists B0 < ∞ such that, for all n, maxj;k An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg(|ZjZk|) ≤ B0.

Condition (A2) can easily be deduced from more primitive assumptions as in Greenshtein and Ritov (2004) but for simplicity we take (A2) as an assumption. Let us review one of the results in Greenshtein and Ritov (2004). For the moment, replace (A1) with the assumption that pnnb for some b. Under these conditions, it follows that

Δnmaxj,kΓ^jkΓjk=OP(lognn).

Hence,

supβB(Ωn)R(β)R^(β)=supβB(Ωn)γT(ΓΓ^)γΔnsupβB(Ωn)γ12=Ωn2OP(lognn).

The latter term is oP (1) as long as Ωn = o((n/log n)1/4). Thus we have:

Theorem 8.1 (Greenshtein and Ritov 2004)

If Ωn = o((n/log n)1/4) then the lasso estimator is persistent.

For future reference, let us state a slightly different version of their result that we will need. We omit the proof.

Theorem 8.2

Let γ > 0 be such that ξ + γ < 1. Let Ωn = O(n(1−ξ−γ)/4). Then, under (B1) and (B2),

P(supβB(Ωn)R^(β)R(β)>1nγ/4)=O(ecnγ/2)
(20)

for some c > 0.

The estimator [beta] n) lies on the boundary of the ball Bn) and is very sensitive to the exact choice of Ωn. A potential improvement—and something that reflects actual practice—is to compute the set of lasso estimators [beta] ([ell]) for 0 ≤ [ell] ≤ Ωn and then select from that set based on cross validation. We now confirm that the resulting estimator preserves persistence. As before we split the data into An external file that holds a picture, illustration, etc.
Object name is nihms109249ig1.jpg and An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg . Construct the lasso estimators {[beta] ([ell]): 0 ≤ [ell] ≤ Ωn}. Choose [l with hat] by cross validation using An external file that holds a picture, illustration, etc.
Object name is nihms109249ig2.jpg . Let [beta] = [beta] ([l with hat] ).

Theorem 8.3

Let γ > 0 be such that ξ + γ < 1. Under (A1), (A2) and (A3), if Ωn = O(n(1−ξ−γ)/4). then the cross validated lasso estimator [beta] is persistent. Moreover,

R(β^)inf0ΩnR(β^())P0.
(21)

Proof

Let β*([ell]) = argminβ[set membership]B([ell])R(β). Define h([ell]) = R(β*([ell])), g([ell]) = R([beta] ([ell])) and c([ell]) = L ([beta] ([ell])). Note that, for any vector b, we can write R(b) = τ2 + bTΣb −2bT ρ where ρ = (An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg(Y X1), …, An external file that holds a picture, illustration, etc.
Object name is nihms109249ig4.jpg (Y Xp))T.

Clearly, h is monotone nonincreasing on [0, Ωn]. We claim that |h([ell] + δ) − h([ell])| ≤ cΩnδ where c depends only on Γ. To see this, let u = β*([ell]), v = β*([ell] + δ) and a = [ell] β*([ell] + δ)/([ell] + δ) so that a [set membership] B([ell]). Then,

h(+δ)h()=R(u)R(a)=R(v)+R(a)R(v)=h(+δ)+2δ+δρTvδ(2+δ)(+δ)2vTvh(+δ)+2δC+δ(2Ωn+δ)C

where C = maxj,kj,k| = O(1).

Next we claim that g([ell]) is Lipschitz on [0, Ωn] with probability tending to 1. Let [beta] ([ell]) = argminβ[set membership][B with circumflex] ([ell])R (β) denote the lasso estimator and set û = [beta] ([ell]) and v = [beta] ([ell] + δ). Let εn = nγ/4. From (20), the following chain of equations hold except on a set of exponentially small probability:

g(+δ)=R(v^)R^(v^)+εnR^(v)+εnR(v)+2εn=h(+δ)+2εnh()+cΩnδ+2εn=R(u)+cΩnδ+2εnR(u^)+cΩnδ+2εn=g()+cΩnδ+2εn.

A similar argument can be applied in the other direction. Conclude that

g(+δ)g()cΩnδ+2εn

except on a set of small probability.

Now let A = {0, δ, 2δ, …, } where m is the smallest integer such that ≥ Ωn. Thus, m ~ Ωn/δn. Choose δ = δn = n−3(1−ξγ)/8. Then Ωnδn → 0 and Ωn/δnn3(1−ξγ)/4. Using the same argument as in the proof of Theorem 3.2,

maxAL^(β^())R(β^())=σn

where σn = oP (1). Then,

R(β(Ωn))R(β^)L^(β^(^))+σnL^(mδn)+σng(mδn)+2σng(Ωn)+2σn+cΩnδnh(Ωn)+2σn+εn+cΩnδn=R(β(Ωn))+2σn+εn+cΩnδn

and persistence follows. To show the second result, let beta = argmin0≤[ell]≤Ωn g([ell]) and [beta with macron above] = argmin[ell][set membership]A g([ell]). Then,

R(β)L^(β)+σnL^(β¯)+σnR(β¯)+2σnR(β)+2σn+cδnΩn

and the claim follows.

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