Abstract
Empirical Bayes methods have been extensively used for microarray data analysis by modeling the large number of unknown parameters as random effects. Empirical Bayes allows borrowing information across genes and can automatically adjust for multiple testing and selection bias. However, the standard empirical Bayes model can perform poorly if the assumed working prior deviates from the true prior. This paper proposes a new rank-conditioned inference in which the shrinkage and confidence intervals are based on the distribution of the error conditioned on rank of the data. Our approach is in contrast to a Bayesian posterior, which conditions on the data themselves. The new method is almost as efficient as standard Bayesian methods when the working prior is close to the true prior, and it is much more robust when the working prior is not close. In addition, it allows a more accurate (but also more complex) non-parametric estimate of the prior to be easily incorporated, resulting in improved inference. The new method’s prior robustness is demonstrated via simulation experiments. Application to a breast cancer gene expression microarray dataset is presented. Our R package rank.Shrinkage provides a ready-to-use implementation of the proposed methodology.
Keywords: Bayesian shrinkage, Confidence intervals, Ranking bias, Robust multiple estimation
1. Introduction
Large-scale technologies, which measure many similar entities in parallel, have emerged as an important tool in biomedical research. For example, the expression level of thousands of genes is compared between cancer and normal tissues in microarray experiments. In genome-wide association studies, the log odds ratio of the association of disease status (disease vs. control) and single-nucleotide polymorphism (SNP) frequency is estimated for thousands or millions of SNPs in a single case–control study. There are two prominent features in large-scale data. First, different parameters (e.g. difference in expression levels between cancer and normal tissues for different genes) are often studied with the same set of subjects and using the same design. Second, a large majority of the underlying parameters are 0. Because of this, the unknown parameters can be profitably modeled as random effects in an empirical Bayes framework. A popular model is to treat the large number of parameters as draws from a spike-and-slab prior distribution that is a mixture of a large mass at 0 and a non-zero component. The parameters in the spike-and-slab prior can be estimated from the many parallel measurements, resulting in an empirical Bayes analysis that borrows information across different genes or SNPs. The empirical Bayes framework automatically adjusts for the large number of hypothesis tests or effect estimates. The application of empirical Bayes to large-scale testing naturally leads to the false discovery rate (FDR) and the local FDR (Benjamini and Yekutieli, 2005; Efron and others, 2001; Newton and others, 2004; Efron, 2008). The application to parallel estimation (e.g. of the log fold changes in expression level) in the microarray context includes Newton and others (2001), Kendziorski and others (2003), and Smyth (2004). There is substantial literature in this area and the reader is referred to Efron (2010) for a summary of the state of the art in the empirical Bayes approach to large-scale inference and for complete references.
This paper focuses on estimating the effect sizes, the log fold change in gene expression level in microarray data, for example. We show that a popular empirical Bayes random effects model can lead to poor performance if the form of the prior is mis-specified; this has important practical implications because in real applications we are never certain of the correct distributional form, especially in the tails of the distribution, which often produce the most interesting observations. Motivated by this sensitivity to the form of the random effects model, we propose a new rank-conditioned inference in which shrinkage and confidence intervals are based on the distribution of the measurement error conditioned on the rank of the data instead of on the data themselves as in a traditional Bayesian posterior. The primary advantage of the rank-conditioned method is that it is almost as efficient as standard Bayesian methods when the working prior is close to the true prior, and it is much more robust when the working prior is not close. In particular, the proposed method provides efficient and valid inference even when the working random effects model substantially deviates from the true model in location. The proposed method can, therefore, substantially improve empirical Bayes inference for microarray studies as well as other large-scale data such as for genome-wide association studies and flow cytometry.
To put the rank-conditioned method in the context of the broader literature, we note the following. First, we condition on rank of the observed data in order to obtain more robust estimation of effect size. This is different from Noma and others (2010), whose aim is to better rank the effect sizes. Second, the idea of combining a Bayes formulation with rank-based likelihood has previously been proposed and studied in other context; for example, Dunson and Taylor (2005) use the idea for estimating quantiles, Gu and Ghosal (2009) for estimating receiver operating characteristic curves, and Hoff (2007) in estimating semi-parametric copula. Large-scale data is an area where this idea can be more profitably used because rank of the observed data and the observed data themselves are closely correlated. We are, therefore, able to take advantage of the robust property of the rank method with little loss of efficiency compared with standard empirical Bayes. Third, improved robustness can also be achieved through a more diffuse prior for the random effects. For example, Do and others (2005) and Kim and others (2009) combine Dirichlet process with spike-and-slab prior in a non-parametric Bayes model for random effects. A more diffuse prior, however, necessarily weakens the effectiveness of shrinkage and information borrowing as seen in the simulation study in Section 4.2.
The paper is organized as follows. Section 2 describes a model for large-scale microarray data analysis. Section 3 presents our proposed ranked-conditioned inference. Section 4 consists of simulation studies assessing the performance of rank-conditioned inference. Section 5 applies the proposed method to a breast cancer microarray dataset. Section 6 is discussion.
2. Empirical Bayes model for large-scale data
For concreteness, throughout the remainder of the paper, we focus on estimating the standardized effect size in case–control microarray experiments; application of our method in other large-scale data, such as genome-wide association studies, is similar.
We begin by describing an empirical Bayes model for the log fold change in gene expression. Let 
and 
be the log expression level of the 
th gene for the 
th subject in the cancer and normal group, respectively. The total number of genes is 
so that 
. We start with the model
 |
where 
is the variance of the 
th gene expression common for the cancer and normal groups, and 
and 
are the respective sample sizes. The quantity 
is the average (log) fold change (Guo and others, 2006; Choe and others, 2005). Let 
be the mean of 
over 
, and similarly let 
be the mean of 
. It then follows that
 |
where
 |
is the standardized log fold change and 
. Note that 
and 
typically do not vary much from gene to gene in a microarray experiment so that variance 
should be relatively constant across 
.
The first stage of our empirical Bayes model is
 |
(2.1) |
where 
independently for 
. In application, the 
in the definition of 
will be replaced by its pooled estimate using 
and 
and 
will then follow a 
-distribution. For simplicity, we shall use normal error model (2.1), since the 
degrees of freedom, 
, is large for the breast cancer data in Section 6. For a smaller 
, a modified version of (2.1) based on a non-central 
-distribution can be used instead. For a genome-wide association study, 
can be the estimated log odds ratio from a logistic regression for the association between disease status and the 
th SNP, 
be the unknown true log odds ratio and 
be the standard error of estimate 
. Next, we will model 
as independent random draws from a prior 
. Given prior 
, the Bayesian inference for 
is based on the posterior distribution of 
given 
with density
 |
(2.2) |
where 
is a 
density. The posterior mean 
is a standard Bayes estimator of 
and the 
and 
quantiles of the posterior distribution provide the 
confidence limits 
and 
.
The prior 
, however, is unknown. Empirical Bayes analysis uses a working prior 
in place of 
with the parameters in 
estimated from data 
usually via maximum likelihood (Morris, 1983). Our parametric working prior 
is a three-component mixture
 |
(2.3) |
where 
is the delta function (point mass) at 0, and 
and 
respectively, model the up- and down-regulated genes. This working prior is the same as that in Noma and others (2010) with one important difference: we use (2.3) to model the distribution of the standardized differences 
instead of the raw differences 
. We show in Section 5 that modeling the standardized differences 
as draws from a common prior leads to a much better fit to a breast cancer microarray dataset.
An important practical advantage of working prior (2.3) is that the posterior distribution 
is also a mixture of the same form as (2.3) (see Noma and others, 2010; Muralidharan, 2010) for analytical formula), which makes programming much easier and computing time manageable for large-scale problems. Spike-and-slab priors such as (2.3) have been used in variable selection and shrinkage estimation (see, e.g. Ishwaran and Rao, 2005) and have played a prominent role in multiple testing (Efron and others, 2001).
3. Rank-conditioned inference
3.1. Rank-conditioned shrinkage
For our basic model (2.1), we have
 |
The Bayesian estimate 
can also be written as
 |
which reflects the fact that the conditional mean of 
, given the observed 
, is no longer 0.
For the dataset 
, let 
be the rank of 
among 
. Our rank-conditioned inference for 
is based on the conditional distribution
 |
(3.1) |
where 
is the realized value of rank 
. The rank-conditioned shrinkage estimator is defined as
 |
(3.2) |
where 
is the conditional mean of the error 
, given that 
has rank 
among 
. Given prior 
, a draw from (3.1), 
, which is error 
conditional on 
, can be generated using the following three steps:
Step 1: Generate 
from density 
independently for 
. Let 
, where 
.
Step 2: Let 
be the rank of 
among 
.
Step 3: Repeat Steps 1–2 until 
. Then output 
.
Theorem 3.1 —
Let
and
independently for
. Let
be defined as in model (2.1). then
is unbiased in the sense that
for any given
and
, when the expectation is evaluated with respect to the joint distribution of
and
.
Proof. —
Theorem 3.1 follows directly from definition (3.2) and (2.1):
Theorem 3.1 says that
corrects the ranking bias, a concept discussed as in Jeffries (2009).
In addition to point estimate
, the proposed method provides a natural confidence interval for
. Let
and
satisfy
(3.3) It follows that
We have, therefore, shown that the interval
(3.4) contains the realized
with
probability, given
when
and
are drawn as in Theorem 3.1.
Conditioning on 
in the rank-conditioned shrinkage estimator (3.2) and confidence limits (3.4) is in fact closely related to conditioning on 
itself in standard Bayes, as a larger 
generally corresponds to a higher rank. More specifically, let 
be the empirical distribution of 
. Suppose that 
, 
, can be modeled as draws from some distribution 
. It then follows from Glivenko–Cantelli theorem that 
converges uniformly to 
, the distribution of 
with 
, 
, and 
. In such a case, conditioning on 
is almost the same as conditioning on 
so long as 
is not close to 0 or 1 (the difference can be more substantial for 
close to 0 or 1). The proposed rank-conditioned inference, however, can be much more robust than standard empirical Bayes against mis-specification of 
. For this, we have the following result.
Theorem 3.2 —
Let
and
independently for
. Let
be defined as in model (2.1). In the case where the
are equal, conditional distribution (3.1) and consequently rank-conditioned estimator (3.2) and confidence limits (3.4) remain the same (and valid) when the true prior density
is replaced by density
for any given constant
.
Proof. —
The proof is straightforward. Let
and
+
as in the three steps above. When the
are equal, the rank of
is not changed when
are all translated by a constant
, so the distribution of
does not change. Theorem 3.2 then follows.
For unequal
, Theorem 3.2 remains approximately valid so long as the variation in
is small. Section 5 demonstrates that the rank-based shrinkage is in general more robust, not just against location shift. This is a unique feature of rank-conditioned shrinkage: the ranking bias
is negative for lower ranked
and positive for higher ranked
even when evaluated under a badly specified prior. In the three steps for generating
at the beginning of this section, the prior
determines which
will be output as
. As such, the effect of a grossly mis-specified
on
remains limited. A grossly mis-specified
can, however, have a much larger distorting impact on Bayes shrinkage estimator
.
Finally, a confidence interval such as (3.4) that adjusts for ranking
can be crucial for valid inference; Benjamini and Yekutieli (2005) show that the unadjusted marginal confidence interval of
can have coverage probability that differs substantially from the nominal coverage for top-ranked parameters selected based on the same data. They propose the false coverage rate controlled confidence interval as a solution to this problem. As shown in Efron (2010, pp. 230–233) however, this interval can differ markedly from the corresponding Bayes interval and can be frighteningly wide. Westfall (2005) suggests constructing empirical Bayes confidence intervals centered at the shrunken estimators; the same idea is used and further developed in Qiu and Gene Hwang (2007) and in Ghosh (2009). Our interval is similar, but is instead based on rank-conditioned shrinkage. It is generally very close to the corresponding Bayes interval when the working prior is close to the true prior.
3.2. Non-parametric update of the parametric prior
In Section 2, a parametric working prior 
is used in empirical Bayes to capture the primary structure of 
. For the rank-conditioned method, we propose a non-parametric update of density 
to density 
by formula
 |
where the posterior density 
is given by (2.2) with 
replaced by 
and with 
replaced by a generic 
. The parameters in 
will take the values of their maximum likelihood estimates. 
can be interpreted as the average of the posterior densities for 
given 
with prior 
. Vardi and others (1985) use a similar update to improve the estimated image density in positron emission tomography. They show that it is one expectation-maximization iteration and therefore always increases the (marginal) likelihood of 
. See also Eggermont and LaRiccia (1997). The use of 
in place of 
does not significantly increase the computational burden for rank-conditioned inference. The density 
could potentially be further updated but the analytical complexity and computational cost will increase drastically.
3.3. Algorithm and implementation
The three-step algorithm for drawing 
in Section 3.1 is greatly simplified for the special case of 
for all 
because the distribution of 
depends only on the rank 
and not on 
. Under this condition, Steps 2 and 3 become
Let
be the order statistics of
. Let
be the
that corresponds to
. Output
for
.
In this way, one round of Steps 1–3 generates a complete and independent set of 
. The 
in (3.2) is now simply 
and the 
and 
in (3.3) are defined by 
irrespective of the value of 
.
For most large-scale problems, the values of error standard deviation 
may not be constant but they are not far apart (say within a factor of 3 or 4) because of the inherent common design structure. For the microarray example in Section 2, 
for standardized difference 
depends only on sample size 
and 
. Therefore, 
does not differ too much unless the number of missing data points varies dramatically between genes. Similarly, in genome-wide association study, each SNP is compared between the same set of cases and controls. For such dataset, we can partition the 
observations, 
, into several sub-groups so that 
for observations within each sub-group varies within a factor of 1.5, for example. The simplified algorithm above can then be applied to each subgroup separately as an approximation. Monte Carlo Markov chain type of algorithm is under investigation to efficiently sample from the rank-conditioned distribution (3.1) without requiring 
to be constant.
4. Assessing performance of rank-conditioned inference
4.1. Simulation study 
This example is adapted from Efron (2010, pp. 230–233). Let 
and 
for all 
for model (2.1). The true 
for random effects 
is (2.3) with 
, 
, 
, 
. These parameter values are chosen to have a moderate Bayes shrinkage effect. Monte Carlo simulation is conducted to compare the Bayes shrinkage estimates 
and rank-conditioned shrinkage estimates 
under five different specifications of working prior 
. These working priors 
have the same parametric form of (2.3) but with possibly different values of 
as given in various rows of Table 1. Parameters not listed are the same as in true prior. For example, 
for all the five working priors. Our simulation study is conducted as follows:
Table 1.
Mean square error of the three methods under different model mis-specification
| Working prior |
 with 
|
 with 
|
 with 
|
|---|---|---|---|
| Same as true prior | 0.675 | 0.677 | 0.677 |
 |
0.763 | 0.710 | 0.681 |
 |
|||
 |
0.762 | 0.685 | 0.683 |
 |
|||
 |
0.702 | 0.686 | 0.683 |
 |
|||
 |
0.699 | 0.681 | 0.679 |
 |
Step 1: Generate 
, 
, from prior 
. Let 
as in model (2.1).
Step 2: Let 
be the order statistics of 
. Let 
be the 
corresponding to 
. The 
can, therefore, refer to different 
for different realizations of 
.
Step 3: Compute empirical Bayes estimate 
under working model 
. Compute the rank-conditioned estimate 
under working model 
and its non-parametric update 
, respectively.
Step 4: Let 
. Calculate the mean square loss
 |
for estimator 
and estimator 
for both 
and 
. We only include the 500 lowest and 500 highest 
in 
because these 
are most interesting in large-scale data analysis.
Steps 1–4 are replicated 1000 times and the mean square error 
for Bayes method and mean square error 
for rank-conditioned inference are estimated by averaging the squared error loss over these replications. The estimated 
and 
values are given in Table 1. In row 1, the working prior is the same as the true prior and the standard Bayes is therefore optimal. The rank-conditional inference under both 
and 
shows little loss of efficiency with almost the same mean square error. In rows 2 and 3, 
and 
in the working prior are shifted away from 
and 
in the true prior. The 
increases noticeably with this model mis-specification. The rank-conditioned inference, especially under 
, proves robust with a much smaller change in 
from row 1. In rows 4 and 5, 
and 
in the working prior are inflated or shrunk from 
and 
in the true prior. Again, rank-conditioned inference is more robust.
4.2. Simulation study 
For simulation 2, we continue to let 
and 
for all 
in model (2.1). The true prior 
of 
now has the form
 |
(4.1) |
where 
is the Gamma distribution with shape 
and scale 
. We take 
so that 
always has mean 
5 and variance 1 and 
always has mean 5 and variance 1 for any 
. For the simulation study, we generate 
and 
, 
, as specified by (2.1) and (4.1). Let 
be the 
th order statistic of 
and let 
be the corresponding 
. For every dataset 
, we compute a point estimate and 90% confidence limits for 
, 
, using three different methods. Method 1 is Bayes estimate 
and confidence limits 
and 
in Section 2 using 
in (2.3) as the working prior. In accordance with empirical Bayes, parameters 
in 
are substituted by their maximum likelihood estimates using data 
under models (2.1) and (2.3). We shall call this parametric Bayes method. Method 2 is also based on Bayes posterior but with a more diffuse prior using Dirichlet process mixture. Let 
be the Dirichlet process with base distribution 
and scaling parameter 1 and let 
be a (random) distribution drawn from this Dirichlet process. Following Do and others (2005) and Dunson (2010), we take the more diffuse prior, 
, as
 |
where 
, 
is generated as
 |
To be consistent with Method 1, 
and 
in 
and 
in 
are substituted by their maximum likelihood estimates under models (2.1) and (2.3) as in Method 1. For the inverse gamma distribution, we choose shape parameter 
and scale parameter 
, where 
is maximum likelihood estimate of variance 
in (2.3), so that the mean of the inverse gamma equals 
. Note that the prior 
is considerably more diffuse and less informative than 
due to the extra variation in 
. We call Method 2 non-parametric DP Bayes. Method 3 is the proposed rank-conditioned inference under 
, the non-parametric update of 
. Again, the parameters in 
are substituted by their maximum likelihood estimates. Let 
be the point estimate and 
be the confidence limits of 
from one of the three methods. Let 1(
) be the indicator function. The mean square error and actual coverage rate for parameter 
, 
, are estimated by averaging
 |
over 1000 replications of 
.
The simulation study is conducted for 
, 
and 
with prior 
given by (4.1); the results are given in Figures 1–3, respectively. In each figure, the left panel shows the estimated root MSE for the lowest 100 ranked genes, 
for 
, and the right panel shows the estimated actual coverage rate of nominal 90% confidence intervals for 
. Figure 1 shows the case where 
; when 
is large, the normal distribution in the working prior 
approximates 
in true prior 
extremely well. As expected, the parametric Bayes performs the best among the three methods with the smallest mean square errors and close (to nominal) actual coverage rates for all 
, 
. The non-parametric Bayes with Dirichlet process prior performs poorly as the mean square errors are large and the actual coverage rates are far off. This is not surprising because an overly diffuse prior 
does not bring the needed shrinkage. Figure 3 shows the case for 
, in which the working prior deviates substantially from the true prior. The parametric Bayes performs poorly with huge MSE and far off actual coverage rates, while the non-parametric Bayes is much superior in both MSE and coverage rates. Figure 2 is for 
, an intermediate case between 
and 
. Neither of the two methods works well especially for 
. Our proposed rank-based inference, however, performs well for all the three cases. In particular, it is only slightly worse than the parametric Bayes for 
when the prior is correctly specified. When the prior is mis-specified as in 
and 
, its mean square errors are the smallest among the three methods and its actual coverage rates are not too far off from the nominal 90%. The rank-conditioned inference therefore achieves robustness against mis-specified prior with minimal loss of efficiency under correctly specified prior. While not shown in the graphs, the superior performance of the rank-conditioned inference is similarly observed for the highest ranked 
such as 
. The difference between the methods is small for middle-ranked 
as their inference are primarily determined by the large mass at 0 which is present in both the true prior 
and working prior 
. Finally, our implementation of Method 2 is based on R package DPpackage (Jara and others, 2011). Two additional simulation studies are included in supplementary material available at Biostatistics online.
Figure 1.
Simulation study of Section 4.2 with the correct prior in (4.1) and parameter 
. The left panel is the root mean square error for parameter estimate 
, 
, and the right panel is the actual coverage rate of confidence intervals for 
at the 90% nominal level. Parametric empirical Bayes and rank-conditional inference perform similarly in this case with smaller mean square error and correct actual coverage rates. The non-parametric Bayes model with Dirichlet prior performs considerably worse compared with the other two methods in both mean square error and actual coverage rate.
Figure 3.
Simulation study of Section 4.2 with the correct prior in (4.1) and parameter 
. The rank-conditional inference is the best in terms of MSE and non-parametric Bayes with Dirichlet prior is the close second best. The parametric Bayes is the distant third, with much larger MSE for 
. For the actual coverage rates, non-parametric Bayes method with Dirichlet prior is the best and the rank-conditioned inference is the close second. The parametric Bayes again is the distant third due to its much lower actual coverage rates.
Figure 2.
Simulation study of Section 4.2 with the correct prior in (4.1) and parameter 
. The rank-conditioned method has the smallest root mean square errors and close actual confidence rates. The parametric Bayes and non-parametric Bayes with Dirichlet prior perform badly due to the large mean square error especially for 
.
5. Application to breast cancer microarray dataset
We now apply our proposed method to the breast cancer data in Wang and others (2005). This was a large Affymetrix-based gene expression profiling study of 
genes on 286 untreated patients with lymph node-negative primary breast cancer. The data are available at http://www.ncbi.nlm.nih.gov/geo/ as dataset GSE2034. We will compare gene expression level between patients who developed distant metastasis (74 subjects) and patients who were relapse-free at 5 years (135 subjects) among the 209 estrogen receptor positive patients. These data were also analyzed in Noma and others (2010). We use the gene expression model described in Section 2 with 
being the standardized sample mean difference in log gene expression level and 
being the true standardized mean difference for the 
th gene. We have 
and 
for all 
as there are no missing values for any gene. It then follows that 
in (2.1) is 
for all 
.
The maximum likelihood estimates of the parameters in the working prior 
obtained by assuming models (2.1) and (2.3) for 
are
| η1 | μ1 | ω1 | η2 | μ2 | ω2 |
|---|---|---|---|---|---|
| 0.0856 | 0.258 | 0.0426 | 0.315 | −0.159 | 0.0470 |
which suggests about 40% of non-zero 
among 
genes. In order to check the fit of the parametric prior 
, we simulated new data from the fitted 
and compared its distribution to that of the original data through the following algorithm. Let 
, where 
and 
. The percentiles of 
and 
are given in the table below, which shows excellent fit of model 
to data 
. As comparison, we also fit the model used in Noma and others (2010), which formulates in terms of the unstandardized log fold change 
. Using notation of this paper, their model is
 |
where 
has form (2.3). The discrepancy between the percentiles of the original data 
and the simulated new data 
is much larger here. Modeling the standardized log fold change therefore provides much better fit to this dataset.
**
 |
Coming back to the model in Section 2 and using the maximum likelihood estimates for 
, 
, 
, 
, 
and 
obtained above, the standard empirical Bayes estimates 
and the corresponding 90% confidence interval for all 
are then computed under 
. Rank-conditioned estimates 
and 90% intervals are also calculated under both 
and the non-parametric update 
. Results for 
that correspond to the five lowest ranked 
and to the five highest ranked 
are given in Figure 4 for the three methods. We make three observations. First, the three methods have a huge shrinkage effect on the raw estimate 
for these top genes. For example, 
but 
and 
(under both 
and 
). Second, results from empirical Bayes and rank-conditioned inference under 
and 
are very similar although the rank-conditioned confidence intervals are a little wider. The same is true for other 
not shown in Figure 4. This is not surprising, given the excellent fit of working prior 
to 
as discussed above. The agreement of the three methods and the robustness properties of the rank-conditioned inference should give us more confidence in the result. Third, an oddity of the rank-conditioned inference is that 
can be slightly larger than 
even though 
by definition. This happens when the difference in rank-conditioned bias for 
and 
as random variables exceeds their observed difference in the observed 
and 
. The same can happen to estimates of other ranks. This is generally a small peculiarity that is appropriately accounted for by the wide confidence intervals.
Figure 4.
Parametric empirical Bayes and rank-conditional inference of 
for the five lowest ranked and the five highest ranked 
for the breast cancer dataset in Section 5.
6. Discussion
We have proposed a rank-conditioned inference that can substantially improve the prior robustness of empirical Bayes inference with little loss of efficiency. More research is needed, however, to further develop and establish the proposed method. For example, in the simulations presented in Section 4, the actual coverage rates for the rank-conditioned intervals, in spite of being a substantial improvement over standard empirical Bayes, are still below the nominal 90% rate for 
and 
. We expect that it is possible to further improve the actual coverage rate by drawing on similar research in the empirical Bayes literature, such as in Morris (1983), Laird and Louis (1987), He (1992), Qiu and Gene Hwang (2007), and Gene Hwang and others (2009). Second, model (2.1) assumes that errors 
are independent, which can be unrealistic in many applications. We are currently working to relax this requirement to accommodate a more general correlation structure. Preliminary results show that the method in this paper continues to work well if the correlation of 
is mild. Details will be reported in a future manuscript. We hope this paper will stimulate more research in robust Bayes inference for large-scale data to meet the pressing analytical need in genomics and genetics.
7. Software
Our R package rank.Shrinkage provides a ready-to-use implementation of the proposed methodology. The R code for the simulation studies is available at https://sites.google.com/site/jiangangliao/.
Supplementary material
Supplementary Material is available at http://biostatistics.oxfordjournals.org.
Funding
L.G.L.’s and T.M.'s work was supported by Penn State Cancer Institute internal grant. Berg's work was supported by Grant Number UL1 TR000127 from the National Center for Advancing Translational Sciences (NCATS). The contents of this paper are solely the responsibility of the authors and do not necessarily represent the official views of the NIH.
Supplementary Material
Acknowledgements
We are grateful for the extremely thorough review and suggestions by an anonymous reviewer and an associate editor, which led to a greatly improved manuscript. Conflict of Interest: None declared.
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