Abstract
Background
In the context of genomic association studies, for which a large number of statistical tests are performed simultaneously, the local False Discovery Rate (lFDR), which quantifies the evidence of a specific gene association with a clinical or biological variable of interest, is a relevant criterion for taking into account the multiple testing problem. The lFDR not only allows an inference to be made for each gene through its specific value, but also an estimate of BenjaminiHochberg's False Discovery Rate (FDR) for subsets of genes.
Results
In the framework of estimating procedures without any distributional assumption under the alternative hypothesis, a new and efficient procedure for estimating the lFDR is described. The results of a simulation study indicated good performances for the proposed estimator in comparison to four published ones. The five different procedures were applied to real datasets.
Conclusion
A novel and efficient procedure for estimating lFDR was developed and evaluated.
Background
The use of current highdensity microarrays for genomic association studies leads to the simultaneous evaluation of a huge number of statistical hypotheses. Thus, one of the main problems faced by the investigator is the selection of genes (or gene products) worthy of further analysis taking multiple testing into account.
Although the oldest extension of the classical type I error rate is the familywise error rate (FWER), which is defined as the probability of falsely rejecting at least one null hypothesis (e.g., the lack of relationship between geneexpression changes and a phenotype), FWERbased procedures are often too conservative, particularly when numerous hypotheses are tested [1]. As an alternative and less stringent error criterion, Benjamini and Hochberg introduced, in their seminal paper [2], the False Discovery Rate (FDR), which is defined as the expected proportion of false discoveries among all discoveries. Here, a discovery refers to a rejected null hypothesis.
Assuming that the test statistics are independent and identically distributed under the null hypothesis, Storey [3] demonstrated that, for a fixed rejection region Γ, which is considered to be the same for every test, the FDR is asymptotically equal to the following posterior probability:
where H is the random variable such that H = 0 if the null hypothesis, noted H_{0}, is true; H = 1 if the alternative hypothesis, noted H_{1}, is true; and T is the test statistic considered for all tested hypotheses. However, one drawback is that the FDR criterion associated with a particular rejection region Γ refers to all the test statistics within the region without distinguishing between those that are close to the boundary and those that are not [4].
For this purpose, Efron [5] introduced a new error criterion called the local False Discovery Rate (lFDR) which can be interpreted as a variant of BenjaminiHochberg's FDR, that gives each tested null hypothesis its own measure of significance. While the FDR is defined for a whole rejection region, the lFDR is defined for a particular value of the test statistic. More formally:
As discussed by Efron [6], the local nature of the lFDR is an advantage for interpreting results from individual test statistics. Moreover, the FDR is the conditional expectation of the lFDR given T ∈ Γ:
In this context, most of the published procedures for estimating lFDR proceed from a twocomponent mixture model approach, in which the marginal distribution of the test statistic can be written:
Here, f_{0 }and f_{1 }are the conditional density functions corresponding to null and alternative hypotheses, respectively, and π_{0 }= Pr(H = 0). Using these notations, lFDR can be expressed as:
A variety of estimators have been proposed that either consider a full modelbased approach (for a few [710]) or estimate an upper bound of lFDR without any assumption for f_{1}. It is worth noting that, in this latter framework, the probability π_{0 }is not identifiable [11]. Thus, from equation (5), only an upper bound estimate can be obtained for lFDR.
Four procedures that do not require a distributional hypothesis for f_{1 }were introduced by Efron [6,12], Aubert et al. [13], Scheid and Spang [14] and Broberg [15]. These methods are based on the separate estimations of π_{0}, f_{0 }and f from the calculated pvalues. For the last three procedures [1315], the pvalues are supposed to be uniformly distributed under the null hypothesis, while Efron's approach estimates f_{0 }from the observed data.
Herein, we describe a novel and efficient procedure for estimating lFDR. While classical approaches are based on the estimation of the marginal density f, we propose directly estimating π_{0 }and 1/f (equation 5) within the same framework.
To situate our procedure among the four published, we briefly recall below their individual principles.
Efron (2004) [12]
For this procedure, the pvalues are transformed into zvalues for which the theoretical distribution (under the null hypothesis) is a standard normal distribution. To take into account that f_{0 }may be different from the theoretical null distribution, the parameters are estimated from the observed distribution of the zvalues as summarized below.
The density f is nonparametrically estimated using a general Poisson linear model, in which log(f(z)) is modeled as a natural spline function with seven degrees of freedom. Then, the null distribution parameters are estimated as follows. The expectation is taken as arg max((z)) and the variance is deduced by quadratically approximating log((z)) for central zvalues (for which f_{1}(z) is supposed to be null). The proportion π_{0 }is then estimated by the ratio of the means calculated from these central zvalues. The lFDR is finally estimated by . It should be noted that in addition to the normality assumption for the zvalues under the null hypothesis, the procedure is also based on the assumptions that central zvalues mainly consist of true null hypotheses and that the proportion (1  π_{0}) of modified genes is small. In particular, Efron recommends using this procedure for π_{0 }> 90%.
Aubert et al. (2004) [13]
Assuming that the pvalues are uniformly distributed under the null hypothesis (f_{0 }= 1), the procedure is based on the separate estimations of π_{0 }and f .
Ordering the pvalues (p_{(1) }≤...≤ p_{(m)}), as Aubert et al. [13] did, a natural estimator of f is:
where is the empirical cumulative distribution function of the pvalues. The resulting estimator for this lFDR is then . However, as noted by Aubert et al. [13], the variance of this estimator is large. A more stable estimator, related to the moving average methodology and corresponding to a generalization of the estimator 6, was given by the authors [13]. To estimate the probability π_{0}, Aubert et al. [13] proposed using an existing procedure, like those proposed by Storey and Tibshirani [16] or Hochberg and Benjamini [17].
Scheid and Spang (2004) [14]
As for the procedure proposed by Aubert et al., the pvalues are supposed to be uniformly distributed under the null hypothesis. Thus, this procedure is based on the separate estimations of π_{0 }and f . The marginal distribution f is estimated by dividing the interval [0, 1] into 100 equidistant bins from which a corresponding histogram is derived. A smoothing spline with seven degrees of freedom is then used to estimate f.
The probability π_{0 }is estimated by a stochastic downhill algorithm (summarized below) with the intention of finding the largest subset of genes that could follow a uniform distribution. A penalized KolmogoroffSmirnoff score related to the uniform distribution is calculated for the whole gene set:
where m is the total number of genes, J is the set of genes under consideration (first, the whole set of genes), F_{J }is the empirical cumulative distribution for the set J, and λ is a tuning parameter adaptively chosen (for details on the choice of, λ see [14]). Then, iteratively, genes are excluded so that the KolmogoroffSmirnoff score decreases. In practice, the procedure stops when the score is not reduced in 2m iterations. The score penalty takes into account the sample size m and avoids overfitting. At the end of the procedure, π_{0 }is estimated by the proportion of the remaining genes. Then, lFDR is estimated by .
Broberg (2005) [15]
The procedure proposed by Broberg to estimate lFDR is also based on the assumption that the pvalues are uniformly distributed under the null hypothesis. Then, as for the two previous methods, the procedure is based on the separate estimations of π_{0 }and f . The marginal density f of the pvalues is estimated by a Poisson regression, similar to the procedure proposed by Efron. To enforce monotony, Broberg proposed using the Pooling Adjacent Violators algorithm (see [15] for details).
The probability π_{0 }is then estimated by min_{p∈[0,1] }(p). Then, lFDR is estimated by .
Limitations of these estimators
Through different estimations of π_{0}, f_{0 }and f, these four procedures attempt to estimate an upper bound of lFDR. However, each of these methods has its own drawback. Efron's procedure [6,12] is restricted to situations in which π_{0 }> 90%. The method of Aubert et al. [13] yields an estimator with a large variance. Sheid and Spang's procedure [14] is based on an iterative algorithm and requires extensive computational time (for large datasets). Finally, Broberg's approach [15] sometimes substantially underestimates lFDR. Our procedure, developed in details under Methods, is based on a polynomial regression under monotony and convexity constraints of the inverse function of the empirical cumulative distribution. Thus, an estimated upper bound of lFDR with small variability can be expected, regardless of the true value of π_{0}.
Results
Here, we compared, through simulations, our method to the four procedures described above. The five procedures are then applied to real datasets.
Simulated data
To compare our new estimator to the four previously published procedures, we performed a simulation study. Data were generated to mimic a twoclass comparison study with normalized logratio measurements for m genes (i = 1,...,m) obtained from 20 experiments corresponding to two conditions (j = 1, 2), each with 10 replicated samples (k = 1,...,10), which corresponds to classical sample sizes for differential geneexpression studies. Two total numbers of genes were considered: one small (m = 500) and one larger (m = 5, 000). In each case, all values were independently sampled from a normal distribution, X_{i,j,k }~ N(μ_{ij}, 1). For the first condition (j = 1), all data were simulated with μ_{i1 }= 0. For the second condition (j = 2), a proportion π_{0 }of genes was simulated with μ_{i2 }= 0 (unmodified genes), while modified genes were simulated using three different configurations: (a) μ_{i2 }= 1 for the first half, μ_{i2 }= 2 for the second half; (b) μ_{i2 }= 0.5 for the first half, μ_{i2 }= 1 for the second half; and (c) μ_{i2 }= 0.5 for the first third, μ_{i2 }= 1 for the second third and μ_{i2 }= 2 for the last third.
In this way, we tried to mimic realistic situations with different patterns. Here, the distribution of modified genes is a simple mixture of two components with small expression differences (configuration (a)) and large expression differences (configuration (b)), or a more complex mixture with three components (configuration (c)).
Four different π_{0 }values were considered. Because Efron's procedure was developed for situations with π_{0 }values greater than 0.90, we used π_{0 }= 0.9 and π_{0 }= 0.98. We also considered two lower values of π_{0 }that correspond to realistic situations not considered by Efron (π_{0 }= 0.8 and π_{0 }= 0.6). In total, 2 × 3 × 4 = 24 different cases were considered.
To evaluate the behavior of the five procedures in the context of dependent data, we also generated datasets with socalled clumpy dependence (that is, datasets for which the measurements on the genes are dependent in small groups, with each group being independent of the others).
We applied the protocol described in [18] and [19] as follows: First, an independent dataset matrix (x_{ijk}) was generated, as described above. Then, for each group of 100 genes, a random vector A = {a_{jk}}, where j = 1, 2 and k = 1,..., 10 was generated from a standard normal distribution. The data matrix (y_{ijk}) was then built so that: with ρ = 0.5. Thus, in each group, the genes have the same correlation, that is to say for i_{1 }≠ i_{2}, . To render the results comparable with those obtained in the independent setting, the expectations μ_{ij }used for generating the matrix (x_{ijk}) were divided by so that the expectations of the random variables Y_{ijk }correspond to those described in configurations (a), (b) and (c) for independent data. We also considered other ρ values that gave similar results (data not shown).
In each case, the pvalues, calculated under the null hypothesis H_{0 }: μ_{i1 }= μ_{i2}, were obtained from the Student's statistic. Then, we estimated lFDR from our procedure, referred to as polfdr, and the four procedures presented in the background section, referred to as locfdr (Efron), LocalFDR (Aubert et al.), twilight (Scheid and Spang), pava.fdr (Broberg). Although these procedures were not designed to estimate the probability π_{0 }independently of lFDR, we also compared the estimators of π_{0 }obtained from the five procedures.
For each case, 1,000 datasets were simulated. To compare the different estimators, we considered three different criteria that are described below.
Criterion 1
Since the main contribution of lFDR is that it gives each tested hypothesis its own measure of significance, a small bias for any value within the whole interval [0, 1] can be preferable to a smaller bias limited to a subset of values within the interval. For this purpose and to assess the amplitude of the bias for the five procedures, we considered the infinity norm of the integrated error over the interval [0, 1] defined as follows:
and estimated by:
where i = 1,...,m are the m pvalues corresponding to the k^{th }dataset (among the 1,000 simulated datasets for each case). Here, the theoretical values lFDR() are calculated from a numerical approximation of the noncentered Student's distribution [20].
The estimated values of b_{1 }for independent data are reported in the Table 1. Although these values were always less than or equal to 0.17 for the polfdr procedure, the highest b_{1 }values for the LocalFDR, pava.fdr, twilight and locfdr procedures were 0.20, 0.21, 0.43 and 0.87, respectively. These results also showed that the locfdr method tended to substantially overestimate lDFR. For example, Figure 1 shows the expected lFDR as a function of p for each estimator with m = 500, π_{0 }= 0.8 and configuration (c) (the figures corresponding to all the other cases are provided in additional files). For these figures, the horizontal scale was logtransformed to better demonstrate the differences between the methods for small pvalues. For dependent datasets, the bias of the five estimators increased. While the bias of our estimator was always less than or equal to 0.17, the highest bias values for the methods pava.fdr, LocalFDR, twilight, locfdr were 0.20, 0.23, 0.41 and 0.87, respectively (see additional files, Table 10).
Table 1. Estimated values of b_{1 }for the five estimators in each independent simulated case.
Figure 1. Expected lFDR as a function of log(p) for each estimator with m = 500, π_{0 }= 0.8 and configuration (c).
Criterion 2
As noted under Background, the five methods were designed to estimate an lFDR upper bound. However, a negative bias can occur in some cases, leading to more false positive results than expected. In this context, we propose investigating with the five procedures the minimal negative bias, denoted b_{2}, over the interval [0, 1]:
and estimated by:
Results for independent datasets (Table 2) indicated that all the estimators have nonnegligible minimal negative biases. However, while b_{2 }was always less than or equal to 0.08 for our method, the maximal b_{2 }values were 0.11, 0.18, 0.21 and 0.43 for the estimators locfdr, LocalFDR, pava.fdr and twilight, respectively. More precisely, while our estimator slightly underestimated lFDR in some cases, when π_{0} was close to 1, the twilight method tended to underestimate lFDR for small pvalues (see Figure 1) and the pava.fdr method tended to substantially underestimate lFDR for all pvalues (for example, see Figure 2). The pava.fdr method underestimation can be attributed to the upper bound of π_{0}, which is estimated by min[(p_{(i)})], because E{min[(p_{(i)})]} ≤ min[E(p_{(i)})}]. Thus, even though this method can sometimes lead to a low bias (because its negative bias compensates for the gap between the upper bound and the true value), this estimator can generate high negative bias (see Figure 2). These results also indicated that even though the locfdr method tended to overestimate lFDR for the majority of pvalues, it also tended to underestimate lFDR for pvalues close to 1.
Table 2. Estimated values of b_{2 }for the five estimators in each independent simulated case.
Figure 2. Expected lFDR as a function of log(p) for each estimator with m = 5000, π_{0 }= 0.6 and configuration (a).
Criterion 3
To evaluate the accuracy of the five procedures at all points simultaneously, we estimated the root mean integrated square error (RMISE) of the five estimators which is defined by:
and estimated by:
As shown in Table 3, these results indicated that, except for the pava.fdr method (which can substantially underestimate lFDR, as shown above), our method gave the lowest RMISE in 15/24 cases. For the 6 cases with π_{0 }close to one (π_{0 }= 0.98), the locfdr method yielded the lowest RMISE. For the last 3 cases, the difference between our method's RMISE and the lowest value (obtained with the twilight estimator) did not exceed 0.4% (case 7). Moreover, these results also indicated that the LocalFDR estimator, despite a small bias in all cases had a higher RMISE than our estimator due to its wide variance.
Table 3. Estimated RMISE for the five estimators in each independent simulated case.
For dependent data, the RMISE of the five estimators increased and the differences were smaller. Our method yielded the lowest RMISE for 7/24 cases (see the Table 12 in additional files).
However, because in practice, some investigators might want to select only genes with low lFDR, we also reported the results obtained with the 3 criteria over the interval [0, 0.2] (See additional files). They showed that our method maintained good performances compared to the four others. Other thresholds for the pvalues were considered (10% and 40%) and gave similar results (data not shown).
To compare the performance of the different estimators of the parameter π_{0 }obtained with the different methods, we evaluated their expectations and their root mean square errors.
Table 4 gives the means of the five estimators of the parameter π_{0 }over the 1,000 simulated independent datasets (results for dependent datasets are provided in additional files, Tables 13–14). The average bias over the 24 simulated datasets was the smallest for our new method (0.1%) with a maximal positive bias of 12% (for m = 5, 000, π_{0 }= 60% and configuration (b)) and a maximal negative bias of 4% (for m = 500, π_{0 }= 98% and configuration (c)). It is worth noting that the method with the highest positive bias was locfdr (29%), while the one with the highest negative bias was pava.fdr (13%).
Table 4. Mean of all estimates of π_{0 }for the five estimators in each independent simulated case.
The estimated root mean square errors for each estimator of the parameter π_{0 }are given in Table 5. Note that the root mean square errors of our estimator were less than or equal to 0.126 for the 24 simulated datasets, while it could reach 0.130, 0.132, 0.145 and 0.292 for locfdr, LocalFDR, twilight and pava.fdr methods, respectively.
Table 5. Mean square error of all estimates of π_{0 }for the five estimators in each independentsimulated case.
Concerning computing time, our procedure was rapid, while the twilight method was cumbersome and impracticably long for large numbers of tested hypotheses. For example, the means of computing times on a personal computer (over 20 simulated datasets) for m = 5, 000, π_{0 }= 0.6 and configuration (c) were 50s, 2s, 1s, 1s and 1s for the methods twilight, LocalFDR, polfdr, pava.fdr and locfdr, respectively. For a larger number tested hypotheses m = 50, 000 (not considered in the simulation study), the means of computing times were 7,261s, 162s, 108s, 2s and 1s, respectively.
Real data
Our method, together with twilight, LocalFDR, locfdr and pava.fdr, was applied to two datasets from genomic breastcancer studies (Hedenfalk et al. [21] and Wang et al. [22]).
Data from Hedenfalk et al. [21]
Hedenfalk et al. [21] investigated the geneexpression changes between hereditary (BRCA1, BRCA2) and nonhereditary breast cancers. The initial dataset consists of 3,226 genes with expression logratios corresponding to the fluorescent intensities from a tumor sample divided by those from a common reference sample. Like Aubert et al. [13], we focused on the comparison of BRCA1 and BRCA2, and used the same pvalues which were calculated for each gene from a twosample ttest.
Figure 3 shows the estimated lFDR as a function of the pvalues for the five estimators. The five procedures yielded different results. For example, the estimated lFDR for 3 different genes are reported in Table 6. These results show clear differences between the five methods. In particular, the locfdr method gave 1 for the three genes, which can be explained by a π_{0 }value smaller than 0.9. Indeed, the estimated π_{0 }values were, respectively, 0.67, 0.67, 0.66, 0.66 and 1 for the polfdr, twilight, LocalFDR, pava.fdr and locfdr methods. Concerning the four remaining procedures, the highest differences for the three genes were respectively 3%, 7% and 5%.
Figure 3. Estimated lFDR as a function of log(p) for each estimator for the Hedenfalk et al. dataset.
Table 6. lFDR estimations for three genes in Hedenfalk et al. data.
Data from Wang et al. [22]
Wang et al. [22] wanted to provide quantitative geneexpression combinations to predict disease outcomes for patients with lymphnode negative breast cancers. Over 22,000 expression measurements were obtained from Affymetrix oligonucleotide microarray U133A GeneChips for 286 samples. The expression values calculated by the Affymetrix GeneChip analysis software MAS5 are available on the GEO website [23] with clinical data. For normalisation, the quantile method [24] was applied on logtransformed data.
Here, we focused on identifying geneexpression changes that distinguish patients who experienced a tumour relapse within 5 years, from patients who continued to be diseasefree after a period of at least 5 years. The pvalues were calculated for each gene from a twosample ttest and the five methods were applied.
Figure 4 shows the estimated lFDR as a function of the pvalues for the 5 estimators. As noted above,FDR can be estimated from lFDR using equation (3) via the mean of the estimated lFDR over the rejection region Γ. When selecting all genes so that the estimated FDR is less than 5%, our method selected 325 genes while the pava.fdr and LocalFDR methods selected 367 and 229 genes, respectively, and the twilight locfdr methods did not select any gene. It is worth noting that these strong differences have substantial consequences on the following analyses. The estimated π_{0 }values were, respectively, 0.711, 0.720, 0.714, 0.723 and 0.914 for the polfdr, pava.fdr, LocalFDR, twilight and locfdr methods.
Figure 4. Estimated lFDR as a function of log(p) for each estimator for the Wang et al. dataset.
Discussion
In the simulations, for independent datasets, the results indicated good performances for our procedure compared to the four previously published methods. Indeed, while the infinity norm b_{1 }was small in every simulated case with our procedure, it could be large for twilight and locfdr procedures. Moreover, despite the fact that the five estimators were designed with conservative biases, the twilight procedure could generate substantial negative bias for small pvalues, the locfdr procedure underestimated the lFDR for pvalues close to 1, and pava.fdr tended to underestimate lFDR for all pvalues. In addition, and compared to LocalFDR, our method gave smaller RMISE in all cases. When considering only the lowest pvalues, the simulation results showed the same trend. In summary, our new estimator exhibited more stable behavior than the four others.
For dependent datasets, simulation results led to similar conclusions. Indeed, correlations between genes do not affect the marginal distribution of the pvalues but increase the variability of the different methods and the bias of the estimators of π_{0}.
It is worth noting that a major assumption underlying our procedure, like twilight, LocalFDR and pava.fdr, relies on the distribution of the pvalues under the null hypothesis. Because the uniformity assumption is sometimes not tenable [12], Efron's procedure estimates the null distribution parameters from the observed marginal distribution. However, a limitation of that approach is the need for additional assumptions concerning the proportion of true null hypotheses. Another way to address the problem of the null distribution is how the pvalues are calculated, notably using sampling methods (for a few [2527]).
Conclusion
Herein, we proposed a novel, simple and efficient procedure for estimating the lFDR. Estimating its value is essential for genomic studies, as it quantifies genespecific evidence for being associated with the clinical or biological variable of interest. Moreover, it enables calculation of the FDR.
As seen from the simulation results, our new estimator performed well in comparison to locfdr, twilight, LocalFDR and pava.fdr. As discussed above, our method yielded a positive bias for lFDR that reflects the conservative estimation of the probability π_{0}. However, this limitation is compensated for by the fact that no assumption is required for f_{1}.
Finally, we think that extending our approach to multidimensional settings could be useful, as recently discussed by Ploner et al. [28], but will require additional investigations.
The R function polfdr that implements the procedure is available on the polfdr website [30].
Methods
As for the procedures proposed by Aubert et al., Scheid and Spang and Broberg, we make the assumption that, under the null hypothesis, the pvalues are uniformly distributed. However, instead of estimating the density f (and then taking the reciprocal of the estimate), we directly estimate the reciprocal of f.
1/f estimation
Let's consider ϕ = F^{1}(p), the inverse cumulative distribution function of the pvalues. Then, ∀p ∈ [0, 1], ϕ(F(p)) = p and 1/f is the first derivative of the function ϕ. Indeed, since ϕ ∘ F is the identity function:
Moreover:
Thus:
Equation 16, illustrated in the Figure 5, is linked to the geometrical relationship between the FDR and lFDR, as noted by Efron [6].
Figure 5. Graph of the null cumulative distribution versus the marginal cumulative distribution.
Because the lFDR (and thus 1/f) is nonnegative, the function ϕ is nondecreasing. Moreover, assuming that lFDR is nondecreasing with p (that is to say that, the closer a pvalue is to one, the greater the probability that the null hypothesis is true), the function ϕ is convex. Then, we propose using a convex 10degree polynomial for ϕ.
Therefore, we consider the following linear formulation to represent the relationship between the pvalues and the empirical cumulative distribution function:
where p = t(p_{(1)},...,p_{(m)}) is the column vector of observed pvalues, , is the vector of the empirical cumulative distribution function of the pvalues, A = t(a_{0},...,a_{d}) is the column vector of the polynomial's coefficients, d is the degree of the polynomial, and E, the error term, is a random vector for which the expectation is 0.
The estimator of the polynomial regression coeffcients' vector A can be obtained by solving the following leastsquare minimization problem with constraints:
where
We impose the constraints CA ≥ 0 on our minimization problem due to the convexity and monotony of ϕ, which can be written: ∀i ∈ {1,...,m}, and . Quadratic programming is used to calculate the solution ([29]). Finally, an estimate of 1/f(p) = ϕ'(p) is deduced from the estimated regression coefficients.
π_{0 }estimation
Classical approaches attempted to estimate π_{0 }from f(1), which is the lowest upper bound of π_{0 }based on the mixture model (4). Indeed, if no assumption is made for f_{1}, π_{0 }is not identifiable and f(1) is the lowest upper bound based on the equation (4). Here, we propose using the same model to estimate π_{0 }that is used to estimate 1/f. Therefore, we consider the reciprocal of the function ϕ. However, due to higher bias and variance at the boundaries of the domain, estimating π_{0 }from a value close (but not equal) to 1 is more appropriate. In order to obtain a less sensitive estimator with respect to ϕ', it is reasonable to estimate π_{0 }at the point where ϕ" is at its minimum:
In practice, we propose setting a = 0.5. Note that the estimation of π_{0 }is not sensitive to the choice of a and other values can be considered.
Authors' contributions
CD, ABH and PB have equally contributed to this work. All authors read and approved the final manuscript.
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Acknowledgements
CD received a postdoctoral grant from the Région IledeFrance (EPIGENIC project). We thank the three anonymous reviewers for their helpful comments that have contributed improving the manuscript.
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