A constrained polynomial regression procedure for estimating the local False Discovery Rate
- Cyril Dalmasso^{1}Email author,
- Avner Bar-Hen^{2} and
- Philippe Broët^{1}
DOI: 10.1186/1471-2105-8-229
© Dalmasso et al; licensee BioMed Central Ltd. 2007
Received: 22 September 2006
Accepted: 29 June 2007
Published: 29 June 2007
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 Benjamini-Hochberg'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 high-density 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 family-wise error rate (FWER), which is defined as the probability of falsely rejecting at least one null hypothesis (e.g., the lack of relationship between gene-expression changes and a phenotype), FWER-based 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:
FDR(Γ) = Pr(H = 0|T ∈ Γ) (1)
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 Benjamini-Hochberg'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:
lFDR(t) = Pr(H = 0|T = t). (2)
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:
FDR(Γ) = E(lFDR(T)|T ∈ Γ). (3)
In this context, most of the published procedures for estimating lFDR proceed from a two-component mixture model approach, in which the marginal distribution of the test statistic can be written:
f(t) = π_{0}f_{0}(t) + (1 - π_{0})f_{1}(t). (4)
A variety of estimators have been proposed that either consider a full model-based approach (for a few [7–10]) 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 p-values. For the last three procedures [13–15], the p-values 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 p-values are transformed into z-values 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 z-values as summarized below.
The density f is non-parametrically 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($\widehat{f}$(z)) and the variance is deduced by quadratically approximating log($\widehat{f}$(z)) for central z-values (for which f_{1}(z) is supposed to be null). The proportion π_{0} is then estimated by the ratio of the means $\overline{f(z)}/\overline{{f}_{0}(z)}$ calculated from these central z-values. The lFDR is finally estimated by $lFDR(z)=\stackrel{\_}{{\pi}_{0}}\stackrel{\_}{{f}_{0}(z)/}\stackrel{\_}{f(z)}$. It should be noted that in addition to the normality assumption for the z-values under the null hypothesis, the procedure is also based on the assumptions that central z-values 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 p-values are uniformly distributed under the null hypothesis (f_{0} = 1), the procedure is based on the separate estimations of π_{0} and f .
where $\widehat{F}$ is the empirical cumulative distribution function of the p-values. The resulting estimator for this lFDR is then $l\stackrel{\_}{FDR}({p}_{(i)})=\frac{m{\widehat{\pi}}_{0}({p}_{(i+1)}-{p}_{(i-1)})}{2}$. 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 p-values 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.
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 Kolmogoroff-Smirnoff 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 $l\stackrel{\_}{FDR}=\stackrel{\_}{{\pi}_{0}}/\widehat{f}$.
Broberg (2005) [15]
The procedure proposed by Broberg to estimate lFDR is also based on the assumption that the p-values 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 p-values 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]}$\widehat{f}$(p). Then, lFDR is estimated by $l\stackrel{\_}{FDR}=\frac{\stackrel{\_}{{\pi}_{0}}}{\widehat{f}}$.
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 two-class comparison study with normalized log-ratio 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 gene-expression 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_{i 1}= 0. For the second condition (j = 2), a proportion π_{0} of genes was simulated with μ_{i 2}= 0 (unmodified genes), while modified genes were simulated using three different configurations: (a) μ_{i 2}= 1 for the first half, μ_{i 2}= 2 for the second half; (b) μ_{i 2}= 0.5 for the first half, μ_{i 2}= 1 for the second half; and (c) μ_{i 2}= 0.5 for the first third, μ_{i 2}= 1 for the second third and μ_{i 2}= 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 so-called 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: ${y}_{ijk}=\sqrt{\rho}{a}_{jk}+\sqrt{1-\rho}{x}_{ijk}$ with ρ = 0.5. Thus, in each group, the genes have the same correlation, that is to say for i_{1} ≠ i_{2}, $Corr({y}_{{i}_{1}j},{y}_{{i}_{2}j})=0.5$. 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 $\sqrt{1-\rho}$ 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 p-values, calculated under the null hypothesis H_{0} : μ_{i 1}= μ_{i 2}, 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
where ${p}_{i}^{(k)}$i = 1,...,m are the m p-values corresponding to the k^{ th }dataset (among the 1,000 simulated datasets for each case). Here, the theoretical values lFDR(${p}_{i}^{(k)}$) are calculated from a numerical approximation of the non-centered Student's distribution [20].
Estimated values of b_{1} for the five estimators in each independent simulated case.
Case | m | π _{0} | Configuration | polfdr | twilight | LocalFDR | pava.fdr | Locfdr |
---|---|---|---|---|---|---|---|---|
1 | 500 | 0.6 | (a) | 0.032 | 0.047 | 0.067 | 0.133 | 0.869 |
2 | (b) | 0.170 | 0.149 | 0.195 | 0.160 | 0.836 | ||
3 | (c) | 0.118 | 0.123 | 0.155 | 0.096 | 0.843 | ||
4 | 0.8 | (a) | 0.062 | 0.131 | 0.041 | 0.116 | 0.695 | |
5 | (b) | 0.071 | 0.097 | 0.105 | 0.061 | 0.599 | ||
6 | (c) | 0.051 | 0.156 | 0.079 | 0.057 | 0.555 | ||
7 | 0.9 | (a) | 0.071 | 0.268 | 0.041 | 0.115 | 0.312 | |
8 | (b) | 0.054 | 0.116 | 0.052 | 0.047 | 0.376 | ||
9 | (c) | 0.050 | 0.315 | 0.049 | 0.095 | 0.265 | ||
10 | 0.98 | (a) | 0.073 | 0.387 | 0.163 | 0.139 | 0.113 | |
11 | (b) | 0.051 | 0.105 | 0.029 | 0.135 | 0.098 | ||
12 | (c) | 0.061 | 0.260 | 0.120 | 0.157 | 0.109 | ||
13 | 5,000 | 0.6 | (a) | 0.035 | 0.038 | 0.026 | 0.212 | 0.869 |
14 | (b) | 0.171 | 0.167 | 0.165 | 0.167 | 0.839 | ||
15 | (c) | 0.118 | 0.129 | 0.117 | 0.065 | 0.843 | ||
16 | 0.8 | (a) | 0.056 | 0.129 | 0.013 | 0.092 | 0.441 | |
17 | (b) | 0.071 | 0.110 | 0.073 | 0.068 | 0.502 | ||
18 | (c) | 0.051 | 0.156 | 0.053 | 0.039 | 0.406 | ||
19 | 0.9 | (a) | 0.083 | 0.268 | 0.039 | 0.056 | 0.183 | |
20 | (b) | 0.033 | 0.123 | 0.036 | 0.032 | 0.297 | ||
21 | (c) | 0.057 | 0.316 | 0.043 | 0.029 | 0.184 | ||
22 | 0.98 | (a) | 0.035 | 0.427 | 0.183 | 0.035 | 0.052 | |
23 | (b) | 0.046 | 0.071 | 0.035 | 0.027 | 0.081 | ||
24 | (c) | 0.034 | 0.293 | 0.141 | 0.035 | 0.047 |
Criterion 2
Estimated values of b_{2} for the five estimators in each independent simulated case.
Case | m | π _{0} | Configuration | polfdr | twilight | LocalFDR | pava.fdr | locfdr |
---|---|---|---|---|---|---|---|---|
1 | 500 | 0.6 | (a) | 0.015 | 0.047 | 0.000 | 0.133 | 0.000 |
2 | (b) | 0.000 | 0.016 | 0.000 | 0.000 | 0.000 | ||
3 | (c) | 0.000 | 0.039 | 0.000 | 0.010 | 0.000 | ||
4 | 0.8 | (a) | 0.057 | 0.131 | 0.000 | 0.116 | 0.000 | |
5 | (b) | 0.000 | 0.071 | 0.000 | 0.024 | 0.000 | ||
6 | (c) | 0.011 | 0.156 | 0.000 | 0.057 | 0.000 | ||
7 | 0.9 | (a) | 0.071 | 0.268 | 0.041 | 0.115 | 0.046 | |
8 | (b) | 0.005 | 0.116 | 0.013 | 0.047 | 0.031 | ||
9 | (c) | 0.040 | 0.315 | 0.049 | 0.095 | 0.050 | ||
10 | 0.98 | (a) | 0.073 | 0.387 | 0.163 | 0.139 | 0.113 | |
11 | (b) | 0.051 | 0.105 | 0.029 | 0.135 | 0.098 | ||
12 | (c) | 0.061 | 0.260 | 0.120 | 0.157 | 0.109 | ||
13 | 5,000 | 0.6 | (a) | 0.011 | 0.019 | 0.000 | 0.212 | 0.000 |
14 | (b) | 0.000 | 0.018 | 0.000 | 0.000 | 0.000 | ||
15 | (c) | 0.000 | 0.041 | 0.000 | 0.000 | 0.000 | ||
16 | 0.8 | (a) | 0.056 | 0.129 | 0.005 | 0.092 | 0.000 | |
17 | (b) | 0.000 | 0.079 | 0.000 | 0.000 | 0.000 | ||
18 | (c) | 0.016 | 0.156 | 0.000 | 0.003 | 0.000 | ||
19 | 0.9 | (a) | 0.083 | 0.268 | 0.039 | 0.056 | 0.001 | |
20 | (b) | 0.000 | 0.123 | 0.021 | 0.000 | 0.000 | ||
21 | (c) | 0.057 | 0.316 | 0.043 | 0.029 | 0.000 | ||
22 | 0.98 | (a) | 0.027 | 0.427 | 0.183 | 0.035 | 0.023 | |
23 | (b) | 0.010 | 0.071 | 0.035 | 0.027 | 0.017 | ||
24 | (c) | 0.018 | 0.293 | 0.141 | 0.035 | 0.021 |
Criterion 3
Estimated RMISE for the five estimators in each independent simulated case.
Case | m | π _{0} | Configuration | polfdr | twilight | LocalFDR | pava.fdr | locfdr |
---|---|---|---|---|---|---|---|---|
1 | 500 | 0.6 | (a) | 0.071 | 0.093 | 0.194 | 0.136 | 0.208 |
2 | (b) | 0.157 | 0.155 | 0.235 | 0.121 | 0.340 | ||
3 | (c) | 0.118 | 0.122 | 0.221 | 0.090 | 0.279 | ||
4 | 0.8 | (a) | 0.067 | 0.085 | 0.187 | 0.122 | 0.144 | |
5 | (b) | 0.095 | 0.094 | 0.201 | 0.087 | 0.193 | ||
6 | (c) | 0.083 | 0.089 | 0.194 | 0.091 | 0.157 | ||
7 | 0.9 | (a) | 0.089 | 0.085 | 0.180 | 0.112 | 0.076 | |
8 | (b) | 0.080 | 0.081 | 0.178 | 0.090 | 0.110 | ||
9 | (c) | 0.075 | 0.088 | 0.183 | 0.106 | 0.078 | ||
10 | 0.98 | (a) | 0.093 | 0.106 | 0.172 | 0.089 | 0.043 | |
11 | (b) | 0.078 | 0.100 | 0.170 | 0.077 | 0.045 | ||
12 | (c) | 0.081 | 0.098 | 0.170 | 0.079 | 0.044 | ||
13 | 5,000 | 0.6 | (a) | 0.036 | 0.040 | 0.061 | 0.191 | 0.234 |
14 | (b) | 0.149 | 0.153 | 0.152 | 0.133 | 0.343 | ||
15 | (c) | 0.101 | 0.113 | 0.117 | 0.037 | 0.278 | ||
16 | 0.8 | (a) | 0.029 | 0.047 | 0.060 | 0.088 | 0.119 | |
17 | (b) | 0.069 | 0.077 | 0.087 | 0.056 | 0.185 | ||
18 | (c) | 0.052 | 0.071 | 0.074 | 0.032 | 0.143 | ||
19 | 0.9 | (a) | 0.048 | 0.056 | 0.060 | 0.054 | 0.056 | |
20 | (b) | 0.041 | 0.050 | 0.065 | 0.037 | 0.099 | ||
21 | (c) | 0.039 | 0.063 | 0.063 | 0.035 | 0.064 | ||
22 | 0.98 | (a) | 0.042 | 0.069 | 0.062 | 0.027 | 0.021 | |
23 | (b) | 0.035 | 0.031 | 0.056 | 0.023 | 0.029 | ||
24 | (c) | 0.039 | 0.052 | 0.060 | 0.025 | 0.023 |
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 p-values 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.
Mean of all estimates of π_{0} for the five estimators in each independent simulated case.
Case | m | π _{0} | Configuration | polfdr | Twilight | LocalFDR | pava.fdr | locfdr |
---|---|---|---|---|---|---|---|---|
1 | 500 | 0.6 | (a) | 0.604 | 0.613 | 0.523 | 0.852 | 0.604 |
2 | (b) | 0.707 | 0.718 | 0.665 | 0.890 | 0.716 | ||
3 | (c) | 0.656 | 0.677 | 0.604 | 0.839 | 0.669 | ||
4 | 0.8 | (a) | 0.787 | 0.806 | 0.721 | 0.849 | 0.791 | |
5 | (b) | 0.841 | 0.860 | 0.792 | 0.915 | 0.849 | ||
6 | (c) | 0.812 | 0.839 | 0.767 | 0.890 | 0.828 | ||
7 | 0.9 | (a) | 0.863 | 0.897 | 0.824 | 0.918 | 0.886 | |
8 | (b) | 0.903 | 0.915 | 0.876 | 0.954 | 0.912 | ||
9 | (c) | 0.888 | 0.907 | 0.842 | 0.934 | 0.899 | ||
10 | 0.98 | (a) | 0.940 | 0.947 | 0.938 | 0.983 | 0.943 | |
11 | (b) | 0.953 | 0.949 | 0.949 | 0.989 | 0.937 | ||
12 | (c) | 0.951 | 0.954 | 0.948 | 0.988 | 0.947 | ||
13 | 5,000 | 0.6 | (a) | 0.614 | 0.613 | 0.469 | 0.851 | 0.616 |
14 | (b) | 0.720 | 0.718 | 0.707 | 0.888 | 0.725 | ||
15 | (c) | 0.670 | 0.676 | 0.604 | 0.838 | 0.680 | ||
16 | 0.8 | (a) | 0.801 | 0.806 | 0.729 | 0.848 | 0.805 | |
17 | (b) | 0.853 | 0.859 | 0.842 | 0.916 | 0.861 | ||
18 | (c) | 0.833 | 0.841 | 0.803 | 0.888 | 0.841 | ||
19 | 0.9 | (a) | 0.877 | 0.903 | 0.857 | 0.918 | 0.900 | |
20 | (b) | 0.920 | 0.929 | 0.914 | 0.954 | 0.929 | ||
21 | (c) | 0.901 | 0.918 | 0.883 | 0.934 | 0.915 | ||
22 | 0.98 | (a) | 0.968 | 0.974 | 0.971 | 0.982 | 0.975 | |
23 | (b) | 0.974 | 0.980 | 0.979 | 0.989 | 0.980 | ||
24 | (c) | 0.972 | 0.978 | 0.975 | 0.986 | 0.978 |
Mean square error of all estimates of π_{0} for the five estimators in each independentsimulated case.
Case | M | π _{0} | Configuration | polfdr | twilight | LocalFDR | pava.fdr | locfdr |
---|---|---|---|---|---|---|---|---|
1 | 500 | 0.6 | (a) | 0.048 | 0.084 | 0.089 | 0.255 | 0.052 |
2 | (b) | 0.126 | 0.145 | 0.088 | 0.292 | 0.130 | ||
3 | (c) | 0.086 | 0.116 | 0.054 | 0.241 | 0.089 | ||
4 | 0.8 | (a) | 0.052 | 0.090 | 0.096 | 0.057 | 0.056 | |
5 | (b) | 0.078 | 0.109 | 0.064 | 0.120 | 0.080 | ||
6 | (c) | 0.065 | 0.099 | 0.067 | 0.096 | 0.065 | ||
7 | 0.9 | (a) | 0.074 | 0.080 | 0.093 | 0.039 | 0.053 | |
8 | (b) | 0.063 | 0.080 | 0.075 | 0.065 | 0.062 | ||
9 | (c) | 0.060 | 0.084 | 0.088 | 0.050 | 0.056 | ||
10 | 0.98 | (a) | 0.077 | 0.076 | 0.069 | 0.040 | 0.064 | |
11 | (b) | 0.067 | 0.072 | 0.053 | 0.041 | 0.071 | ||
12 | (c) | 0.064 | 0.066 | 0.056 | 0.041 | 0.060 | ||
13 | 5,000 | 0.6 | (a) | 0.023 | 0.029 | 0.132 | 0.251 | 0.024 |
14 | (b) | 0.124 | 0.121 | 0.109 | 0.288 | 0.127 | ||
15 | (c) | 0.075 | 0.081 | 0.015 | 0.238 | 0.083 | ||
16 | 0.8 | (a) | 0.017 | 0.032 | 0.073 | 0.049 | 0.021 | |
17 | (b) | 0.061 | 0.066 | 0.046 | 0.116 | 0.065 | ||
18 | (c) | 0.043 | 0.050 | 0.014 | 0.089 | 0.047 | ||
19 | 0.9 | (a) | 0.039 | 0.031 | 0.045 | 0.021 | 0.019 | |
20 | (b) | 0.034 | 0.042 | 0.027 | 0.055 | 0.035 | ||
21 | (c) | 0.029 | 0.036 | 0.023 | 0.036 | 0.024 | ||
22 | 0.98 | (a) | 0.025 | 0.025 | 0.013 | 0.012 | 0.018 | |
23 | (b) | 0.024 | 0.023 | 0.009 | 0.015 | 0.018 | ||
24 | (c) | 0.023 | 0.024 | 0.011 | 0.014 | 0.018 |
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 breast-cancer studies (Hedenfalk et al. [21] and Wang et al. [22]).
Data from Hedenfalk et al. [21]
Hedenfalk et al. [21] investigated the gene-expression changes between hereditary (BRCA1, BRCA2) and non-hereditary breast cancers. The initial dataset consists of 3,226 genes with expression log-ratios 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 p-values which were calculated for each gene from a two-sample t-test.
lFDR estimations for three genes in Hedenfalk et al. data.
p-value | Rank | polfdr | twilight | LocalFDR | pava.fdr | locfdr |
---|---|---|---|---|---|---|
0.00041 | 36 | 0.05 | 0.03 | 0.02 | 0.03 | 1 |
0.01294 | 297 | 0.16 | 0.13 | 0.18 | 0.20 | 1 |
0.30534 | 1604 | 0.73 | 0.75 | 0.77 | 0.78 | 1 |
Data from Wang et al. [22]
Wang et al. [22] wanted to provide quantitative gene-expression combinations to predict disease outcomes for patients with lymph-node 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 log-transformed data.
Here, we focused on identifying gene-expression changes that distinguish patients who experienced a tumour relapse within 5 years, from patients who continued to be disease-free after a period of at least 5 years. The p-values were calculated for each gene from a two-sample t-test and the five methods were applied.
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 p-values, the locfdr procedure underestimated the lFDR for p-values close to 1, and pava.fdr tended to underestimate lFDR for all p-values. In addition, and compared to LocalFDR, our method gave smaller RMISE in all cases. When considering only the lowest p-values, 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 p-values 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 p-values 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 p-values are calculated, notably using sampling methods (for a few [25–27]).
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 gene-specific 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 p-values 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
Because the lFDR (and thus 1/f) is non-negative, the function ϕ is non-decreasing. Moreover, assuming that lFDR is non-decreasing with p (that is to say that, the closer a p-value is to one, the greater the probability that the null hypothesis is true), the function ϕ is convex. Then, we propose using a convex 10-degree polynomial for ϕ.
where p = t(p_{(1)},...,p_{(m)}) is the column vector of observed p-values, $\tilde{F(p)}=\left({\tilde{F(p)}}^{0},\mathrm{...},{\tilde{F(p)}}^{d}\right)$, $\tilde{F(p)}$ is the vector of the empirical cumulative distribution function of the p-values, 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.
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}, ${\varphi}^{\u2033}(i/m)={\displaystyle {\sum}_{k=2}^{d}\left\{k(k-1){\left(\frac{i}{m}\right)}^{k-2}\times {a}_{k}\right\}}\ge 0$ and ${\varphi}^{\prime}(i/m)={\displaystyle {\sum}_{k=1}^{d}\left\{k{\left(\frac{i}{m}\right)}^{k-1}\times {a}_{k}\right\}}\ge 0$. 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
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.
Declarations
Acknowledgements
CD received a post-doctoral grant from the Région Ile-de-France (EPIGENIC project). We thank the three anonymous reviewers for their helpful comments that have contributed improving the manuscript.
Authors’ Affiliations
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