 Research
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ConRegR: Extrapolative recalibration of the empirical distribution of pvalues to improve false discovery rate estimates
Biology Direct volume 6, Article number: 27 (2011)
Abstract
Background
False discovery rate (FDR) control is commonly accepted as the most appropriate error control in multiple hypothesis testing problems. The accuracy of FDR estimation depends on the accuracy of the estimation of pvalues from each test and validity of the underlying assumptions of the distribution. However, in many practical testing problems such as in genomics, the pvalues could be underestimated or overestimated for many known or unknown reasons. Consequently, FDR estimation would then be influenced and lose its veracity.
Results
We propose a new extrapolative method called Constrained Regression Recalibration (ConRegR) to recalibrate the empirical pvalues by modeling their distribution to improve the FDR estimates. Our ConRegR method is based on the observation that accurately estimated pvalues from true null hypotheses follow uniform distribution and the observed distribution of pvalues is indeed a mixture of distributions of pvalues from true null hypotheses and true alternative hypotheses. Hence, ConRegR recalibrates the observed pvalues so that they exhibit the properties of an ideal empirical pvalue distribution. The proportion of true null hypotheses (π_{0}) and FDR are estimated after the recalibration.
Conclusions
ConRegR provides an efficient way to improve the FDR estimates. It only requires the pvalues from the tests and avoids permutation of the original test data. We demonstrate that the proposed method significantly improves FDR estimation on several gene expression datasets obtained from microarray and RNAseq experiments.
Reviewers
The manuscript was reviewed by Prof. Vladimir Kuznetsov, Prof. Philippe Broet, and Prof. Hongfang Liu (nominated by Prof. Yuriy Gusev).
Background
In highthroughput biological data analysis, multiple hypothesis testing is employed to address certain biological problems. Appropriate tests are chosen for the data, and the pvalues are then computed under some distributional assumptions. Due to the large number of tests performed, error rate controls (which focus on the occurrence of false positives) are commonly used to measure the statistical significance. False discovery rate (FDR) control is accepted as the most appropriate error control. Other useful error rate controls include conditional FDR (cFDR) [1], positive FDR (pFDR) [2] and local FDR (lFDR) [3] which have similar interpretations as that of FDR. However, appropriate FDR estimation depends on the precise pvalues from each test and the validity of the underlying assumptions of the distribution.
The pvalues from multiple hypothesis testing, for n hypotheses, can be described by a mixture model g(p) (1) with two components: one component g_{0}(p) originates from true null hypotheses and follows uniform distribution U(0, 1) [4], and the other component g_{1}(p) results from true alternative hypotheses and follows a distribution confined to the pvalues close to 0 [5]. The mixing parameter, π_{0}, is the proportion of true null hypotheses in the data. More precisely,
where g_{0}(p) = 1 denotes the probability density function of a uniform distribution over (0, 1) and g_{1}(p) ≈ 0 for p close to 1. Therefore, g(p) will be close to a constant (i.e. π_{0}) for p close to 1.
FDR in multiple hypothesis testing for a given pvalue threshold α is estimated as
π_{0} can be estimated from this mixture model in equation (1) as [2]
where β is typically chosen to be 0.25, 0.5 or 0.75. These estimates are reasonable under the uniform distribution assumption of a component in this mixture model [6].
However, in many applied testing problems, the pvalues could be underestimated or overestimated for many known or unknown reasons. The violation of pvalue distribution assumptions may lead to inaccurate FDR estimation. There are many factors influencing FDR estimation in the analysis of highthroughput biological data such as microarray and sequencing studies. Dependence among the test statistics is one of the major factors [7, 8]. Usually in microarray data, there are many groups of genes having similar expression patterns and the test statistics (for example, tstatistic) are not independent within one group. The global effects in the array may also influence the dependence in the data. For example, batch and cluster effects [9, 10] always occur in the experiments and sometimes they may be the major cause of incorrectly estimated FDR.
Further, due to the "large p, small n" problem [11] for the gene expression data, some parameters such as mean and variance for each gene cannot be well estimated, or the test assumptions are not satisfied or the distribution of the statistic under null hypotheses may not be accurate. Therefore, many applied testing methods modified the standard testing methods (for example, modifying tstatistic to moderated tstatistic [12]) to increase their usability. As the modified test statistics only approximately follow some known distribution, the approximate pvalue estimation may influence the FDR estimation. Resampling strategies may better estimate the underlying distributions of the test statistics. However, due to small sample size and data correlation, the limited number of permutations and resampling bias [13] also influence the FDR estimation.
To address the above problems, we propose a novel extrapolative recalibration procedure called Constrained Regression Recalibration (ConRegR) which models the empirical distribution of pvalues in multiple hypothesis testing and recalibrates the imprecise pvalue calculation to better approximated pvalues to improve the FDR estimation. Our approach focuses on pvalues as the pvalues from true null hypotheses are expected to follow the uniform distribution and the interference from the distribution of pvalues from alternative hypotheses is expected to be minimal towards p = 1. In contrast, the estimation of the empirical null distributions of test statistics may not be accurate as their parametric form may not be known beforehand and their accuracy may depend on the data and the resampling strategy used. ConRegR first maps the observed pvalues to predefined uniformly distributed pvalues preserving their rank order and estimates the recalibration mapping function by performing constrained polynomial regression to the k highest pvalues. The constrained polynomial regression is implemented by quadratic programming solvers. Finally, the pvalues will be recalibrated using the normalized recalibration function. FDR is estimated using the recalibrated pvalues and the can be determined during ConRegR procedure. We demonstrate that our ConRegR procedure can significantly improve the estimation of FDR on simulated data, and also the environmental stress response time course microarray datasets in yeast and a human RNAseq dataset.
Methods
Under the null hypotheses, the pvalues are uniformly distributed. Hence, ConRegR first generates the uniformly distributed pvalues within [0, 1] range.
Uniformly distributed pvalue generation
Let p_{ i } denotes the pvalue of the i^{th} test (i = 1, ..., n), without loss of generality, we assume p_{1} ≥ p_{2} ≥ ... ≥ p_{ n }. If we choose a suitable k < n such that the i^{th} null hypothesis is most likely true, then p_{1}, ..., p_{ k }correspond to the order statistics of k independent uniformly distributed random variables provided p_{ i } 's i(i = 1, ..., k) are correctly estimated.
Let denote the ideal pvalues under , and suppose is known. can be defined as
Therefore, are uniformly distributed over .
Then
Using (3), (2) becomes
Since k is usually large, k/(k  1) is almost 1, therefore in (4) can be approximated as
We can estimate the recalibration function f(·), to be described below, between and and apply it to all input pvalues to output the recalibrated pvalues, i.e.
By StoneWeierstrass theorem [14], polynomial functions can well approximate any continuous function in the interval [0,1]. Therefore we use polynomial regression to estimate the recalibration function f(·) satisfying appropriate boundary and monotone constraints.
Constrained Regression Recalibration (ConRegR)
Let and x_{ i } = p_{ i } (i = 1 ... k), and the recalibration polynomial function f(·) is defined as follows,
The constraints f (0) = 0, f (1) = 1 and f' (x) > 0 should be imposed to ensure the orders of the pvalues remain the same after the transformation. Furthermore, the constraint for either f" (x) > 0 or f" (x) < 0 indicates the function f should also be a monotonic convex or monotonic concave function to deal with the situations with underestimated or overestimated pvalues separately and helps in good extrapolation. The constraints f (0) = 0 and f (1) = 1 can be easily met by scaling and shifting the regression function. Therefore, the regression function only depends on the other two constraints which can be combined into one constraint during the regression procedure.
Quadratic programming (QP) [15] is employed to estimate the regression function as follows: Let y = (y_{1}, ..., y_{ k } ) ^{T} , β = (β_{0}, ..., β_{ t })^{T}and
Equation (7) can be rewritten more succinctly as
and the constrains for the first and second order derivatives of f (X) will be Aβ ≥ b where b = (0, ..., 0) ^{T} is a 2l × 1 vector and
is a 2l × (t + 1) matrix, where a_{1}, ..., a_{ l } are l randomly generated numbers following U(0, 1) to guarantee
this constraint is valid in (0, 1), and c is chosen to be 0 (or 1) if f is desired to be convex (or concave respectively).
The least squares procedure for (8) will minimize
Minimizing (9) under Aβ ≥ b is equivalent to minimizing
under Aβ ≥ b, where Q = X^{T} X and q = X^{T} y. Therefore, the constrained polynomial regression problem can be reformulated as a quadratic programming problem.
Two further modifications
We use QuadProg package in R to solve the quadratic programming problem [16]. Due to floating point errors [17], Q = X^{T} X tends to be positive semidefinite instead of being positive definite. To get around this, we add a sufficiently small positive value (λ = 10^{10}) to the diagonal of Q to guarantee
Q' = Q + λ I_{t+1 }is positive definite and Q' replaces Q in (10).
Furthermore, the polynomial function may not accurately fit the data due to the limitation of the polynomial maximal power (usually set the maximal power t = 10). We can add the fraction of the power (i.e. a noninteger power) to increase the accuracy of the fit. For example, let , where m = 1, 2 or more.
Computational procedure
For any given k, after applying ConRegR, the estimation of and its variation (error) are given by
and
where MAD denotes the median absolute deviation. The final regression function and optimal k(k_{ best } ) are determined by examining and over k. Figure 1 illustrates how to choose k_{ best } from the function . Ideally, is not expected to change over a range of k (as shown by the blue dashed line in Figure 1) such that p_{1}, ..., p_{ k } are most likely to be from null hypotheses. If k is too large, p_{1}, ..., p_{ k } may contain too many pvalues from alternate hypotheses and may be wrongly estimated to be close to 1, in an extreme case if k is chosen to be n then = 1. However, the extrapolation in recalibration procedure may be unreliable if only a small number of pvalues (i.e. small k) are used for the regression and may fluctuate near the real π_{0} (the red curve in Figure 1). Therefore, we aim to choose optimal k(k_{ best } ) as a tradeoff to include just enough pvalues from null hypotheses for the regression to achieve good extrapolation. The k that gives stable estimate and the last minimum of is chosen to be the k_{ best } . The regression function, extrapolation and corresponding to k = k_{ best } are chosen for recalibrating pvalues and reestimating FDR.
The following is the computational procedure for a given in descending order:

1.
For each (v is the interval over k and default setting is v = [n/100]), let .

2.
Use equation (5) to compute .

3.
Use quadratic programming to obtain regression function h_{ k } , where c can be predefined or estimated by checking whether more than half of points for are above the diagonal (line from origin to (1, 1)) (c = 1) or below the diagonal (c = 0).

4.
Transform h_{ k } to to satisfy constraints f_{ k } (0) = 0, and f_{ k } (1) = 1.

5.
Repeat steps 24 for all k, and compute the and for each k. Let k_{ best } be the maximal of k which locally minimizes under the constraint of small , where the cutoff of and local minimization criteria should be predefined.

6.
Choose the final regression function f (.) under k_{ best } and output recalibrated pvalues.

7.
Reestimate the FDR using recalibrated pvalues and .
Rcode for ConRegR is attached as Additional file 1.
Results
Dependence simulation
Data dependence is one of the major causes for over or underestimated pvalues. We simulated an expression data, with dependence, Z = (z_{ ij } )(i = 1, ..., n, j = 1, ..., r) with n (n = 10000) genes and r(r = 10) replicates using the formula as follows,
where b_{ i } denotes the biological effect, d_{ ij } denotes the dependence effect. Set b_{ i } = 1, if i ≤ n(1  π_{0}) and b_{ i } = 0 if i > n(1  π_{0}). d_{ i } . = (1, 1, 1, 0, 0, 0, 0, 1, 1, 1) if and d_{ i } . = (1, 1, 1, 0, 0, 0, 0, 1, 1, 1) if ). ε_{ ij } ~ N(0, 1) is the background noise.
To compare the result, we also simulated a data set with no dependence using the same procedure but with the dependence effect d_{ ij } = 0. One sample ttest was performed to generate pvalues. Figure 2 shows the pvalue density histograms for π_{0} = 0.7 and π_{0} = 0.9. As can be seen in the plots B and D in Figure 2, the pvalue histograms from independent data have constant frequency for p ≥ 0.5 and the density near 1 indicates the . However, the pvalue histograms from dependent data (the plots A and C in Figure 2) do not have such constant frequency and pvalue density increases as pvalue increases in the neighborhood of 1. The density near 1 exceeds the respective π_{0}.
ConRegR used the above pvalues as input and output the recalibrated pvalues. The results are summarized in Figure 3. For the independent data sets, the algorithm chose k = 0.71n for π_{0} = 0.7 and k = 0.64n for π_{0} = 0.9 since it locally minimized under . The pvalues do not significantly change after regression. As such, the regression curves almost overlap with the diagonals, and the input pvalue histogram and the output pvalue histogram are very similar to each other. The FDR estimation errors (the absolute difference between FDR estimated by pvalues and real FDR) also do not significantly change after applying ConRegR and the estimation of FDR is very close to the real FDR. However, for the dependent data sets, the algorithm chose k = 0.62n for π_{0} = 0.7 and k = 0.88n for π_{0} = 0.9. The regression curves are all below the diagonals and the output pvalue histograms after applying ConRegR appears more like the ones obtained for the independent data. The accuracy of estimated FDR after applying ConRegR is substantially improved.
To study more complicated dependency situations, we generated dependent datasets with random dependence effect [8] as follows,
where ρ is the correlation constant (here we set ρ = 0.5) which determines the correlation coefficient between genes. Here b_{ i } denotes the biological effect, and d_{ j } denotes the random dependence effect. Set b_{ i } = 1, if i ≤ n(1  π_{0}) and b_{ i } = 0 if i > n(1  π_{0}), and d_{ j } ~ N(0, 1). Let ε_{ ij } N(0, 1) be the background noise. The result for π_{0} = 0.7 and π_{0} = 0.9 are shown in Additional File 2, Figure S1. Similar to the simulations of fixed dependence effect, the estimated FDR after applying ConRegR is closer to real FDR. The results of our procedure for 100 repeated simulations are summarized in the boxandwhisker plots in Figure 4. As shown in this figure, for the independent data sets, the FDR estimation errors (the mean absolute difference between real FDR and the FDR estimated by pvalues using BenjaminiHochberg method) after applying ConRegR is slightly higher. However, it is still acceptable since most simulation resulted in errors below 0.05. For the dependent data sets with fixed and random dependence effects, the FDR estimation errors after applying ConRegR are significantly less than those without applying ConRegR. The FDR estimation for π_{0} = 0.9 is even closer to real FDR after applying ConRegR compared with the result for π_{0} = 0.7 because of more pvalues used for regression and less number of pvalues for extrapolating in datasets of π_{0} = 0.9.
Combined pvalues simulation
In many analyses, more than one dataset are involved and a metaanalysis by combining pvalues from different studies or datasets is needed to estimate the overall significance for each gene. For example, (i) to find genes which are significant in at least one experiment, minimal pvalues will be of interest; (ii) to identify genes which are significant across all the experiments, the maximal pvalues will be of interest; and (iii) in order to detect genes which are significant on average, the product of pvalues will be appropriate. The distribution of combined pvalues will not be uniform even under true null hypotheses [18]. For currently used metaanalysis methods, such as "minimal", "maximal" or "product", we can obtain the transformation functions to recalibrate the combined pvalues to satisfy the condition of pvalues are uniform distributed under true null hypotheses. However, for other more complicated metaanalysis methods, the transformation function cannot be determined accurately leading to under or overestimation of significance, and ConRegR can provide the polynomial function approximation for the unknown transformation.
Suppose for gene i, the pvalues p_{ ij } (j = 1, 2, ..., L) follow the uniform distribution over (0, 1), then 1  (1  p_{min}) ^{L} ~ U(0, 1) and , where p_{min} = min(p_{i1}, p_{i2}, ..., p_{ iL }) and p_{max} = max(p_{i1}, p_{i2}, ..., p_{ iL }). For the pvalues from "product" method, according to Fisher's method [19].
For each metaanalysis method, we simulated two data sets Z^{0} = (δ_{ ij } ), Z = (z_{ ij } )(i = 1, ..., n, j = 1, ..., r) with n(n = 10000) genes and r(r = 10) repeats based on the formula as follows,
where b_{ i } (b_{ i } = 1, if i ≤ n(1  π_{0}) and b_{ i } = 0 if i > n(1  π_{0})) denotes the biological effect, both δ_{ ij ~ } N(0, 1) and ε_{ ij } N(0, 1) are the background noise. The individual pvalues are computed from twosample ttest and the combined pvalues are calculated by L(L = 3) simulations.
To compare the results, we also included two other transformation methods, "square" and "square root". All methods are listed in Table 1.
The two pvalue histograms for each π_{0} = 0.7 and π_{0} = 0.9, and for each of five different methods are plotted in Figure 5. It can be seen from Figure 5 that the pvalue histograms after theoretical transformation have constant frequency after 0.5 and the pvalue density near 1 indicates the . However, the pvalue histograms from "Min", "Square", "Prod" shifted towards 0 and the pvalue histograms from "Max", "Sqroot" shifted towards 1.
ConRegR used the above combined pvalues as input and the results are shown in Additional File 2, Figure S2 (π_{0} = 0.7) and Additional File 1, Figure S3 (π_{0} = 0.9). From Figure S2 and Figure S3, the regression curves are monotonic concave functions for "Min", "Square", "Prod" and monotonic convex functions for "Max", "Sqroot". The histograms after applying ConRegR are also very similar to the theoretical transformed pvalue histograms. The FDR estimation improved significantly after applying ConRegR. It shows that the estimated FDR after applying ConRegR is more likely to be the real FDR. The results of using our procedure for 100 repeated simulations are summarized in Figure 6. The FDR estimation errors after applying ConRegR are significantly less than those obtained without applying ConRegR.
Yeast Environmental Response Data
Yeast environmental stress response data generated by [20, 21] for nearly 6000 genes of yeast (S. cerevisiae) was aimed at understanding how yeast adopts or reacts to various stresses present in its environment. We selected 10 datasets: (1) Heat shock from 25°C to 37°C response; (2) Hydrogen peroxide treatment; (3) Menadione exposure; (4) DTT exposure response; (5) Diamide treatment response; (6) Hyperosmotic shock response; (7) Nitrogen source depletion; (8) Diauxic shift study; and, (910) two nearly identical experiments on stationary phase. We used Limma (Linear Models for Microarray Data) [12] package in R to compute pvalues for responsiveness of genes for each dataset.
The pvalue distribution for each dataset is shown in Additional File 2, Figure S4. As can be seen in Figure S4, the majority of the pvalue histograms do not have similar frequency after p = 0.5, and the density near p = 1 is less than π_{0} = 0.5. This implies that the pvalues were underestimated and the number of significantly responsive genes under these environmental stresses should be less than observed. We applied ConRegR on the pvalues of each dataset. Our result shows that the histograms of recalibrated pvalues obtained by applying ConRegR are better than without recalibration, and π_{0} estimations are all above 0.5 (Figure S4).
We use a true positive set of 270 genes from [22] to compute true FDR (FDR^{r} ). This is the intersection of core environmental stress response genes obtained by coregulation study in [21] and the yeast orthologs of S. pombe stress response genes. These 270 genes have been used as the true positive sets in other studies [23, 24]. The true FDR is calculated based on this 270 gene list and we calculated the improvement of FDR estimation (FDR_{ im } ) for each dataset after applying ConRegR. The FDR_{ im } is defined as followed:
where (respectively, ) is the estimated FDR by recalibration (respectively, input) pvalues for gene i(i = 1 ... n); and is the true FDR for gene i.
The improvements in FDR estimation for all 10 datasets are shown in Figure 7. After applying ConRegR, FDR estimation improved by 15% to 25% which means that the FDR estimation will be closer to the real FDR.
We performed the metaanalysis for 10 datasets to detect the core environmental stress response genes using "maximal" method. The combined pvalues are computed by the maximal pvalues across 10 datasets, and then transferred to meta analysis pvalues by transformation function in Table 1. The pvalue density histograms for metaanalysis before and after applying ConRegR are shown in Figure 8. The metaanalysis pvalues show better distribution after first applying ConRegR to each dataset and then perform the meta analysis. And FDR estimation improved by 38.5% after applying ConRegR.
Significance Analysis of Differential Expression from RNAseq Data
The nextgeneration sequencing technologies have been used for gene expression measurement. In [25], the authors compared RNAseq and Affymetrix microarray experiments and claimed that the sequencing data identified many more differentially expressed genes between human kidney and liver tissue samples than microarray data using the same FDR cutoff. In total, 11,493 significant genes were identified by RNAseq (3380 more genes than Affymetrix), only 6534 (56.9%) genes were also identified by Affymetrix experiments. By checking the pvalue histograms for RNAseq dataset, we found that majority of pvalues are very significant and its frequencies are very nonuniform for p > 0.5. However, the pvalue histogram for Affymetrix datasets is close to uniform for p > 0.5 (Additional File 2, Figure S5).
We applied ConRegR to recalibrate the pvalues obtained from RNAseq datasets and reestimated the FDR. We found 9481 significantly differentially expressed genes (only 1368 more genes than affymetrix) at FDR ≤ 0.1%. Among them, 6266 genes (66.1%) were also identified by Affymetrix experiments. There is an increase of 9.2% overlap after application of ConRegR (Figure 9). The FDR estimation is improved by 20% after applying ConRegR if we used significant genes identified by affymetrix experiments as the true positive set.
Discussion
ConRegR focuses on the uniformity of pvalues under null hypotheses and uses constrained polynomial regression to recalibrate the empirical pvalue distribution to more welldefined pvalue distribution. Therefore, the FDR estimation can be improved after the recalibration since the assumption of FDR estimation is that the input pvalues should follow such an ideal empirical pvalue distribution under null hypothesis. If the input pvalues follow the properties of ideal empirical pvalues distribution, the regression function tends to be diagonal line (i.e., y = x) and the pvalues do not change considerably after recalibration.
Though our method is discussed in the context of global FDR control, it is equally applicable to the other FDR like controls such as local FDR. Our method does not provide any new FDR control, but inputs better calibrated pvalues to the existing FDR estimators to improve their efficacy.
In ConRegR, the cutoff of and local minimization criteria can be changed for choosing the suitable k after checking the plots of and . From the combined pvalues simulation, the regression function may not fit the data well for the "Square" case. The fractional power, such as 1/2, can be added in the polynomial function to improve the fit.
The assumption that the regression function is convex or concave can be useful to deal with the imprecise pvalues whose distribution is biased towards 1 or 0 respectively. These are the most common cases of the pvalues being underestimated or overestimated. However, in some cases, the pvalues can be underestimated in one range of pvalues and overestimated in the remaining range of pvalues. These pvalue distributions may have peak or valley in the middle of the pvalue range or even have multiple peaks. The regression function will then no longer be convex nor concave. Regression function to handle this situation is currently under study. Our ConRegR can be generalized to an iterative weighted least squares method (e.g. decreasing weights from 1 to n). The weight program in the current version of the ConRegR is assigning a weight of 1 for all pvalues from 1 to k and a weight of 0 for the rest. Furthermore, different optimization schemes also need to be experimented. These will be explored in our future work. The distribution of pvalues from multiple testing can be modeled by the mixture of uniform distribution and some other welldefined distribution such as Beta distribution [5]. The parametric recalibration method is under development. The discrete pvalues from some nonparametric tests cannot be modeled by mixture model and new procedure should be explored to resolve this kind of problem.
Reviewers' comments
Reviewer's report 1
Prof. Vladimir Kuznetsov, Bioinformatics Institute, A*STAR, Singapore
Review
Major: Summary. Mathematical part of the manuscript is major, but it is not convicted. Choosing the model (2) is not justified. The used model assumes a uniform distribution of the tail of FDR distribution is not correct due to experimental data; the most empirical FDR distribution exhibits the nonhomogeneous fat tail as it was showed in Suppl. 1.
Authors' response: Thanks for your comment. The pvalue distribution under true null hypotheses follows the uniform distribution if the data satisfy all the assumptions of the hypothesis testing method [4]. However, the FDR distribution may not have similar property. The model (2) is based on the properties of the pvalue distribution, but not on the FDR distribution. For experimental data, many factors influence those assumptions and the pvalue distribution may not have uniformly distributed tail which is central to the paper. We clarified this in paragraphs 2, 3 & 4 in page 2.
The final FDR distribution of combined pvalue function is not be uniform "even under true null hypotheses", because it is a mixture distribution of the samples from different populations. If the samples are taken from significantly different distributions with different sample size and possible batch effect (not removed), does not allowed you to combine the datasets. If the samples are taken from the same distribution, why should not the pvalue distributions be uniform (assuming that they were uniform in the individual datasets before merging)?
Authors' response : In combined pvalues simulation, we assume pvalues follow the uniform distribution for each dataset under true null hypotheses. For each gene, the overall pvalue was obtained by combining its pvalues across experiments using different combination methods which are typical nonparametric metaanalysis methods. The combined pvalue distributions are not uniform which explains why another distribution is used to derive the overall pvalue [18]. We clarified this in paragraphs 1 and 2 in page 9.
Questionable statements
p.3
"Dependence among the test statistics is one of the major factors". It would be nice if you give examples of the test statistics you are mentioning.
Authors' response : We added the "tstatistic" as an example in text (paragraph 1 in page 3). "large p, small n problem". Could you describe the meaning this problem?
Authors' response : "large p, small n" problem is the problem that the number of variables (p) is much bigger than sample size (n) which always occurs in gene expression studies. We added one reference [11]for this problem (paragraph 2 in page 3). The n and p used in this context are different from that used in the discussion throughout the paper.
"ConRegR first maps the observed pvalues to uniformly distributed pvalues". From the context of this paragraph it is unclear where the set of 'uniformly distributed pvalues' is obtained from.
Authors' response : We changed to "predefined uniformly distributed pvalues" (paragraph 3 in page 3).
p.4
"Let pi' denote the ideal pvalues". How did you decide that the ideal distribution of the pvalues is a linear function, as defined in eq2 and is data and testindependent?
Authors' response: is the "ideal" pvalues. By our definition in equation 2 , their distribution is exactly uniform distribution. The "ideal" pvalues are predefined by equation 5 and not related to any data or test. We clarified this in paragraph 3 in page 4.
"the recalibration functions f() between p':(1..k) and p:(1..k) and apply it to all input pvalues (i.e p:(1..n))". However, you mentioned before (p.3) that the distribution of p:(1..n) is a mixture of functions. Hence, how technically valid is mapping the whole set p:(1..n) with a function empirically estimated on a limited subset p:(1..k) ? How do you ensure that the parameters of mapping function f() estimated on the subset (1..k) are also valid in the subset (k+1..n) ?
Authors' response : This is the reason that we emphasized throughout the manuscript that our procedure is extrapolative in nature. The constraints and the choice of k in our procedure help the extrapolation reasonable as shown in our simulations and real data. This is discussed in the methods section. However, we cannot guarantee that the mapping function is 100% valid for all the data.
"By StoneWeirstrass theorem, polynomial functions can well approximate (a small typo here) any continuous functions". Before you defined p:(1..n) as a discrete set. How do you ensure that the function defined on this set continuous? Why quadratic programming is chosen for polynomial approximation, but not linear? Why not other function, e.g. cubic splines, sum of exponents with least squares fitting?
Authors' response : Since pvalues are continuous, the mapping function from pvalues to pvalues should be continuous function. The discrete set p(1..n) is only a sample set from pvalues. We used quadratic programming because the constraints are needed in approximation. We clarified it on last three paragraphs in page 4. We agree that different optimization procedures also can be chosen. But they are subject of our future study.
p.5
"where ai,(i = 1,...,l) are l randomly generated numbers following U(0,1)". Why do they need to be generated randomly from U(0,1)? Since eq9 and eq10 are solved for beta under A * beta >= b, the solution beta depends on the random constrains A. "we add a sufficiently small positive value (delta(Q) = 1010) to the diagonal of Q". Since Q = XXT, this is equivalent to delta(X) = sqrt(delta(Q)) = 10^{5}.
Authors' response : Since the range of × is (x_{ k }, 1), we generated a_{ i } from U(0,1) to guarantee the constraint A * beta >= b is valid in (0,1). The points of constraints can also be chosen uniformly throughout the range [0,1]. As we chose 10000 points for constraints, the uniform selection or random selection may not make significant difference. We clarified it on paragraph 3 in page 5. The small positive value δ = 10^{10} is not the determinant value of Q. we changed the notation to λ (paragraph 2 in page 6).
p.6
"the polynomial function may not accurately fit the data due to the limitation of the polynomial maximal power (usually set the maximal power t = 10)". How do you ensure that your approximating polynomial is not overparameterized?
Authors' response : Since we approximate the polynomial function with monotonic convex/concave constraints, the approximation is not overparameterized. In Additional File 2, figures S2 and S3, the quadratic and cubic polynomial functions were well fitted by a polynomial of t = 10.
"estimated pi0(k) may oscillate near the real pi0". What is the source of the oscillations? Are they periodic?
Authors' response : The oscillation is due to the extrapolation may be unreliable if less number of pvalue are used for regression. It may not be periodic. To better reflect it, we changed the word to "fluctuate". We clarified it on paragraph 4 in page 6.
"the k closest to 1 that gives stable estimate". What is the definition of stability in this context? What numerical criterion do you use to detect the stability?
Authors' response : We used error to define the stability of the estimation, smaller error implies more stable estimate. In results, we used error < 0.05 to detect the stability. We clarified it on paragraph 1 in page 7 and paragraph 2 in page 8.
p.7
"As can be seen in the panels B and D in Figure 2". On Figure 2 the panels are not named. On Figure 3 all the axes need to be labeled and legends for the blue and the red lines should be added. Overall, the description of the results presented on Figure 3 is not comprehensible.
Authors' response : We changed "panels B and D" to "plots B and D" (paragraph 3 in page 7) and updated in Figure 2. We added the labels and legends for Figure 3.
p.8
"Combined pvalues simulation". The whole subsection needs to be described in the "Methods" section, since it introduces novel numerical techniques. In this subsection when you state "FDR improved by XX%", it could be good to show the particular FDR values before the improvement and after the improvement as well. The pvalue distributions for the sets of experimental data would better be shown in the paper, rather than in supplementary materials.
Authors' response : The methods section described the ConRegR method and the combined pvalue simulation subsection shows the simulation result by applying ConRegR. So we think it is suitable in the result section. The particular FDR values before and after applying ConRegR have large numbers and it is not possible to summarize by one table. So we used improvement percentage to show the efficacy of ConRegR. The Figure S1 now becomes Figure 5in the paper.
"In many analyses ...the product of pvalues will be appropriate". Where do these examples come from? Especially, since p1 * p2 < min(p1, p2), it's not logical that the pvalue of significance across, at least, half of the experiments is lower than the pvalue in all the experiments.
Authors' response : The product of pvalues is based on Fisher's method for combining independent tests of significance (reference [19]). The combined pvalues for product and minimal method are not comparable before transformation.
"The distribution of combined pvalues will not be uniform even under true null hypotheses". If the samples are taken from significantly different distributions (i.e the batch effect was not removed), what allowed you to combine the datasets? If the samples are taken from the same distribution, why should not the pvalue distributions be uniform (assuming that they were uniform in the individual datasets before merging)?
Authors' response : Please see our response to the 2nd comment.
Reviewer's report 2
Prof. Philippe Broet, JE2492, University ParisSud, France
This reviewer provided no comments for publication.
Reviewer's report 3
Prof. Hongfang Liu, Department of Biostatistics, Bioinformatics, and Biomathematics, Georgetown University Medical Center, NW, USA
This reviewer provided no comments for publication.
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Acknowledgements
The authors thank Edison T. Liu for his constant encouragement and support during this work. We appreciate Huaien Luo and Cyril Dalmasso for their valuable comments. We thank the reviewers Prof. Vladimir Kuznetsov, Prof. Philippe Broet and Prof. Hongfang Liu for their valuable comments which improve the manuscript. This work was supported by the Genome Institute of Singapore, Biomedical Research Council, Agency for Science, Technology and Research (A*STAR), Singapore.
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RKMK and KPC proposed the project. JL and RKMK developed the model, implemented the algorithm and wrote the paper. All the authors evaluated the results and approved the manuscript.
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Li, J., Paramita, P., Choi, K.P. et al. ConRegR: Extrapolative recalibration of the empirical distribution of pvalues to improve false discovery rate estimates. Biol Direct 6, 27 (2011). https://doi.org/10.1186/17456150627
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DOI: https://doi.org/10.1186/17456150627