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- Summary
- Introduction
- Truncated Product and Rank Truncation Product Methods
- Soft Truncation Threshold
- Gamma Method
- Simulation Results
- Difference in Interpretation of TPM and RTP Rejections
- Discussion
- References

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Pharm Stat. Author manuscript; available in PMC 2008 October 17.

Published in final edited form as:

PMCID: PMC2569904

NIHMSID: NIHMS45076

Dmitri V. Zaykin,^{1,}^{*} Lev A. Zhivotovsky,^{2} Wendy Czika,^{3} Susan Shao,^{3} and Russell D. Wolfinger^{3}

The publisher's final edited version of this article is available at Pharm Stat

See other articles in PMC that cite the published article.

In large-scale genomics experiments involving thousands of statistical tests, such as association scans and microarray expression experiments, a key question is: Which of the *L* tests represent true associations (TAs)? The traditional way to control false findings is via individual adjustments. In the presence of multiple TAs, *p*-value combination methods offer certain advantages. Both Fisher’s and Lancaster’s combination methods use an inverse gamma transformation. We identify the relation of the shape parameter of that distribution to the implicit threshold value; *p*-values below that threshold are favored by the inverse gamma method (GM). We explore this feature to improve power over Fisher’s method when *L* is large and the number of TAs is moderate. However, the improvement in power provided by combination methods is at the expense of a weaker claim made upon rejection of the null hypothesis – that there are some TAs among the *L* tests. Thus, GM remains a global test. To allow a stronger claim about a subset of *p*-values that is smaller than *L*, we investigate two methods with an explicit truncation: the rank truncated product method (RTP) that combines the first *K* ordered *p*-values, and the truncated product method (TPM) that combines *p*-values that are smaller than a specified threshold. We conclude that TPM allows claims to be made about subsets of *p*-values, while the claim of the RTP is, like GM, more appropriately about all *L* tests. GM gives somewhat higher power than TPM, RTP, Fisher, and Simes methods across a range of simulations.

We consider a problem encountered in large scale experiments such as genome-wide association scans. An experiment like this is usually preliminary in the sense that the primary goal is to identify a set of promising genetic variants, possibly predictive of a trait such as a disease. Results of the experiment are typically ordered by a measure of association such as a *p*-value, and a certain number of best-ranking results become candidates for a follow-up study. A follow-up study may consist of efforts to replicate the putative associations in independent samples, and additional genetic markers can be typed in the vicinity of the original associations. It is generally understood that the TAs may be only loosely clumped among the best-ranking results, and some may not be captured at all. Thus, statistical methods that allow making simultaneous claims about sets of results are important. A recent comprehensive overview of association scans, their goals, design and interpretation of results, including multiple testing issues is given in Hirschhorn and Daly [1].

Zaykin and Zhivotovsky [2] investigated the problem of TA ranks by examining the distribution of their relative position among the results ordered by a measure of association. They considered multiple effects including situations with highly correlated test statistics. The correlation in those experiments was due to linkage disequilibrium (LD) between alleles of neighboring genetic polymorphisms. Neither monotone decay nor block-structured LD was found to affect the distribution of ranks to a substantial degree. This is due to the local nature of LD that drives the correlation in association studies, in contrast to the recombination-driven correlation in linkage studies, which extends throughout large chromosomal segments. In the presence of multiple TAs, two questions regarding the ranks can be asked: (1) what is the distribution of the rank associated with the best-ranking TA – this is an issue of finding at least one of the multiple TAs; and (2) what is the distribution of the rank of the worst-ranking TA, which is an issue of finding all of the multiple TAs. These questions are related to the distribution of the first and the last order statistics associated with the TAs. In the first case, the ranks are improved, subject to the argument that in the context of genetically determined traits, magnitudes of each of the effects tend to decrease, as the number of effects increases. While each individual effect and the associated power might be small, discovery and the ability to score weakly predictive variants can be utilized to create highly predictive profiles involving multiple effects.

Regarding the practical issue of detecting multiple TAs, methods that combine *p*-values are especially promising. Many common methods for combining *p*-values are of the following form. First, take a transformation *H* of *p*-values, *T _{i}* =

In research published before modern computing power was prominent, the form of *H* was typically dictated by the ease with which the resulting distribution *F* could be evaluated. Nowadays any form of *H* can be used with the aid of a simple Monte Carlo evaluation: (1) Calculate *T*_{0} = Σ*H*(*p _{i}*), using the observed set of

For any choice of transformation *H,* combined *p*-value
statisticscan also be computed on subsets of all *L p*-values. The truncated product method (TPM) of Zaykin et al. [3] uses only *p*-values that are below some threshold *τ*. Significance is assessed by evaluating the distribution of their product, thus the method brings together the features of Fisher’s and the binomial methods. In contrast, the rank truncation product method (RTP; [6], [7], [8]) takes the product of the first *K p* -values and evaluates its distribution. When *K*=*L*, this method reduces to Fisher’s method.

The choice of the transformation *H* is highly relevant in our context of extreme heterogeneity among effects, where only a proportion of them might be non-zero. Methods such as TPM or RTP specify an explicit threshold such that large *p*-values above the threshold are penalized. When the TPM threshold *τ* is equal to the minimum *p*-value, *p*_{(1)}, or when *K*=1 in the RTP method, both of the methods reduce to the Śidák correction, 1− (1 − *p*_{(1)})* ^{L}* (keeping in mind that with the exception of

Even without an explicit truncation, any form of *H* implicitly imposes what we call the “soft truncation threshold” (STT), a feature important in situations when there is pronounced heterogeneity of effects. Consider Fisher’s method as an example. Elston [9] noted that in the limit (*L* → ∞) Fisher’s method favors *p*-values that are below 1/*e* ≈ 0.368. That is, if all *p*-values are equal to 1/*e*, the combined *p ^{c}* is also 1/

The threshold of Fisher’s method is somewhat smaller than 1/*e* for any finite value of *L*. The example of *L*=2 is illustrative. In this case, Fisher’s combined *p*-value can be found as
${p}^{c}=1-{\displaystyle \underset{w}{\overset{1}{\int}}}{\displaystyle \underset{t}{\overset{1}{\int}}}(1/v)\mathrm{dv\phantom{\rule{0.16667em}{0ex}}dt=w[1-ln(w)]}$, where *w* is the product of the two p-values. Thus, the STT for *L*=2 can be found by determining the value of *p* such that *p*^{2} [1 − ln(*p*^{2})] = *p*. This value is 0.284668 < 1/*e*. For larger values of *L*, it is convenient to take the log of the product and find the value numerically by solving 1− *F*[*L*, 1; − *L* ln(*x*)] = *x*, where *F* denotes the Gamma(*L*, 1) CDF evaluated at − *L*ln(*x*). For example, with *L*=100, the STT is ≈ 0.356; thus, the value is approaching 1/*e*. A notable feature of this threshold is its invariance in the presence of correlations between *p*-values. First, we note that under the independence of *p*-values, the value 1/*e* can be found by a Central Limit Theorem argument, as the solution for *x* in
$1-\mathrm{\Phi}[\sqrt{L}(-ln(x)-1)]=1/2$, where Φ (·) is the standard normal CDF. Assume the correlations are of the kind as to give convergence of *Y* = −Σ ln( *p _{i}*) to the normal distribution for correlated

Fisher’s method is in essence to add up a series of two-degree-of-freedom chi-squares; however, the degrees of freedom do not need to be the same. An explicit weighting of *p*-values was given by Good [11]. Lancaster [12] suggested that varying degrees of freedom can be used to weight the respective *p*-values and proposed a weighted modification where the degrees of freedom can be taken to be equal to the sample sizes of the combined experiments. In Lancaster’s method,
$Y={\displaystyle \sum _{i=1}^{L}}{G}_{{d}_{i}/2,2}^{-1}(1-{p}_{i})$, where the transformation *H* is
${G}_{{d}_{i}/2,2}^{-1}(1-{p}_{i})$, the inverse of the Gamma(*d _{i}*/

The following modification (which we call the “Gamma Method”, GM) exploits the same feature of the Gamma distribution as Lancaster’s weighting method - that a sum of gamma-distributed random variables with the same rate still has a gamma distribution. The rate does not affect the combined *p ^{c}*, and we will assume the rate to be 1. Let the transformation

$$\begin{array}{l}Y=\sum _{i=1}^{L}{G}_{a,1}^{-1}(1-{p}_{i})\\ {p}^{c}=1-{G}_{La,1}(Y)\end{array}$$

Thus, the method utilizes the common shape value (*a*) for the gamma distribution. A specific value is chosen to explicitly control the STT. When *a* is large, the gamma CDF becomes the normal inverse method, which has an STT of 0.5. The procedure becomes Fisher’s method when *a*=1, with the STT as discussed above. For a given STT=*x*, the large *L* value of *a* can be found by solving
$a={G}_{a,1}^{-1}(1-x)$. For example, *a*=0.0137 gives STT=0.05; *a*=0.0383 gives STT=0.1; and *a*=1 gives STT=1/*e*. The finite *L* values of *a* for a given value of STT=*x* are found by solving
$x=1-{G}_{La,1}\left(L{G}_{a,1}^{-1}[1-x]\right)$. For STT=0.05 with *L*=10, the value of *a* is 0.0346.

Table 1 gives results of a power evaluation in experiments with *L*=5000 and using one-degree-of-freedom chi-square tests. The number of TAs ranges from 10 to 200 with various powers specified for the individual effects at the level of 5%. The number of simulations to obtain a set of power values for each row was at least 10000. The STT values of 0.05 and 0.10 for the GM in the table are for infinite *L*. It appears from these experiments that GM with a low STT provides good overall power across different combinations of parameters. The method is most successful in situations with relatively low proportions of TAs when there is still advantage in using combination tests over family-wise error rate (FWER) adjustments such as the Śidák correction. We found that values of STT lower than 0.05 created difficulties with the numerical evaluation of the Gamma distribution by the DCDFLIB routines [13], possibly because the distribution becomes concentrated at zero when the parameter *a* is too small. In these cases, we observed somewhat inflated type-I error rates. The chosen values (STT of 0.05 and 0.10), or values higher than that were verified to give a correct proportion of rejections under H_{0}, i.e. when the number of TAs=0.

The overall power of GM for the chosen combinations of power for individual TAs and the numbers of TA is consistently higher than the power of other methods. However, there is arbitrariness in the choice of parameters of the three methods. For example, it is possible to increase power for all three methods in the settings with the number of TAs=200, by increasing *K*, *τ*, and the STT. A more expanded set of these values was applied to the setting with the number of TAs=200 at 20% power. Note that for this setting, the best test in Table 1 is FCP. By examining a wide range of parameters, we found that GM at STT=0.25 gave the best power (0.800) among all methods;. TPM power kept increasing up to *τ*=1 (0.756 power); and the best value of *K* for the RTP was found to be *K*=1300 (0.786 power). The increments used were 0.05 for STT and *τ*, and 100 for *K*. Next, we repeated this experiment for the setting with TAs=200 at 25% power. For GM the maximum power was 0.954 at STT=0.2; for TPM, the maximum power was 0.908 at *τ*=0.15; for RTP the maximum power was 0.945 at *K*=1300. Thus, in both extended cases the GM provided a higher power, and the difference is highly statistically significant by the Mann-Whitney test.

Although the power of RTP is very close to that of the GM, there appears to be a greater uncertainty about the choice of the parameter *K*. When the power is low, the best value of *K* tends to be much higher (we will discuss this phenomenon in more detail below). On the other hand, the best choice of the STT parameter for GM appears to be around 0.25 when the number of TAs is equal to 200. It is still affected by the proportion of TAs to false associations (FAs), but not so much by the power associated with the TAs. To illustrate this, Figure 1 shows combination test power values for the GM when the individual power varies from 10% (lower graph) to 30% (upper graph). The best STT value is between 0.20 (for 30% power) and 0.30 (for 10% power). Note that because both Fisher’s and the inverse normal (Stouffer’s) methods are particular cases of the GM, the *X* axis in Figure 1 includes Fisher’s method at *X* = 1/*e* and the inverse normal method at STT = 0.5.

The Simes test [14] has been included in Table 1 for comparison. There are certain advantages in using this test, for example, it is robust in the presence of correlations [15]. However, a feature of this test is that the overall *p*-value cannot be smaller than the minimum *p*-value, *p*_{(1)}, while methods that combine *p*-values explicitly can give an overall *p* that is smaller than *p*_{(1)}. In our case of relatively small power to detect the individual TAs, this yields a higher overall power.

Although the simulation results for GM in Table 1 appear very promising, we focus attention for the remainder of this report on TPM and RTP in order to address some fundamental issues of interpretation. In Table 1 and the extended results described previously, RTP exhibits higher power than TPM. However, there are important differences in the way the null hypothesis of RTP and TPM are formulated, and the claims one can make upon the rejection of the null. The claim of the TPM is that upon rejection of the null, there are one or more TAs among the effects represented by *p*-values that are at least as small as the truncation threshold, *τ*. We are careful not to assign a degree of certainty or probability to this event, and still give the corresponding *p*-value a standard frequentist interpretation. On the other hand, one cannot make a similar claim with the RTP: i.e. that there is at least one TA among the effects represented by the first *K p*-values.

An intuitive explanation for this phenomenon is as follows. The requirement that the TA *p*-values should rank above *K* would be satisfied more often when the FA *p*-values happened to be unusually small. Thus, if we evaluate the proportion of rejections among experiments where all TAs rank above the *K*, we would expect it to be above the nominal level. However, there is no similar dependency among the TAs and the FAs for the TPM with any truncation threshold *τ* – the event that all TA *p*-values are larger than *τ* simply has an effect of increasing *L* without affecting the value of the *p*-value product.

To verify these assertions, we conducted the following additional simulations. To evaluate RTP under the above scenario, we generated 5 true effects for each simulation experiment with *L*=25 and individual power 70% for a TA, under the condition that the first 5 smallest *p*-values represent FAs. We rejected simulated samples where this was not the case. Next, we took the product of the first *K*=5 *p*-values and obtained the computed *p ^{c}*. In these simulations, RTP still had 30% power to reject the hypothesis that there are no true effects among the total of

It is possible to modify TPM for interpretation on a narrower rejection set, and Appendix 1 describes details of one possible approach. This is reminiscent of an issue with Benjamini and Hochberg’s [16] false discovery rate (FDR) where FDR = *E*[*F*/(*T*+*F*) | *T*+*F* > 0] Pr(*T*+*F* > 0), denoting True and False discoveries by *T* and *F*. The difference with TPM is that we don’t necessarily know what Pr(*T*+*F* > 0) is, and in fact this probability can decrease to *α* as *L* increases. This would happen in settings such as association genome-wide scans, where most added hypotheses are FAs, so that the prior proportion of TAs would approach zero. In this case, *E*[*F*/(*T*+*F*) | *T*+*F* > 0] approaches 1, and Pr(*T*+*F* > 0) approaches *α*. Then across the sets of *L* experiments, FDR rejects *α*% of the time, as it should, because it becomes the Simes test which is known to maintain “weak control” of the FWER. The value of Pr(*T*+*F* > 0) depends on power and also on the prior Pr(H_{0}) (in other words, on the population proportion of FAs). Thus, we cannot easily obtain the FDR conditional on having made rejections, *E*[*F*/(*T*+*F*) | *T*+*F* > 0], which would be interpreted as the “proportion of false rejections among rejections”. Nevertheless, in certain settings such as microarray expression experiments, where the proportion of TAs is large and does not decrease with *L*, Pr(*T*+*F* > 0) may approach 1 and the conditional interpretation of FDR is warranted.

Care must also be taken in interpreting the value of *K* used with RTP. Closely related to RTP is the Set Association Method by Hoh et al. [17] proposed for large-scale genome association experiments. The set association approach is to take the sum of the first *K* largest association test statistics,
${S}_{K}={\sum}_{i=1}^{k}{T}_{i}$, to form an overall test that includes the *K* most significant genetic markers. Hoh et al. proposed finding the value of *K* that maximizes the significance of *S _{i}*, obtained from its Monte Carlo permutation distribution. While the method results in a powerful test under models with multiple contributing factors, there is a temptation to interpret the value of

In situations with multiple TAs that are a small proportion relative to the overall number of tests, *L*, truncation *p*-value combination methods become more powerful, although the claims associated with these methods are less specific. The objective of these approaches is to combine a subset of promising *p*-values, while taking into account all *L* tests. Focusing on a subset of small *p*-values can be achieved by combining the *K* smallest *p*-values such as in RTP, or by combining *p*-values that are smaller than a threshold *τ*, as in TPM. While RTP is often able to provide higher power than TPM (at *τ* < 1), TPM allows one to make stronger claims. In the event of rejection, the TPM claim is that there are genuine effects among those represented by *p*-values that are smaller than *τ*. The RTP claim remains the same as that of Fisher’s combination test: that there are genuine effects among all *L* tests. This is less specific than the claim of TPM, despite the fact that a certain value *K*<*L* led to a rejection.

The RTP and TPM formulations involve an explicit threshold, the *p*-value rank or the *p*-value magnitude, respectively. An alternative method based on the inverse gamma transformation (GM) was found to provide an overall combination test that was more powerful in simulations. A parameter of this method allows a control of what we call the soft truncation threshold (STT) – the value that favors *p*-values that are smaller than the STT. GM provides high power at fixed values of the STT across settings where both the power and the number of TAs are allowed to vary. In principle, it is possible to bring together the advantages of the TPM and the GM. Such a method would consider a sum of transformed *p*-values among *p*-values that are smaller than a specified threshold.

Adjustments for individual *p*-values can be based on any of the combination methods via the closure principle [19]. A computational shortcut is available for TPM as described in Appendix 2 of Zaykin et al. [3]. The same closure shortcut is applied for GM for *p*-values as small as the STT.

This research was supported in part by the Intramural Research Program of the National Institutes of Health, National Institute of Environmental Health Sciences.

We here describe a TPM modification, where the H_{0} rejection claim pertains to the subset of experiments where at least one *p*-value is ≤ *τ*. At the value of *L*=100, such as in the above simulations, the probability of at least one *p*-value falling below *τ*=0.05 is 99.4%, which is almost a certainty. In general, however, the standard TPM claim is that the probability of rejection under H_{0} is *α*, including experiments where *p*_{(1)} > *τ*. In the case when *p*_{(1)} >*τ*, H_{0} is accepted, and then the overall rate of rejections across all experiments, including these, is *α*. However, it is easy to make a modification so that in the case of the H_{0} rejection, the claim excludes subsets of experiments with *p*_{(1)} > *τ*, which under the H_{0} happens with probability 1− (1−*τ*)* ^{L}*. Denote the product of all

$$\begin{array}{l}Pr(W\le w)=\sum _{k=1}^{L}\left(\begin{array}{c}L\\ k\end{array}\right){(1-\tau )}^{L-k}\\ \times \left(w\sum _{s=0}^{k-1}\frac{{(kln\tau -lnw)}^{s}}{s!}I(w\le {\tau}^{k})+{\tau}^{k}I(w>{\tau}^{k})\right)\end{array}$$

where *I* (·) is the indicator function. For *τ* =1, this formula gives Fisher’s combined *p*-value:
$Pr(W\le w\tau =1)=w{\displaystyle \sum _{s=0}^{L-1}}\frac{{(-lnw)}^{s}}{s!}$. The value of Pr(*W* ≤ *w*) cannot exceed 1− (1−*τ*)* ^{L}*, and is equal to it when

$$\begin{array}{l}Pr(W\le w)=Pr(W<w{p}_{(1)}\tau )Pr({p}_{(1)}\tau )& =Pr(Ww{p}_{(1)}\tau )\left(1-{[1-\tau ]}^{L}\right)\end{array}$$

Therefore, the TPM *p*-value concerning the subsets with at least one p-value ≤ *τ* is

$${p}^{}$$

As an example, suppose the TPM *p*-value is 0.01, *τ*=0.05, and *L*=25. Then *p** = 0.014, which is larger than 0.01. However, the modification gives a stronger interpretation: *p** is the combined probability under the null for the subset of experiments where at least one *p*-value is less than *τ*.

Given the ordered set of random *p*-values, {*P*_{1}, *P*_{2},…, *P _{L}*}, it is possible to obtain the distribution of the product of the first

Now we will give an alternative form of the distribution that covers all cases. The distribution of the *K*-th smallest *p*-value itself is *P _{K}* ~ Beta(

$$\begin{array}{l}{W}_{K}=\left(\frac{{P}_{1}}{{P}_{2}}\right){\left(\frac{{P}_{2}}{{P}_{3}}\right)}^{2}\dots {\left(\frac{{P}_{k-1}}{{P}_{k}}\right)}^{K-1}\\ {U}_{i}={P}_{i-1}/{P}_{i}\\ {P}_{K}^{K}={U}_{1}{U}_{2}^{2}\dots {U}_{K-1}^{K-1}{P}_{K}\end{array}$$

The variables ${U}_{1},{U}_{2}^{2},\dots ,{U}_{K-1}^{K-1},{U}_{K}$ are independent with distributions

$$\begin{array}{l}{U}_{i}~\text{Beta}(i,1)\\ {U}_{i}^{i}~\text{Uniform}(0,1)\end{array}$$

The variable
${P}_{K}^{K}$ is the *K*-th power of the Beta(*K*, *L*−*K*+1) random variable with the density
${P}_{K}^{K}~\left(\begin{array}{c}L\\ K\end{array}\right){(1-{x}^{1/K})}^{L-K}$. Then the product distribution is

$$\begin{array}{l}Pr({W}_{K}\le w)=1-\underset{w}{\overset{1}{\int}}\left(\begin{array}{c}L\\ K\end{array}\right){(1-{x}^{1/K})}^{L-K}\\ \times \underset{w}{\overset{1}{\int}}\underset{\frac{w}{{x}_{2}}}{\overset{1}{\int}}\underset{\frac{w}{{x}_{2}{x}_{3}}}{\overset{1}{\int}}\dots \underset{\frac{w}{{x}_{2}\dots {x}_{K-1}}}{\overset{1}{\int}}d{x}_{1}\dots d{x}_{K}\\ =1-\left(\begin{array}{c}L\\ K\end{array}\right)\underset{w}{\overset{1}{\int}}{(1-{x}^{1/K})}^{L-K}\\ \times \left[1-\frac{w}{x}\sum _{j=0}^{K-2}\frac{{\left(ln\frac{x}{w}\right)}^{j}}{j!}\right]dx\end{array}$$

In practice, this integral requires a numerical evaluation. Figure 2 shows a bias resulting from interpreting the value of *K* that minimizes Pr(*W _{K}* ≤

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