 # True and false discoveries with independent e-values

In this note we use e-values (a non-Bayesian version of Bayes factors) in the context of multiple hypothesis testing assuming that the base tests produce independent e-values. Our simulation studies and theoretical considerations suggest that, under this assumption, our new algorithms are superior to the known algorithms using independent p-values and to our recent algorithms using e-values that are not necessarily independent.

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## 1 Introduction

Our recent paper  gives a generic procedure for turning e-merging functions into discovery matrices and applies it to arithmetic mean. Using arithmetic mean is very natural in the case of arbitrary dependence between the base e-values, at least in the symmetric case, since arithmetic mean essentially dominates any e-merging function [6, Theorem 5.1]. But in this note we will show that in the case of independent e-values we can greatly improve on arithmetic mean.

## 2 Discovery matrices for independent e-values

To make our exposition self-contained, we start from basic definitions (see our previous papers [4, 5, 6] exploring e-values for further information).

An e-variable

on a probability space

is a nonnegative extended random variable

such that . A measurable function for an integer is an ie-merging function if, for any probability space and any independent e-variables on it, the extended random variable is an e-variable. We will only consider ie-merging functions that are increasing in each argument and are symmetric (do not depend on the order of their arguments).

Important examples of ie-merging functions  are

 Un(e1,…,eK):=1(Kn)∑{k1,…,kn}⊆{1,…,K}ek1…ekn,n∈{1,…,K}. (1)

We will refer to them as the U-statistics (they are the standard U-statistics with product as kernel). The statistics play a special role since they belong to the narrower class of e-merging functions, meaning that is an e-variable whenever are e-variables (not necessarily independent).

Multiple hypothesis testing using was explored in [5, 6], and in this note we will mainly concentrate on . It will be convenient to generalize (1) to the case ; namely, we set

 Un(e1,…,eK):=UK(e1,…,eK),n>K

(we are mostly interested in the case and ).

Let us fix the underlying sample space , which is simply a measurable space. Let be the set of all probability measure on the sample space. A simple hypothesis is and a (composite) hypothesis is . An e-variable w.r. to a hypothesis is an extended random variable such that for all . It is clear that any ie-merging function transforms independent e-variables w.r. to (i.e., independent e-variables w.r. to any ) to an e-variable w.r. to .

An e-value is a value taken by an e-variable. Let us fix , hypotheses , and independent e-variables w.r. to , respectively. (The e-variables are required to be independent under any .) An e-test is a family , , of nonnegative extended random variables such that for all .

Let us say that a measurable function is a discovery matrix if there exists an e-test , , such that, for all and all ,

 (∀R⊆{1,…,K}:(|R|=r&mink∈REk(ω)≥maxk∉REk(ω))⟹|{k∈R∣Q∉Hk}|≥j)∨(EQ(ω)≥Dr,j(E1(ω),…,EK(ω))), (2)

where and stand for “and” and “or”, respectively. To emphasize that we interpret as a matrix, we write its values as . The intuition behind (2) is that if is large and we reject hypotheses with largest , we can count on at least true discoveries.

Algorithm 1 is one way of constructing a discovery matrix based on a family of ie-merging functions , . It uses the notation , where and is a symmetric function of arguments, to mean the value of on the sequence of , , arranged in any order. The algorithm is an obvious modification of Algorithm 2 in ; now we apply it to arbitrary ie-merging functions (such as ) rather than just to arithmetic mean (i.e., ). As in , the e-values are assumed to be ordered, without loss of generality.

The validity of Algorithm 1 can be demonstrated by the argument in the proof of Theorem 2.1 in . It is clear that, in the case of , the assumption of independence of can be relaxed to the assumption that all covariances , , are nonpositive.

The discovery matrix constructed in Algorithm 1 does not depend on the probability spaces , hypotheses , or e-variables , and in this sense is universal (in the terminology of [6, Section 5]).

## 3 A toy simulation study

In this section we run Algorithm 1 applied to and . Slightly generalizing the explanation in [6, Appendix B in Working Paper 27], we can see that the discovery matrix can be computed in time . For , the time can be improved from to [6, Appendix B in Working Paper 27]. For , we can easily improve the time to by noticing that

This is sufficient to cope with the case that we usually use in our simulation studies. Figure 1: Left panel: the discovery matrix for the U1 statistic (i.e., arithmetic mean) for 100 false and 100 true null hypotheses. Right panel: the U2 analogue.

We generate the base e-values as in Section 3 of 

: the null hypothesis is

, , the first observations are generated from , the last from , all independently, and the base e-variables are the likelihood ratios

 E(x):=exp(−(x+3)2/2)exp(−x2/2)=exp(−3x−9/2).

The results are shown in Figure 1 (whose left panel is identical to the left panel of Figure 2 in ); they are much better for . Each panel shows the lower triangular matrix , the left for and the right for . The colour scheme used in this figure is inspired by Jeffreys’s [3, Appendix B] (as in ):

• The entries with below 1 are shown in dark green; there is no evidence that there are at least true discoveries among hypotheses with the largest e-values.

• The entries are shown in green. For them the evidence is poor.

• The entries are shown in yellow. The evidence is substantial.

• The entries are shown in red. The evidence is strong.

• The entries are shown in dark red. The evidence is very strong.

• Finally, the entries are shown in black, and for them the evidence is decisive. Figure 2: Left panel: the discovery p-matrix for the GWGS procedure. Right panel: the U2 discovery matrix e-to-p calibrated.

It is interesting that after the crude e-to-p calibration our method produces p-values that look even better than the p-values produced by the GWGS procedure (in the terminology of ) designed specifically for p-values: see Figure 2.

In Figure 2 we use what we called Fisher’s scale in , but now we extend it by two further thresholds, one of which is , as advocated by . Our colour scheme is:

• P-values above are shown in green; they are not significant.

• P-values between and are shown in yellow; they are significant but not highly significant.

• P-values between and are shown in red; they are highly significant (but fail to attain the more stringent criterion of significance advocated in ).

• P-values between and are shown in dark red.

• P-values below are shown in black; they can be regarded as providing decisive evidence against the null hypothesis (to use Jeffreys’s expression).

## 4 An attempt of a theoretical explanation

We start from an alternative representation of , which will shed some light on the expected performance of our algorithm.

Let , be the arithmetic mean of , be the quadratic mean of , and

 var(e):=1KK∑k=1(ek−M1)2=M22−M21

be the sample variance of

.

###### Lemma 4.1.

For any ,

 U2(e)=M21−1K−1var(e). (3)
###### Proof.

By definition,

 U2(e) =1K(K−1)/2∑i
###### Corollary 4.2.

For any ,

 var(e)≤(K−1)M21.

For some the equality holds as equality.

###### Proof.

The first statement follows from , and an example for the second one is . ∎

According to Corollary 4.2,

 rvar(e):=var(e)(K−1)M21,

which we will call the relative (sample) variance of , is a dimensionless quantity in the interval . When , we set . The relative variance is zero if and only if all coincide, and it is 1 if and only if all but one are zero.

Using the notion of relative variance, we can rewrite (3) as

 U2(e)=M21(1−rvar(e)).

We can see the method of this paper based on has a potential for improving on the method of , but the best it can achieve is squaring the entries of the discovery matrix. An entry is squared if the multiset of e-values on which the infimum in the algorithm of  is attained consists of a single value. Otherwise we suffer as the e-values become more diverse.

## 5 Conclusion

The most natural direction of further research is to find computationally efficient procedures for computing discovery matrices based on , .

### Acknowledgments

We are grateful to Yuri Gurevich for useful discussions. In our simulation studies we used Python and R, including the package hommel .

V. Vovk’s research has been partially supported by Astra Zeneca and Stena Line. R. Wang is supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2018-03823, RGPAS-2018-522590).

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