On methods for correcting for the look-elsewhere effect in searches for new physics

Preprint typeset in JINST style - HYPER VERSION On methods f or correcting f or the look-else where effect in sear c hes f or ne w ph ysics S. Algeri 1 , 2 ∗ , D .A. van Dyk 1 , J. Conrad 1 , 2 † , B. Anderson 2 1 Statistics Section, Department of Mathematics, Imperial Colle ge London, South K ensington Campus, London SW7 2AZ, United Kingdom 2 The Oskar Klein Centr e for Cosmoparticle Physics, AlbaNova, SE-106 91 Stoc kholm, Sweden E-mail: s.algeri14@imperial.ac.uk A B S T R A C T : The search for new significant peaks ov er a energy spectrum often in v olves a statis- tical multiple hypothesis testing problem. Separate tests of hypothesis are conducted at dif ferent locations o ver a fine grid producing an ensemble of local p-values, the smallest of which is re- ported as evidence for the new resonance. Unfortunately , controlling the false detection rate (type I error rate) of such procedures may lead to excessi vely stringent acceptance criteria. In the recent physics literature, two promising statistical tools hav e been proposed to overcome these limitations. In 2005, a method to “find needles in haystacks" was introduced by Pilla et al. [1], and a second method was later proposed by Gross and V itells [2] in the context of the “look-elsewhere ef fect" and trial factors. W e show that, although the two methods exhibit similar performance for large sample sizes, for relativ ely small sample sizes, the method of Pilla et al. leads to an artificial in- flation of statistical po wer that stems from an increase in the false detection rate. This method, on the other hand, becomes particularly useful in multidimensional searches, where the Monte Carlo simulations required by Gross and V itells are often unfeasible. W e apply the methods to realistic simulations of the Fermi Large Area T elescope data, in particular the search for dark matter annihi- lation lines. Further , we discuss the counter-intuiti ve scenario where the look-else where corrections are more conservati ve than much more computationally efficient corrections for multiple hypoth- esis testing. Finally , we provide general guidelines for na vigating the tradeoffs between statistical and computational ef ficiency when selecting a statistical procedure for signal detection. K E Y W O R D S : Analysis and statistical methods, Data analysis, Dark Matter detectors. ∗ Corresponding author . † W allenberg Academy Fellow . Contents 1. Introduction 1 2. T ype I error , local power and good tests of hypothesis 3 3. Signal detection via multiple h ypothesis testing 5 4. Needles in haystacks and look elsewhere effect 6 5. Simulation studies 9 6. A pplication to realistic data 13 7. A sequential approach 15 8. Discussion 20 9. Acknowledgement 20 A. A ppendix 21 A.1 Cov ariance function of { C ? P L ( θ ) , θ ∈ Θ } 21 A.2 Geometric constant ξ 0 in the calculation of p PL 21 1. Introduction In High Ener gy Physics (HEP) the statistical evidence for new physics is determined using p- v alues, i.e., the probability of observing a signal as strong or stronger than the one observed if the proposed new physics does not e xist. If the location of the resonance in question is kno wn, the p-v alue can be easily obtained with classical methods such as the Likelihood Ratio T est (LR T), using the asymptotic distribution provided under the conditions specified in W ilks or Chernoff ’ s theorems [3, 4]. Unfortunately , the most realistic scenario in volv es signals with unknown locations, leading to what is kno wn in the statistics literature as a non-identifiability problem [5]. T o tackle this difficulty , physicists traditionally considered multiple hypothesis testing: they scan the energy spectrum 1 ov er a predetermined number of locations (or grid points), and sequen- tially test for resonance in each location [6, 7]. As discussed in detail in Section 3, when the number of grid points is large, the detection threshold for the resulting local p-v alues becomes more anti- conserv ativ e than the overall significance, which translates into a higher number of false discoveries 1 The search of a ne w source emission can occur ov er the spectrum of the mass, energy or an y other physical charac- teristic; for simplicity , we will refer to it as energy spectrum. – 1 – than expected. This is typically the case when the discretization of the search range is chosen fine enough to approximate the continuum of the energy window considered. W e discuss the details of this phenomenon in Sections 2 and 3. The situation is particularly problematic in the more realistic case of correlated tests. For in- stance, if the signal is dispersed ov er a wide energy range, its detection in a particular location may be correlated with that in nearby grid points. Unlike the case of uncorrelated tests in which the local significances can be determined exaclty , in presence of correlation, we can only deter- mine upper bounds for these significances, such as those provided by the Bonferroni’ s correction. Unfortunately , such bounds may often be excessi vely conservati ve [8, 9]. W e focus on the prob- lem of finding a single, or fe w peaks above background rather than multiple signals, and thus appealing methods such as T ukey’ s multiple comparisons [10] or the popular F alse Disco very Rate (FDR) [11–13] do not apply in this scenario. In order to ov ercome some of the limitations arising in multiple hypothesis testing, two promis- ing methods have been recently proposed in physics literature. The first (henceforth PL) was intro- duced in 2005 [1] and refined in [14]. Its methodology relies on the Score function and is purported to be more powerful than the usual Likelihood Ratio T est (LR T) approach. Unfortunately , the math- ematical implementation of the method is not straightforward, which strongly limited its diffusion within the physics community . This is one of the main motiv ations of this w ork. Specifically one of the questions we aim to address is if, despite its technical difficulties, PL provides some advantages in practical applications. It turns out that PL is particularly helpful for multi-dimensional signal searches. The second approach (hereinafter GV) belongs to the class of LR T -based methods. It was first introduced in 2010 [2], and recently extended [15] to compare non-nested models. In contrast to PL, GV enjoys easy implementation, which has led to a wide range of applications in v arious searches for ne w physics including in the disco very of the Higgs boson [6, 7, 16, 17]. From a theoretical perspecti ve, both approaches require an approximation of tail probabilities of the form P ( sup Y t > c ) , where Y t is either a χ 2 or a Gaussian process. These approximations compute the dis- tribution of the relev ant test statistic ev aluated at each possible signal location in the large-sample limit. GV formalizes the problem in terms of the number of times the process Y t , when viewed as a function of the signal location, passes upward through the threshold c ; this is called the number of “upcrossings”. PL, on the other hand, in volv es the so-called tube formulae, where an approxima- tion of P ( sup Y t > c ) is obtained as the ratio between the volume of a tube built around the manifold associated with sup Y t on the unit sphere, and the v olume of the unit sphere itself. Although we de- scribe both methods more fully in Section 4, we do not focus on their mathematical details, but rather emphasize their computational implementation; readers are directed to [1, 2, 14, 15, 18–20] for technical de velopment. While either GV or PL can be used to control the false detection rate and ensure sufficient statistical po wer , they can be computationally expensi ve in complex models. GV specifically , may easily become unfeasible in the multidimensional scenario. Multiple hypothesis testing procedures, on the other hand, can be much quicker , but are often ov erly conserv ati ve in terms of the false detection rate when the number of tests is large. Perhaps counter-intuiti vely , ho wever , situations do occur where multiple hypothesis testing lead to the same or ev en less conserv ati ve inference than GV and PL. Not surprisingly , this depends on the number of tests conducted, i.e., GV and PL bounds on p-values are less likely to be larger than the Bonferroni’ s bound as the number of – 2 – tests increases. In the absence of specific guidelines as to the optimal number of tests to conduct, and in order to optimize computational speed while adhering to a prescribed false-positi ve rate as closely as possible, we summarize our findings as a simple algorithm that implements a sequential selection of the statistical procedure. Although it is well known that choosing a statistical procedure on the basis of its outcome can detrimentally effect the statistical significance, an effect called “flip- flopping” by Feldman and Cousins [29], we show that our sequential pr ocedur e is immune to this ef fect. The remainder of this paper is or ganized as follows: in Section 2 we re view the background of hypothesis testing, we define the auxiliary concepts of goodness of a test and local power , which are used for our comparison of PL and GV . In Section 3, we revie w the multiple hypothesis testing approach for signal detection and we underline the respectiv e disadvantages in terms of significance requirements. In Section 4, we pro vide a simplified o vervie w of the technical results of PL and GV . In Section 5, a suite of simulation studies is used to highlight the performance of the two methods in terms of approximation to the tail probabilities, false detection rate and statistical po wer . W e show that both solutions exhibit adv antages and suffer limitations, not only in terms of computational requirements and statistical power , but most importantly , in terms of the specific conditions they require of the models being tested. An application to a realistic data simulation is conducted in Section 6. The sequential approach is discussed in Section 7 and discussion in Section 8. 2. T ype I error , local power and good tests of hypothesis Consider the framew ork of a classical detection problem. Suppose N e vent counts are observed ov er a predetermined energy band Y . W e are interested in knowing if some of these ev ents are due to a new emission source or if they all can be attributable to the background and its random fluctuations. W e further assume that if there is no new source, the energy y of the N e vents can be modeled using a probability density function (pdf) f ( y , φ ) ov er Y where φ is a potentially unkno wn free parameter . Whereas, if the new resonance is present, ev ents associated with it have energy distribution g ( y , θ ) ov er Y , and we let θ ∈ Θ with Θ representing the search windo w for the new resonance over the energy range. T ypically Θ ≡ Y , but in principle one could consider Θ ⊂ Y . Thus, we can write the full model for N counts as ( 1 − η ) f ( y , φ ) + η g ( y , θ ) , (2.1) where η is the source strengh, and positiv e v alues of η indicate the presence of the new signal. From a statistical perspecti ve, the search for ne w physics corresponds to a test of hypothesis in which the null hypothesis , H 0 , which stipulates that only background counts are observed, is tested against the alternative hypothesis , H 1 , which stipulates a proportion η of the observed counts are due to ne w physics. Notationally this test is written H 0 : η = 0 versus H 1 : η > 0 . (2.2) The test is then conducted by specifying an opportune test statistic T , whose observed v alue t obs is calculated on the av ailable data, and a detection is claimed if t obs exceeds a specified detection threshold t α . The latter is determined by controlling the probability of a type I err or or the false – 3 – detection rate, which we allow to be no larger than a predetermined lev el α . For obvious reasons, it is sensible to choose α sufficiently small, and it is common practice in physics to adopt a 3, 4 or 5 σ thresholds i.e., α = 1 − Φ ( x ) x = 3 , 4 , 5 , (2.3) where Φ ( · ) is the cumulativ e density function (cdf) of a standard normal distrib ution. If t obs > t α a discov ery is claimed, whereas if t obs ≤ t α we conclude that there is no sufficient e vidence to claim detection of a ne w signal. An equi valent formulation of a test of hypothesis can be made in terms of a p-value i.e., the probability of observing a value of T that, under the hypothesis of no signal emission ( H 0 ), is greater than t obs . Formally p-v alue = P ( T ≥ t obs | η = 0 ) . (2.4) The p-v alue is then compared to the tar get probability of a type I error , α . In this case, a discov ery is claimed if p-v alue < α , whereas the ne w resonance is not detected if p-value ≥ α . In addition to the type I error , another important property of a test of hypothesis is its statistical power i.e., the probability of detecting the new signal when it is present. For the test in ( 2.2) we can write α = P ( T > t α | η = 0 ) Po wer ( η , θ ) = P ( T > t α | η , θ ) , η > 0 . (2.5) The goal is to construct a good detection test, that is, a test with the probability of false detection, equal to or smaller than the predetermined le vel α , but with the po wer as large as possible. Consequently , if two or more tests with the same lev el α are to be compared, the test with higher power is preferred. As specified in (2.5), for the model in (2.1) the power depends on both the signal strength η and its location θ . For η , the detection power can be summarized using upper limits as discussed in [21], whereas in this paper , we focus on the power with respect to the source location. This is of particular importance when the dispersion of the signal depends on its position (as in our examples in Section 5), and widely spread source signals are expected to be more difficult to detect, i.e., exhibit lower statistical po wer . Hereafter , we refer to the power at a fixed location θ as the local power , and we say that a test is uniformly more powerful locally than another test with the same le vel α , if, for fixed η , its local power is greater than or equal to that of the other test, for e very possible θ in the ener gy range Θ . W e in vestigate the goodness and the local power of PL and GV in Section 4. T ypically , the exact distribution of the test statistic T cannot be specified explicitely , and clas- sical statistical methods rely on its asymptotic distribution. It follows that the resulting p-values, α , and power are also asymptotic quantities. In this paper, we mainly consider the asymptotic distributions of v arious test statistics and thus, the p-values, α le vels and powers that we quote are implicitly asymptotic quantities. The only exceptions are the values quoted in the simulation studies in Section 5. There, the distribution of reference is the simulated distribution of T , and we refer to the quantities of interest as simulated false detection rate and simulated po wer . – 4 – 3. Signal detection via multiple hypothesis testing As anticipated in Section 1, the statistical detection of new physics can often be viewed as a multiple hypothesis testing problem. An ensemble of R tests are conducted simultaneously , any of which can result in a false detection. While the individual tests are designed to control their specific false detection rate, the ov erall probability of ha ving at least one false detection increases as R increases, leading to a higher rate of false disco veries than e xpected. For the test in (2.2), a natural choice of the test statistic T is the LR T . Define LRT θ = − 2 log L ( 0 , ˆ φ 0 , - ) L ( ˆ η 1 , ˆ φ 1 , θ ) , (3.1) where L ( η , φ , θ ) is the likelihood function under (2.1). Notice that under H 0 (i.e., η = 0), the parameter θ has no meaning and no value. The numerator and denominator of (3.1) are the maxi- mum likelihood achiev able under H 0 and H 1 respecti vely , with ˆ φ 0 being the Maximum Likelihood Estimate (MLE) of φ under H 0 and ˆ φ 1 and ˆ η 1 the MLEs under H 1 . Under H 0 , the distribution of the data does not depend on θ . Because this violates a key assumption of both W ilks or Chernof f ’ s theorems [3, 4], the distribution of LR T is not kno wn and we cannot directly compute the p-v alue for (2.2). T o o vercome this difficulty , a naïve approach in volves the discretization of the ener gy range Θ into R search regions, resulting in a grid of fixed values Θ G = { θ 1 , . . . , θ R } . R simultaneous LR Ts are then conducted for the hypotheses in (2.2), fixing θ in (3.1) to be equal to each of the θ r ∈ Θ G . In this w ay , a set of R local p-values is produced, and the smallest, namely p L , is compared with the established target probability of type I error , α L . Notice that α L corresponds to the false detection rate for a specific test among the R av ailable, and thus is the local significance. Howe ver , we must take account of the fact that R hypotheses are being tested simultaneously and must also consider the chance of having at least one false detection among the ensemble of R tests, namely the global significance, α G . If the R tests are independent, i.e., detecting a signal in a giv en energy location does not depend on its detection in other locations, it can be easily sho wn [8] that α G = 1 − ( 1 − α L ) R , (3.2) and the resulting adjusted (global) p-v alue [8, 9] is p G = 1 − ( 1 − p L ) R . (3.3) Consider a toy example in which we have, 50 grid points ov er the energy spectrum Y and 50 uncorrelated tests at the 5 σ significance le vel, the chance of ha ving at least one false detection among the 50 tests, i.e., the overall false detection rate, is α G = 1 . 4 · 10 − 5 which corresponds to 4 . 18 σ significance. This is approximately 50 times larger than the α L = 2 . 87 · 10 − 7 associated with 5 σ . Con versely , if the R tests are correlated, as in the case of disperse source emission, controlling for the false detection rate is more problematic. In this scenario, contrary to (3.2), an e xact general relationship between α L and α G cannot be established, since the specific correlation structure varies – 5 – on a case-by-case basis. Thus, the only general statement that we can make is α G ≤ R α L . (3.4) The adjusted p-value corresponding to ( 3.4) is kno wn as the Bonferroni correction [8], specifically , p BF = R p L (3.5) which bounds p G in that p G ≤ p BF . In particular , p BF is a first order approximation of p G , and thus the two p-values are equi valent when dealing with strong signals, i.e., when p L → 0. This is reflected in the toy e xample abov e, where p BF is equal to p G , and also leads to 4 . 18 σ significance. (Recall α G α L ≈ 50 in the toy example.) Despite their easy implementation, these procedures are often dismissed by practitioners be- cause, in addition to the stringent requirements to control for the overall false detection rate, they artificially depend on the number of tests R . This is particularly troublesome gi ven the typically arbitrary nature of setting R when discretizing the ener gy spectrum Θ . W e discuss belo w , howe ver , practical situations in which these methods pro vide reasonable inference and occasionally perform better than the often preferred look-else where corrections of GV and PL. 4. Needles in haystacks and look elsewhere effect In this section we consider methods that directly address problems associated with parameters that are only present under H 1 . Rather than constructing R tests, these methods consider a single test of hypothesis and a single global p-value. The ke y element of these methods is to consider new test statistics, which are not affected by the non-identifiability of the parameters. The two methods we consider follo w a similar ov erall strategy which we no w summarize. Consider the model in (2.1). W e denote the MLE of the parameters η and φ by ˆ φ θ , ˆ η θ for each fixed v alue θ ∈ Θ , and we specify a local test statistic C ( y , ˆ φ θ , ˆ η θ , θ ) for the test in (2.2). For bre vity , we write C ( y , ˆ φ θ , ˆ η θ , θ ) as C ( θ ) . In practice, for each fixed value θ r ∈ Θ G , we compute c ( θ 1 ) , . . . , c ( θ R ) , where c ( θ r ) corresponds to the observed value of C ( θ ) with θ = θ r . The collection of values { c ( θ 1 ) , . . . , c ( θ R ) } can be viewed as a realization of a stochastic process { C ( θ ) , θ ∈ Θ } , and a global test statistic, for (2.2) is C = sup θ ∈ Θ C ( θ ) . (4.1) Because we only observe C ( θ ) for θ r ∈ Θ G , the observed v alue of C is c ( ˆ θ ) = max θ r ∈ Θ G c ( θ r ) (4.2) where ˆ θ is the v alue θ r ∈ Θ G where this maximum is attained, and which corresponds to our estimate of the signal location. Finally , the global p-value of the test is obtained by approximating the tail probability P ( C > c ( ˆ θ )) (4.3) – 6 – under H 0 . The choice of the statistic C and the approximation method for computing ( 4.3) are the main characteristics dif ferentiating the approaches of PL and GV . T o deri ve C , PL [1, 14] considers the Score process { C ? PL ( θ ) , θ ∈ Θ } , with C ? PL ( θ ) = N ∑ i = 1  f ( y i , φ ) g ( y i , θ ) − 1  (4.4) being the Score function of (2.1) under H 0 and the generic local statistic C ( θ ) above is replaced by the normalized Score function, C PL ( θ ) = C ? PL ( θ ) p NW ( θ , θ ) (4.5) where W ( θ , θ † ) is the cov ariance function of { C ? PL ( θ ) , θ ∈ Θ } . The functional form of W ( θ , θ ) depends on whether the free parameter under H 0 , φ , is kno wn or not (see Appendix A.1). The stochastic process of interest is { C PL ( θ ) , θ ∈ Θ } and we let C PL = sup θ ∈ Θ C PL ( θ ) and c PL ( ˆ θ ) be its observed value. In order to simplify notation we drop the dependence of c PL ( ˆ θ ) on ˆ θ and write simply , c PL . The corresponding global p-value is P ( C PL > c PL ) ; [14] prov e that, under H 0 , C PL con verges to the supremum of a mean zero Gaussian process as N → ∞ . The approximation, p PL , of P ( C PL > c PL ) is obtained through so-called tube formulae for Gaussian processes [20]. In particular, the supremum of the Gaussian (large-sample) limiting process of { C PL ( θ ) , θ ∈ Θ } is approximated via an appropriate one-dimensional manifold over a unit sphere; a tube is then constructed around the manifold and the ratio of the volume of the tube and of a unit sphere is used to approximate P ( C PL > c PL ) . If θ is one-dimensional, the approximation to P ( C PL > c PL ) is p PL = ξ 0 2 π P ( χ 2 2 ≥ c 2 PL ) + 1 2 P ( χ 2 1 ≥ c 2 PL ) , (4.6) which becomes more precise as c PL → ∞ , and where in general P ( χ 2 s ≥ q ) = 1 − P ( χ 2 s < q ) , with P ( χ 2 s < q ) being the cumulati ve density distribution of a χ 2 random v ariable with s degrees of freedom ev aluated at q . The quantity ξ 0 in (4.6) is the volume of the one-dimensional manifold (see Appendix A.2 for more details). Instead of the Score function, GV [2] focuses on the LR T in (3.1), and thus C GV ( θ ) = LRT ( θ ) . For the specific case of (2.2), H 0 is on the boundary of the parameter space, and thus under H 0 the LR T process conv erges asymptotically to a 1 2 χ 2 1 + 1 2 δ ( 0 ) random process [2, 15]. W ith this choice, and again, dropping the dependence on ˆ θ , we let C GV = sup θ ∈ Θ C GV ( θ ) and c GV be its observed v alue depending on the data. The global p-value P ( C GV > c GV ) , is approximated by p GV = P ( χ 2 1 > c GV ) 2 + E [ U ( c 0 ) | H 0 ] e − c GV − c 0 2 . (4.7) which becomes more precise as c GV → ∞ and where c 0 is a small threshold such that c 0 << c GV , and U ( c 0 ) is the number of times the LR T process, when viewed as a function of θ , crosses from belo w c 0 to abov e c 0 ; this is called the number of upcrossings. An illustrative example is shown in Figure 1. In (4.7), E [ U ( c 0 ) | H 0 ] is the expected number of upcrossings under H 0 of the (large- sample) LR T process, and is estimated via a Monte Carlo simulation of size M as described in Algorithm 1. – 7 – Search Region LR T process 1 10 20 30 40 50 60 70 80 90 100 c 0 Figure 1: Upcrossings (red crosses) of the threshold c 0 by the LR T process. Algorithm 1. • For m = 1 , . . . , M : (1) - Simulate a large number (e.g., 1,000) of observ ations from f ( y , ˆ φ 0 ) ; (2) - for each θ r ∈ Θ G calculate LRT ( θ r ) as in ( 3.1); (3) - for each r ∈ [ 1; R − 1 ] count how many times LRT ( θ r ) < c 0 and LRT ( θ r + 1 ) ≥ c 0 , i.e., the number of upcrossings of c 0 by the LR T process under H 0 for simulation m , namely , U m ( c 0 ) . • Estimate E [ U ( c 0 ) | H 0 ] with 1 M ∑ M m = 1 U m ( c 0 ) . The threshold c 0 is typically chosen to be small enough so that a reliable estimate of E [ U ( c 0 ) | H 0 ] can be obtained with a small Monte Carlo simulation size M , but large enough so that the effect of the resolution R of Θ G on the number of upcrossings is negligible (see [2]). Although (4.6) and (4.7) both hold when c PL and c GV are large, when they are small, the right hand sides of (4.6) and (4.7) provide upper bounds for the respecti ve tail probabilities. GV’ s global p-v alue, p GV , is always greater than or equal to the smallest local p-value, p L , introduced in Section 3. Thus GV always leads to an equal or smaller number of false discoveries than one would have using multiple hypothesis testing when no correction is applied. This can be – 8 – easily sho wn by noticing that for the test in ( 2.2) p L = 1 2 P ( χ 2 1 > L RT θ ? ) (4.8) where LRT θ ? is calculated according to (3.1) with θ = θ ? . Notice that θ ? ≡ ˆ θ , i.e., the location where the smallest p-value is observed is also where the observed local LR T statistic, achieves its maximum. Thus, the LRT θ ? coincides with the observed value c GV of the GV test statistic C GV . It follo ws by (4.7) and (4.8) that the inequality p GV ≥ p L always holds. 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 y Density η = 0 γ = 1.9 γ = 1.6 γ = 1.3 γ = 1 γ = 0.7 γ = 0.4 γ = 0.1 2 4 6 8 10 0.0 0.5 1.0 1.5 y Density η = 0 M χ = 9 M χ = 7.5 M χ = 6 M χ = 4.5 M χ = 3 M χ = 1.5 Figure 2: Left panel: probability density functions for Example I under H 0 (blue line) and H 1 (orange lines) with η = 0 . 2 and γ = 0 . 1 , 0 . 4 , 0 . 7 , 1 , 1 . 3 , 1 . 6 , 1 . 9. Right panel: probability density functions for Example II under H 0 (blue line) with τ = 1 . 4 and H 1 (orange lines) with η = 0 . 2 and M χ = 1 . 5 , 3 , 4 . 5 , 6 , 7 . 5 , 9. Another fundamental dif ference between the multiple h ypothesis testing approach in Section 3 and the methods discussed in this section is the le vel at which the optimization occurs. In the former , the p L is the minimum of set of local p-v alues p L = min θ r ∈ Θ G p ( θ r ) , and the result, is e ventually corrected afterwards according to (3.3) or (3.5). Conv ersely , as ex- pressed in (4.2) in PL and GV , the optimization occurs with respect to the statistic C ( θ ) , and a correction for p L is eventually generated intrinsically , by approximating the tail probability of the test statistic C . 5. Simulation studies A fundamental result in probability theory states that the Score test and the LR T are asymptotically equi valent when the number of events is large (i.e., for large sample sizes). As shown in [1], the – 9 – same can be prov en for the C PL and C GV of PL and GV , respecti vely , and thus, we e xpect the asymptotic equality between p L and p G V to hold for p P L , at least for large sample sizes. Unfortunately , as one might expect, the asymptotic equiv alence does not necessarily hold for small sample sizes, i.e., when only a few counts are a vailable. In order to inv estigate this scenario, we consider two examples. In Example I, we refer to the toy model in [1] where a Breit-W igner resonance is superimposed on a linear background. The full model is ( 1 − η ) 1 + 0 . 3 y 2 . 6 + η 0 . 1 k γ π ( 0 . 01 + ( y − γ ) 2 ) (5.1) where k γ is a normalizing constant, y ∈ [ 0; 2 ] and γ ∈ ( 0; 2 ] . Notice that the null model has no free parameters and thus PL can be directly applied with no further adjustment of the cov ariance function (see Section 4). In Example II, the background is power -law distributed with unknown parameter τ . The signal component is modeled as a Gaussian bump with dispersion proportional to the signal location. Specifically , the full model is ( 1 − η ) 1 k τ y τ + 1 + η k M χ exp  − ( y − M χ ) 2 0 . 02 M 2 χ  (5.2) with k τ and k M χ normalizing constants, y ∈ [ 1; 10 ] , τ > 0 and M χ ∈ [ 1; 10 ] . Owing to the unkno wn parameter τ under H 0 , we must use the extended theory in [14] for PL. The pdfs used in Example I and II are plotted in Fig. 2. For both examples, we ev aluate the false detection rate (or type I error), and the local power as described in Section 2, and examine how it depends on the number of e vents; specifically , we considered sample sizes of 10 , 50 , 100 , 200 and 500. The false detection rate and local power are obtained via Monte Carlo simulations from the null model ( η = 0) and from the alternative model with η = 0 . 2, respectively . Although τ is unknown in Example II, it can be estimated with the MLE ˆ τ under H 0 . The simulations are then drawn from (5.2) with τ = ˆ τ . This simulation procedure is kno wn in the statistical literature as the parametric bootstrap [22]. In principle, the observed sample used to compute ˆ τ could either come from the null or from the alternativ e model. Thus, in order to ev aluate the consistency of PL and GV in both situations, two further sub-cases are needed. In Example IIa, we draw the “observ ed" sample from (5.2) with η = 0 and τ = 1 . 4, i.e., in absence of ne w physics. In Example IIb, we draw the “observed" sample with η = 0 . 2, τ = 1 . 4 and M χ = 9. Results of the simulation studies appear in Fig. 3. Its columns correspond to Example I, Example IIa and Example IIb, respectiv ely . In the first ro w , we report the simulated detection rates; the simulated test statistics C PL and C GV (where θ is either γ or M χ ) were calculated for each of 100 , 000 datasets generated from the null model. These values were then compared to the nominal thresholds at 3 σ , obtained, as in (5.3) and (5.4), by setting p PL and p GV in (4.6) and (4.7) equal to 1 − Φ ( 3 ) = 0 . 0013 and solving for c PL and c GV respecti vely , i.e., 1 − Φ ( 3 ) = ξ 0 2 π P ( χ 2 2 ≥ c 2 PL ) + 1 2 P ( χ 2 1 ≥ c 2 PL ) (5.3) 1 − Φ ( 3 ) = P ( χ 2 1 > c GV ) 2 + E [ U ( c 0 ) | H 0 ] e − c GV − c 0 2 . (5.4) – 10 – ● ● ● ● ● ● ● Example I γ T ype I Error (log−scale) 0.001 0.002 0.005 0.01 0.02 0.04 3 σ 0.1 0.4 0.7 1 1.3 1.6 1.9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Example IIa M χ 0.001 0.002 0.005 0.01 0.02 0.04 3 σ 0.1 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Example IIb M χ 0.001 0.002 0.005 0.01 0.02 0.04 3 σ 0.1 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● γ P ower 0 0.1 0.3 0.5 0.7 0.9 1 0.1 0.4 0.7 1 1.3 1.6 1.9 ● ● ● ● ● ● ● ● ● ● ● ● ● M χ 0 0.1 0.3 0.5 0.7 0.9 1 1.5 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● ● ● ● M χ 0 0.1 0.3 0.5 0.7 0.9 1 1.5 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● ● ● ● ● γ P ower (adjusted) 0 0.1 0.3 0.5 0.7 0.9 1 0.1 0.4 0.7 1 1.3 1.6 1.9 ● ● ● ● ● ● ● ● ● ● ● ● ● M χ 0 0.1 0.3 0.5 0.7 0.9 1 1.5 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● ● ● ● M χ 0 0.1 0.3 0.5 0.7 0.9 1 1.5 3 4.5 6 7.5 9 ● ● ● ● ● ● ● ● ● N=10 PL N=50 PL N=100 PL N=200 PL N=500 PL N=10 GV N=50 GV N=100 GV N=200 GV N=500 GV Figure 3: Simulated probability of type I error (top row), power (middle row) and adjusted power (bottom ro w) for Example I (first column), Example IIa (second column) and Example IIb (third column) with different sample size N ov er 100 , 000 simulations. The gray symbols corresponds to PL and the blue symbols to GV . Shaded areas indicate regions expected to contain 68% (dark gray) and 95% (light gray) of the symbols if the nominal type I error of 0.0013 holds. – 11 – In the second ro w of Fig. 3, we plot the local power functions; the procedure is the same as for the simulated false detection rates e xcept the 100 , 000 datasets were generated from the alternati ve models with η = 0 . 2 with different values for the location parameters γ and M χ . In the third row , we e valuate an adjusted version of the local power; the simulated values of C PL and C GV are the same as used in the plots in the second ro w , b ut instead of comparing them with the nominal thresholds c PL and c G V , we compared them with their empirical (bootstrap) thresholds. The empirical threshold correspond to the 0 . 9987 quantiles of the 100 , 000 simulated values of C PL and C GV generated for the first ro w of Fig. 3, i.e., the empirical distributions of the test statistic under H 0 . Looking at the first ro w of Fig. 3, the simulated f alse detection rates associated with GV are alw ays consistent with the nominal 3 σ error rate. This is not the case for PL. Although the false detection curves appear to approach the desired value as the sample size increases, they are alw ays higher than expected. Looking at the second ro w of Fig. 3, on the other hand, the simulated local power of PL is alw ays higher than that of GV , at least the for the smaller samples sizes. The difference between the local power functions decreases when the sample size increases, leading to two identical curv es at 500 counts. These results are, howe ver , not sufficient to determine weather PL or GV is better . In particular , we recall our definition of good test as a test of hypothesis which makes the power as high as possible while keeping the false detection rate less than or equal to α G , which in our examples is set to 0.0013. In this sense, the increased power of PL is artificial; it is due to an increase of the probability of a type I error , and thus does not satisfy our goodness requirements. Con versely , GV seems to fit in our definition of a good test of hypothesis: the false detection rate is equal to or smaller than expected, and its local po wer function approaches that of PL as the sample size increases. As specified in (4.6), p PL is a v alid approximation to P ( C PL ( ˆ θ ) > c PL ) asymptotically , i.e., for large v alues of c PL . The higher than expected type I error rate of PL in our simulations, howe ver , does not appear to be the result of c PL being too small. As described in [1], the error rate of p PL as an approximation to P ( C PL ( ˆ θ ) > c PL ) is in the order of o ( c − 1 e − c 2 / 2 ) . In our three examples the values for c PL solving (5.3) are 3.896, 3.939 and 3.937 respectively , leading to an approximation error of the order of 10 − 4 . Thus, the high false detection rate of PL is unlikely to be due to an underestimation of the 3 σ nominal thresholds. Instead, it indicates that e ven a sample size of 500 is not sufficiently large to guarantee the con vergence of C P L to the supremum of a mean zero Gaussian process, as discussed in Section 4. This, howe ver , does not in validate the utility of PL for large sample sizes as sho wn in [1, 14]. A more detailed comparison of the detection power of PL and GV can be done by correcting the false detection rate (as in the third ro w of Fig. 3). Specifically , we can use the empirical detection threshold when ev aluating the local power of the two procedures. This guarantees a f alse detection rate of 0.0013 (3 σ significance). GV has a lower chance of T ype I error than the adjusted PL, i.e., the adjusted PL has probability 0.0013 of T ype I error , which bounds that of GV , see first ro w of Fig. 3. Despite this, for all three examples and for all signal locations (values of γ or M χ ) considered, GV is equally or more po werful than PL when using the empirical threshold. Thus, the e vidence from this simulation indicates that for small sample sizes, GV is uniformly locally more po werful than PL. Comparing the local power functions in the second and third ro ws of Fig. 3 with the pdfs in Fig. 2, we see that, for Example I, the detection po wer of the testing procedures is affected by both the specific location of the signal and its spread over the search region. The power is higher when – 12 – the resonance is narro wly dispersed and is located in a region with lo w background. In Example II, only the location of the source emission seems to affect the po wer . In particular , detection appears to be more dif ficult in high background areas of the spectrum, and thus the strength of the signal is weaker with respect to the background sources. These issues are overcome if at least 500 counts are a vailable; in this case both procedure exhibit maximum detection power regardless the location or dispersion of the signal. Fe w computational difficulties arose when implementing PL and GV . For PL, the most prob- lematic step is the calculation of the geometric constant ξ 0 in ( 4.6), which is computed via (A.3) for Example I and via (A.4) for Example II. This in volv es the numerical computation of nested in- tegrals and it can significantly slow do wn the testing procedure for complicated models. In the case of Examples I and II, small ranges over the energy spectra Y ( [ 0; 2 ] and [ 1; 10 ] respecti vely) were chosen in order to speed up the computation of these integrals, which tended to di ver ge numerically ov er larger energy bands. In presence of nuisance parameters under the null model, such as τ in Example II, the calculation of ξ 0 required by (A.4) is particularly complicated and considerably slo wer than that required by (A.3). The main difficulty with GV is associated with Step 2 of Algorithm 1 in Section 4, which in- volv es a multidimensional constrained optimization that must be repeated M times o ver a grid, Θ G , of size R . In Example II for instance, where R is set to 50, at each of the M = 100 , 000 Monte Carlo simulations, 50 two-dimensional constrained optimizations are implemented simoultaneously . If the nuisance parameter under H 1 , θ , is one-dimensional, the necessary computation can easily be accomplished by choosing a small threshold c 0 as described in Section 4 and in more detail in [2]. Unfortunately , using GV is more complicated when θ is multidimensional. A possible solution is proposed in [30] in which, the number of upcrossings of the LR T process is replaced by the concept of Euler characteristics, which unfortunately does not enjoy the advantages av ailable with the c 0 threshold. As discussed by the authors, the higher the number of dimensions, the higher the chances the χ 2 approximation may fail as the number of regions with weak background increases. Further , increasing the dimensions, the computational effort for each Monte Carlo simulation in- creases drastically . Larger sample sizes are needed for each simulation in order to guarantee χ 2 distribution. This, combined with the Monte Carlo simulation size needed for adequate accuracy , may lead to impractical CPU requirement. In this scenario, provided there is suf ficient data to ensure an appropriate type I error rate, the numerical integrations required by PL may be prefer- able. Some examples of multidimensional case are discussed in both [1, 14]; specifically , in [1], the analysis in our Example I is further extended to a tw o dimensional search. 6. Application to realistic data As a practical application, we perform the testing procedures discussed in Section 3 and 4 on a simulated observ ation of a monochromatic feature by the Fermi Large Area T elescope (LA T). The existence of such a feature within the LA T energy windo w would be an indication of new physics; of particular interest, it could result from the self-annihilation of a dark matter particle, and has con- sequently been the subject of several recent studies [24–26]. W e consider emission resulting from the self-annihilation of a particle making up the substantial dark matter mass of the V irgo galaxy cluster (distributed according to [27]). W e further specify that the particle hav e a mass of 35 GeV – 13 – Signal Signal Method Location Strength Sig. Unadjusted local 35.82 0.042 5 . 920 σ Bonferroni 35.82 0.042 5 . 152 σ Gross & V itells 35.82 0.042 5 . 192 σ Pilla et al. 35.82 0.042 ∗ 5 . 531 σ ∗ Obtained afterwards via MLE by fixing the signal location to its PL estimate (see text). T able 1: Summary of multiple hypothesis testing, GV and PL on the Fermi LA T simulation. For the multiple hypothesis testing case, the smallest of R = 80 (undadjusted local) p-values, Bonferroni’ s bound on the global p-v alue, along with GV and PL, are reported with their respectiv e statistic. Signal Location (GeV) Significance ( σ ) 10 40 70 100 130 160 190 220 250 280 310 340 0 1 2 3 4 5 6 GV PL Unadjusted local p−values Bonferroni adjusted local p−v alues Figure 4: Unadjusted local p-values (orange diamonds), Bonferroni adjusted local p-values (green dots), PL global p-v alue (gray dotted line) and GV global p-value (blue dashed line) for the Fermi LA T simulation. The Bonferroni’ s bound on the global p-value is only slightly more conservati ve than the GV p-v alue and a direct-to-photon thermally-averaged annihilation cross section of 1 × 10 − 23 cm 2 . Competing with this signal, we introduce a simple astrophysical background corresponding to isotropic emis- sion following a spectral power -law with inde x 2 . 4, i.e., τ = 1 . 4. Both signal and background mod- els are then simulated for a fi ve-year observation period using the gtobssim package, av ailable at http://fermi.gsfc.nasa.gov/ssc/data/analysis/software , which takes into account details of the instrument and orbit. The setup yields, on av erage, 64 signal and 2391 background e vents. The full model is the same as in Example II i.e., as giv en in ( 5.2); results of the sev eral methods – 14 – Unadjusted Bonferroni Gross local adj. local & V itells Bkg only 97056 37 2907 T ime (secs) 0.974 0.000 136.282 Bkg + sig 10496 45210 44294 T ime (secs) 1.061 0.000 137.532 T able 2: Summary on the analysis of 100,000 simulated datasets from Example II in Section 5. W e report the number of times each testing method is used by the sequential approach to make a final decision at 3 σ , and the respectiv e a verage computational times. The first two lines refer to the background only simulations and whereas the last two lines correspond to the background + signal simulations. are shown in T able 1 and Fig. 4. In the multiple hypothesis testing analysis, the smallest of the local p-values is reported along with the respectiv e estimates for the signal strength and location. As discussed in Section 4, the latter are equiv alent to those obtained with GV . The test statistic of PL, C PL ( ˆ θ ) , is constructed under the assumption that η = 0, and thus does not depend on the signal strength. Ho wev er, it does depend on the location of the source emission, and thus the estimation of η under H 1 must be conducted once the signal location has been estimated (through MLE for instance). In our analysis, the PL estimate for the source location is equiv alent to both that of GV and of the local p-values methods; it follo ws that the resulting MLE for the signal strength is the same for all methods. The local p-value approach leads to the largest significance of 5 . 920 σ , followed by PL 5 . 531 σ , GV 5 . 192 σ and finally Bonferroni with 5 . 152 σ . Although PL provides the most significant of the global p-values, it is dif ficult to interpret this result gi ven PL ’ s higher than expected rate of false detections in the simulation study . The Bonferroni adjusted local p-v alue, over the set of 80 simultaneous tests, it is only slightly more conservati ve than GV . The disparity between the two is expected to gro w , howe ver , as the number of grid points over the ener gy spectrum increases. 7. A sequential approach The PL and GV methods are typically used to overcome the over -conservati veness of the Bonfer- roni’ s bound. Thus, one might expect the global p-v alues p G V and p P L to be smaller or equal to p BF . Unfortunately , this is not always true; for the specific case of GV , combining (4.7) and (4.8), we hav e p G V = p L + E [ U ( c ) | H 0 ] ≤ p L + p B F = ( R + 1 ) p L . (7.1) Where E [ U ( c G V ) | H 0 ] = E [ U ( c 0 ) | H 0 ] e − c GV − c 0 2 is the expected number of upcrossings of the ob- served value for the test statistic c G V , i.e., c G V = LRT θ ? in (4.8). Since the expected number of upcrossings abov e c G V is bounded by the expected number of times the LR T process takes a value greather than c G V , i.e., R p L = p B F , and giv en the asymptotic equiv alence of GV and PL for large sample size (see Section 4), we hav e p P L ≈ p G V ≤ R + 1 R p B F ≈ p B F for large R . (7.2) – 15 – For small R , the bound in ( 7.2) allo ws Bonferroni to pro vide a sharper bound than either GV or PL. A more formal justification of 7.1 and 7.2 can be found in [28]. Based on this and the results of the previous sections, it is possible to establish general guide- lines for selecting the appropriate statistical testing procedure. The goal is to adhere a prescribed false-positi ve rate as closely as possible while minimizing computational effort. This can be accom- plished by combining the simplicity of multiple hypothesis testing with the rob ustness of global p- v alues in a multi-stage procedure. Specifically , Fig. 5 summarizes a simple step-by-step algorithm where multiple hypothesis testing methods are implemented first, and the more time-consuming GV and PL are implemented only if simpler methods exhibit poor type I error rates and/or po wer . Define a fine grid Θ G of size R Compute p L No new signal detected Compute p BF Is the search in 1D? Is the number of events large? New signal detected Compute p GV Compute p PL p L < α G p L ≥ α G p BF ≥ α G p BF < α G no yes no yes yes p GV ≥ α G p GV < α G p PL ≥ α G p PL < α G Figure 5: Outline of the sequential approach. General guidelines for statistical signal detections in HEP . Θ G is the grid of possible signal-search locations; its resolution is giv en by R . p L is the minimum of the local p-values and p BF its Bonferroni adjusted counterpart. α G is the predeter- mined false detection rate. p PL and p GV are the global p-v alues provided by PL [1, 14] and GV [2] respecti vely . Dashed arrows indicate that two actions are equally valid, and dotted lines lead to the final conclusion in terms of e vidence in fa vor of the ne w resonance. W e focus on the case of a one-dimensional search. In which, p L ≤ p P L ≈ p G V / p B F , (7.3) where the approximation sign in the last inequality allo ws the situation discussed abov e where – 16 – T ype I error Power Unadjusted local 0.03033 0.89502 Bonferroni adj. local 0.00040 0.45211 Gross & V itells 0.00089 0.53159 Sequential approach 0.00087 0.53161 T able 3: Probability of type I error and po wer of the testing methods and sequential approach implemented on 100,000 simulated datasets from Example II in Section 5. p G V ≥ p B F . Despite this possibility , the bound in (7.3) is an approximation for lar ge R , where R + 1 R ≈ 1. In order to implement the sequential approach, we first calculate the R unadjusted local p- v alues over the grid Θ G ; the minimum of these p-v alues is denoted by p L . From (7.3), if we observ e p L > α G we fail to reject reject H 0 with any of the procedures and we can immediately conclude that we cannot reject H 0 . On the other hand, if p L ≤ α G , a correction for the simultaneous R tests is needed, and because of its easy implementation, we compute p B F . Whereas, if p B F < α G , then all methods reject H 0 , and we can claim evidence in fa vor of the ne w source. Con versely , if p B F ≥ α G we should implement a method that is typically less conservati ve than Bonferroni’ s correction, when dealing with large significances (e.g. 3 σ , 4 σ , 5 σ ), such as GV or PL. Specifically , on the basis of the simulations in Section 5, GV appears to be preferable for small sample sizes, as it provides a false-positi ve rate less than or equal to α G . For large sample sizes, PL and GV are equi valent, and the decision between GV and PL depends on the details of the models compared. As discussed in Section 5, PL requires e xtensi ve numerical inte gration which can di ver ge for large search windows Θ , while GV requires a small number of Monte Carlo simulations which might become troublesome for complicated models. Finally , if p G V < α G (or p P L < α G ) we can claim e vidence in support of the new resonance, whereas if p G V ≥ α G (or p P L ≥ α G ) we cannot claim that a signal has been detected. The sequential approach inv olves choosing a procedure based on the characteristics of the data. Thus, one might be concerned about possible “flip-flopping” similar to that described by Feldman and Cousins in [29] in the context of confidence intervals. As argued belo w , howe ver , this is not the case for the sequential approach illustrated in Fig. 5. By virtue of (7.3), both the type I error and the power of the sequential approach are approximately equiv alent to those of GV (or PL) for large values of R . For clarity , we hereinafter suppose GV is used rather than PL in the sequential approach. The statistical results follow in exactly the same way howe ver , if PL is used for large sample sizes. Let ˜ α be the false detection rate associated with the sequential approach, and consider the e vents B F 0 = { Reject H 0 at le vel α G with Bonferroni } G V 0 = { Reject H 0 at le vel α G with GV } . As in (2.5) we use P ( ·| η = 0 ) to denote the probability that one ev ent occurs gi ven that the null hypothesis is true, i.e., in absence of the signal. Because the sequential approach rejects H 0 when – 17 – either Bonferroni or GV does so, it follo ws that ˜ α = P ( B F 0 or G V 0 | η = 0 ) = P ( B F 0 | η = 0 ) + P ( G V 0 | η = 0 ) − P ( B F 0 and G V 0 | η = 0 ) = P ( B F 0 | η = 0 ) + P ( G V 0 | η = 0 ) − P ( G V 0 | B F 0 , η = 0 ) P ( B F 0 | η = 0 ) . By the ordering of the p-v alues in ( 7.3), if H 0 is rejected by Bonferroni, then it is typically rejected by GV and thus, P ( G V 0 | B F 0 , η = 0 ) ≈ 1 , from which it follows that ˜ α ≈ P ( G V 0 | η = 0 ) , where P ( G V 0 | η = 0 ) is the false detection rate of GV . The power of the sequential approach can be obtained in a similar manner by considering the e vents L 1 = { Reject H 0 at le vel α G with local p-v alues } G V 1 = { Reject H 0 at le vel α G with GV } , and e valuating probabilities of the type P ( ·| η , θ ) defined in (2.5). T o illustrate its statistical properties, we apply the combined approach to a set of 100,000 sim- ulated datasets from the model in Example II with τ fixed at 1.4. For each dataset we first simulate 2000 background only ev ents and then we simulate 30 additional ev ents from a Gaussian source centered at 9 GeV . For both the 100,000 background only datasets and the 100,000 background plus source datasets we compute unadjusted local p-v alues, Bonferroni’ s corrections, and GV . T able 2 reports the number of times each of the testing procedures considered is selected by the sequential approach to make a final decision at the 3 σ significance lev el. The av erage computational times for each method are also reported. In the presence of source emission, the most computationally expensi ve method GV was used only about 44% of the time, leading to a computational gain of about 89 days over the 100,000 simulations. Con versely , in absence of the signal, GV was used about 2 . 9% of the time, leading to a computational gain of about 155 days. In order to assess the robustness of the method with respect to the desired statistical properties, we computed the false discov ery rate and the power using nominal lev els at 3 σ significance. The results are presented in T able 3. As discussed above, the sequential approach exhibits statistical properties which are approximately equiv alent to those of GV (or PL). As e xpected, the small discrepancies between the two methods are due to the f act that in 0 . 375% of the replications p G V > p B F . When removing these cases from the analysis, both the probability of a T ype I error and the po wer of the sequential approach coincide with those of GV . Finally , Fig. 6 displays the p-v alues computed with each procedure on each of the 100,000 simulated background-only datasets. Ideally a p-value will follo w a uniform distrib ution on the unit interv al under repeated sampling of data under H 0 : this insures that the method will have the tar get T ype I error rate. In the QQ-plots in Fig. 6, the p-values will fall along the 45 ◦ line if they follow a uniform distrib ution. If they deviate abov e this line, the procedure is conserv ativ e and if they de viate below the procedure will exhibit too many false positiv es. As expected, the unadjusted local p-v alues are always smaller than their expected values assuming uniform distribution, whereas both – 18 – ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Unadjusted local p−value Expected Obser v ed ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0.005 0.01 0.001 0.005 0.01 3 σ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Bonferr oni adjusted Expected Obser v ed ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0.005 0.01 0.001 0.005 0.01 3 σ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Gross and Vitells Expected Obser v ed ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0.005 0.01 0.001 0.005 0.01 3 σ ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Combined solution Expected Obser v ed ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 0.005 0.01 0.001 0.005 0.01 3 σ Figure 6: QQ-plots for the unadjusted local, Bonferroni’ s bound and GV p-v alues computed for the 100,000 simulated background-only datasets from Example II of Section 5. Each dataset considers 2000 background only e vents. The p-values selected via the sequential procedure in Fig. 5 are also reported. Each set of p-v alues is compared with the expected quantiles of a Uniform distribution on [ 0 , 1 ] . The inlayed plots in each panel magnify the important range of the p-value distributions near zero. Bonferroni and GV are conserv ative. The sequential approach leads to an intermediate situation in which the p-values are ov er-conserv ativ e up to the significance lev el α G adopted at each step of the – 19 – algorithm in Fig. 5 (3 σ in Fig. 6), whereas the p-values become under-conserv ativ e abov e α G , i.e., only for uninteresting cases. 8. Discussion In this article we in vestigate the performance of four different testing procedures for the statistical detection of new particles: the multiple hypothesis testing approach based on local p-values [6, 7], its Bonferroni adjusted counterpart, the LR T -based approach of Gross and V itells [2], GV , and the Score-based approach of Pilla et al. [1, 14], PL. T o the best of our knowledge, ours is the first application in a realistic scientific problem of PL in [14], i.e., in presence of nuisance parameters under H 0 . W e show analytically that local p-values are strongly affected by the arbitrary choice of the grid resolution, R , ov er the energy range where the tests are conducted. Specifically , when R is suf- ficiently large, the unadjusted p-v alues provide a higher number of false detections than expected, whereas the Bonferroni’ s bound on the global p-value may lead to over conservati ve inference if R is large. Ho wever , as shown in our realistic data analysis, if R is only moderately large ( R = 80 in our case) Bonferroni represents a reasonable choice. Additionally , cases may arise where Bon- ferroni’ s bound leads to less stringent acceptance criteria than GV and PL. Thus, in order to make final conclusions and to take adv antage of the easy implementation of the Bonferroni correction, it should al ways be used as a preliminary tool in statistical signal detection as described in Section 7. If the number of search regions R is quite lar ge, a good trade-of f is provided by both PL and GV which produce global p-v alues as a measure of the e vidence for a new source of emission. Although, PL and GV lead to the same conclusions for large sample sizes, based on our simulations, for small samples sizes PL may produce a higher number of false detections than expected. This strongly compromises the reliability of PL when only a few ev ents are av ailable, and thus GV is preferable in this case. From a computational perspectiv e, dif ficulties may arise with both methods when dealing with comple x models; these stem from the required numerical i ntegrations of PL and the Monte Carlo simulations and multidimensional optimization of GV . The latter are not required by PL since the procedure does not require estimation of the signal strength. PL requires a higher lev el of mathematical complexity to compute the geometric constants in volved. This is e xacerbated when free parameters are present under the null model, and the methodology must be extended as in [14]. On the other hand, PL can automatically be implemented when the nuisance parameter under the alternative hypothesis is multidimensional, whereas the existing multi variate counterpart of GV [30] relies on the computation of Euler characteristics, which does not enjoy the simplicity and computational ef ficiency of the one-dimensional case. Section 7 summarizes the methods and provides step-by-step guidelines for a sequential ap- proach for statistical signal detection in High Energy Physics. The sequential approach preserves both false detection rate and power , while allowing considerable gains in terms of implementation and computational time relati ve to other methods. 9. Acknowledgement JC thanks the support of the Knut and Alice W allenberg foundation and the Swedish Research – 20 – Council. DvD acknowledges support from a W olfson Research Merit A ward (WM110023) pro- vided by the British Royal Society and from Marie-Curie Career Integration (FP7-PEOPLE-2012- CIG-321865) and Marie-Skodowska-Curie RISE (H2020-MSCA-RISE-2015-691164) Grants both provided by the European Commission. A. Appendix A.1 Covariance function of { C ? P L ( θ ) , θ ∈ Θ } If the nuisance parameter under H 0 , φ , is known, the cov ariance function W ( θ , θ † ) in ( 4.5) of { C ? P L ( θ ) , θ ∈ Θ } is given by W ( θ , θ † ) = Z Θ g ( y , θ ) g ( y , θ † ) f ( y , , , , φ ) d θ − 1 . (A.1) Con versely , if φ is unknown, it is replaced by its MLE under H 0 in (4.4) and the cov ariance func- tion W ( θ , θ † ) is modified accordingly . For illustration, we consider the case where φ is one- dimensional and W ( θ , θ † ) is given by W ( θ , θ † ) = W φ ( θ , θ † ) − W ( θ | ˆ φ 0 ) W ( θ † | ˆ φ 0 ) I ( ˆ φ 0 ) , (A.2) where ˆ φ 0 is the MLE of φ under H 0 , I ( ˆ φ 0 ) is the Fisher information ∂ 2 log f ( y , φ ) ∂ 2 φ under H 0 e valuated at ˆ φ 0 , and W ( θ | ˆ φ 0 ) = R g ( y , θ ) ∂ log f ( y , φ ) ∂ φ | φ = ˆ φ 0 d y . The multi-dimensional generalization of (A.2) is described in [14]. A.2 Geometric constant ξ 0 in the calculation of p PL If the nuisance parameter under H 0 , φ , is kno wn, the geometric constant ξ 0 in (4.6) is gi ven by ξ 0 = Z Θ s     W ( θ , θ † ) ∂ 2 W ( θ , θ † ) ∂ θ ∂ θ † − ∂ W ( θ , θ † ) ∂ θ ∂ W ( θ , θ † ) ∂ θ †     θ † = θ W ( θ , θ ) d θ . (A.3) Whereas, if φ is unkno wn, ξ 0 is gi ven by ξ 0 = Z Θ r ∂ 2 ρ ? ( θ , θ † ) ∂ θ ∂ θ †     θ † = θ d θ with ρ ? ( θ , θ † ) = W ( θ , θ † ) p W ( θ , θ ) W ( θ † , θ † ) . 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