text: [30] H. Wiener, Structural determination of paraﬃn boiling points, J. Am. Chem. Soc. 69(1) (1947), 2636–2638. [31] W. Yan, B.-Y. Yang, and Y.-N. Yeh, The behavior of Wiener indices and polynomials of graphs under ﬁve graph decorations, Appl. Math. Letters 20(3) (2007), 290–295. 17

label: reference

text: H.

text: Wiener,

text: Structural

text: determination

text: of

text: paraﬃn

text: boiling

text: points,

text: J.

text: Am.

text: Chem.

text: Soc.

text: W.

text: Yan,

text: B.-Y.

text: Yang,

text: and

text: Y.-N.

text: Yeh,

text: The

text: behavior

text: of

text: Wiener

text: indices

text: and

text: polynomials

text: of

text: graphs

text: under

text: ﬁve

text: graph

text: decorations,

text: Appl.

text: Math.

text: Letters


text: (cid:0) (cid:1) − The layout of the article is as follows. In Section , we determine that n 1 is 2 the maximum modulus among all Wiener roots of connected graphs of order n, and we describe the extremal graphs which have a root of this maximum modulus. In contrast, we then show that for trees, the rate of growth of the maximum modulus is Θ(n). Finally, we determine that −2 is the minimum modulus among all nonzero Wiener roots of connected n 2 graphs of order n, and again we are able to describe the extremal graphs (which are trees). In Section , we prove that the closure of the collection of all real Wiener roots of connected −∞ graphs is ( , 0], and that for trees, the closure of all real Wiener roots still contains −∞ − ( , 1]. In Section , we study the real and imaginary parts of Wiener roots. We show that there are Wiener roots of trees with arbitrarily large imaginary part, and that there are Wiener roots of trees with arbitrarily large positive real part. This work culminates in a proof of the fact that the collection of complex Wiener roots of all graphs is not contained in any half-plane (a region in the complex plane consisting of all points on one side of a line and no points on the other side). Finally, we prove that almost all graphs have all real Wiener roots, and we ﬁnd purely imaginary Wiener roots. Throughout, we compare and contrast our results with what is known about the roots of other graph polynomials. 2 Bounding the Modulus of Wiener Roots Bounding the modulus of roots of various graph polynomials has been a central point of interest in the literature. For example, Sokal [25] settled an outstanding problem in proving that the modulus of a chromatic root (a root of the chromatic polynomial) of a graph of maximum degree ∆ is at most a constant times ∆. This implies that the modulus of any chromatic root of a graph of order n is at most Cn for some positive constant C; the precise value of the smallest such constant is still unknown. The independence (cid:0) (cid:1) polynomial of a graph G is the generating function for the number of independent sets of each cardinality in G. Among all graphs of order n and ﬁxed independence number β, − − the maximum modulus of an independence root is (n/β)β 1 + O nβ 2 [7]. For the all- terminal reliability polynomial, it is suspected that the collection of all roots is bounded in | − | ≤ modulus (as no roots have been found outside of the disk z 1 1.2). Currently the best known upper bound on the modulus of the roots for all graphs of order n is linear in n as shown in [6]. For the independence polynomials, the all-terminal-reliability polynomials and the chromatic polynomials, the extremal graphs that yield roots of maximum modulus are not known. In contrast we completely determine the extremal graphs for which the Wiener polynomials, of all connected graphs of order n, have maximum modulus of any root. To do so, we will make use of the classical Enestr¨om-Kakeya Theorem [19]. Theorem 2.1 (The Enestr¨om-Kakeya Theorem). If f(x) = a + a x + . . . + a xn has 0 1 n positive real coeﬃcients, then all complex roots of f lie in the annulus (cid:110) (cid:111) (cid:110) (cid:111) ≤ | | ≤ r z R ≤ ≤ − ≤ ≤ − where r = min ai : 0 i n 1 and R = max ai : 0 i n 1 . ai+1 ai+1 As a straightforward application of the Enestr¨om-Kakeya Theorem, consider the path ≥ (cid:88) Pn of order n 3. The reduced(cid:99)Wiener polynomial of Pn is given by − n 1 − − W (P ; x) = (n i)xi 1. n i=1 3 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: The

text: layout

text: of

text: the

text: article

text: is

text: as

text: follows.

text: In

text: Section

text: we

text: determine

text: that

text: n

text: is

text: the

text: maximum

text: modulus

text: among

text: all

text: Wiener

text: roots

text: of

text: connected

text: graphs

text: of

text: order

text: n,

text: and

text: we

text: describe

text: the

text: extremal

text: graphs

text: which

text: have

text: a

text: root

text: of

text: this

text: maximum

text: modulus.

text: In

text: contrast,

text: we

text: then

text: show

text: that

text: for

text: trees,

text: the

text: rate

text: of

text: growth

text: of

text: the

text: maximum

text: modulus

text: is

text: Θ(n).

text: Finally,

text: we

text: determine

text: that

text: −2

text: is

text: the

text: minimum

text: modulus

text: among

text: all

text: nonzero

text: Wiener

text: roots

text: of

text: connected

text: n

text: graphs

text: of

text: order

text: n,

text: and

text: again

text: we

text: are

text: able

text: to

text: describe

text: the

text: extremal

text: graphs

text: (which

text: are

text: trees).

text: In

text: Section

text: we

text: prove

text: that

text: the

text: closure

text: of

text: the

text: collection

text: of

text: all

text: real

text: Wiener

text: roots

text: of

text: connected

text: graphs

text: is

text: and

text: that

text: for

text: trees,

text: the

text: closure

text: of

text: all

text: real

text: Wiener

text: roots

text: still

text: contains

text: In

text: Section

text: we

text: study

text: the

text: real

text: and

text: imaginary

text: parts

text: of

text: Wiener

text: roots.

text: We

text: show

text: that

text: there

text: are

text: Wiener

text: roots

text: of

text: trees

text: with

text: arbitrarily

text: large

text: imaginary

text: part,

text: and

text: that

text: there

text: are

text: Wiener

text: roots

text: of

text: trees

text: with

text: arbitrarily

text: large

text: positive

text: real

text: part.

text: This

text: work

text: culminates

text: in

text: a

text: proof

text: of

text: the

text: fact

text: that

text: the

text: collection

text: of

text: complex

text: Wiener

text: roots

text: of

text: all

text: graphs

text: is

text: not

text: contained

text: in

text: any

text: half-plane

text: (a

text: region

text: in

text: the

text: complex

text: plane

text: consisting

text: of

text: all

text: points

text: on

text: one

text: side

text: of

text: a

text: line

text: and

text: no

text: points

text: on

text: the

text: other

text: side).

text: Finally,

text: we

text: prove

text: that

text: almost

text: all

text: graphs

text: have

text: all

text: real

text: Wiener

text: roots,

text: and

text: we

text: ﬁnd

text: purely

text: imaginary

text: Wiener

text: roots.

text: Throughout,

text: we

text: compare

text: and

text: contrast

text: our

text: results

text: with

text: what

text: is

text: known

text: about

text: the

text: roots

text: of

text: other

text: graph

text: polynomials.

text: Bounding

text: the

text: Modulus

text: of

text: Wiener

text: Roots

text: Bounding

text: the

text: modulus

text: of

text: roots

text: of

text: various

text: graph

text: polynomials

text: has

text: been

text: a

text: central

text: point

text: of

text: interest

text: in

text: the

text: literature.

text: For

text: example,

text: Sokal

text: settled

text: an

text: outstanding

text: problem

text: in

text: proving

text: that

text: the

text: modulus

text: of

text: a

text: chromatic

text: root

text: (a

text: root

text: of

text: the

text: chromatic

text: polynomial)

text: of

text: a

text: graph

text: of

text: maximum

text: degree

text: is

text: at

text: most

text: a

text: constant

text: times

text: This

text: implies

text: that

text: the

text: modulus

text: of

text: any

text: chromatic

text: root

text: of

text: a

text: graph

text: of

text: order

text: n

text: is

text: at

text: most

text: Cn

text: for

text: some

text: positive

text: constant

text: C;

text: the

text: precise

text: value

text: of

text: the

text: smallest

text: such

text: constant

text: is

text: still

text: unknown.

text: The

text: independence

text: polynomial

text: of

text: a

text: graph

text: G

text: is

text: the

text: generating

text: function

text: for

text: the

text: number

text: of

text: independent

text: sets

text: of

text: each

text: cardinality

text: in

text: G.

text: Among

text: all

text: graphs

text: of

text: order

text: n

text: and

text: ﬁxed

text: independence

text: number

text: β,

text: the

text: maximum

text: modulus

text: of

text: an

text: independence

text: root

text: is

text: (n/β)β

text: O

text: nβ

text: For

text: the

text: all-

text: terminal

text: reliability

text: polynomial,

text: it

text: is

text: suspected

text: that

text: the

text: collection

text: of

text: all

text: roots

text: is

text: bounded

text: in

text: modulus

text: (as

text: no

text: roots

text: have

text: been

text: found

text: outside

text: of

text: the

text: disk

text: z

text: Currently

text: the

text: best

text: known

text: upper

text: bound

text: on

text: the

text: modulus

text: of

text: the

text: roots

text: for

text: all

text: graphs

text: of

text: order

text: n

text: is

text: linear

text: in

text: n

text: as

text: shown

text: in

text: For

text: the

text: independence

text: polynomials,

text: the

text: all-terminal-reliability

text: polynomials

text: and

text: the

text: chromatic

text: polynomials,

text: the

text: extremal

text: graphs

text: that

text: yield

text: roots

text: of

text: maximum

text: modulus

text: are

text: not

text: known.

text: In

text: contrast

text: we

text: completely

text: determine

text: the

text: extremal

text: graphs

text: for

text: which

text: the

text: Wiener

text: polynomials,

text: of

text: all

text: connected

text: graphs

text: of

text: order

text: n,

text: have

text: maximum

text: modulus

text: of

text: any

text: root.

text: To

text: do

text: so,

text: we

text: will

text: make

text: use

text: of

text: the

text: classical

text: Enestr¨om-Kakeya

text: Theorem

text: Theorem

text: (The

text: Enestr¨om-Kakeya

text: Theorem).

text: If

text: f(x)

text: a

text: a

text: x

text: a

text: xn

text: has

text: n

text: positive

text: real

text: coeﬃcients,

text: then

text: all

text: complex

text: roots

text: of

text: f

text: lie

text: in

text: the

text: annulus

text: r

text: z

text: R

text: where

text: r

text: min

text: ai

text: i

text: n

text: and

text: R

text: max

text: ai

text: i

text: n

text: As

text: a

text: straightforward

text: application

text: of

text: the

text: Enestr¨om-Kakeya

text: Theorem,

text: consider

text: the

text: path

text: Pn

text: of

text: order

text: n

text: The

text: polynomial

text: of

text: Pn

text: is

text: given

text: by

text: n

text: W

text: (P

text: x)

text: (n

text: i)xi

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:16) (cid:16) ≥ (cid:17) (cid:17) (cid:18) (cid:19) which is ne(cid:99)gative for n 6, and √ √ W T ; − 1 + √1 n + 8 = 7 2 − 1 n2 + (−36 2 + 29)n + 63, n 2 4 2 (cid:99) (cid:16) (cid:16) (cid:17) (cid:16) (cid:17) (cid:17) which is positive for all n. By the Intermediate Value Theorem, W (T ; x), and hence n − − ≥ W (T ; x), has a root in the interval 1 + √1 n + 7, 1 + √1 n + 8 for all n n 2 2 6. Having obtained an upper bound on the modulus of any Wiener root of a graph (or tree) of order n, we now turn to a lower bound. Of course, the Wiener root of smallest modulus among all connected graphs of order n is 0, as 0 is a root of every Wiener polynomial, but what about the nonzero root of smallest modulus? Equivalently, what is the smallest modulus of a root of the reduced Wiener polynomial of a connected graph of order n? Below we prove a lower bound on the ratio of successive coeﬃcients of the Wiener polynomial which allows us to apply the Enestr¨om-Kakeya Theorem once more. ≥ ≥ Lemma 2.7. Let G be a connected graph of order n 3 and diameter D 2. Then d ≥ 2 k − − d n k 1 k+1 ≤ for 1 k < D. ∈ { − } { } Proof. Let k 1, . . . , D 1 . We count ordered pairs (w, u, v ), where d(u, v) = k and either w is joined to u and at distance k + 1 from v, or w is joined to v and at S | | distance k + 1 from u; let be the set of all such ordered pairs. Clearly, S is at most − − − k { } k (n (k + 1))d = (n k 1)d , as for any pair of vertices u, v at distance k, the k + 1 k k vertices on any shortest path between u and v cannot serve as the point w. On the other hand, for any pair of vertices x and y at distance k + 1, ﬁx a path P of this length between them, with vertex x adjacent to x on P , and y adjacent to y on P 1 1 (it may be the case that x = y , if k = 1). Then we can form two distinct ordered pairs, { } { } 1 1 S (x, x , y ) and (y, x, y ), both of which belong to . Working over all pairs of vertices 1 1 k at distance k+1, we generate 2d pairs through this process, and we claim that they are k+1 all distinct. Note ﬁrst of all that the ﬁrst coordinate in any ordered pair generated in this manner is adjacent to precisely one element of the second coordinate. Suppose that two { } { (cid:48) (cid:48)} S { } pairs x, y and x , y at distance k + 1 yield the same member of , say (x, x , y ). (cid:48) (cid:48) k (cid:48) 1 Then either x = x or y = x. Without loss of generality, assume that x = x. Since x (cid:48) (cid:48) (cid:48) (cid:54) (cid:48) is adjacent to x and d(x , y ) = k + 1 > 1, it follows that y = x , and thus y = y. We 1 1 conclude that − − ≥ |S | ≥ (n k 1)d 2d . k k k+1 ≥ Corollary 2.8. Among all connected graphs of order n 3, the graph K − has nonzero 1,n 1 Wiener root of mimimum modulus n−2 2. M(cid:0)or≥eo(cid:1)ver, K1,n−1 is the unique graph of order n with a Wiener root of this modulus for all n 3. − − Proof. Since W (K − ; x) = (n 1)x + n 1 x2, the star K − has a single nonzero − 1,n 1 2 1,n 1 Wiener root −2 . n 2 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: which

text: is

text: for

text: n

text: and

text: W

text: T

text: √1

text: n

text: n

text: which

text: is

text: positive

text: for

text: all

text: n.

text: By

text: the

text: Intermediate

text: Value

text: Theorem,

text: W

text: (T

text: x),

text: and

text: hence

text: n

text: W

text: (T

text: x),

text: has

text: a

text: root

text: in

text: the

text: interval

text: √1

text: n

text: √1

text: n

text: for

text: all

text: n

text: n

text: Having

text: obtained

text: an

text: upper

text: bound

text: on

text: the

text: modulus

text: of

text: any

text: Wiener

text: root

text: of

text: a

text: graph

text: (or

text: tree)

text: of

text: order

text: n,

text: we

text: now

text: turn

text: to

text: a

text: lower

text: bound.

text: Of

text: course,

text: the

text: Wiener

text: root

text: of

text: smallest

text: modulus

text: among

text: all

text: connected

text: graphs

text: of

text: order

text: n

text: is

text: as

text: is

text: a

text: root

text: of

text: every

text: Wiener

text: polynomial,

text: but

text: what

text: about

text: the

text: nonzero

text: root

text: of

text: smallest

text: modulus?

text: Equivalently,

text: what

text: is

text: the

text: smallest

text: modulus

text: of

text: a

text: root

text: of

text: the

text: reduced

text: Wiener

text: polynomial

text: of

text: a

text: connected

text: graph

text: of

text: order

text: n?

text: Below

text: we

text: prove

text: a

text: lower

text: bound

text: on

text: the

text: ratio

text: of

text: successive

text: coeﬃcients

text: of

text: the

text: Wiener

text: polynomial

text: which

text: allows

text: us

text: to

text: apply

text: the

text: Enestr¨om-Kakeya

text: Theorem

text: once

text: more.

text: Lemma

text: Let

text: G

text: be

text: a

text: connected

text: graph

text: of

text: order

text: n

text: and

text: diameter

text: D

text: Then

text: d

text: k

text: d

text: n

text: k

text: for

text: k

text: D.

text: Proof.

text: Let

text: k

text: D

text: We

text: count

text: ordered

text: pairs

text: (w,

text: u,

text: v

text: where

text: d(u,

text: v)

text: k

text: and

text: either

text: w

text: is

text: joined

text: to

text: u

text: and

text: at

text: distance

text: k

text: from

text: v,

text: or

text: w

text: is

text: joined

text: to

text: v

text: and

text: at

text: S

text: distance

text: k

text: from

text: u;

text: let

text: be

text: the

text: set

text: of

text: all

text: such

text: ordered

text: pairs.

text: Clearly,

text: S

text: is

text: at

text: most

text: k

text: k

text: (n

text: (k

text: (n

text: k

text: as

text: for

text: any

text: pair

text: of

text: vertices

text: u,

text: v

text: at

text: distance

text: k,

text: the

text: k

text: k

text: k

text: vertices

text: on

text: any

text: shortest

text: path

text: between

text: u

text: and

text: v

text: cannot

text: serve

text: as

text: the

text: point

text: w.

text: On

text: the

text: other

text: hand,

text: for

text: any

text: pair

text: of

text: vertices

text: x

text: and

text: y

text: at

text: distance

text: k

text: ﬁx

text: a

text: path

text: P

text: of

text: this

text: length

text: between

text: them,

text: with

text: vertex

text: x

text: adjacent

text: to

text: x

text: on

text: P

text: and

text: y

text: adjacent

text: to

text: y

text: on

text: P

text: (it

text: may

text: be

text: the

text: case

text: that

text: x

text: y

text: if

text: k

text: Then

text: we

text: can

text: form

text: two

text: distinct

text: ordered

text: pairs,

text: S

text: (x,

text: x

text: y

text: and

text: (y,

text: x,

text: y

text: both

text: of

text: which

text: belong

text: to

text: Working

text: over

text: all

text: pairs

text: of

text: vertices

text: k

text: at

text: distance

text: we

text: generate

text: pairs

text: through

text: this

text: process,

text: and

text: we

text: claim

text: that

text: they

text: are

text: all

text: distinct.

text: Note

text: ﬁrst

text: of

text: all

text: that

text: the

text: ﬁrst

text: coordinate

text: in

text: any

text: ordered

text: pair

text: generated

text: in

text: this

text: manner

text: is

text: adjacent

text: to

text: precisely

text: one

text: element

text: of

text: the

text: second

text: coordinate.

text: Suppose

text: that

text: two

text: S

text: pairs

text: x,

text: y

text: and

text: x

text: y

text: at

text: distance

text: k

text: yield

text: the

text: same

text: member

text: of

text: say

text: (x,

text: x

text: y

text: k

text: Then

text: either

text: x

text: x

text: or

text: y

text: x.

text: Without

text: loss

text: of

text: generality,

text: assume

text: that

text: x

text: x.

text: Since

text: x

text: is

text: adjacent

text: to

text: x

text: and

text: d(x

text: y

text: k

text: it

text: follows

text: that

text: y

text: x

text: and

text: thus

text: y

text: y.

text: We

text: conclude

text: that

text: |S

text: (n

text: k

text: k

text: k

text: Corollary

text: Among

text: all

text: connected

text: graphs

text: of

text: order

text: n

text: the

text: graph

text: K

text: has

text: nonzero

text: Wiener

text: root

text: of

text: mimimum

text: modulus

text: is

text: the

text: unique

text: graph

text: of

text: order

text: n

text: with

text: a

text: Wiener

text: root

text: of

text: this

text: modulus

text: for

text: all

text: n

text: Proof.

text: Since

text: W

text: (K

text: x)

text: (n

text: n

text: the

text: star

text: K

text: has

text: a

text: single

text: nonzero

text: Wiener

text: root

text: −2

text: n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Average Power and λ-power in Multiple Testing Scenar- ios when the Benjamini-Hochberg False Discovery Rate Procedure is Used Grant Izmirlian E-mail: izmirlig@mail.nih.gov Biometry Research Group, Division of Cancer Prevention; National National Cancer Insti- tute; BG 9609 RM 5E130 MSC 9789; 9609 Medical Center Dr; Bethesda, MD 20892-9789 USA; This article is a U.S. Government work and is in the public domain in the U.S.A. Summary. We discuss several approaches to deﬁning power in studies designed around the Benjamini-Hochberg (BH) false discovery rate (FDR) procedure. We focus primarily on the average power and the λ-power, which are the expected true positive fraction and the probability that the true positive fraction exceeds λ, respectively. We show that the average power converges as the number of simultaneous tests tends to inﬁnity, to a limit that is nearly equivalent to the power introduced independently by Jung (2005) and by Liu and Hwang (2007). Furthermore, we prove a CLT which allows asymptotic approximation to the λ-power. Moreover, we prove SLLNs and CLTs for all quantities connected to the BH-FDR procedure: the positive call fraction, true positive fraction, and false discovery fraction, with full characterization of almost sure limits and limits in distribution. We discuss ramiﬁcations of the CLT for the false discovery fraction, introducing a procedure which allows tighter control of the false discovery fraction than the BH-FDR that can be used at the design and analysis steps. We conduct a large simulation study covering a fairly substantial portion of the space of possible inputs. We show its application in design of a biomarker study, a micro-array experiment and a GWAS study. Keywords: Multiple testing; false discovery rate; average power; k power; LLN; CLT

text: Average

label: title

text: Power

text: and

text: λ-power

text: in

text: Multiple

text: Testing

text: Scenar-

text: ios

text: when

text: the

text: Benjamini-Hochberg

text: False

text: Discovery

text: Rate

text: Procedure

text: is

text: Used

text: Grant

label: author

text: Izmirlian

text: E-mail:

text: izmirlig@mail.nih.gov

label: footer

text: Biometry

text: Research

text: Group,

text: Division

text: of

text: Cancer

text: Prevention;

text: National

text: National

text: Cancer

text: Insti-

text: tute;

text: BG

text: RM

text: MSC

text: Medical

text: Center

text: Dr;

text: Bethesda,

text: MD

text: USA;

text: This

text: article

text: is

text: a

text: U.S.

text: Government

text: work

text: and

text: is

text: in

text: the

text: public

text: domain

text: in

text: the

text: U.S.A.

text: Summary.

text: We

label: abstract

text: discuss

text: several

text: approaches

text: to

text: deﬁning

text: power

text: in

text: studies

text: designed

text: around

text: the

text: Benjamini-Hochberg

text: (BH)

text: false

text: discovery

text: rate

text: (FDR)

text: procedure.

text: We

text: focus

text: primarily

text: on

text: the

text: average

text: power

text: and

text: the

text: λ-power,

text: which

text: are

text: the

text: expected

text: true

text: positive

text: fraction

text: and

text: the

text: probability

text: that

text: the

text: true

text: positive

text: fraction

text: exceeds

text: λ,

text: respectively.

text: We

text: show

text: that

text: the

text: average

text: power

text: converges

text: as

text: the

text: number

text: of

text: simultaneous

text: tests

text: tends

text: to

text: inﬁnity,

text: to

text: a

text: limit

text: that

text: is

text: nearly

text: equivalent

text: to

text: the

text: power

text: introduced

text: independently

text: by

text: Jung

text: and

text: by

text: Liu

text: and

text: Hwang

text: Furthermore,

text: we

text: prove

text: a

text: CLT

text: which

text: allows

text: asymptotic

text: approximation

text: to

text: the

text: λ-power.

text: Moreover,

text: we

text: prove

text: SLLNs

text: and

text: CLTs

text: for

text: all

text: quantities

text: connected

text: to

text: the

text: BH-FDR

text: procedure:

text: the

text: positive

text: call

text: fraction,

text: true

text: positive

text: fraction,

text: and

text: false

text: discovery

text: fraction,

text: with

text: full

text: characterization

text: of

text: almost

text: sure

text: limits

text: and

text: limits

text: in

text: distribution.

text: We

text: discuss

text: ramiﬁcations

text: of

text: the

text: CLT

text: for

text: the

text: false

text: discovery

text: fraction,

text: introducing

text: a

text: procedure

text: which

text: allows

text: tighter

text: control

text: of

text: the

text: false

text: discovery

text: fraction

text: than

text: the

text: BH-FDR

text: that

text: can

text: be

text: used

text: at

text: the

text: design

text: and

text: analysis

text: steps.

text: We

text: conduct

text: a

text: large

text: simulation

text: study

text: covering

text: a

text: fairly

text: substantial

text: portion

text: of

text: the

text: space

text: of

text: possible

text: inputs.

text: We

text: show

text: its

text: application

text: in

text: design

text: of

text: a

text: biomarker

text: study,

text: a

text: micro-array

text: experiment

text: and

text: a

text: GWAS

text: study.

text: Keywords:

text: Multiple

text: testing;

text: false

text: discovery

text: rate;

text: average

text: power;

text: k

text: power;

text: LLN;

text: CLT


text: − P{ ≤ } − lim m 1 T = P γf , ξ = 0 = (1 r) γf a.s. , (4.5) →∞ m i i m − P{ | ≤ } − l→im∞(cid:2) Jm1 Tm(cid:3) = ξi = 0 Pi γf = (1 r)f a.s. (4.6) m and E − − lim J 1 T = (1 r)f (4.7) →∞ m m m − Remark 4.4. Because T = J S then by Theorem 4.3.1 and its corollary m− m −m 4.3, we obtain the identity (1 r)f = 1 rπ /γ, which can be rearranged to obtain ∞ 1, an expression for the limiting positive call fraction: rπ ∞ γ = −1, . (4.8) 1 f 0 Next we point out the connection between the IST average power, π and the ∞ 1, Jung/Liu power, πJL. Towards this end, let 1 (cid:48) − c = F¯ 1(γf) . (4.9) 0,n so that (cid:48) π = F¯ (c ) , (4.10) ∞ A,n 1, (cid:48) and recall that πJL = F¯ (c). To investigate the connection between c and 1 A,n c, we begin by writing out the deﬁning expression for the limiting positive call fraction, γ, given in Theorem 4.1, γ = G(γf), expanding G in terms of F and A,n F , obtaining: 0,n − (cid:48) (cid:48) γ = (1 r)F¯ (c ) + rF¯ (c ) (4.11) 0,n A,n Plugging in the alternate expression for γ given in expression 4.8 on the left hand side above, we have (cid:48) rF¯ (c ) − (cid:48) (cid:48) A−,n = (1 r)F¯ (c ) + rF¯ (c ) (4.12) 0,n A,n 1 f 0 Inverting both sides, rearranging slightly, taking the complement, and recalling − that f = (1 r)f, gives 0 − (cid:48) − (1 r)F¯ (c ) 0,n (1 r)f = − (cid:48) (cid:48) (4.13) (1 r)F¯ (c ) + rF¯ (c ) 0,n A,n 11 ##LTLine## ##LTLine## ##LTLine##

label: equation

text: P{

text: lim

text: m

text: T

text: P

text: γf

text: ξ

text: r)

text: γf

text: a.s.

text: m

text: i

text: i

text: m

text: P{

text: ξi

text: Pi

text: γf

text: r)f

text: a.s.

text: m

text: and

text: E

text: lim

text: J

text: T

text: r)f

text: m

text: m

text: m

text: Remark

text: Because

text: T

text: J

text: S

text: then

text: by

text: Theorem

text: and

text: its

text: corollary

text: m−

text: m

text: −m

text: we

text: obtain

text: the

text: identity

text: r)f

text: rπ

text: /γ,

text: which

text: can

text: be

text: rearranged

text: to

text: obtain

text: an

text: expression

text: for

text: the

text: limiting

text: positive

text: call

text: fraction:

text: rπ

text: γ

text: f

text: Next

text: we

text: point

text: out

text: the

text: connection

text: between

text: the

text: IST

text: average

text: power,

text: π

text: and

text: the

text: Jung/Liu

text: power,

text: πJL.

text: Towards

text: this

text: end,

text: let

text: c

text: F¯

text: so

text: that

text: π

text: F¯

text: (c

text: A,n

text: and

text: recall

text: that

text: πJL

text: F¯

text: (c).

text: To

text: investigate

text: the

text: connection

text: between

text: c

text: and

text: A,n

text: c,

text: we

text: begin

text: by

text: writing

text: out

text: the

text: deﬁning

text: expression

text: for

text: the

text: limiting

text: positive

text: call

text: fraction,

text: γ,

text: given

text: in

text: Theorem

text: γ

text: G(γf),

text: expanding

text: G

text: in

text: terms

text: of

text: F

text: and

text: A,n

text: F

text: obtaining:

text: γ

text: r)F¯

text: (c

text: rF¯

text: (c

text: A,n

text: Plugging

text: in

text: the

text: alternate

text: expression

text: for

text: γ

text: given

text: in

text: expression

text: on

text: the

text: left

text: hand

text: side

text: above,

text: we

text: have

text: rF¯

text: (c

text: A−,n

text: r)F¯

text: (c

text: rF¯

text: (c

text: A,n

text: f

text: Inverting

text: both

text: sides,

text: rearranging

text: slightly,

text: taking

text: the

text: complement,

text: and

text: recalling

text: that

text: f

text: r)f,

text: gives

text: r)F¯

text: (c

text: r)f

text: r)F¯

text: (c

text: rF¯

text: (c

text: A,n

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 6. Excerpted results from a simulation study modelling micro-array studies, m = 54,675, for various E values of eﬀect sizes, values of [M ], FDR, and n. Shown are the λ-power at λ = 75% and at 90% from m CLT and from simulations, λ (π ∞) and sample size ratio. The IST average power is also shown for S/M 1, comparison. Eﬀ Sz E[Mm] n FDR π1,∞ λ75-pwr λˆ75-pwr λ90-pwr λˆ90-pwr λeq SS Ratio 0.60 100 100 0.15 0.683 0.112 0.098 0.000 0.000 0.656 1.420 0.60 100 110 0.15 0.763 0.602 0.611 0.002 0.000 0.728 1.309 0.60 100 120 0.15 0.826 0.963 0.953 0.039 0.037 0.786 1.217 0.60 100 130 0.15 0.874 1.000 0.998 0.233 0.248 0.832 1.138 0.60 100 140 0.15 0.910 1.000 1.000 0.627 0.661 0.869 1.071 0.60 1000 70 0.15 0.662 0.000 0.000 0.000 0.000 0.654 1.500 0.60 1000 80 0.15 0.761 0.758 0.754 0.000 0.000 0.750 1.312 0.60 1000 90 0.15 0.835 1.000 1.000 0.000 0.000 0.822 1.178 0.60 1000 100 0.15 0.887 1.000 1.000 0.121 0.110 0.874 1.070 0.60 1000 110 0.15 0.924 1.000 1.000 0.997 0.993 0.911 0.973 0.60 2000 60 0.15 0.640 0.000 0.000 0.000 0.000 0.635 1.500 0.60 2000 70 0.15 0.752 0.557 0.549 0.000 0.000 0.744 1.343 0.60 2000 80 0.15 0.832 1.000 1.000 0.000 0.000 0.823 1.188 0.60 2000 90 0.15 0.888 1.000 1.000 0.063 0.054 0.879 1.056 0.60 2000 100 0.15 0.927 1.000 1.000 1.000 1.000 0.918 0.960 0.80 100 60 0.15 0.717 0.266 0.311 0.000 0.000 0.687 1.367 0.80 100 70 0.15 0.835 0.981 0.984 0.058 0.055 0.795 1.200 0.80 100 80 0.15 0.909 1.000 1.000 0.612 0.625 0.868 1.075 0.80 1000 40 0.15 0.653 0.000 0.000 0.000 0.000 0.646 1.500 0.80 1000 50 0.15 0.816 1.000 1.000 0.000 0.000 0.803 1.220 0.80 1000 60 0.15 0.907 1.000 1.000 0.762 0.780 0.894 1.017 0.80 2000 40 0.15 0.747 0.404 0.393 0.000 0.000 0.739 1.350 0.80 2000 50 0.15 0.875 1.000 1.000 0.001 0.001 0.866 1.100 1.00 100 40 0.15 0.725 0.320 0.346 0.000 0.000 0.694 1.350 1.00 100 50 0.15 0.886 1.000 0.999 0.338 0.352 0.844 1.120 1.00 1000 30 0.15 0.754 0.592 0.593 0.000 0.000 0.743 1.333 1.00 1000 40 0.15 0.915 1.000 1.000 0.940 0.942 0.902 1.175 1.00 2000 30 0.15 0.830 1.000 1.000 0.000 0.000 0.821 1.200 19 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Excerpted

label: caption

text: results

text: from

text: a

text: simulation

text: study

text: modelling

text: micro-array

text: studies,

text: m

text: for

text: various

text: E

text: values

text: of

text: eﬀect

text: sizes,

text: values

text: of

text: [M

text: FDR,

text: and

text: n.

text: Shown

text: are

text: the

text: λ-power

text: at

text: λ

text: and

text: at

text: from

text: m

text: CLT

text: and

text: from

text: simulations,

text: λ

text: (π

text: and

text: sample

text: size

text: ratio.

text: The

text: IST

text: average

text: power

text: is

text: also

text: shown

text: for

text: S/M

text: comparison.

text: Eﬀ

label: table

text: Sz

text: E[Mm]

text: n

text: FDR

text: π1,∞

text: λeq

text: SS

text: Ratio

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Table 8. Excerpted results from a simulation study modelling GWA studies, m = 1,000,000. IST average E power, J/L power, lower bound and simulated average power for a selection of eﬀect sizes, values of [M ], m FDR, and n. Eﬀ Sz E[Mm] n FDR π1,∞ λ75-pwr λˆ75-pwr λ90-pwr λˆ90-pwr λeq SS Ratio 0.08 400 7800 0.01 0.612 0.000 0.000 0.000 0.000 0.604 1.413 0.08 400 8400 0.01 0.690 0.011 0.012 0.000 0.000 0.677 1.320 0.08 400 9000 0.01 0.757 0.624 0.627 0.000 0.000 0.741 1.239 0.08 400 9600 0.01 0.813 0.999 0.999 0.000 0.000 0.794 1.168 0.08 400 10200 0.01 0.858 1.000 1.000 0.012 0.011 0.838 1.105 0.08 400 10800 0.01 0.893 1.000 1.000 0.341 0.347 0.873 1.048 0.08 400 11400 0.01 0.921 1.000 1.000 0.933 0.927 0.901 0.998 0.28 400 650 0.01 0.626 0.000 0.000 0.000 0.000 0.617 1.394 0.28 400 700 0.01 0.704 0.037 0.040 0.000 0.000 0.690 1.303 0.28 400 750 0.01 0.770 0.807 0.827 0.000 0.000 0.753 1.223 0.28 400 800 0.01 0.824 1.000 1.000 0.000 0.000 0.805 1.153 0.28 400 850 0.01 0.868 1.000 1.000 0.036 0.031 0.848 1.089 0.28 400 900 0.01 0.902 1.000 1.000 0.543 0.548 0.882 1.034 0.28 400 950 0.01 0.928 1.000 1.000 0.981 0.972 0.908 0.984 0.68 400 120 0.01 0.676 0.003 0.001 0.000 0.000 0.663 1.325 0.68 400 160 0.01 0.912 1.000 1.000 0.784 0.785 0.892 1.019 0.08 1000 7500 0.01 0.652 0.000 0.000 0.000 0.000 0.645 1.374 0.08 1000 8000 0.01 0.716 0.017 0.008 0.000 0.000 0.707 1.292 0.08 1000 8500 0.01 0.771 0.926 0.938 0.000 0.000 0.760 1.220 0.08 1000 9000 0.01 0.818 1.000 1.000 0.000 0.000 0.806 1.156 0.08 1000 9500 0.01 0.856 1.000 1.000 0.000 0.000 0.844 1.098 0.08 1000 10000 0.01 0.888 1.000 1.000 0.123 0.130 0.875 1.046 0.08 1000 10500 0.01 0.913 1.000 1.000 0.921 0.913 0.901 1.151 0.28 1000 600 0.01 0.623 0.000 0.000 0.000 0.000 0.617 1.408 0.28 1000 640 0.01 0.689 0.000 0.000 0.000 0.000 0.681 1.325 0.28 1000 680 0.01 0.747 0.432 0.474 0.000 0.000 0.737 1.250 0.28 1000 720 0.01 0.797 1.000 1.000 0.000 0.000 0.785 1.185 0.28 1000 760 0.01 0.838 1.000 1.000 0.000 0.000 0.826 1.125 0.28 1000 800 0.01 0.872 1.000 1.000 0.007 0.006 0.860 1.072 0.28 1000 840 0.01 0.900 1.000 1.000 0.516 0.506 0.888 1.151 0.28 1000 880 0.01 0.923 1.000 1.000 0.995 0.993 0.910 0.980 0.68 1000 120 0.01 0.752 0.546 0.586 0.000 0.000 0.741 1.242 21 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Excerpted

text: results

text: from

text: a

text: simulation

text: study

text: modelling

text: GWA

text: studies,

text: m

text: IST

text: average

text: E

text: power,

text: J/L

text: power,

text: lower

text: bound

text: and

text: simulated

text: average

text: power

text: for

text: a

text: selection

text: of

text: eﬀect

text: sizes,

text: values

text: of

text: [M

text: m

text: FDR,

text: and

text: n.

text: Eﬀ

text: Sz

text: E[Mm]

text: n

text: FDR

text: π1,∞

text: λeq

text: SS

text: Ratio

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: allowed to vary from 0.6 to 1.5 in increments of 0.1. At each conﬁguration, a range of sample sizes were chosen to result in powers between 60% and 95% as mentioned E above. This resulted in 10,020 conﬁgurations of the parameters, f, [M ], θ, and m n (full set of paramter combinations). The job took roughly 12 hours on the NIH Biowulf cluster. Table 1 tabulates the IST average power, the Jung/Liu power and the simulated mean of the TPF at 28 diﬀerent parameter settings excerpted from the full set of 10,020 parameter combinations. Over the full set of parameter settings, the both the IST, π , and Jung/Liu, πJL, powers are very close to the simulated average 1,∞ 1 power. The diﬀerence between the IST power, π , and the simulated power was ∞ 1, less than 0.022%, 0.077% and 0.19% at 50%, 90% and 99% of the parameter settings, respectively. The Jung/Liu power diﬀered from the IST power by less than 0.15%, 0.5% and 0.92% at 50%, 90% and 99% of the parameter settings, respectively. As remarked earlier, the Jung/Liu power is actually the average power at the oracle ≤ threshold, but for such small values of r 0.046 there is not much gain in power to be had. The lower bound comes within roughly 10% of the simulated power, with diﬀerences with the simulated power less than 2.1%, 4.5%, 6.2%, 8.5% and 15% at 20%, 40%, 50%, 60% and 80% of the parameter settings, respectively. Table 2 displays, at threshhold 0.75 and at threshhold 0.90, the λ power as de- rived from the CLT 4.6 and estimated from simulation replicates (hatted version), respectively, excerpted from the full set of 10,020 parameter combinatons as before. In the last column is the ratio of the sample size required for λ -power to the 90 original sample size. First, we note that when restricted to powers strictly between 50% and 100%, occuring at 1,066 paramter combinations, the CLT approximate- and simulated- λ -power were within the following relative error of one another 75 (median over parameter conditions (lower quartile, upper quartile)): 0.4% (0.2%, 1.2%), with 2.4% over 5%. Corresponding results for the simulated and CLT ap- proximate λ -power for powers strictly between 50% and 100% occuring at 1476 of 90 the parameter values, were within the following relativer error of one another 0.5% (0.2%, 1.3%), with 1.8% over 5%. Also noteworthy is corroboration in ordering of the average power and λ -power based upon the size of k relative to 100λ . All val- k eq ues of λ are less than 90%, but some are between 75% and 90%, and the ordering eq of average power and λ powers is in accordance with expression 3.8. Furthermore the discrepancy between the average power and the λ-power is reﬂective diﬀerence between λ and λ . This trend is echoed in the magnitude of the sample size ratio, eq with magnitude increasing in the discrepency between λ and 0.90. The relatively eq rapid rise in sample size, n, of all λ-powers is an indication of the degree to which the distribution of the TPF, S /M , is spiked. m m 6.2. GWA Studies The last simulation study we considered experiments typical of GWA studies with m = 1,000,000 simultaneous tests. We attempted to cover a broad spectrum of parameters spanning the domain typical of GWA study designs. The false discovery rate, f, was ranged over the values 0.5%, 1%, 5% and 10%. The expected number of E tests with non-zero means, [M ] = mr, was varied from 400 to 1000 in increments m of 200 representing values of r ranging from 4e-04 to 0.001. The eﬀect size, θ, was 27

text: allowed

text: to

text: vary

text: from

text: to

text: in

text: increments

text: of

text: At

text: each

text: conﬁguration,

text: a

text: range

text: of

text: sample

text: sizes

text: were

text: chosen

text: to

text: result

text: in

text: powers

text: between

text: and

text: as

text: mentioned

text: E

text: above.

text: This

text: resulted

text: in

text: conﬁgurations

text: of

text: the

text: parameters,

text: f,

text: [M

text: θ,

text: and

text: m

text: n

text: (full

text: set

text: of

text: paramter

text: combinations).

text: The

text: job

text: took

text: roughly

text: hours

text: on

text: the

text: NIH

text: Biowulf

text: cluster.

text: Table

text: tabulates

text: the

text: IST

text: average

text: power,

text: the

text: Jung/Liu

text: power

text: and

text: the

text: simulated

text: mean

text: of

text: the

text: TPF

text: at

text: diﬀerent

text: parameter

text: settings

text: excerpted

text: from

text: the

text: full

text: set

text: of

text: parameter

text: combinations.

text: Over

text: the

text: full

text: set

text: of

text: parameter

text: settings,

text: the

text: both

text: the

text: IST,

text: π

text: and

text: Jung/Liu,

text: πJL,

text: powers

text: are

text: very

text: close

text: to

text: the

text: simulated

text: average

text: power.

text: The

text: diﬀerence

text: between

text: the

text: IST

text: power,

text: π

text: and

text: the

text: simulated

text: power

text: was

text: less

text: than

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: The

text: Jung/Liu

text: power

text: diﬀered

text: from

text: the

text: IST

text: power

text: by

text: less

text: than

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: As

text: remarked

text: earlier,

text: the

text: Jung/Liu

text: power

text: is

text: actually

text: the

text: average

text: power

text: at

text: the

text: oracle

text: threshold,

text: but

text: for

text: such

text: small

text: values

text: of

text: r

text: there

text: is

text: not

text: much

text: gain

text: in

text: power

text: to

text: be

text: had.

text: The

text: lower

text: bound

text: comes

text: within

text: roughly

text: of

text: the

text: simulated

text: power,

text: with

text: diﬀerences

text: with

text: the

text: simulated

text: power

text: less

text: than

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: Table

text: displays,

text: at

text: threshhold

text: and

text: at

text: threshhold

text: the

text: λ

text: power

text: as

text: de-

text: rived

text: from

text: the

text: CLT

text: and

text: estimated

text: from

text: simulation

text: replicates

text: (hatted

text: version),

text: respectively,

text: excerpted

text: from

text: the

text: full

text: set

text: of

text: parameter

text: combinatons

text: as

text: before.

text: In

text: the

text: last

text: column

text: is

text: the

text: ratio

text: of

text: the

text: sample

text: size

text: required

text: for

text: λ

text: -power

text: to

text: the

text: original

text: sample

text: size.

text: First,

text: we

text: note

text: that

text: when

text: restricted

text: to

text: powers

text: strictly

text: between

text: and

text: occuring

text: at

text: paramter

text: combinations,

text: the

text: CLT

text: approximate-

text: and

text: simulated-

text: λ

text: -power

text: were

text: within

text: the

text: following

text: relative

text: error

text: of

text: one

text: another

text: (median

text: over

text: parameter

text: conditions

text: (lower

text: quartile,

text: upper

text: quartile)):

text: with

text: over

text: Corresponding

text: results

text: for

text: the

text: simulated

text: and

text: CLT

text: ap-

text: proximate

text: λ

text: -power

text: for

text: powers

text: strictly

text: between

text: and

text: occuring

text: at

text: of

text: the

text: parameter

text: values,

text: were

text: within

text: the

text: following

text: relativer

text: error

text: of

text: one

text: another

text: with

text: over

text: Also

text: noteworthy

text: is

text: corroboration

text: in

text: ordering

text: of

text: the

text: average

text: power

text: and

text: λ

text: -power

text: based

text: upon

text: the

text: size

text: of

text: k

text: relative

text: to

text: All

text: val-

text: k

text: eq

text: ues

text: of

text: λ

text: are

text: less

text: than

text: but

text: some

text: are

text: between

text: and

text: and

text: the

text: ordering

text: eq

text: of

text: average

text: power

text: and

text: λ

text: powers

text: is

text: in

text: accordance

text: with

text: expression

text: Furthermore

text: the

text: discrepancy

text: between

text: the

text: average

text: power

text: and

text: the

text: λ-power

text: is

text: reﬂective

text: diﬀerence

text: between

text: λ

text: and

text: λ

text: This

text: trend

text: is

text: echoed

text: in

text: the

text: magnitude

text: of

text: the

text: sample

text: size

text: ratio,

text: eq

text: with

text: magnitude

text: increasing

text: in

text: the

text: discrepency

text: between

text: λ

text: and

text: The

text: relatively

text: eq

text: rapid

text: rise

text: in

text: sample

text: size,

text: n,

text: of

text: all

text: λ-powers

text: is

text: an

text: indication

text: of

text: the

text: degree

text: to

text: which

text: the

text: distribution

text: of

text: the

text: TPF,

text: S

text: /M

text: is

text: spiked.

text: m

text: m

text: GWA

text: Studies

text: The

text: last

text: simulation

text: study

text: we

text: considered

text: experiments

text: typical

text: of

text: GWA

text: studies

text: with

text: m

text: simultaneous

text: tests.

text: We

text: attempted

text: to

text: cover

text: a

text: broad

text: spectrum

text: of

text: parameters

text: spanning

text: the

text: domain

text: typical

text: of

text: GWA

text: study

text: designs.

text: The

text: false

text: discovery

text: rate,

text: f,

text: was

text: ranged

text: over

text: the

text: values

text: and

text: The

text: expected

text: number

text: of

text: E

text: tests

text: with

text: non-zero

text: means,

text: [M

text: mr,

text: was

text: varied

text: from

text: to

text: in

text: increments

text: m

text: of

text: representing

text: values

text: of

text: r

text: ranging

text: from

text: to

text: The

text: eﬀect

text: size,

text: θ,

text: was


text: allowed to vary from 0.08 to 0.68 in increments of 0.2. At each conﬁguration, a range of sample sizes were chosen to result in powers between 50% and 98% as mentioned E above. This resulted in 512 conﬁgurations of the parameters, f, [M ], θ, and n m (full set of parameter combinations). The job took roughly 34 hours on the NIH Biowulf cluster. Table 3 tabulates the IST average power, the Jung/Liu power and the simulated mean of the TPF at 32 diﬀerent parameter settings excerpted from the full set of 512 parameter combinations. Over the full set of parameter settings, the both the IST, π , and Jung/Liu, πJL, powers are very close to the simulated average power. 1,∞ 1 The diﬀerence between the IST power, π , and the simulated power was less than ∞ 1, 0.031, 0.087 and 0.147 at 50%, 90% and 99% of the parameter settings, respectively. The Jung/Liu power diﬀered from the IST power by 0.0038%, 0.0081% and 0.011% at 50%, 90% and 99% of the parameter settings, respectively. As remarked earlier, the Jung/Liu power is actually the average power at the oracle threshold, but for ≤ such small values of r 0.001 the gain in power is now less than 1%. The lower bound comes within roughly 10% of the simulated power, with diﬀerences with the simulated power less than 0.83%, 3.8%, 6.9%, 9.6% and 15% at 20%, 40%, 50%, 60% and 80% of the parameter settings, respectively. Table 4 displays, at threshhold 0.75 and at threshhold 0.90, the λ power as derived from the CLT 4.6 and estimated from simulation replicates (hatted version), respectively, excerpted from the full set of 512 parameter combinatons as before. In the last column is the ratio of the sample size required for λ -power to the 90 original sample size. First, we note that when restricted to powers strictly between 50% and 100%, occuring at 68 paramter combinations, the CLT approximate- and simulated- λ -power were within the following relative error of one another (median 75 over parameter conditions (lower quartile, upper quartile)): 0.5% (0.2%, 1.3%), with 1.5% over 5%. Corresponding results for the simulated and CLT approximate λ - 90 power for powers strictly between 50% and 100% occuring at 87 of the parameter values, were within the following relativer error of one another 0.6% (0.2%, 1.2%), with 1.1% over 5%. Also noteworthy is corroboration in ordering of the average power and λ -power based upon the size of k relative to 100λ . All values of k eq λ are less than 90%, but some are between 75% and 90%, and the ordering of eq average power and λ powers is in accordance with expression 3.8. Furthermore the discrepancy between the average power and the λ-power is reﬂective diﬀerence between λ and λ . This trend is echoed in the magnitude of the sample size ratio, eq with magnitude increasing in the disc√repency between λ and 0.90. Notice that eq over values considered the ranges of nθ are comparable among the the micro- array, GWAS and biomarker simulation studies. Therefore, the “all” or “nothing” rapid rise in the λ-powers with increasing sample size here must be solely due to the distribution of the TPF, S /M , being even more dramatically spiked, since m m the number of simultaneous tests, m, is in this case, considerably larger. 7. Discussion √ We proved LLNs for the PCF, FDF and TPF as well as CLTs for m scaled versions of them. Our LLN result for the TPF allowed characterization of the large 28 ##LTLine## ##LTLine##

text: allowed

text: to

text: vary

text: from

text: to

text: in

text: increments

text: of

text: At

text: each

text: conﬁguration,

text: a

text: range

text: of

text: sample

text: sizes

text: were

text: chosen

text: to

text: result

text: in

text: powers

text: between

text: and

text: as

text: mentioned

text: E

text: above.

text: This

text: resulted

text: in

text: conﬁgurations

text: of

text: the

text: parameters,

text: f,

text: [M

text: θ,

text: and

text: n

text: m

text: (full

text: set

text: of

text: parameter

text: combinations).

text: The

text: job

text: took

text: roughly

text: hours

text: on

text: the

text: NIH

text: Biowulf

text: cluster.

text: Table

text: tabulates

text: the

text: IST

text: average

text: power,

text: the

text: Jung/Liu

text: power

text: and

text: the

text: simulated

text: mean

text: of

text: the

text: TPF

text: at

text: diﬀerent

text: parameter

text: settings

text: excerpted

text: from

text: the

text: full

text: set

text: of

text: parameter

text: combinations.

text: Over

text: the

text: full

text: set

text: of

text: parameter

text: settings,

text: the

text: both

text: the

text: IST,

text: π

text: and

text: Jung/Liu,

text: πJL,

text: powers

text: are

text: very

text: close

text: to

text: the

text: simulated

text: average

text: power.

text: The

text: diﬀerence

text: between

text: the

text: IST

text: power,

text: π

text: and

text: the

text: simulated

text: power

text: was

text: less

text: than

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: The

text: Jung/Liu

text: power

text: diﬀered

text: from

text: the

text: IST

text: power

text: by

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: As

text: remarked

text: earlier,

text: the

text: Jung/Liu

text: power

text: is

text: actually

text: the

text: average

text: power

text: at

text: the

text: oracle

text: threshold,

text: but

text: for

text: such

text: small

text: values

text: of

text: r

text: the

text: gain

text: in

text: power

text: is

text: now

text: less

text: than

text: The

text: lower

text: bound

text: comes

text: within

text: roughly

text: of

text: the

text: simulated

text: power,

text: with

text: diﬀerences

text: with

text: the

text: simulated

text: power

text: less

text: than

text: and

text: at

text: and

text: of

text: the

text: parameter

text: settings,

text: respectively.

text: Table

text: displays,

text: at

text: threshhold

text: and

text: at

text: threshhold

text: the

text: λ

text: power

text: as

text: derived

text: from

text: the

text: CLT

text: and

text: estimated

text: from

text: simulation

text: replicates

text: (hatted

text: version),

text: respectively,

text: excerpted

text: from

text: the

text: full

text: set

text: of

text: parameter

text: combinatons

text: as

text: before.

text: In

text: the

text: last

text: column

text: is

text: the

text: ratio

text: of

text: the

text: sample

text: size

text: required

text: for

text: λ

text: -power

text: to

text: the

text: original

text: sample

text: size.

text: First,

text: we

text: note

text: that

text: when

text: restricted

text: to

text: powers

text: strictly

text: between

text: and

text: occuring

text: at

text: paramter

text: combinations,

text: the

text: CLT

text: approximate-

text: and

text: simulated-

text: λ

text: -power

text: were

text: within

text: the

text: following

text: relative

text: error

text: of

text: one

text: another

text: (median

text: over

text: parameter

text: conditions

text: (lower

text: quartile,

text: upper

text: quartile)):

text: with

text: over

text: Corresponding

text: results

text: for

text: the

text: simulated

text: and

text: CLT

text: approximate

text: λ

text: power

text: for

text: powers

text: strictly

text: between

text: and

text: occuring

text: at

text: of

text: the

text: parameter

text: values,

text: were

text: within

text: the

text: following

text: relativer

text: error

text: of

text: one

text: another

text: with

text: over

text: Also

text: noteworthy

text: is

text: corroboration

text: in

text: ordering

text: of

text: the

text: average

text: power

text: and

text: λ

text: -power

text: based

text: upon

text: the

text: size

text: of

text: k

text: relative

text: to

text: All

text: values

text: of

text: k

text: eq

text: λ

text: are

text: less

text: than

text: but

text: some

text: are

text: between

text: and

text: and

text: the

text: ordering

text: of

text: eq

text: average

text: power

text: and

text: λ

text: powers

text: is

text: in

text: accordance

text: with

text: expression

text: Furthermore

text: the

text: discrepancy

text: between

text: the

text: average

text: power

text: and

text: the

text: λ-power

text: is

text: reﬂective

text: diﬀerence

text: between

text: λ

text: and

text: λ

text: This

text: trend

text: is

text: echoed

text: in

text: the

text: magnitude

text: of

text: the

text: sample

text: size

text: ratio,

text: eq

text: with

text: magnitude

text: increasing

text: in

text: the

text: disc√repency

text: between

text: λ

text: and

text: Notice

text: that

text: eq

text: over

text: values

text: considered

text: the

text: ranges

text: of

text: nθ

text: are

text: comparable

text: among

text: the

text: the

text: micro-

text: array,

text: GWAS

text: and

text: biomarker

text: simulation

text: studies.

text: Therefore,

text: the

text: “all”

text: or

text: “nothing”

text: rapid

text: rise

text: in

text: the

text: λ-powers

text: with

text: increasing

text: sample

text: size

text: here

text: must

text: be

text: solely

text: due

text: to

text: the

text: distribution

text: of

text: the

text: TPF,

text: S

text: /M

text: being

text: even

text: more

text: dramatically

text: spiked,

text: since

text: m

text: m

text: the

text: number

text: of

text: simultaneous

text: tests,

text: m,

text: is

text: in

text: this

text: case,

text: considerably

text: larger.

text: Discussion

text: We

text: proved

text: LLNs

text: for

text: the

text: PCF,

text: FDF

text: and

text: TPF

text: as

text: well

text: as

text: CLTs

text: for

text: m

text: scaled

text: versions

text: of

text: them.

text: Our

text: LLN

text: result

text: for

text: the

text: TPF

text: allowed

text: characterization

text: of

text: the

text: large

text: ##LTLine##

text: ##LTLine##


text: named above. Glueck et al. (2008) provided an explicit formula for the average power in a ﬁnite number, m of simultaneous test, but its complexity grows as the factorial of the number of simultaneous tests, and this is clearly intractable in the realm of micro-array studies and GWAS where there are tens of thousands or even a million simultaneous tests in question. The λ-Power As we shall see, the true positive fraction, S /M , converges almost surely to its m m mean value as the number of simultaneous tests tends to inﬁnitiy. However, using the average power, which is based upon its expected value, makes sense only when the width of the distribution of S /M is very narrow. To have more deﬁnitive m m control over the true positive fraction, some authors have introduced the“K-power”. This was originally introduced in a model where the number of non-null distributed tests was ﬁxed and was deﬁned as the probability that the number of true positives exceeded a given integral threshhold, k. In the current setting in which the number of non-null distributed tests is a binomial random variable this no longer makes sense. We introduce instead the λ-power, which is the probability that the true ∈ positive fraction, Sm/Mm exceeds a given(cid:26)threshold(cid:27), λ (0, 1): Definition 3.2. We deﬁne the λ-power: P Sm ≥ π (λ) = λ . (3.4) S/M M m We will also use the term “λ -power” to denote π (k/100), the λ-power at k S/M threshhold k/100. The associated quantile function is denoted: − λ (π) = π 1 (π). (3.5) S/M S/M As mentioned above, the λ-power becomes especially meaningful in experiments for which there are a small to interimediate number of simultaneous tests and for which the distribution of the TPF, S /M , becomes non-negligibly dispersed. m m Remark 3.3. Because the distribution of the TPF is nearly symetric for even relatively small values of m > 50, the mean and median nearly coincide. Thus ≈ π (λ) 1/2 when λ = π (3.6) ∞ S/M 1, Because the λ-power takes the values 1 when λ = 0 and 0 when λ = 1 and is − continuous by assumption, there exists a quantile, λ = π 1 (π ), at which the λ eq ∞ S/M 1, power equals the average power: π (λ ) = π . (3.7) eq ∞ S/M 1, 7 ##LTLine##

text: named

text: above.

text: Glueck

text: et

text: al.

text: provided

text: an

text: explicit

text: formula

text: for

text: the

text: average

text: power

text: in

text: a

text: ﬁnite

text: number,

text: m

text: of

text: simultaneous

text: test,

text: but

text: its

text: complexity

text: grows

text: as

text: the

text: factorial

text: of

text: the

text: number

text: of

text: simultaneous

text: tests,

text: and

text: this

text: is

text: clearly

text: intractable

text: in

text: the

text: realm

text: of

text: micro-array

text: studies

text: and

text: GWAS

text: where

text: there

text: are

text: tens

text: of

text: thousands

text: or

text: even

text: a

text: million

text: simultaneous

text: tests

text: in

text: question.

text: The

label: section

text: λ-Power

text: As

text: we

text: shall

text: see,

text: the

text: true

text: positive

text: fraction,

text: S

text: /M

text: converges

text: almost

text: surely

text: to

text: its

text: m

text: m

text: mean

text: value

text: as

text: the

text: number

text: of

text: simultaneous

text: tests

text: tends

text: to

text: inﬁnitiy.

text: However,

text: using

text: the

text: average

text: power,

text: which

text: is

text: based

text: upon

text: its

text: expected

text: value,

text: makes

text: sense

text: only

text: when

text: the

text: width

text: of

text: the

text: distribution

text: of

text: S

text: /M

text: is

text: very

text: narrow.

text: To

text: have

text: more

text: deﬁnitive

text: m

text: m

text: control

text: over

text: the

text: true

text: positive

text: fraction,

text: some

text: authors

text: have

text: introduced

text: the“K-power”.

text: This

text: was

text: originally

text: introduced

text: in

text: a

text: model

text: where

text: the

text: number

text: of

text: non-null

text: distributed

text: tests

text: was

text: ﬁxed

text: and

text: was

text: deﬁned

text: as

text: the

text: probability

text: that

text: the

text: number

text: of

text: true

text: positives

text: exceeded

text: a

text: given

text: integral

text: threshhold,

text: k.

text: In

text: the

text: current

text: setting

text: in

text: which

text: the

text: number

text: of

text: non-null

text: distributed

text: tests

text: is

text: a

text: binomial

text: random

text: variable

text: this

text: no

text: longer

text: makes

text: sense.

text: We

text: introduce

text: instead

text: the

text: λ-power,

text: which

text: is

text: the

text: probability

text: that

text: the

text: true

text: positive

text: fraction,

text: Sm/Mm

text: exceeds

text: a

text: λ

text: Definition

text: We

text: deﬁne

text: the

text: λ-power:

text: P

text: Sm

text: π

text: (λ)

text: λ

text: S/M

text: M

text: m

text: We

text: will

text: also

text: use

text: the

text: term

text: “λ

text: -power”

text: to

text: denote

text: π

text: the

text: λ-power

text: at

text: k

text: S/M

text: threshhold

text: The

text: associated

text: quantile

text: function

text: is

text: denoted:

text: λ

text: (π)

text: π

text: (π).

text: S/M

text: S/M

text: As

text: mentioned

text: above,

text: the

text: λ-power

text: becomes

text: especially

text: meaningful

text: in

text: experiments

text: for

text: which

text: there

text: are

text: a

text: small

text: to

text: interimediate

text: number

text: of

text: simultaneous

text: tests

text: and

text: for

text: which

text: the

text: distribution

text: of

text: the

text: TPF,

text: S

text: /M

text: becomes

text: non-negligibly

text: dispersed.

text: m

text: m

text: Remark

text: Because

text: the

text: distribution

text: of

text: the

text: TPF

text: is

text: nearly

text: symetric

text: for

text: even

text: relatively

text: small

text: values

text: of

text: m

text: the

text: mean

text: and

text: median

text: nearly

text: coincide.

text: Thus

text: π

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text: – 16 – cavity in our case, the magnetopause standoﬀ distance in Ganymede’s magnetosphere), Hall MHD simulations are necessary to capture the correct global structure of the magnetosphere. 4. Summary and Discussions In this work, we performed a multi-ﬂuid Hall MHD simulation to study the cometary plasma environment in the inner coma region of comet 67P/Churyumov-Gerasimenko. With the same model set up as Huang et al. (2016), the Hall MHD simulation shows a very diﬀerent picture: the inner coma is no longer symmetric and low magnetic ﬁeld regions can form outside the global diamagnetic cavity and the solution is time-dependent. The only diﬀerence between the Hall MHD simulation and the classical MHD simulations by Huang et al. (2016) is that the Hall velocity term is considered in the magnetic induction equation, which means that the current can aﬀect the evolution of the magnetic ﬁeld. It is well known that the Hall eﬀect is important in magnetic reconnections (Birn et al. 2001), partially due to relative weak magnetic ﬁeld and strong currents near the magnetic null point. Hall MHD simulations of the magnetospheres of planets and moons typically do not show signiﬁcant diﬀerences compared to the classical MHD simulations except in the regions where magnetic reconnections occur like the dayside magnetopause or the nightside magnetotail. Dorelli et al. (2015) reported that Hall eﬀect is important in Ganymede’s magnetosphere because the magnetopause standoﬀ distance is in the order of 10 times larger than the ion inertial length. One would not expect the Hall MHD simulations dramatically to change the simulated inner coma environment for a comet because magnetic ﬁeld reconnections have only been reported on the nightside (Huang et al. 2016), but our simulations show that in fact the results change dramatically. The diamagnetic cavity is a unique feature in the cometary environment, which is not shared by other planets or moons in the solar system, and it has received lots of attention since the Giotto

text: cavity

text: in

text: our

text: case,

text: the

text: magnetopause

text: standoﬀ

text: distance

text: in

text: Ganymede’s

text: magnetosphere),

text: Hall

text: MHD

text: simulations

text: are

text: necessary

text: to

text: capture

text: the

text: correct

text: global

text: structure

text: of

text: the

text: magnetosphere.

font: OFENEJ+TeXGyreTermes-Bold

text: Summary

text: and

text: Discussions

text: In

text: this

text: work,

text: we

text: performed

text: a

text: multi-ﬂuid

text: Hall

text: MHD

text: simulation

text: to

text: study

text: the

text: cometary

text: plasma

text: environment

text: in

text: the

text: inner

text: coma

text: region

text: of

text: comet

text: With

text: the

text: same

text: model

text: set

text: up

text: as

text: Huang

text: et

text: al.

text: the

text: Hall

text: MHD

text: simulation

text: shows

text: a

text: very

text: diﬀerent

text: picture:

text: the

text: inner

text: coma

text: is

text: no

text: longer

text: symmetric

text: and

text: low

text: magnetic

text: ﬁeld

text: regions

text: can

text: form

text: outside

text: the

text: global

text: diamagnetic

text: cavity

text: and

text: the

text: solution

text: is

text: time-dependent.

text: The

text: only

text: diﬀerence

text: between

text: the

text: Hall

text: MHD

text: simulation

text: and

text: the

text: classical

text: MHD

text: simulations

text: by

text: Huang

text: et

text: al.

text: is

text: that

text: the

text: Hall

text: velocity

text: term

text: is

text: considered

text: in

text: the

text: magnetic

text: induction

text: equation,

text: which

text: means

text: that

text: the

text: current

text: can

text: aﬀect

text: the

text: evolution

text: of

text: the

text: magnetic

text: ﬁeld.

text: It

text: is

text: well

text: known

text: that

text: the

text: Hall

text: eﬀect

text: is

text: important

text: in

text: magnetic

text: reconnections

text: (Birn

text: et

text: al.

text: partially

text: due

text: to

text: relative

text: weak

text: magnetic

text: ﬁeld

text: and

text: strong

text: currents

text: near

text: the

text: magnetic

text: point.

text: Hall

text: MHD

text: simulations

text: of

text: the

text: magnetospheres

text: of

text: planets

text: and

text: moons

text: typically

text: do

text: not

text: show

text: signiﬁcant

text: diﬀerences

text: compared

text: to

text: the

text: classical

text: MHD

text: simulations

text: except

text: in

text: the

text: regions

text: where

text: magnetic

text: reconnections

text: occur

text: like

text: the

text: dayside

text: magnetopause

text: or

text: the

text: nightside

text: magnetotail.

text: Dorelli

text: et

text: al.

text: reported

text: that

text: Hall

text: eﬀect

text: is

text: important

text: in

text: Ganymede’s

text: magnetosphere

text: because

text: the

text: magnetopause

text: standoﬀ

text: distance

text: is

text: in

text: the

text: order

text: of

text: times

text: larger

text: than

text: the

text: ion

text: inertial

text: length.

text: One

text: would

text: not

text: expect

text: the

text: Hall

text: MHD

text: simulations

text: dramatically

text: to

text: change

text: the

text: simulated

text: inner

text: coma

text: environment

text: for

text: a

text: comet

text: because

text: magnetic

text: ﬁeld

text: reconnections

text: have

text: only

text: been

text: reported

text: on

text: the

text: nightside

text: (Huang

text: et

text: al.

text: but

text: our

text: simulations

text: show

text: that

text: in

text: fact

text: the

text: results

text: change

text: dramatically.

text: The

text: diamagnetic

text: cavity

text: is

text: a

text: unique

text: feature

text: in

text: the

text: cometary

text: environment,

text: which

text: is

text: not

text: shared

text: by

text: other

text: planets

text: or

text: moons

text: in

text: the

text: solar

text: system,

text: and

text: it

text: has

text: received

text: lots

text: of

text: attention

text: since

text: the

text: Giotto


text: – 17 – Fig. 5.— The U component for the cometary ion velocity at the surface close to the mag- y [ ] [ ] netic reconnection surface, which is deﬁned by 3 points: 480 0 0 km, 500 20 33 5 km and , , , , . [ − ] 460 0 33 5 km. The cometary ions move to the +y direction on the right side while they move , , . to the -y direction on the left side, near the magnetic null point denoted by the red cross mark, as indicated by the two black arrows. ##LTFigure## ##LTFigure##

text: Fig.

text: The

text: U

text: component

text: for

text: the

text: cometary

text: ion

text: velocity

text: at

text: the

text: surface

text: close

text: to

text: the

text: mag-

text: y

font: IYZZES+rntxmi

font: OEVBYD+txsys

text: netic

text: reconnection

text: surface,

text: which

text: is

text: deﬁned

text: by

text: points:

text: km,

text: km

text: and

font: OMNSCT+rtxmi

text: km.

text: The

text: cometary

text: ions

text: move

text: to

text: the

text: +y

text: direction

text: on

text: the

text: right

text: side

text: while

text: they

text: move

text: to

text: the

text: -y

text: direction

text: on

text: the

text: left

text: side,

text: near

text: the

text: magnetic

text: point

text: denoted

text: by

text: the

text: red

text: cross

text: mark,

text: as

text: indicated

text: by

text: the

text: two

text: black

text: arrows.

text: ##LTFigure##

text: ##LTFigure##