text: Property 3: Obtain second-order unbiased Taylor series MSE estimator of EBLUP without any bias correction; that is, E[M (Aˆ )] = i;approx i − M (θˆEB) + o(m 1). i i Property 4: Produce a strictly positive second-order unbiased single para- metric bootstrap MSE estimator without any bias-correction. Note that the variance component in the FH model (1) is not area speciﬁc, but to satisfy the above properties simultaneously for a given area, we pro- pose an area speciﬁc estimator of A. This introduces an area speciﬁc bias, − but interestingly the order of bias is O(m 1), same as the bias of the ML estimator of A but higher than that of REML in the higher-order asymptotic sense. This seems to be a reasonable approach as our main targets are area speciﬁc parameters and not the global parameter A. Obviously, if A is the main target, we would recommend a standard variance component method. We stress that in general none of the existing methods for estimating A satisﬁes any of all the four properties simultaneously. In Section 2, we propose a new adjusted maximum likelihood estima- tor of A that satisﬁes all the four desirable properties listed above. The balanced case has been heavily studied in the literature. We consider the balanced case in Section 3 and show how our results are related to the ones in the literature. In Section 4, using a real life data from the U.S. Census Bureau, we demonstrate superior performances of our proposed estimators and MSE estimators over the competing estimators. A Monte Carlo sim- ulation study, described in Section 5, shows that the proposed estimators outperform competing estimators. All the technical proofs are deferred to the Appendix. 2 A New Adjusted Maximum Likelihood Estima- tor of A The residual maximum likelihood estimator of A is deﬁned as: Aˆ = arg max L (A), RE ∈ ∞ RE A [0, ) (cid:18) (cid:19) where L (A) is the residual likelihood of A given by RE | (cid:48) − |− | |− −1 (cid:48) LRE(A) = X V 1X 12 V 12 exp y P y , 2 6 ##LTLine## ##LTLine## ##LTLine##

text: Property

text: Obtain

text: second-order

text: unbiased

text: Taylor

text: series

text: MSE

text: estimator

text: of

text: EBLUP

text: without

text: any

text: bias

text: correction;

text: that

text: is,

text: E[M

text: (Aˆ

text: i;approx

text: i

text: M

text: (θˆEB)

text: o(m

text: i

text: i

text: Property

text: Produce

text: a

text: strictly

text: positive

text: second-order

text: unbiased

text: single

text: para-

text: metric

text: bootstrap

text: MSE

text: estimator

text: without

text: any

text: bias-correction.

text: Note

text: that

text: the

text: variance

text: component

text: in

text: the

text: FH

text: model

text: is

text: not

text: area

text: speciﬁc,

text: but

text: to

text: satisfy

text: the

text: above

text: properties

text: simultaneously

text: for

text: a

text: given

text: area,

text: we

text: pro-

text: pose

text: an

text: area

text: speciﬁc

text: estimator

text: of

text: A.

text: This

text: introduces

text: an

text: area

text: speciﬁc

text: bias,

text: but

text: interestingly

text: the

text: order

text: of

text: bias

text: is

text: O(m

text: same

text: as

text: the

text: bias

text: of

text: the

text: ML

text: estimator

text: of

text: A

text: but

text: higher

text: than

text: that

text: of

text: REML

text: in

text: the

text: higher-order

text: asymptotic

text: sense.

text: This

text: seems

text: to

text: be

text: a

text: reasonable

text: approach

text: as

text: our

text: main

text: targets

text: are

text: area

text: speciﬁc

text: parameters

text: and

text: not

text: the

text: global

text: parameter

text: A.

text: Obviously,

text: if

text: A

text: is

text: the

text: main

text: target,

text: we

text: would

text: recommend

text: a

text: standard

text: variance

text: component

text: method.

text: We

text: stress

text: that

text: in

text: general

text: of

text: the

text: existing

text: methods

text: for

text: estimating

text: A

text: satisﬁes

text: any

text: of

text: all

text: the

text: four

text: properties

text: simultaneously.

text: In

text: Section

text: we

text: propose

text: a

text: new

text: adjusted

text: maximum

text: likelihood

text: estima-

text: tor

text: of

text: A

text: that

text: satisﬁes

text: all

text: the

text: four

text: desirable

text: properties

text: listed

text: above.

text: The

text: balanced

text: case

text: has

text: been

text: heavily

text: studied

text: in

text: the

text: literature.

text: We

text: consider

text: the

text: balanced

text: case

text: in

text: Section

text: and

text: show

text: how

text: our

text: results

text: are

text: related

text: to

text: the

text: ones

text: in

text: the

text: literature.

text: In

text: Section

text: using

text: a

text: real

text: life

text: data

text: from

text: the

text: U.S.

text: Census

text: Bureau,

text: we

text: demonstrate

text: superior

text: performances

text: of

text: our

text: proposed

text: estimators

text: and

text: MSE

text: estimators

text: over

text: the

text: competing

text: estimators.

text: A

text: Monte

text: Carlo

text: sim-

text: ulation

text: study,

text: described

text: in

text: Section

text: shows

text: that

text: the

text: proposed

text: estimators

text: outperform

text: competing

text: estimators.

text: All

text: the

text: technical

text: proofs

text: are

text: deferred

text: to

text: the

text: Appendix.

text: A

text: New

text: Adjusted

text: Maximum

text: Likelihood

text: Estima-

text: tor

text: of

text: A

text: The

text: residual

text: maximum

text: likelihood

text: estimator

text: of

text: A

text: is

text: deﬁned

text: as:

text: Aˆ

text: arg

text: max

text: L

text: (A),

text: RE

text: RE

text: A

text: where

text: L

text: (A)

text: is

text: the

text: residual

text: likelihood

text: of

text: A

text: given

text: by

text: RE

text: −1

text: LRE(A)

text: X

text: V

text: V

text: exp

text: y

text: P

text: y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: − − − (cid:48) − − (cid:48) − with P = V 1 V 1X(X V 1X) 1X V 1. Note that Aˆ does not satisfy RE any of the four desirable properties listed in the introduction. In an eﬀort to ﬁnd a likelihood-based estimator of A that satisﬁes all the four desirable properties, we deﬁne the followed adjusted maximum likeli- hood estimator of A: Aˆ = arg max h (A)L (A), i ∈ ∞ i RE A [0, ) where h (A) is a factor to be suitably chosen so that all the four desirable i properties are satisﬁed. We ﬁrst ﬁnd h (A) so that the resulting estimator of A results in a nearly i unbiased estimator of B that also protects EBLUP from overshrinking. In i other words, we ﬁrst ﬁnd the adjustment factor h (A) that simultaneously i satisﬁes Properties 1 and 2. Interestingly, it turns out that such a adjusted maximum likelihood estimator also satisﬁes Properties 3 and 4. Using Lemma 1 in Appendix A and Taylor series expansion, we have 2D2 − Var(Bˆ ) = i − + o(m 1), (2) i (A + D )4tr[V 2] i for large m. We restrict ourselves to the class of estimators of A that satisﬁes (2). Using Lemma 1 and(cid:20) Taylor series expansion, w(cid:21)e have ∂B ∂ log h (A) 1 ∂2B 2 − E(Bˆ ) = B + i i + i − + o(m 1). (3) i i ∂A ∂A 2 ∂A2 tr[V 2] Thus, Property 1 is satisﬁed if we have ∂B ∂ log h (A) 1 ∂2B i i i + = 0. (4) ∂A ∂A 2 ∂A2 Now the diﬀerential equation (12) simpliﬁes to: ∂ log h (A) 1 i = . (5) ∂A A + D i 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: with

text: P

text: V

text: V

text: V

text: V

text: Note

text: that

text: Aˆ

text: does

text: not

text: satisfy

text: RE

text: any

text: of

text: the

text: four

text: desirable

text: properties

text: listed

text: in

text: the

text: introduction.

text: In

text: an

text: eﬀort

text: to

text: ﬁnd

text: a

text: likelihood-based

text: estimator

text: of

text: A

text: that

text: satisﬁes

text: all

text: the

text: four

text: desirable

text: properties,

text: we

text: deﬁne

text: the

text: followed

text: adjusted

text: maximum

text: likeli-

text: hood

text: estimator

text: of

text: A:

text: Aˆ

text: arg

text: max

text: h

text: (A)L

text: (A),

text: i

text: i

text: RE

text: A

text: where

text: h

text: (A)

text: is

text: a

text: factor

text: to

text: be

text: suitably

text: chosen

text: so

text: that

text: all

text: the

text: four

text: desirable

text: i

text: properties

text: are

text: satisﬁed.

text: We

text: ﬁrst

text: ﬁnd

text: h

text: (A)

text: so

text: that

text: the

text: resulting

text: estimator

text: of

text: A

text: results

text: in

text: a

text: nearly

text: i

text: unbiased

text: estimator

text: of

text: B

text: that

text: also

text: protects

text: EBLUP

text: from

text: overshrinking.

text: In

text: i

text: other

text: words,

text: we

text: ﬁrst

text: ﬁnd

text: the

text: adjustment

text: factor

text: h

text: (A)

text: that

text: simultaneously

text: i

text: satisﬁes

text: Properties

text: and

text: Interestingly,

text: it

text: turns

text: out

text: that

text: such

text: a

text: adjusted

text: maximum

text: likelihood

text: estimator

text: also

text: satisﬁes

text: Properties

text: and

text: Using

text: Lemma

text: in

text: Appendix

text: A

text: and

text: Taylor

text: series

text: expansion,

text: we

text: have

text: Var(Bˆ

text: i

text: o(m

text: i

text: (A

text: D

text: i

text: for

text: large

text: m.

text: We

text: restrict

text: ourselves

text: to

text: the

text: class

text: of

text: estimators

text: of

text: A

text: that

text: satisﬁes

text: Using

text: Lemma

text: Taylor

text: series

text: expansion,

text: have

text: ∂B

text: log

text: h

text: (A)

text: E(Bˆ

text: B

text: i

text: i

text: i

text: o(m

text: i

text: i

text: ∂A

text: ∂A

text: tr[V

text: Thus,

text: Property

text: is

text: satisﬁed

text: if

text: we

text: have

text: ∂B

text: log

text: h

text: (A)

text: i

text: i

text: i

text: ∂A

text: ∂A

text: Now

text: the

text: diﬀerential

text: equation

text: simpliﬁes

text: to:

text: log

text: h

text: (A)

text: i

text: ∂A

text: A

text: D

text: i

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: ∗ (cid:48) · · · ∗ ∗ ∼ where θ = x βˆ(Aˆ , , Aˆ ) + v with v N(0, Aˆ ). Note i i 1;MG m;MG i i i;MG that the new bootstrap MSE estimator does not require any bias correction. The following theorem states that our proposed adjusted maximum like- lihood estimator of A satisﬁes all the four desirable properties. − Theorem 1. Under the regularity conditions R1 R7, we have, for large m, (i)Bias(Bˆi;MG) = o(1); V ar(Bˆi;MG) = (A+Di2)D4ti2r[V −2] + o(m−1); ≤ (ii) 0 < inf ≥ Bˆ sup ≥ Bˆ < 1, for m > p + 2; m 1 i;MG m 1 i;MG − − (iii)E(Mˆ ) M (θˆEB ) = o(m 1); i;MG i i;MG − − (iv) E(Mˆ boot ) M (θˆEB ) = o(m 1). i;MG i i;MG For proof of Theorem 1, see Appendix B. · · · 3 The balanced case: D = D, i = 1, , m i In this section, we show how the proposed adjusted maximum likelihood estimator of A is related to the problem of simultaneous estimation of several independent normal means, a topic for intense research activities, especially in the 60’s, 70’s and 80’s, since the introduction of the celebrated James- Stein estimator (James and Stein 1961). | i∼nd · · · (cid:80) Let y θ N(θ , 1), i = 1, , m. James and Stein (1961) showed that ≥ i i i for m 3 the maximum likelihood (also unbiased) estimator of θ is inadmis- i − sible under the s(cid:80)um of squared error loss function L(θˆ, θ) = m (θˆ θ )2 − j=1 j j and is dominated by the James-Stein estimator: θˆJS = (1 Bˆ )y , where − i JS i Bˆ = (m 2)/ m y2. That is,   JS j=1 j (cid:88) (cid:88)     m m − | ≤ − | ∀ ∈ E (θˆJS θ )2 θ E (y θ )2 θ , θ Rm, (8) j j j j j=1 j=1 where Rm is the m-dimensional Euclidean space, with strict inequality holding for at least one point θ. The dominance result, however, does not hold for individual components. Efron and Morris (1973) oﬀered an empirical Bayesian justiﬁcation of the ∼ · · · iid James-Stein estimator under the prior θ N(0, A), i = 1, , m. Their i 9 ##LTLine##

text: where

text: θ

text: x

text: βˆ(Aˆ

text: Aˆ

text: v

text: with

text: v

text: Aˆ

text: Note

text: i

text: i

text: m;MG

text: i

text: i

text: i;MG

text: that

text: the

text: new

text: bootstrap

text: MSE

text: estimator

text: does

text: not

text: require

text: any

text: bias

text: correction.

text: The

text: following

text: theorem

text: states

text: that

text: our

text: proposed

text: adjusted

text: maximum

text: like-

text: lihood

text: estimator

text: of

text: A

text: satisﬁes

text: all

text: the

text: four

text: desirable

text: properties.

text: Theorem

text: Under

text: the

text: regularity

text: conditions

text: we

text: have,

text: for

text: large

text: m,

text: (i)Bias(Bˆi;MG)

text: V

text: ar(Bˆi;MG)

text: (ii)

text: inf

text: Bˆ

text: sup

text: Bˆ

text: for

text: m

text: p

text: m

text: i;MG

text: m

text: i;MG

text: (iii)E(Mˆ

text: M

text: (θˆEB

text: o(m

text: i;MG

text: i

text: i;MG

text: (iv)

text: E(Mˆ

text: boot

text: M

text: (θˆEB

text: o(m

text: i;MG

text: i

text: i;MG

text: For

text: proof

text: of

text: Theorem

text: see

text: Appendix

text: B.

text: The

text: balanced

text: case:

text: D

text: D,

text: i

text: m

text: i

text: In

text: this

text: section,

text: we

text: show

text: how

text: the

text: proposed

text: adjusted

text: maximum

text: likelihood

text: estimator

text: of

text: A

text: is

text: related

text: to

text: the

text: problem

text: of

text: simultaneous

text: estimation

text: of

text: several

text: independent

text: normal

text: means,

text: a

text: topic

text: for

text: intense

text: research

text: activities,

text: especially

text: in

text: the

text: and

text: since

text: the

text: introduction

text: of

text: the

text: celebrated

text: James-

text: Stein

text: estimator

text: (James

text: and

text: Stein

text: i∼nd

text: Let

text: y

text: θ

text: N(θ

text: i

text: m.

text: James

text: and

text: Stein

text: showed

text: that

text: i

text: i

text: i

text: for

text: m

text: the

text: maximum

text: likelihood

text: (also

text: unbiased)

text: estimator

text: of

text: θ

text: is

text: inadmis-

text: i

text: sible

text: under

text: the

text: of

text: squared

text: error

text: loss

text: function

text: L(θˆ,

text: θ)

text: m

text: (θˆ

text: θ

text: j

text: j

text: and

text: is

text: dominated

text: by

text: the

text: James-Stein

text: estimator:

text: θˆJS

text: Bˆ

text: )y

text: where

text: i

text: JS

text: i

text: Bˆ

text: (m

text: m

text: That

text: is,

text: JS

text: j

text: m

text: m

text: E

text: (θˆJS

text: θ

text: θ

text: E

text: (y

text: θ

text: θ

text: θ

text: Rm,

text: j

text: j

text: j

text: j

text: where

text: Rm

text: is

text: the

text: m-dimensional

text: Euclidean

text: space,

text: with

text: strict

text: inequality

text: holding

text: for

text: at

text: least

text: one

text: point

text: θ.

text: The

text: dominance

text: result,

text: however,

text: does

text: not

text: hold

text: for

text: individual

text: components.

text: Efron

text: and

text: Morris

text: oﬀered

text: an

text: empirical

text: Bayesian

text: justiﬁcation

text: of

text: the

text: iid

text: James-Stein

text: estimator

text: under

text: the

text: prior

text: θ

text: A),

text: i

text: m.

text: Their

text: i

text: ##LTLine##


text: 16 P. DING, T. J. VANDERWEELE AND J. M. ROBINS The four probabilities (p , p , p , p ) satisfy the conditions in Lemma S5, Therefore, (2) 11 10 01 00 holds. Replacing the probabilities in (2) by their deﬁnitions above, we have | | ≤ pr(A = 1 U > u, Z = z )pr(A = 1 U u, Z = z ) ≤ | 1 | ≤ 0 1 pr(A = 1 U > u, Z = z )pr(A = 1 U u, Z = z ) | 0 | 1 ⇐⇒ pr(A = 1 U > u, z ) ≤ pr(A = 1 U > u, z ) | ≤ 1 | ≤ 0 , pr(A = 1 U u, z ) pr(A = 1 U u, z ) 1 0 and | | ≤ pr(A = 0 U > u, Z = z )pr(A = 0 U u, Z = z ) ≤ | 1 | ≤ 0 1 pr(A = 0 U > u, Z = z )pr(A = 0 U u, Z = z ) | 0 | 1 ⇐⇒ pr(A = 0 U > u, z ) ≤ pr(A = 0 U > u, z ) | ≤ 1 | ≤ 0 . pr(A = 0 U u, z ) pr(A = 0 U u, z ) 1 0 Therefore, for both a = 1 and 0 and for all values of u, | pr(A = a U > u, Z = z) | ≤ (S6) pr(A = a U u, Z = z) is non-increasing in z. Because of the independence of Z and U, we have | F (u A = a, Z = z) ≤ | pr(U u, A = a Z = z) = | pr(A = a Z = z) ≤ | ≤ (cid:26) pr(U u)pr(A = a U (cid:27)u, Z = z) = ≤ | ≤ | pr(U u)pr(A = a U u, Z = z) + pr(U > u)pr(A = a U > u, Z = z) | − pr(U > u) × pr(A = a U > u, Z = z) 1 = 1 + ≤ | ≤ . pr(U u) pr(A = a U u, Z = z) | Therefore, F (u A = a, Z = z) is a non-increasing function of (S6), and the conclusion (cid:3) holds. Lemmas S5 and S6 above hold under the assumption of no additive interaction, and the fol- lowing two lemmas state similar results under the assumption of no multiplicative interaction. ≥ ≥ LEMMA S7. If p max(p , p ), min(p , p ) p , and p p = p p , then 11 10 01 10 01 00 11 00 10 01 − − − − ≥ (1 p )(1 p ) ≤ p p p + p 0, − 11 − 00 1. 11 10 01 00 (1 p )(1 p ) 10 01 Proof of Lemma S7. Using the same notation in the proof of Lemma S5, p p = p p im- ≥ ≥ ≥ 11 00 10 01 plies RR = RR RR , with RR 1, RR 1, and RR 1. Therefore, 11 10 01 10 01 11 − − − − − − ≥ p p p + p = p (RR RR RR RR + 1) = p (RR 1)(RR 1) 0, 11 10 01 00 00 10 01 10 01 00 10 01 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: DING,

text: T.

text: J.

text: VANDERWEELE

text: AND

text: J.

text: M.

text: ROBINS

text: The

text: four

text: probabilities

text: (p

text: p

text: p

text: p

text: satisfy

text: the

text: conditions

text: in

text: Lemma

text: Therefore,

text: holds.

text: Replacing

text: the

text: probabilities

text: in

text: by

text: their

text: deﬁnitions

text: above,

text: we

text: have

label: equation

text: pr(A

text: U

text: u,

text: Z

text: z

text: )pr(A

text: U

text: u,

text: Z

text: z

text: pr(A

text: U

text: u,

text: Z

text: z

text: )pr(A

text: U

text: u,

text: Z

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: and

text: pr(A

text: U

text: u,

text: Z

text: z

text: )pr(A

text: U

text: u,

text: Z

text: z

text: pr(A

text: U

text: u,

text: Z

text: z

text: )pr(A

text: U

text: u,

text: Z

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: pr(A

text: U

text: u,

text: z

text: Therefore,

text: for

text: both

text: a

text: and

text: and

text: for

text: all

text: values

text: of

text: u,

text: pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: is

text: non-increasing

text: in

text: z.

text: Because

text: of

text: the

text: independence

text: of

text: Z

text: and

text: U,

text: we

text: have

text: F

text: (u

text: A

text: a,

text: Z

text: z)

text: pr(U

text: u,

text: A

text: a

text: Z

text: z)

text: pr(A

text: a

text: Z

text: z)

text: pr(U

text: u)pr(A

text: a

text: U

text: Z

text: z)

text: pr(U

text: u)pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: pr(U

text: u)pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: pr(U

text: u)

text: pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: pr(U

text: u)

text: pr(A

text: a

text: U

text: u,

text: Z

text: z)

text: Therefore,

text: F

text: (u

text: A

text: a,

text: Z

text: z)

text: is

text: a

text: non-increasing

text: function

text: of

text: and

text: the

text: conclusion

text: holds.

text: Lemmas

text: and

text: above

text: hold

text: under

text: the

text: assumption

text: of

text: no

text: additive

text: interaction,

text: and

text: the

text: fol-

text: lowing

text: two

text: lemmas

text: state

text: similar

text: results

text: under

text: the

text: assumption

text: of

text: no

text: multiplicative

text: interaction.

text: LEMMA

text: If

text: p

text: max(p

text: p

text: min(p

text: p

text: p

text: and

text: p

text: p

text: p

text: p

text: then

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: Proof

text: of

text: Lemma

text: Using

text: the

text: same

text: notation

text: in

text: the

text: proof

text: of

text: Lemma

text: p

text: p

text: p

text: p

text: im-

text: plies

text: RR

text: RR

text: RR

text: with

text: RR

text: RR

text: and

text: RR

text: Therefore,

text: p

text: p

text: p

text: p

text: p

text: (RR

text: RR

text: RR

text: RR

text: p

text: (RR

text: ##LTLine##

text: ##LTLine##

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text: 18 P. DING, T. J. VANDERWEELE AND J. M. ROBINS causal effects for the treated and control populations: { } − { } − ACEtrue = E Y (1) E Y (0) = fACEtrue + (1 f)ACEtrue. 1 0 Analogously, with a s(cid:90)calar instrumental(cid:90)variable, the adjusted estimator for the whole population can be written as − − ACEadj = µ (z)F (dz) µ (z)F (dz) = fACEadj + (1 f)ACEadj, 1 0 1 0 (cid:90) (cid:90) and with a general instrumental variable, − − ACEadj = ν (π)F (dπ) ν (π)F (dπ) = fACEadj + (1 f)ACEadj. 1 0 1 0 LEMMA S9. With a scalar instrumental variable Z, the differences between the adjusted and unadjusted estimators are { } ACEadj − ACEunadj = −cov Π(Z−), µ0(Z) , 1 f(1 f) { } ACEadj − ACEunadj = −cov Π(Z−), µ1(Z) , 0 f(1 f) { } { } − −cov Π(Z), µ (Z) − cov Π(Z), µ (Z) ACEadj ACEunadj = − 0 1 . 1 f f With a general instrumental variable Z, the above formulas hold if we replace Π(Z) by Π and | | µ (Z) = E(Y A = a, Z) by ν (Π) = E(Y A = a, Π). a a − Proof of Lemma S9. The difference ACEadj ACEunadj is equal to 1 (cid:90) − ACEadj ACEunadj 1 (cid:90) (cid:90) | − | = E(Y A = 0) µ (z)F (dz A = 1) 0 (cid:82) (cid:82) | − | = µ (z)F (dz A = 0) µ (z)F (dz A = 1) 0 0 { − } µ (z) 1(cid:104) Π(z) F (dz) − µ (z)Π(z)F (dz) (cid:105) 0 0 = − 1 f f 1 (cid:104) { − } { } − { (cid:105) } { − } = − E µ (Z)(1 Π(Z)) E Π(Z) E µ (Z)Π(Z) E 1 Π(Z) 0 0 f(1 f) 1 { } { } − { } = − E µ (Z) E Π(Z) E µ (Z)Π(Z) 0 0 f(1 f) { } −cov Π(Z), µ (Z) = − 0 . f(1 f) − Similarly, the difference ACEadj ACEunadj is equal to 0 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: P.

text: DING,

text: T.

text: J.

text: VANDERWEELE

text: AND

text: J.

text: M.

text: ROBINS

text: causal

text: effects

text: for

text: the

text: treated

text: and

text: control

text: populations:

text: ACEtrue

text: E

text: Y

text: E

text: Y

text: fACEtrue

text: f)ACEtrue.

text: Analogously,

text: with

text: a

text: the

text: adjusted

text: estimator

text: for

text: the

text: whole

text: population

text: can

text: be

text: written

text: as

text: ACEadj

text: µ

text: (z)F

text: (dz)

text: µ

text: (z)F

text: (dz)

text: fACEadj

text: f)ACEadj,

text: and

text: with

text: a

text: general

text: instrumental

text: variable,

text: ACEadj

text: ν

text: (π)F

text: (dπ)

text: ν

text: (π)F

text: (dπ)

text: fACEadj

text: f)ACEadj.

text: LEMMA

text: With

text: a

text: scalar

text: instrumental

text: variable

text: Z,

text: the

text: differences

text: between

text: the

text: adjusted

text: and

text: unadjusted

text: estimators

text: are

text: ACEadj

text: ACEunadj

text: −cov

text: Π(Z−),

text: f)

text: ACEadj

text: ACEunadj

text: −cov

text: Π(Z−),

text: f)

text: −cov

text: Π(Z),

text: µ

text: (Z)

text: cov

text: Π(Z),

text: µ

text: (Z)

text: ACEadj

text: ACEunadj

text: f

text: f

text: With

text: a

text: general

text: instrumental

text: variable

text: Z,

text: the

text: above

text: formulas

text: hold

text: if

text: we

text: replace

text: Π(Z)

text: by

text: Π

text: and

text: µ

text: (Z)

text: E(Y

text: A

text: a,

text: Z)

text: by

text: ν

text: (Π)

text: E(Y

text: A

text: a,

text: Π).

text: a

text: a

text: Proof

text: of

text: Lemma

text: The

text: difference

text: ACEadj

text: ACEunadj

text: is

text: equal

text: to

text: ACEadj

text: ACEunadj

text: E(Y

text: A

text: µ

text: (z)F

text: (dz

text: A

text: µ

text: (z)F

text: (dz

text: A

text: µ

text: (z)F

text: (dz

text: A

text: µ

text: (z)

text: Π(z)

text: F

text: (dz)

text: µ

text: (z)Π(z)F

text: (dz)

text: f

text: f

text: E

text: µ

text: Π(Z))

text: E

text: Π(Z)

text: E

text: µ

text: (Z)Π(Z)

text: E

text: Π(Z)

text: f)

text: E

text: µ

text: (Z)

text: E

text: Π(Z)

text: E

text: µ

text: (Z)Π(Z)

text: f)

text: −cov

text: Π(Z),

text: µ

text: (Z)

text: f)

text: Similarly,

text: the

text: difference

text: ACEadj

text: ACEunadj

text: is

text: equal

text: to

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Instrumental variables as bias ampliﬁers 21 | Proof of Theorem 3. Because of the independence of Z and U, we have pr(A = 1 Z = z) = { } | { } β(z)E γ(U) and pr(A = 1 U = u) = E β(Z) γ(u) are non-decreasing in z and u. Accord- ing to Lemma S8, the multiplicative model of A also implies that for both a = 1 and 0 and for | ≥ all z and u, ∂F (u A = a, Z = z)/∂z 0. Following exactly the same steps of the proof of (cid:3) Theorem 2, we can prove Theorem 3. Proof of Corollary 2. For binary and independent Z and U, monotonicity, no multiplicative interaction, and Lemma S7 imply − − p p ≤ (1 p )(1 p ) ≤ 11 00 = 1 1, − 11 − 00 1. (S9) p p (1 p )(1 p ) 10 01 10 01 (cid:3) With the above results in (S9), the rest of the proof is the same as the proof of Corollary 1. Proof of Theorem 4. First, we consider the treatment effect on the population under treat- ≥ | ment. Taking U = Y (0) in Lemma S3, we have ACEunadj ACEtrue, because A Y (0) { | } { | 1 } Y (0), pr A = 1 Y (0) is non-decreasing in Y (0), and E Y A = 0, Y (0) = Y (0) is non- { | } ≤ ≥ decreasing in Y (0). The condition cov Π, E(Y A = 0, Π) 0 implies that ACEadj ≥ ≥ 1 ACEunadj according to Lemma S9. Therefore, ACEadj ACEunadj ACEtrue. 1 1 Second, we take U = Y (1) in Lemma S4, and by a similar argument as above we have ≥ ≥ ACEadj ACEunadj ACEtrue. 0 0 − The conclusion holds because ACEtrue = fACEtrue + (1 f)ACEtrue and ACEadj = fACEadj + (1 − f)ACEadj. 1 0 (cid:3) 1 0 { } Proof of Theorem 5. Under the additive model of A given Π and U = Y (1), Y (0) , we have | { } the following results. First, pr(A = 1 Π) = Π is increasing in Π. Second, Π Y (1), Y (0) implies (cid:90) (cid:90) { | } | | pr A = 1 Π, Y (1) = y = pr(A = 1 Π, U)F (dy y ) 1 0 1 (cid:90) { } | = Π + δ(y ) + η(y ) F (dy y ) (cid:101) 1 0 0 1 | ≡ = Π + δ(y ) + η(y )F (dy y ) Π + δ(y ). 1 0 0 1 1 Denote the inﬁmum and supremum of Y (0) given Y (1) = y by y (y ) and y , with the 1 0 1 0 later not depending on y according to Condition (c) of Theorem 5. Applying integration or 1 summation by parts, we have (cid:90) (cid:26) (cid:27) (cid:101) (cid:90) |(cid:26) | (cid:27) − dη(y ) | δ(y ) = δ(y ) + η(y )F (y y ) y0=y0 0 F (y y ) dy 1 1 0 0 1 y0=y0(y1) dy0 0 1 0 − dη(y ) | 0 = δ(y ) + η(y ) F (y y ) dy . (cid:101) 1 0 dy 0 1 0 0 The function δ(y ) is non-decreasing in y , because 1 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Instrumental

text: variables

text: as

text: bias

text: ampliﬁers

text: Proof

text: of

text: Theorem

text: Because

text: of

text: the

text: independence

text: of

text: Z

text: and

text: U,

text: we

text: have

text: pr(A

text: Z

text: z)

text: β(z)E

text: γ(U)

text: and

text: pr(A

text: U

text: u)

text: E

text: β(Z)

text: γ(u)

text: are

text: non-decreasing

text: in

text: z

text: and

text: u.

text: Accord-

text: ing

text: to

text: Lemma

text: the

text: multiplicative

text: model

text: of

text: A

text: also

text: implies

text: that

text: for

text: both

text: a

text: and

text: and

text: for

text: all

text: z

text: and

text: u,

text: ∂F

text: (u

text: A

text: a,

text: Z

text: z)/∂z

text: Following

text: exactly

text: the

text: same

text: steps

text: of

text: the

text: proof

text: of

text: Theorem

text: we

text: can

text: prove

text: Theorem

text: Proof

text: of

text: Corollary

text: For

text: binary

text: and

text: independent

text: Z

text: and

text: U,

text: monotonicity,

text: no

text: multiplicative

text: interaction,

text: and

text: Lemma

text: imply

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: p

text: With

text: the

text: above

text: results

text: in

text: the

text: rest

text: of

text: the

text: proof

text: is

text: the

text: same

text: as

text: the

text: proof

text: of

text: Corollary

text: Proof

text: of

text: Theorem

text: First,

text: we

text: consider

text: the

text: treatment

text: effect

text: on

text: the

text: population

text: under

text: treat-

text: ment.

text: Taking

text: U

text: Y

text: in

text: Lemma

text: we

text: have

text: ACEunadj

text: ACEtrue,

text: because

text: A

text: Y

text: Y

text: pr

text: A

text: Y

text: is

text: non-decreasing

text: in

text: Y

text: and

text: E

text: Y

text: A

text: Y

text: Y

text: is

text: non-

text: decreasing

text: in

text: Y

text: The

text: condition

text: cov

text: Π,

text: E(Y

text: A

text: Π)

text: implies

text: that

text: ACEadj

text: ACEunadj

text: according

text: to

text: Lemma

text: Therefore,

text: ACEadj

text: ACEunadj

text: ACEtrue.

text: Second,

text: we

text: take

text: U

text: Y

text: in

text: Lemma

text: and

text: by

text: a

text: similar

text: argument

text: as

text: above

text: we

text: have

text: ACEadj

text: ACEunadj

text: ACEtrue.

text: The

text: conclusion

text: holds

text: because

text: ACEtrue

text: fACEtrue

text: f)ACEtrue

text: and

text: ACEadj

text: fACEadj

text: f)ACEadj.

text: Proof

text: of

text: Theorem

text: Under

text: the

text: additive

text: model

text: of

text: A

text: given

text: Π

text: and

text: U

text: Y

text: Y

text: we

text: have

text: the

text: following

text: results.

text: First,

text: pr(A

text: Π)

text: Π

text: is

text: increasing

text: in

text: Π.

text: Second,

text: Π

text: Y

text: Y

text: implies

text: pr

text: A

text: Π,

text: Y

text: y

text: pr(A

text: Π,

text: U)F

text: (dy

text: y

text: Π

text: δ(y

text: η(y

text: F

text: (dy

text: y

text: Π

text: δ(y

text: η(y

text: )F

text: (dy

text: y

text: Π

text: δ(y

text: Denote

text: the

text: inﬁmum

text: and

text: supremum

text: of

text: Y

text: given

text: Y

text: y

text: by

text: y

text: (y

text: and

text: y

text: with

text: the

text: later

text: not

text: depending

text: on

text: y

text: according

text: to

text: Condition

text: (c)

text: of

text: Theorem

text: Applying

text: integration

text: or

text: summation

text: by

text: parts,

text: we

text: have

text: dη(y

text: δ(y

text: δ(y

text: η(y

text: )F

text: (y

text: y

text: F

text: (y

text: y

text: dy

text: dη(y

text: δ(y

text: η(y

text: F

text: (y

text: y

text: dy

text: dy

text: The

text: function

text: δ(y

text: is

text: non-decreasing

text: in

text: y

text: because

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: 24 P. DING, T. J. VANDERWEELE AND J. M. ROBINS (cid:101) ≥ { | } ≥ { | } Because OR 1, we have pr Y (0) = 1 Y (1) = 1 pr Y (0) = 1 Y (1) = 0 , which ≥Y { | } implies that δ 1. Therefore, pr A = 1 Π, Y (1) is multiplicati(cid:101)ve and non-decreasing in Π { | } and Y (1). Third, we can similarly s(cid:101)how that pr A = 1 Π, Y (0) is multiplicative and non- { | } decreasin{g in Π|and Y (0). F}ourth, pr A = 1 Y (1) = y1 = αfδy1 is non-decreasing in y1. Fifth, pr A = 1 Y (0) = y0 = αfηy0 is non-decreasing in y0. (cid:3) The rest part is the same as the proof of Theorem 6. Proof of Theorem 7.. In Figure 4, Z and U are two independent confounders for the relation- | | ship between A and Y . Because pr(A = 1 Z = z, U = u) and E(Y A = a, Z = z, U = u) (cid:82) are non-decreasing in z and u for both a = 0 and 1, Lemmas S2–S4 imply that the unadjusted estimator, ACEunadj, is larger than or equal to ACEtrue, ACEtrue and ACEtrue. | 1 0 | The independence between Z and U implies pr(A = 1 Z = z) = pr(A = 1 Z = z, U = | | u)F (du), and the monotonicity of pr(A = 1 Z = z, U = u) in z implies that pr(A = 1 Z = (cid:3) z) is non-decreasing in z. The rest of the proof is identical to the proof of Theorem 1. APPENDIX 3. EXTENSIONS TO OTHER CAUSAL MEASURES · Appendix 3 1. Distributional Causal Effects Sometimes we are also interested in estimating the distributional causal effects (Ju & Geng, 2010) for the treatment, control and whole populations: { | } − { | } DCEtrue(y) = pr Y (1) > y A = 1 pr Y (0) > y A = 1 , 1 { | } − { | } DCEtrue(y) = pr Y (1) > y A = 0 pr Y (0) > y A = 0 , 0 { } − { } DCEtrue(y) = pr Y (1) > y pr Y (0) > y . The unadjusted estimator is | − | DCEunadj(y) = pr(Y > y A = 1) pr(Y > y A = 0). The adjusted estimators for the treatment, co(cid:90)ntrol and whole populations are (cid:90) | − | | DCEadj(y) = pr(Y > y A = 1) pr(Y > y A = 0, z)F (dz A = 1), 1 (cid:90) (cid:90) | | − | DCEadj(y) = pr(Y > y A = 1, z)F (dz A = 0) pr(Y > y A = 0), 0 | − | DCEadj(y) = pr(Y > y A = 1, z)F (dz) pr(Y > y A = 0, z)F (dz). If the outcome is binary, then the distributional causal effects at y < 1 are the average causal ≥ effects, and zero at y 1. All results about distributional causal effects reduce to average causal effects for binary outcome. For a general outcome, the distributional causal effects are the av- erage causal effects on the dichotomized outcome I = I(Y > y). Therefore, if we replace the y outcome Y by I in Theorems 1–3, the results about Z-Bias hold for distributional effects. For y | instance, the condition that pr(Y > y A = a, U = u) is non-decreasing in u for all a is the → same as requiring a non-negative sign on the arrow U Y , according to the theory of signed

text: P.

text: DING,

text: T.

text: J.

text: VANDERWEELE

text: AND

text: J.

text: M.

text: ROBINS

text: Because

text: OR

text: we

text: have

text: pr

text: Y

text: Y

text: pr

text: Y

text: Y

text: which

text: ≥Y

text: implies

text: that

text: δ

text: Therefore,

text: pr

text: A

text: Π,

text: Y

text: is

text: and

text: non-decreasing

text: in

text: Π

text: and

text: Y

text: Third,

text: we

text: can

text: similarly

text: that

text: pr

text: A

text: Π,

text: Y

text: is

text: multiplicative

text: and

text: non-

text: decreasin{g

text: in

text: Π|and

text: Y

text: F}ourth,

text: pr

text: A

text: Y

text: is

text: non-decreasing

text: in

text: Fifth,

text: pr

text: A

text: Y

text: is

text: non-decreasing

text: in

text: The

text: rest

text: part

text: is

text: the

text: same

text: as

text: the

text: proof

text: of

text: Theorem

text: Proof

text: of

text: Theorem

text: In

text: Figure

text: Z

text: and

text: U

text: are

text: two

text: independent

text: confounders

text: for

text: the

text: relation-

text: ship

text: between

text: A

text: and

text: Y

text: Because

text: pr(A

text: Z

text: z,

text: U

text: u)

text: and

text: E(Y

text: A

text: a,

text: Z

text: z,

text: U

text: u)

text: are

text: non-decreasing

text: in

text: z

text: and

text: u

text: for

text: both

text: a

text: and

text: Lemmas

text: imply

text: that

text: the

text: unadjusted

text: estimator,

text: ACEunadj,

text: is

text: larger

text: than

text: or

text: equal

text: to

text: ACEtrue,

text: ACEtrue

text: and

text: ACEtrue.

text: The

text: independence

text: between

text: Z

text: and

text: U

text: implies

text: pr(A

text: Z

text: z)

text: pr(A

text: Z

text: z,

text: U

text: u)F

text: (du),

text: and

text: the

text: monotonicity

text: of

text: pr(A

text: Z

text: z,

text: U

text: u)

text: in

text: z

text: implies

text: that

text: pr(A

text: Z

text: z)

text: is

text: non-decreasing

text: in

text: z.

text: The

text: rest

text: of

text: the

text: proof

text: is

text: identical

text: to

text: the

text: proof

text: of

text: Theorem

text: APPENDIX

text: EXTENSIONS

text: TO

text: OTHER

text: CAUSAL

text: MEASURES

text: Appendix

text: Distributional

text: Causal

text: Effects

text: Sometimes

text: we

text: are

text: also

text: interested

text: in

text: estimating

text: the

text: distributional

text: causal

text: effects

text: (Ju

text: Geng,

text: for

text: the

text: treatment,

text: control

text: and

text: whole

text: populations:

text: DCEtrue(y)

text: pr

text: Y

text: y

text: A

text: pr

text: Y

text: y

text: A

text: DCEtrue(y)

text: pr

text: Y

text: y

text: A

text: pr

text: Y

text: y

text: A

text: DCEtrue(y)

text: pr

text: Y

text: y

text: pr

text: Y

text: y

text: The

text: unadjusted

text: estimator

text: is

text: DCEunadj(y)

text: pr(Y

text: y

text: A

text: pr(Y

text: y

text: A

text: The

text: adjusted

text: estimators

text: for

text: the

text: treatment,

text: and

text: whole

text: populations

text: are

text: DCEadj(y)

text: pr(Y

text: y

text: A

text: pr(Y

text: y

text: A

text: z)F

text: (dz

text: A

text: DCEadj(y)

text: pr(Y

text: y

text: A

text: z)F

text: (dz

text: A

text: pr(Y

text: y

text: A

text: DCEadj(y)

text: pr(Y

text: y

text: A

text: z)F

text: (dz)

text: pr(Y

text: y

text: A

text: z)F

text: (dz).

text: If

text: the

text: outcome

text: is

text: binary,

text: then

text: the

text: distributional

text: causal

text: effects

text: at

text: y

text: are

text: the

text: average

text: causal

text: effects,

text: and

text: zero

text: at

text: y

text: All

text: results

text: about

text: distributional

text: causal

text: effects

text: reduce

text: to

text: average

text: causal

text: effects

text: for

text: binary

text: outcome.

text: For

text: a

text: general

text: outcome,

text: the

text: distributional

text: causal

text: effects

text: are

text: the

text: av-

text: erage

text: causal

text: effects

text: on

text: the

text: dichotomized

text: outcome

text: I

text: I(Y

text: y).

text: Therefore,

text: if

text: we

text: replace

text: the

text: y

text: outcome

text: Y

text: by

text: I

text: in

text: Theorems

text: the

text: results

text: about

text: Z-Bias

text: hold

text: for

text: distributional

text: effects.

text: For

text: y

text: instance,

text: the

text: condition

text: that

text: pr(Y

text: y

text: A

text: a,

text: U

text: u)

text: is

text: non-decreasing

text: in

text: u

text: for

text: all

text: a

text: is

text: the

text: same

text: as

text: requiring

text: a

text: non-negative

text: sign

text: on

text: the

text: arrow

text: U

text: Y

text: according

text: to

text: the

text: theory

text: of

text: signed


text: 4 P. DING, T. J. VANDERWE(cid:90)ELE AND J. M. ROBINS (cid:90) | − | ACEtrue = E(Y A = 1) m (u)F (du A = 1), 1 0 (cid:90) (cid:90) | − | ACEtrue = m (u)F (du A = 0) E(Y A = 0), 0 1 − ACEtrue = m (u)F (du) m (u)F (du). 1 0 The unadjusted estimator is the naive comparison between the treatment and control means | − | ACEunadj = E(Y A = 1) E(Y A = 0). | Deﬁne µ (z) = E(Y A = a, Z = z) as the conditional mean of the outcome given the treat- a ment and instrumental variable. Because the instrumental variable Z is also a pretreatment co- variate unaffected by the treatment, the usual stra(cid:90)tegy to adjust for all pretreatment covariates suggests using the adjusted estimator for the population under treatment | − | ACEadj = E(Y A = 1) µ (z)F (dz A = 1), 1 0 (cid:90) for the population under control | − | ACEadj = µ (z)F (dz A = 0) E(Y A = 0), 0 1 (cid:90) (cid:90) and for the whole population − ACEadj = µ (z)F (dz) µ (z)F (dz). 1 0 Surprisingly, for linear structural equation models on (Z, U, A, Y ), previous theory demon- strated that the magnitudes of the biases of the adjusted estimators are no smaller than the unad- justed ones (Pearl, 2010, 2011, 2013; Wooldridge, 2016). The goal of the rest of our paper is to show that this phenomenon exists in more general scenarios. 3. SCALAR INSTRUMENTAL VARIABLE AND SCALAR CONFOUNDER We ﬁrst give a theorem for a scalar instrumental variable Z and a scalar confounder U. THEOREM 1. In the causal diagram of Figure 1(b) with scalar Z and U, if [(a)] | | 1. pr(A = 1 Z = z) is non-decreasing in z, pr(A = 1 U = u) is non-decreasing in u, and | E(Y A = a, U = u) is non-decreasing in u for both a = 0 and 1; | 2. E(Y A = a, Z = z) is non-increasing in z for both a = 0 and 1, then             ACEadj ACEunadj ACEtrue 1 ≥ ≥ 1 ACEadj ACEunadj ACEtrue . (1) ACE0adj ACEunadj ACE0true Inequalities among vectors as in (1) should be interpreted as component-wise relationships. Intuitively, the monotonicity in Condition (a) of Theorem 1 requires non-negative dependence → → → structures on arrows Z A, U A and U Y in the causal diagram of Figure 1(b). Because

text: P.

text: DING,

text: T.

text: J.

text: AND

text: J.

text: M.

text: ROBINS

text: ACEtrue

text: E(Y

text: A

text: m

text: (u)F

text: (du

text: A

text: ACEtrue

text: m

text: (u)F

text: (du

text: A

text: E(Y

text: A

text: ACEtrue

text: m

text: (u)F

text: (du)

text: m

text: (u)F

text: (du).

text: The

text: unadjusted

text: estimator

text: is

text: the

text: naive

text: comparison

text: between

text: the

text: treatment

text: and

text: control

text: means

text: ACEunadj

text: E(Y

text: A

text: E(Y

text: A

text: Deﬁne

text: µ

text: (z)

text: E(Y

text: A

text: a,

text: Z

text: z)

text: as

text: the

text: conditional

text: mean

text: of

text: the

text: outcome

text: given

text: the

text: treat-

text: a

text: ment

text: and

text: instrumental

text: variable.

text: Because

text: the

text: instrumental

text: variable

text: Z

text: is

text: also

text: a

text: pretreatment

text: co-

text: variate

text: unaffected

text: by

text: the

text: treatment,

text: the

text: usual

text: to

text: adjust

text: for

text: all

text: pretreatment

text: covariates

text: suggests

text: using

text: the

text: adjusted

text: estimator

text: for

text: the

text: population

text: under

text: treatment

text: ACEadj

text: E(Y

text: A

text: µ

text: (z)F

text: (dz

text: A

text: for

text: the

text: population

text: under

text: control

text: ACEadj

text: µ

text: (z)F

text: (dz

text: A

text: E(Y

text: A

text: and

text: for

text: the

text: whole

text: population

text: ACEadj

text: µ

text: (z)F

text: (dz)

text: µ

text: (z)F

text: (dz).

text: Surprisingly,

text: for

text: linear

text: structural

text: equation

text: models

text: on

text: (Z,

text: U,

text: A,

text: Y

text: previous

text: theory

text: demon-

text: strated

text: that

text: the

text: magnitudes

text: of

text: the

text: biases

text: of

text: the

text: adjusted

text: estimators

text: are

text: no

text: smaller

text: than

text: the

text: unad-

text: justed

text: ones

text: (Pearl,

text: Wooldridge,

text: The

text: goal

text: of

text: the

text: rest

text: of

text: our

text: paper

text: is

text: to

text: show

text: that

text: this

text: phenomenon

text: exists

text: in

text: more

text: general

text: scenarios.

text: SCALAR

text: INSTRUMENTAL

text: VARIABLE

text: AND

text: SCALAR

text: CONFOUNDER

text: We

text: ﬁrst

text: give

text: a

text: theorem

text: for

text: a

text: scalar

text: instrumental

text: variable

text: Z

text: and

text: a

text: scalar

text: confounder

text: U.

text: THEOREM

text: In

text: the

text: causal

text: diagram

text: of

text: Figure

text: with

text: scalar

text: Z

text: and

text: U,

text: if

text: [(a)]

text: pr(A

text: Z

text: z)

text: is

text: non-decreasing

text: in

text: z,

text: pr(A

text: U

text: u)

text: is

text: non-decreasing

text: in

text: u,

text: and

text: E(Y

text: A

text: a,

text: U

text: u)

text: is

text: non-decreasing

text: in

text: u

text: for

text: both

text: a

text: and

text: E(Y

text: A

text: a,

text: Z

text: z)

text: is

text: non-increasing

text: in

text: z

text: for

text: both

text: a

text: and

text: then

text: ACEadj

text: ACEunadj

text: ACEtrue

text: ACEadj

text: ACEunadj

text: ACEtrue

text: ACEunadj

text: Inequalities

text: among

text: vectors

text: as

text: in

text: should

text: be

text: interpreted

text: as

text: component-wise

text: relationships.

text: Intuitively,

text: the

text: monotonicity

text: in

text: Condition

text: (a)

text: of

text: Theorem

text: requires

text: non-negative

text: dependence

text: structures

text: on

text: arrows

text: Z

text: A,

text: U

text: A

text: and

text: U

text: Y

text: in

text: the

text: causal

text: diagram

text: of

text: Figure

text: Because


text: Instrumental variables as bias ampliﬁers 7 { } U = Y (1), Y (0) (cid:126)(cid:126) (cid:32)(cid:32) (cid:47)(cid:47) (cid:47)(cid:47) (cid:47)(cid:47) Z Π = Π(Z) A Y Fig. 2: Directed Acyclic Graph for Z-Bias With General Instrument and Confounder bias is more general than the one in Theorem 1. Analogous to Section , we will present more transparent sufﬁcient conditions for Z-Bias to aid interpretation. In the following, we use the distributional association measure (Cox & Wermuth, 2003; Ma et al., 2006; Xie et al., 2008), i.e., random variable V has a non-negative distributional association | ≤ on random variable W , if the conditional distribution satisﬁes ∂F (w v)/∂v 0 for all v and w. If the random variables are discrete, then partial differentiation is replaced by differencing between adjacent levels (Cox & Wermuth, 2003). { } If there is no additive interaction between Π and Y (1), Y (0) on A, then we have the fol- lowing results. THEOREM 5. In the causal diagram of Figure 2, (1) holds if | { } { } · · [(a)]pr(A = 1 Π, U) = Π + δ Y (1) + η Y (0) with δ( ) and η( ) being non-decreasing; { } | Y (1), Y (0) have non-negative distributional associations on each other, i.e., ∂F (y ≤ | ≤ 1 y )/∂y 0 and ∂F (y y )/∂y 0 for all y and y ; the essential supremum of Y (1) given 0 0 0 1 1 1 0 Y (0) does not depend on Y (0), and the essential supremum of Y (0) given Y (1) does not depend on Y (1). | { } 123... Remark 1. If we impose an additive model pr(A = 1 Π, U) = h(Π) + δ Y (1) + { } | { } η Y (0) , then independence of Π and U implies that pr(A = 1 Π) = h(Π) + E[δ Y (1) ] + { } { } { } E[η Y (0) ] = Π. Therefore, we must have h(Π) = Π and E[δ Y (1) ] + E[η Y (0) ] = 0. When the outcome is binary, the distributional association between Y (1) and Y (0) becomes their odds ratio (Xie et al., 2008), and non-negative distributional association between Y (1) and Y (0) is equivalent to { } { } pr Y (1) = 1, Y (0) = 1 pr Y (1) = 0, Y (0) = 0 ≥ OR = { } { } 1. Y pr Y (1) = 1, Y (0) = 0 pr Y (1) = 0, Y (0) = 1 We can further relax the model assumption of A given Π and U by allowing for non-negative interaction between Y (1) and Y (0) on A. COROLLARY 3. In the causal diagram of Figure 2 with a binary outcome Y , (1) holds if | ≥ ≥ [(a)] pr(A = 1 Π, U) = α + Π + δY (1) + ηY (0) + θY (1)Y (0) with δ, η, θ 0; OR 1. Y | 12.. Remark 2. If we have an additive model of A given Π and U, pr(A = 1 Π, U) = h(Π) + g(U), then the functional form g(U) = α + δY (1) + ηY (0) + θY (1)Y (0) imposes no re- | striction for binary outcome. Furthermore, pr(A = 1 Π) = Π implies that h(Π) = Π and { } − { } − { } − { } E g(U) = 0, i.e., α = δE Y (1) ηE Y (0) θE Y (1)Y (0) . Therefore, the additive model in Condition (1) of Corollary 3 is | − { } − { } − { } pr(A = 1 Π, U) = Π + δ[Y (1) E Y (1) ] + η[Y (0) E Y (0) ] + θ[Y (1)Y (0) E Y (1)Y (0) ]. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Instrumental

text: variables

text: as

text: bias

text: ampliﬁers

text: U

text: Y

text: Y

font: default

text: Z

text: Π

text: Π(Z)

text: A

text: Y

text: Fig.

text: Directed

label: caption

text: Acyclic

text: Graph

text: for

text: Z-Bias

text: With

text: General

text: Instrument

text: and

text: Confounder

text: bias

text: is

text: more

text: general

text: than

text: the

text: one

text: in

text: Theorem

text: Analogous

text: to

text: Section

text: we

text: will

text: present

text: more

text: transparent

text: sufﬁcient

text: conditions

text: for

text: Z-Bias

text: to

text: aid

text: interpretation.

text: In

text: the

text: following,

text: we

text: use

text: the

text: distributional

text: association

text: measure

text: (Cox

text: Wermuth,

text: Ma

text: et

text: al.,

text: Xie

text: et

text: al.,

text: i.e.,

text: random

text: variable

text: V

text: has

text: a

text: non-negative

text: distributional

text: association

text: on

text: random

text: variable

text: W

text: if

text: the

text: conditional

text: distribution

text: satisﬁes

text: ∂F

text: (w

text: v)/∂v

text: for

text: all

text: v

text: and

text: w.

text: If

text: the

text: random

text: variables

text: are

text: discrete,

text: then

text: partial

text: differentiation

text: is

text: replaced

text: by

text: differencing

text: between

text: adjacent

text: levels

text: (Cox

text: Wermuth,

text: If

text: there

text: is

text: no

text: additive

text: interaction

text: between

text: Π

text: and

text: Y

text: Y

text: on

text: A,

text: then

text: we

text: have

text: the

text: fol-

text: lowing

text: results.

text: THEOREM

text: In

text: the

text: causal

text: diagram

text: of

text: Figure

text: holds

text: if

text: [(a)]pr(A

text: Π,

text: U)

text: Π

text: δ

text: Y

text: η

text: Y

text: with

text: δ(

text: and

text: η(

text: being

text: non-decreasing;

text: Y

text: Y

text: have

text: non-negative

text: distributional

text: associations

text: on

text: each

text: other,

text: i.e.,

text: ∂F

text: (y

text: y

text: )/∂y

text: and

text: ∂F

text: (y

text: y

text: )/∂y

text: for

text: all

text: y

text: and

text: y

text: the

text: essential

text: supremum

text: of

text: Y

text: given

text: Y

text: does

text: not

text: depend

text: on

text: Y

text: and

text: the

text: essential

text: supremum

text: of

text: Y

text: given

text: Y

text: does

text: not

text: depend

text: on

text: Y

text: Remark

text: If

text: we

text: impose

text: an

text: additive

text: model

text: pr(A

text: Π,

text: U)

text: h(Π)

text: δ

text: Y

text: η

text: Y

text: then

text: independence

text: of

text: Π

text: and

text: U

text: implies

text: that

text: pr(A

text: Π)

text: h(Π)

text: E[δ

text: Y

text: E[η

text: Y

text: Π.

text: Therefore,

text: we

text: must

text: have

text: h(Π)

text: Π

text: and

text: E[δ

text: Y

text: E[η

text: Y

text: When

text: the

text: outcome

text: is

text: binary,

text: the

text: distributional

text: association

text: between

text: Y

text: and

text: Y

text: becomes

text: their

text: odds

text: ratio

text: (Xie

text: et

text: al.,

text: and

text: non-negative

text: distributional

text: association

text: between

text: Y

text: and

text: Y

text: is

text: equivalent

text: to

text: pr

text: Y

text: Y

text: pr

text: Y

text: Y

text: OR

text: Y

text: pr

text: Y

text: Y

text: pr

text: Y

text: Y

text: We

text: can

text: further

text: relax

text: the

text: model

text: assumption

text: of

text: A

text: given

text: Π

text: and

text: U

text: by

text: allowing

text: for

text: non-negative

text: interaction

text: between

text: Y

text: and

text: Y

text: on

text: A.

text: COROLLARY

text: In

text: the

text: causal

text: diagram

text: of

text: Figure

text: with

text: a

text: binary

text: outcome

text: Y

text: holds

text: if

text: [(a)]

text: pr(A

text: Π,

text: U)

text: α

text: Π

text: δY

text: ηY

text: θY

text: with

text: δ,

text: η,

text: θ

text: OR

text: Y

text: Remark

text: If

text: we

text: have

text: an

text: additive

text: model

text: of

text: A

text: given

text: Π

text: and

text: U,

text: pr(A

text: Π,

text: U)

text: h(Π)

text: g(U),

text: then

text: the

text: functional

text: form

text: g(U)

text: α

text: δY

text: ηY

text: θY

text: imposes

text: no

text: re-

text: striction

text: for

text: binary

text: outcome.

text: Furthermore,

text: pr(A

text: Π)

text: Π

text: implies

text: that

text: h(Π)

text: Π

text: and

text: E

text: g(U)

text: i.e.,

text: α

text: δE

text: Y

text: ηE

text: Y

text: θE

text: Y

text: Therefore,

text: the

text: additive

text: model

text: in

text: Condition

text: of

text: Corollary

text: is

text: pr(A

text: Π,

text: U)

text: Π

text: δ[Y

text: E

text: Y

text: η[Y

text: E

text: Y

text: θ[Y

text: E

text: Y

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 P. DING, T. J. VANDERWEELE AND J. M. ROBINS { } If there is no multiplicative interaction of Π and Y (1), Y (0) on Z, then we have the follow- ing results. THEOREM 6. In the causal diagram of Figure 2, (1) holds if we replace Condition (a) of Theorem 5 by | { } { } · · [(a’)]pr(A = 1 Π, U) = Πδ Y (1) η Y (0) with δ( ) and η( ) being non-decreasing. 1. COROLLARY 4. In the causal diagram of Figure 2 with a binary outcome Y , (1) holds if we replace Condition (a) of Corollary 3 by | ≥ [(a’)]pr(A = 1 Π, U) = αΠδY (1)ηY (0)θY (1)Y (0) with δ, η, θ 1. 5. ILLUSTRATIONS · 5 1. Numerical Examples Myers et al. (2011) simulated binary (Z, U, A, Y ) to investigate Z-Bias. They generated (Z, U) according to pr(Z = 1) = 0.5 and pr(U = 1) = γ . The ﬁrst set of their generative models is 0 additive, | | pr(A = 1 U, Z) = α + α U + α Z, pr(Y = 1 U, A) = β + β U + β A, (3) 0 1 2 0 1 2 where the coefﬁcients are all positive. The second set of their generative models is multiplica- tive, | | pr(A = 1 U, Z) = α αUαZ, pr(Y = 1 U, A) = β βUβA, (4) 0 1 2 0 1 2 where the coefﬁcients in (3) and (4) are all positive. They use simulation to show that Z- Bias arises under these models. In fact, in the above models, Z and U have monotonic effects on A without additive or multiplicative interactions, and U acts monotonically on Y , given A. Therefore, Corollaries 1 and 2 imply that Z-Bias must occur. The qualitative conclusion follows immediately from our theory. However, our theory does not make statements about the magnitude of the bias, and for more details about the magnitude and ﬁnite sample properties, see Myers et al. (2011). We further use three numerical examples to illustrate the role of the no-interaction assump- tions required by Theorems 2 and 3 and Corollaries 1 and 2. Recall the conditional probability of | the treatment A, p = pr(A = 1 Z = z, U = u), and deﬁne the conditional probabilities of the zu | outcome Y as r = pr(Y = 1 A = a, U = u), for z, a, u = 0, 1. Table 1 gives three examples, au where monotonicity on the conditional distributions of A and Y hold, and there are both additive and multiplicative interactions. In all cases, the instrumental variable Z is Bernoulli(p = 0.5), and the confounder U is another independent Bernoulli(π = 0.5). In Case 1, the weaker con- dition (2) holds, and our theory implies that Z-Bias arises. In Case 2, neither the condition in Theorem 1 or (2) holds, but Z-Bias still arises. Our conditions are only sufﬁcient but not neces- sary. In Case 3, neither the condition in Theorem 1 or (2) holds, and Z-Bias does not arise. Finally, for binary (Z, U, A, Y ) we use Monte Carlo to compute the volume of the Z-Bias space, i.e., the parameter space of p, π, p ’s and r ’s in which the adjusted estimator has higher zu au bias than the unadjusted estimator. We randomly draw these ten probabilities from independent Uniform(0, 1) random variables, and for each draw of these probabilities we compute the average causal effect ACEtrue, the unadjusted estimator ACEunadj and the adjusted estimator ACEadj.

text: P.

text: DING,

text: T.

text: J.

text: VANDERWEELE

text: AND

text: J.

text: M.

text: ROBINS

text: If

text: there

text: is

text: no

text: multiplicative

text: interaction

text: of

text: Π

text: and

text: Y

text: Y

text: on

text: Z,

text: then

text: we

text: have

text: the

text: follow-

text: ing

text: results.

text: THEOREM

text: In

text: the

text: causal

text: diagram

text: of

text: Figure

text: holds

text: if

text: we

text: replace

text: Condition

text: (a)

text: of

text: Theorem

text: by

label: list

text: [(a’)]pr(A

text: Π,

text: U)

text: Πδ

text: Y

text: η

text: Y

text: with

text: δ(

text: and

text: η(

text: being

text: non-decreasing.

text: COROLLARY

text: In

text: the

text: causal

text: diagram

text: of

text: Figure

text: with

text: a

text: binary

text: outcome

text: Y

text: holds

text: if

text: we

text: replace

text: Condition

text: (a)

text: of

text: Corollary

text: by

text: [(a’)]pr(A

text: Π,

text: U)

text: αΠδY

text: with

text: δ,

text: η,

text: θ

text: ILLUSTRATIONS

text: Numerical

text: Examples

text: Myers

text: et

text: al.

text: simulated

text: binary

text: (Z,

text: U,

text: A,

text: Y

text: to

text: investigate

text: Z-Bias.

text: They

text: generated

text: (Z,

text: U)

text: according

text: to

text: pr(Z

text: and

text: pr(U

text: γ

text: The

text: ﬁrst

text: set

text: of

text: their

text: generative

text: models

text: is

text: additive,

text: pr(A

text: U,

text: Z)

text: α

text: α

text: U

text: α

text: Z,

text: pr(Y

text: U,

text: A)

text: β

text: β

text: U

text: β

text: A,

text: where

text: the

text: coefﬁcients

text: are

text: all

text: positive.

text: The

text: second

text: set

text: of

text: their

text: generative

text: models

text: is

text: multiplica-

text: tive,

text: pr(A

text: U,

text: Z)

text: α

text: αUαZ,

text: pr(Y

text: U,

text: A)

text: β

text: βUβA,

text: where

text: the

text: coefﬁcients

text: in

text: and

text: are

text: all

text: positive.

text: They

text: use

text: simulation

text: to

text: show

text: that

text: Z-

text: Bias

text: arises

text: under

text: these

text: models.

text: In

text: fact,

text: in

text: the

text: above

text: models,

text: Z

text: and

text: U

text: have

text: monotonic

text: effects

text: on

text: A

text: without

text: additive

text: or

text: multiplicative

text: interactions,

text: and

text: U

text: acts

text: monotonically

text: on

text: Y

text: given

text: A.

text: Therefore,

text: Corollaries

text: and

text: imply

text: that

text: Z-Bias

text: must

text: occur.

text: The

text: qualitative

text: conclusion

text: follows

text: immediately

text: from

text: our

text: theory.

text: However,

text: our

text: theory

text: does

text: not

text: make

text: statements

text: about

text: the

text: magnitude

text: of

text: the

text: bias,

text: and

text: for

text: more

text: details

text: about

text: the

text: magnitude

text: and

text: ﬁnite

text: sample

text: properties,

text: see

text: Myers

text: et

text: al.

text: We

text: further

text: use

text: three

text: numerical

text: examples

text: to

text: illustrate

text: the

text: role

text: of

text: the

text: no-interaction

text: assump-

text: tions

text: required

text: by

text: Theorems

text: and

text: and

text: Corollaries

text: and

text: Recall

text: the

text: conditional

text: probability

text: of

text: the

text: treatment

text: A,

text: p

text: pr(A

text: Z

text: z,

text: U

text: u),

text: and

text: deﬁne

text: the

text: conditional

text: probabilities

text: of

text: the

text: zu

text: outcome

text: Y

text: as

text: r

text: pr(Y

text: A

text: a,

text: U

text: u),

text: for

text: z,

text: a,

text: u

text: Table

text: gives

text: three

text: examples,

text: au

text: where

text: monotonicity

text: on

text: the

text: conditional

text: distributions

text: of

text: A

text: and

text: Y

text: hold,

text: and

text: there

text: are

text: both

text: additive

text: and

text: multiplicative

text: interactions.

text: In

text: all

text: cases,

text: the

text: instrumental

text: variable

text: Z

text: is

text: Bernoulli(p

text: and

text: the

text: confounder

text: U

text: is

text: another

text: independent

text: Bernoulli(π

text: In

text: Case

text: the

text: weaker

text: con-

text: dition

text: holds,

text: and

text: our

text: theory

text: implies

text: that

text: Z-Bias

text: arises.

text: In

text: Case

text: neither

text: the

text: condition

text: in

text: Theorem

text: or

text: holds,

text: but

text: Z-Bias

text: still

text: arises.

text: Our

text: conditions

text: are

text: only

text: sufﬁcient

text: but

text: not

text: neces-

text: sary.

text: In

text: Case

text: neither

text: the

text: condition

text: in

text: Theorem

text: or

text: holds,

text: and

text: Z-Bias

text: does

text: not

text: arise.

text: Finally,

text: for

text: binary

text: (Z,

text: U,

text: A,

text: Y

text: we

text: use

text: Monte

text: Carlo

text: to

text: compute

text: the

text: volume

text: of

text: the

text: Z-Bias

text: space,

text: i.e.,

text: the

text: parameter

text: space

text: of

text: p,

text: π,

text: p

text: ’s

text: and

text: r

text: ’s

text: in

text: which

text: the

text: adjusted

text: estimator

text: has

text: higher

text: zu

text: au

text: bias

text: than

text: the

text: unadjusted

text: estimator.

text: We

text: randomly

text: draw

text: these

text: ten

text: probabilities

text: from

text: independent

text: random

text: variables,

text: and

text: for

text: each

text: draw

text: of

text: these

text: probabilities

text: we

text: compute

text: the

text: average

text: causal

text: effect

text: ACEtrue,

text: the

text: unadjusted

text: estimator

text: ACEunadj

text: and

text: the

text: adjusted

text: estimator

text: ACEadj.


text: 2 15 d f d g e e g 12 c c a 9 d h e 6 x b z y 3 p r Fig. 3. Fabricated prototypes of the capacitor-loaded SSP-TL and conven- q s tional SSP-TL. 0 0 0.2 0.4 0.6 0.8 1 a corrugation depth (e.g. h = 11 mm), the dispersion curve Fig. 2. Dispersion Curves of (a) conventional SSP-TL and (b) capacitor- becomes almost ﬂat when it is close to the cutoff frequency. loaded SSP-TL. (E.C: Equivalent Circuit) This frequency, in analogy to the plasma frequency of surface plasmons in optics, is inversely proportional to the corrugation excited by the currents along the strip units, C2 and C3 models depth h. Also, the group velocity around this frequency is the electric ﬁeld pointing to the inﬁnity (in x direction), and extremely small, which gives rise to a strongly localized wave C represents the local electric ﬁeld excited between the gap in analogy to the strong light-electron interaction in optics. The 1 (mostly in y direction). small group velocity also leads to a sharp magnitude response, To analyze the unit cell of Fig. 1(a), one adds a periodic which is very useful in high-selectivity ﬁlters. boundary condition on both sides of the unit cell, and subse- Although a high corrugation depth h is good for ﬁeld quently computes the dispersion curve using the software CST conﬁnement, it inevitably increases the line width w that is not Microwave Studio. One may consider an example, in which preferred in a compact system. To overcome this drawback, the substrate is Rogers 4003C ((cid:15) = 3.38, tan δ = 0.0027) we propose the capacitor-loaded SSP-TL in Fig. 1(b). The r with thickness of 1.52 mm, and the strip dimensions in additional loaded capacitors along the surface brings in an Fig. 1(a) are a = 1 mm, w = 5 mm, d = 5 mm and h = 3 mm. extra capacitance, in equivalence to the extra capacitance The calculated dispersion curve is shown as the blue squares brought by the increased corrugation depth in conventional in Fig. 2. Note that it exhibits a typical SSP response, which SSP-TLs. Therefore, the proposed SSP-TL increases the ﬁeld stays close to the air line at low frequencies but goes away conﬁnement capability and meanwhile maintains a small line at high frequencies. The circuit parameters of Fig. 1(a) are width. The capacitor-loaded unit cell can be modeled by calculated as L1 = 0.8 nH, L2 = 0.14 nH, L3 = 0.2 nH, adding the extra capacitance ∆C to the original circuit, as C = 22.5 fF, C = 161 fF and C = 178 fF, by using shown in Fig. 1(b). One may use this circuit model to 1 2 3 the approach provided in [3]. One may further calculate the obtain the dispersion curves in Fig. 2. Note that, the SSP-TL dispersion curve of the equivalent circuit to verify its validity. loaded with different capacitances achieves almost the same One ﬁrstly computes the scattering parameters (S and S ) dispersion curves as the conventional SSP-TL with different 11 21 of the circuit mod(cid:20)el, and sub(cid:18)sequently obtains(cid:19)t(cid:21)he dispersion depths. relation by III. EXPERIMENTAL VALIDATION − (cid:61) 1 − 1 S2 + S2 To validate the proposed capacitor-loaded SSP-TL, we ex- β = cosh 1 11 21 , (1) d 2S perimentally implement the one with ∆C of 0.2 pF on a 1.52- 21 mm-thick Rogers 4003C substrate, in correspondence with the (cid:61) where [.] denotes the operation of taking the imaginary part. purple-line dispersion curve of Fig. 2. In comparison, we also It should be noted that the above equation is obtained under implement the conventional SSP-TL with a corrugation depth the condition of symmetry (S = S ) and reciprocity (S = h = 8 mm, exhibiting a similar dispersion curve to the pro- 11 22 21 S ), and hence valid for symmetrical and reciprocal cases posed SSP-TL. Fig. 3 shows the fabricated prototypes of the 12 only. The computed dispersion curve using (1) is displayed two SSP-TLs. Tapered lines are employed for a smooth transi- as the blue dotted line in Fig. 2, which well follows the one tion and mode conversion between the 50Ω-CPW line (width: obtained by CST full-wave simulation. 12 mm, gap: 0.5 mm) and SSP [4]. As shown in the zoomed To increase the ﬁeld conﬁnement or localization, one should ﬁgure of Fig. 3(a), the proposed SSP-TL is loaded with 0.2 pF further bend the dispersion curve away from the air line. One chip capacitors (Murata GJM1555C1HR20WB01D). The line way to achieve this is to increase the corrugation depth h, parameters are w = 5 mm, h = 3 mm, a = 1 mm, and which is well validated by the dispersion curves of Fig. 2. d = 5 mm. In contrast, the conventional SSP-TL has a much When h increases from 3 mm to 11 mm, the dispersion wider line width (i.e. w = 10 mm), due to the increased curves gradually bend away from the air line, giving rise to corrugation depth h = 8 mm. Thus, the capacitor-loaded SSP- a much slower trapped wave. Note that, in the case of a high TL reduces the line width by half. b ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: d

text: f

text: d

text: g

text: e

text: e

text: g

text: c

text: c

text: a

text: d

text: h

text: e

text: x

text: b

text: z

text: y

text: p

text: r

text: Fig.

text: Fabricated

text: prototypes

text: of

text: the

text: capacitor-loaded

text: SSP-TL

text: and

text: conven-

text: q

text: s

text: tional

text: SSP-TL.

text: a

text: corrugation

text: depth

text: (e.g.

text: h

text: mm),

text: the

text: dispersion

text: curve

text: Fig.

text: Dispersion

text: Curves

text: of

text: (a)

text: conventional

text: SSP-TL

text: and

text: (b)

text: capacitor-

text: becomes

text: almost

text: ﬂat

text: when

text: it

text: is

text: close

text: to

text: the

text: cutoff

text: frequency.

text: loaded

text: SSP-TL.

text: (E.C:

text: Equivalent

text: Circuit)

text: This

text: frequency,

text: in

text: analogy

text: to

text: the

text: plasma

text: frequency

text: of

text: surface

text: plasmons

text: in

text: optics,

text: is

text: inversely

text: proportional

text: to

text: the

text: corrugation

text: excited

text: by

text: the

text: currents

text: along

text: the

text: strip

text: units,

text: and

text: models

text: depth

text: h.

text: Also,

text: the

text: group

text: velocity

text: around

text: this

text: frequency

text: is

text: the

text: electric

text: ﬁeld

text: pointing

text: to

text: the

text: inﬁnity

text: (in

text: x

text: direction),

text: and

text: extremely

text: small,

text: which

text: gives

text: rise

text: to

text: a

text: strongly

text: localized

text: wave

text: C

text: represents

text: the

text: local

text: electric

text: ﬁeld

text: excited

text: between

text: the

text: gap

text: in

text: analogy

text: to

text: the

text: strong

text: light-electron

text: interaction

text: in

text: optics.

text: The

text: (mostly

text: in

text: y

text: direction).

text: small

text: group

text: velocity

text: also

text: leads

text: to

text: a

text: sharp

text: magnitude

text: response,

text: To

text: analyze

text: the

text: unit

text: cell

text: of

text: Fig.

text: one

text: adds

text: a

text: periodic

text: which

text: is

text: very

text: useful

text: in

text: high-selectivity

text: ﬁlters.

text: boundary

text: condition

text: on

text: both

text: sides

text: of

text: the

text: unit

text: cell,

text: and

text: subse-

text: Although

text: a

text: high

text: corrugation

text: depth

text: h

text: is

text: good

text: for

text: ﬁeld

text: quently

text: computes

text: the

text: dispersion

text: curve

text: using

text: the

text: software

text: CST

text: conﬁnement,

text: it

text: inevitably

text: increases

text: the

text: line

text: width

text: w

text: that

text: is

text: not

text: Microwave

text: Studio.

text: One

text: may

text: consider

text: an

text: example,

text: in

text: which

text: preferred

text: in

text: a

text: compact

text: system.

text: To

text: overcome

text: this

text: drawback,

text: the

text: substrate

text: is

text: Rogers

text: tan

text: δ

text: we

text: propose

text: the

text: capacitor-loaded

text: SSP-TL

text: in

text: Fig.

text: The

text: r

text: with

text: thickness

text: of

text: mm,

text: and

text: the

text: strip

text: dimensions

text: in

text: additional

text: loaded

text: capacitors

text: along

text: the

text: surface

text: brings

text: in

text: an

text: Fig.

text: are

text: a

text: mm,

text: w

text: mm,

text: d

text: mm

text: and

text: h

text: mm.

text: extra

text: capacitance,

text: in

text: equivalence

text: to

text: the

text: extra

text: capacitance

text: The

text: calculated

text: dispersion

text: curve

text: is

text: shown

text: as

text: the

text: blue

text: squares

text: brought

text: by

text: the

text: increased

text: corrugation

text: depth

text: in

text: conventional

text: in

text: Fig.

text: Note

text: that

text: it

text: exhibits

text: a

text: typical

text: SSP

text: response,

text: which

text: SSP-TLs.

text: Therefore,

text: the

text: proposed

text: SSP-TL

text: increases

text: the

text: ﬁeld

text: stays

text: close

text: to

text: the

text: air

text: line

text: at

text: low

text: frequencies

text: but

text: goes

text: away

text: conﬁnement

text: capability

text: and

text: meanwhile

text: maintains

text: a

text: small

text: line

text: at

text: high

text: frequencies.

text: The

text: circuit

text: parameters

text: of

text: Fig.

text: are

text: width.

text: The

text: capacitor-loaded

text: unit

text: cell

text: can

text: be

text: modeled

text: by

text: calculated

text: as

text: nH,

text: nH,

text: nH,

text: adding

text: the

text: extra

text: capacitance

text: ∆C

text: to

text: the

text: original

text: circuit,

text: as

text: C

text: fF,

text: C

text: fF

text: and

text: C

text: fF,

text: by

text: using

text: shown

text: in

text: Fig.

text: One

text: may

text: use

text: this

text: circuit

text: model

text: to

text: the

text: approach

text: provided

text: in

text: One

text: may

text: further

text: calculate

text: the

text: obtain

text: the

text: dispersion

text: curves

text: in

text: Fig.

text: Note

text: that,

text: the

text: SSP-TL

text: dispersion

text: curve

text: of

text: the

text: equivalent

text: circuit

text: to

text: verify

text: its

text: validity.

text: loaded

text: with

text: different

text: capacitances

text: achieves

text: almost

text: the

text: same

text: One

text: ﬁrstly

text: computes

text: the

text: scattering

text: parameters

text: (S

text: and

text: S

text: dispersion

text: curves

text: as

text: the

text: conventional

text: SSP-TL

text: with

text: different

text: of

text: the

text: circuit

text: and

text: dispersion

text: depths.

text: relation

text: by

text: III.

text: EXPERIMENTAL

text: VALIDATION

text: To

text: validate

text: the

text: proposed

text: capacitor-loaded

text: SSP-TL,

text: we

text: ex-

text: β

text: cosh

text: d

text: perimentally

text: implement

text: the

text: one

text: with

text: ∆C

text: of

text: pF

text: on

text: a

text: mm-thick

text: Rogers

text: substrate,

text: in

text: correspondence

text: with

text: the

text: where

text: denotes

text: the

text: operation

text: of

text: taking

text: the

text: imaginary

text: part.

text: purple-line

text: dispersion

text: curve

text: of

text: Fig.

text: In

text: comparison,

text: we

text: also

text: It

text: should

text: be

text: noted

text: that

text: the

text: above

text: equation

text: is

text: obtained

text: under

text: implement

text: the

text: conventional

text: SSP-TL

text: with

text: a

text: corrugation

text: depth

text: the

text: condition

text: of

text: symmetry

text: (S

text: S

text: and

text: reciprocity

text: (S

text: h

text: mm,

text: exhibiting

text: a

text: similar

text: dispersion

text: curve

text: to

text: the

text: pro-

text: S

text: and

text: hence

text: valid

text: for

text: symmetrical

text: and

text: reciprocal

text: cases

text: posed

text: SSP-TL.

text: Fig.

text: shows

text: the

text: fabricated

text: prototypes

text: of

text: the

text: only.

text: The

text: computed

text: dispersion

text: curve

text: using

text: is

text: displayed

text: two

text: SSP-TLs.

text: Tapered

text: lines

text: are

text: employed

text: for

text: a

text: smooth

text: transi-

text: as

text: the

text: blue

text: dotted

text: line

text: in

text: Fig.

text: which

text: well

text: follows

text: the

text: one

text: tion

text: and

text: mode

text: conversion

text: between

text: the

text: line

text: (width:

text: obtained

text: by

text: CST

text: full-wave

text: simulation.

text: mm,

text: gap:

text: mm)

text: and

text: SSP

text: As

text: shown

text: in

text: the

text: zoomed

text: To

text: increase

text: the

text: ﬁeld

text: conﬁnement

text: or

text: localization,

text: one

text: should

text: ﬁgure

text: of

text: Fig.

text: the

text: proposed

text: SSP-TL

text: is

text: loaded

text: with

text: pF

text: further

text: bend

text: the

text: dispersion

text: curve

text: away

text: from

text: the

text: air

text: line.

text: One

text: chip

text: capacitors

text: (Murata

text: The

text: line

text: way

text: to

text: achieve

text: this

text: is

text: to

text: increase

text: the

text: corrugation

text: depth

text: h,

text: parameters

text: are

text: w

text: mm,

text: h

text: mm,

text: a

text: mm,

text: and

text: which

text: is

text: well

text: validated

text: by

text: the

text: dispersion

text: curves

text: of

text: Fig.

text: d

text: mm.

text: In

text: contrast,

text: the

text: conventional

text: SSP-TL

text: has

text: a

text: much

text: When

text: h

text: increases

text: from

text: mm

text: to

text: mm,

text: the

text: dispersion

text: wider

text: line

text: width

text: (i.e.

text: w

text: mm),

text: due

text: to

text: the

text: increased

text: curves

text: gradually

text: bend

text: away

text: from

text: the

text: air

text: line,

text: giving

text: rise

text: to

text: corrugation

text: depth

text: h

text: mm.

text: Thus,

text: the

text: capacitor-loaded

text: SSP-

text: a

text: much

text: slower

text: trapped

text: wave.

text: Note

text: that,

text: in

text: the

text: case

text: of

text: a

text: high

text: TL

text: reduces

text: the

text: line

text: width

text: by

text: half.

text: b

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 3 0 45 1 h 30 c d 0.8 −20 15 0.6 d 0 e 0.4 −40 0 -15 -30 0.2 −60 −10 -45 0 -45 -30 -15 0 15 30 45 -45 -30 -15 0 15 30 45 z (mm) z (mm) −80 −20 1 f f e g h 0.8 g 0.6 2 3 4 5 6 7 8 9 10 a 0.4 u u 0.2 Fig. 4. The measured scattering parameters of the two fabricated SSP-TL prototypes in Fig. 3. 0 -45 -30 -15 0 15 30 45 -45 -30 -15 0 15 30 45 z (mm) z (mm) a b The two fabricated prototypes are measured using an Ag- Fig. 5. Cross-section distribution of the electric ﬁeld of the SSP-TLs in (a) ilent PNA network analyzer (E5071C) within the frequency Fig. 3(a) and (b) Fig. 3(b) at 4 GHz. − range 2 10 GHz. The measured responses of the scattering parameters are shown in Fig. 4. Note that the capacitor-loaded 0 SSP-TL has a similar response as the conventional SSP-TL, in spite of a slight discrepancy in the cutoff frequency, possibly h g due to the tolerances brought by PCB fabrication and the ± commercial chip capacitors ( 0.05 pF). Also note that, the −20 proposed SSP-TL has a very sharp magnitude response around 5.8 GHz, which exhibits a highly selective ﬁltering feature. It b also exhibits a wide stop band from 5.8 GHz to 10 GHz. To compare the ﬁeld conﬁnement capability of the two −40 c SSP-TLs, their normalized electric ﬁelds in the cross section d − (x z plane) at 4 GHz are plotted in Fig. 5. Note from the e 2D distributions that the proposed SSP-TL has a better ﬁeld f −60 conﬁnement. To quantify this improvement, we plot the 1D z- 2 3 4 5 6 7 8 a distribution across the maximum point. Then one may deﬁne the conﬁned region using the width when the ﬁeld drops to 0.3 Fig. 6. Measured magnitude responses of the fabricated prototype of Fig. 3(a) − (about 10 dB) of the maximum strength. Note that this width with different loading capacitances. is 10.9 mm in the proposed SSP-TL, and is 16.7 mm in the conventional one. Thus, the conﬁnement capability, inversely proportional to the conﬁned region, is improved by 53%. REFERENCES The proposed SSP-TL loaded with different capacitances are [1] W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon sub- also fabricated and measured. The scattering parameters are wavelength optics.” Nature, vol. 424, no. 6950, pp. 824–30, 2003. shown in Fig. 6. Note that the cutoff frequency (in analogy [2] J. B. Pendry, L. Mart´ın-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces.” Science, vol. 305, no. 5685, to the plasma frequency in optics) and the operation band pp. 847–8, 2004. are completely reconﬁgurable by changing the capacitances. [3] A. Kianinejad, Z. N. Chen, and C. W. Qiu, “Design and modeling of spoof High-selectivity feature is still preserved while reconﬁguring surface plasmon modes-based microwave slow-wave transmission line,” the operation band. The static capacitors may be replaced by IEEE Trans. on Microw. Theory Tech., vol. 63, no. 6, pp. 1817–1825, June 2015. tunable capacitors to make it truly reconﬁgurable, which will [4] H. F. Ma, X. Shen, Q. Cheng, W. X. Jiang, and T. J. Cui, “Broadband and be further explored in our future works. high-efﬁciency conversion from guided waves to spoof surface plasmon polaritons,” Laser Photonics Rev., vol. 8, no. 1, pp. 146–151, 2014. IV. CONCLUSION [5] X. Shen, T. J. Cui, D. Martincano, and F. J. Garciavidal, “Conformal surface plasmons propagating on ultrathin and ﬂexible ﬁlms,” Proc. Nat. In this paper, a capacitor-loaded SSP-TL has been proposed. Acad. Sci., vol. 110, no. 1, pp. 40–45, 2012. [6] Y. Liang, H. Yu, H. C. Zhang, C. Yang, and T. J. Cui, “On-chip sub- It featured reconﬁgurability in the dispersion control. It also terahertz surface plasmon polariton transmission lines in CMOS.” Sci. exhibits a small line width in comparison with the conventional Rep., vol. 5, p. 14853, 2015. SSP-TL. The principle was illustrated and several experimental [7] X. Gao, L. Zhou, Z. Liao, H. F. Ma, and T. J. Cui, “An ultra-wideband surface plasmonic ﬁlter in microwave frequency,” Appl. Phys. Lett., vol. examples were provided. The results completely validated the 104, no. 19, p. 191603, 2014. proposed SSP-TLs. [8] A. Kianinejad, Z. N. Chen, L. Zhang, and W. Liu, “Spoof plasmon- b c )mm( x dleiF dezilamroN ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: h

text: c

text: d

text: d

text: e

text: z

text: (mm)

text: z

text: (mm)

text: f

text: f

text: e

text: g

text: h

text: g

text: a

text: u

text: u

text: Fig.

text: The

text: measured

text: scattering

text: parameters

text: of

text: the

text: two

text: fabricated

text: SSP-TL

text: prototypes

text: in

text: Fig.

text: z

text: (mm)

text: z

text: (mm)

text: a

text: b

text: The

text: two

text: fabricated

text: prototypes

text: are

text: measured

text: using

text: an

text: Ag-

text: Fig.

text: Cross-section

text: distribution

text: of

text: the

text: electric

text: ﬁeld

text: of

text: the

text: SSP-TLs

text: in

text: (a)

text: ilent

text: PNA

text: network

text: analyzer

text: within

text: the

text: frequency

text: Fig.

text: and

text: (b)

text: Fig.

text: at

text: GHz.

text: range

text: GHz.

text: The

text: measured

text: responses

text: of

text: the

text: scattering

text: parameters

text: are

text: shown

text: in

text: Fig.

text: Note

text: that

text: the

text: capacitor-loaded

text: SSP-TL

text: has

text: a

text: similar

text: response

text: as

text: the

text: conventional

text: SSP-TL,

text: in

text: spite

text: of

text: a

text: slight

text: discrepancy

text: in

text: the

text: cutoff

text: frequency,

text: possibly

text: h

text: g

text: due

text: to

text: the

text: tolerances

text: brought

text: by

text: PCB

text: fabrication

text: and

text: the

text: commercial

text: chip

text: capacitors

text: pF).

text: Also

text: note

text: that,

text: the

text: proposed

text: SSP-TL

text: has

text: a

text: very

text: sharp

text: magnitude

text: response

text: around

text: GHz,

text: which

text: exhibits

text: a

text: highly

text: selective

text: ﬁltering

text: feature.

text: It

text: b

text: also

text: exhibits

text: a

text: wide

text: stop

text: band

text: from

text: GHz

text: to

text: GHz.

text: To

text: compare

text: the

text: ﬁeld

text: conﬁnement

text: capability

text: of

text: the

text: two

text: c

text: SSP-TLs,

text: their

text: normalized

text: electric

text: ﬁelds

text: in

text: the

text: cross

text: section

text: d

text: (x

text: z

text: plane)

text: at

text: GHz

text: are

text: plotted

text: in

text: Fig.

text: Note

text: from

text: the

text: e

text: distributions

text: that

text: the

text: proposed

text: SSP-TL

text: has

text: a

text: better

text: ﬁeld

text: f

text: conﬁnement.

text: To

text: quantify

text: this

text: improvement,

text: we

text: plot

text: the

text: z-

text: a

text: distribution

text: across

text: the

text: maximum

text: point.

text: Then

text: one

text: may

text: deﬁne

text: the

text: conﬁned

text: region

text: using

text: the

text: width

text: when

text: the

text: ﬁeld

text: drops

text: to

text: Fig.

text: Measured

text: magnitude

text: responses

text: of

text: the

text: fabricated

text: prototype

text: of

text: Fig.

text: (about

text: dB)

text: of

text: the

text: maximum

text: strength.

text: Note

text: that

text: this

text: width

text: with

text: different

text: loading

text: capacitances.

text: is

text: mm

text: in

text: the

text: proposed

text: SSP-TL,

text: and

text: is

text: mm

text: in

text: the

text: conventional

text: one.

text: Thus,

text: the

text: conﬁnement

text: capability,

text: inversely

text: proportional

text: to

text: the

text: conﬁned

text: region,

text: is

text: improved

text: by

text: REFERENCES

text: The

text: proposed

text: SSP-TL

text: loaded

text: with

text: different

text: capacitances

text: are

text: W.

text: L.

text: Barnes,

text: A.

text: Dereux,

text: and

text: T.

text: W.

text: Ebbesen,

text: “Surface

text: plasmon

text: sub-

text: also

text: fabricated

text: and

text: measured.

text: The

text: scattering

text: parameters

text: are

text: wavelength

text: optics.”

text: Nature,

text: vol.

text: no.

text: pp.

text: shown

text: in

text: Fig.

text: Note

text: that

text: the

text: cutoff

text: frequency

text: (in

text: analogy

text: J.

text: B.

text: Pendry,

text: L.

text: Mart´ın-Moreno,

text: and

text: F.

text: J.

text: Garcia-Vidal,

text: “Mimicking

text: surface

text: plasmons

text: with

text: structured

text: surfaces.”

text: Science,

text: vol.

text: no.

text: to

text: the

text: plasma

text: frequency

text: in

text: optics)

text: and

text: the

text: operation

text: band

text: pp.

text: are

text: completely

text: reconﬁgurable

text: by

text: changing

text: the

text: capacitances.

text: A.

text: Kianinejad,

text: Z.

text: N.

text: Chen,

text: and

text: C.

text: W.

text: Qiu,

text: “Design

text: and

text: modeling

text: of

text: spoof

text: High-selectivity

text: feature

text: is

text: still

text: preserved

text: while

text: reconﬁguring

text: surface

text: plasmon

text: modes-based

text: microwave

text: slow-wave

text: transmission

text: line,”

text: the

text: operation

text: band.

text: The

text: static

text: capacitors

text: may

text: be

text: replaced

text: by

text: IEEE

text: Trans.

text: on

text: Microw.

text: Theory

text: Tech.,

text: vol.

text: no.

text: pp.

text: June

text: tunable

text: capacitors

text: to

text: make

text: it

text: truly

text: reconﬁgurable,

text: which

text: will

text: H.

text: F.

text: Ma,

text: X.

text: Shen,

text: Q.

text: Cheng,

text: W.

text: X.

text: Jiang,

text: and

text: T.

text: J.

text: Cui,

text: “Broadband

text: and

text: be

text: further

text: explored

text: in

text: our

text: future

text: works.

text: high-efﬁciency

text: conversion

text: from

text: guided

text: waves

text: to

text: spoof

text: surface

text: plasmon

text: polaritons,”

text: Laser

text: Photonics

text: Rev.,

text: vol.

text: no.

text: pp.

text: IV.

text: CONCLUSION

text: X.

text: Shen,

text: T.

text: J.

text: Cui,

text: D.

text: Martincano,

text: and

text: F.

text: J.

text: Garciavidal,

text: “Conformal

text: surface

text: plasmons

text: propagating

text: on

text: ultrathin

text: and

text: ﬂexible

text: ﬁlms,”

text: Proc.

text: Nat.

text: In

text: this

text: paper,

text: a

text: capacitor-loaded

text: SSP-TL

text: has

text: been

text: proposed.

text: Acad.

text: Sci.,

text: vol.

text: no.

text: pp.

text: Y.

text: Liang,

text: H.

text: Yu,

text: H.

text: C.

text: Zhang,

text: C.

text: Yang,

text: and

text: T.

text: J.

text: Cui,

text: “On-chip

text: sub-

text: It

text: featured

text: reconﬁgurability

text: in

text: the

text: dispersion

text: control.

text: It

text: also

text: terahertz

text: surface

text: plasmon

text: polariton

text: transmission

text: lines

text: in

text: CMOS.”

text: Sci.

text: exhibits

text: a

text: small

text: line

text: width

text: in

text: comparison

text: with

text: the

text: conventional

text: Rep.,

text: vol.

text: p.

text: SSP-TL.

text: The

text: principle

text: was

text: illustrated

text: and

text: several

text: experimental

text: X.

text: Gao,

text: L.

text: Zhou,

text: Z.

text: Liao,

text: H.

text: F.

text: Ma,

text: and

text: T.

text: J.

text: Cui,

text: “An

text: ultra-wideband

text: surface

text: plasmonic

text: ﬁlter

text: in

text: microwave

text: frequency,”

text: Appl.

text: Phys.

text: Lett.,

text: vol.

text: examples

text: were

text: provided.

text: The

text: results

text: completely

text: validated

text: the

text: no.

text: p.

text: proposed

text: SSP-TLs.

text: A.

text: Kianinejad,

text: Z.

text: N.

text: Chen,

text: L.

text: Zhang,

text: and

text: W.

text: Liu,

text: “Spoof

text: plasmon-

text: b

text: c

text: )mm(

text: x

text: dleiF

text: dezilamroN

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