On Instrumental Variables Estimation of Causal Odds Ratios

0.33 MB·English

by  Stijn Vansteelandt

StatisticalScience
2011,Vol.26,No.3,403–422
DOI:10.1214/11-STS360
(cid:13)c InstituteofMathematicalStatistics,2011
On Instrumental Variables Estimation of
Causal Odds Ratios
Stijn Vansteelandt, Jack Bowden, Manoochehr Babanezhad and Els Goetghebeur
2
1 Abstract. Inference for causal effects can benefit from the availability
0 of aninstrumentalvariable(IV)which,bydefinition,isassociated with
2 the given exposure, but not with the outcome of interest other than

n through a causal exposure effect. Estimation methods for instrumental
a
variables are now well established for continuous outcomes, but much
J
less so for dichotomous outcomes. In this article we review IV estima-
2
1 tion of so-called conditional causal odds ratios which express the effect
of an arbitrary exposure on a dichotomous outcome conditional on the

]
E exposurelevel, instrumental variable and measured covariates. In addi-
M tion, we propose IV estimators of so-called marginal causal odds ratios
which express the effect of an arbitrary exposure on a dichotomous
.
t outcome at the population level, and are therefore of greater public
a
t health relevance. We explore interconnections between the different es-
s
[ timators and supportthe results with extensive simulation studies and

three applications.
1
v Key words and phrases: Causaleffect, causal oddsratio, instrumental
7
variable, marginal effect, Mendelian randomization, logistic structural
8
4 mean model.
2
.
1
0 1. INTRODUCTION one has available all prognostic factors of the expo-
2 surethatarealso associated withtheoutcomeother
Most causal analyses of observational data rely
1
than via apossibleexposureeffectonoutcome.Con-
: heavily on the untestable assumption of no unmea-
v cerns about the validity of this assumption plague
sured confounders. According to this assumption,
i
X observational data analyses and increase the uncer-
tainty surrounding many study results (Greenland,
r Stijn Vansteelandt is Associate Professor, Department
a 2005). This is especially true in settings where the
of Applied Mathematics and Computer Science, Ghent
data analysis is based on registry data or focuses
University, B-9000 Gent, Belgium e-mail:
on research questions different from those conceived
[email protected]. Jack Bowden is Research
at the time of data collection. Substantial progress
Fellow, MRC Biostatistics Unit, Robinson Way,
Cambridge, United Kingdom e-mail: can sometimes be made in settings where measure-
[email protected]. Manoochehr ments are available on a so-called instrumental vari-
Babanezhad is Assistant Professor, Department of able(IV).Thisisaprognosticfactoroftheexposure,
Statistics, Golestan University, Golestan, Gorgan, Iran which is not associated with the outcome, except
e-mail: [email protected]. Els Goetghebeur is via a possible exposure effect on outcome (Angrist,
Professor, Department of Applied Mathematics and 1990; McClellan and Newhouse, 1994; Angrist, Im-
Computer Science, Ghent University, B-9000 Gent, bens and Rubin, 1996; Herna´n and Robins, 2006).
Belgium e-mail: [email protected].
An instrumental variable Z for the effect of expo-
sure X on outcome Y thus satisfies the following
This is an electronic reprint of the original article
published by the Institute of Mathematical Statistics in properties: (a) Z is associated with X; (b) Z affects
Statistical Science, 2011, Vol. 26, No. 3, 403–422. This the outcome Y only through X (i.e., often referred
reprint differs from the original in pagination and to as the exclusion restriction); (c) the association
typographic detail. between Z and Y is unconfounded (i.e., often re-
1
2 VANSTEELANDT,BOWDEN, BABANEZHADANDGOETGHEBEUR
ferredtoastherandomization assumption)(Hern´an Estimation methods for IVs are now well estab-
andRobins,2006).Forinstance,inthedataanalysis lishedfor continuous outcomes. Thecase of dichoto-
section, we will estimate the effect of Cox-2 treat- mous outcomes has received more limited attention.
ment(versusnonselectiveNSAIDs)ongastrointesti- It turns out to be much harder because of the need
nal bleeding, thereby allowing for the possibility of foradditionalmodelingandbecauseofdifficultiesto
unmeasuredvariablesU confoundingtheassociation specify congenial model parameterizations (see Sec-
between X and Y, by choosing the physician’s pre- tions 2.2 and 3). This paper therefore combines dif-
scribing preference for Cox-2 (versus nonselective ferent, scattered developments in the biostatistical,
NSAIDs) as an instrumental variable (Brookhart epidemiological andeconometricliteratureandaims
and Schneeweiss, 2007). Because this is associated toimprovetheclarityandcomparabilityofthesede-
with Cox-2 treatment [i.e., (a)], it would qualify as velopmentsbycastingthemwithinacommoncausal
an IV if it were reasonable that the physician’s pre- language based on counterfactuals.
scribing preference can only affect a patient’s gas- Traditional econometric approaches have their
trointestinal bleeding through his/her prescription rootsinstructuralequationstheoryandhavethereby
[i.e., (b)] and is not otherwise associated with that largely focusedontheestimation ofconditional cau-
patient’sgastrointestinalbleeding[i.e.,(c)].Assump- sal effects, where rather than employing counterfac-
tion (b) could fail, however, if preferential prescrip- tuals to define causal effects, conditioning is made
tion of Cox-2 were correlated with other treatment on all common causes, U, of exposure X and out-
preferences that have their own impact on gastroin- come Y (see Blundell and Powell, 2003, for a re-
testinal bleeding; the latter assumption could fail if view). By this conditioning, one can assign a causal
patients with high risk of bleeding are more often interpretation to association measures such as
seen with physicians who prefer Cox-2 (Hern´an and odds(Y =1X =x+1,C,U)
| .
Robins, 2006). In this article, we will more gener-
odds(Y =1X =x,C,U)
ally assumethattheinstrumentalvariables assump- |
Thiscan beseenbynotingthatthisoddsratiomea-
tions (a), (b)and(c) hold conditional ona (possibly
surecan—undera consistency assumption that Y =
empty) set of measured covariates C.
Y(x) if X =x—equivalently be written as (Pearl,
IVs have a long tradition in econometrics and are
1995)
becoming increasingly popular in biostatistics and
epidemiology. This is partly because the plausibil- odds Y(x+1)=1C,U
(1) { | },
ity of a measured variable as an IV can sometimes odds Y(x)=1C,U
{ | }
be partially justified on the basis of the study de-
where Y(x) denotes the (possibly) counterfactual
sign or biological theory. For instance, in random-
outcome following an intervention setting X at the
ized encouragement designs whereby, say, pregnant
exposurelevel x andwherefor any V,W, odds(W =
women who smoke are randomly assigned to inten-
1V) P(W =1V)/P(W =0V).Effectmeasure(1)
sified encouragement to quit smoking or not, ran- | ≡ | |
thus compares the odds of “success” if the exposu-
domization could qualify as an IV for assessing the
reX wereuniformlysettox+1versusxwithinstra-
effects of smokingon low birth weight (Permutt and
taofC andU.Because U isunmeasured,thesestra-
Hebel, 1989), since it guarantees the validity of IV
ta are not identified, which makes (1) less appealing
assumption (c). The growing success of IV meth-
as an effect measure and of limited use for policy
ods in biostatistics and epidemiology can, however,
making. Its interpretation is especially hindered in
be mainly attributed to applications in genetic epi-
view of noncollapsibility of the odds ratio (Green-
demiology (Smith and Ebrahim, 2004). Here, the
land, Robins and Pearl, 1999), following which the
random assortment of genes transferred from par-
magnitude of conditional odds ratios changes with
ents to offspring resembles the use of randomization
the conditioning sets, even in the absence of con-
in experiments and is therefore often referred to as
founding or effect modification. Similar limitations
“Mendelian randomization” (Katan, 1986). Build-
are inherent to the so-called treatment effect on the
ing on this idea, genetic variants may sometimes
treated at the IV level z of exposure x (Tan, 2010),
qualify as an IV for estimating the relationship be-
odds Y(x)=1X(z)=x
tween a genetically affected exposure and a disease (2) { | },
odds Y(0)=1X(z)=x
outcome, although violations of the necessary con-
{ | }
ditions may occur (see Didelez and Sheehan, 2007, andtoso-calledlocalorprincipalstratificationcausal
and Lawlor et al., 2008, for rigorous discussions). odds ratios (Hirano et al., 2000; Frangakis and Ru-
IV ESTIMATION OF CAUSALODDSRATIOS 3
bin2002;Abadie,2003;ClarkeandWindmeyer,2009; fection within those who acquired it (Vansteelandt
seeBowden etal.,2010,forareview).Foradichoto- et al., 2009).
mous instrumental variable Z and dichotomous ex- While the comparison in (4) could alternatively
posure X taking values 0 and 1, the latter measure be expressed as a risk difference or relative risk, our
the association between instrumental variable and focus throughout will be limited to odds ratios be-
outcome within the nonidentifiable principal stra- cause models for other association measures do not
tum of subjects for whom an increase in the instru- guarantee probabilities within the unitinterval, and
mental variable induces an increase in the exposure; mightnotbeapplicableundercase–controlsampling
that is, (Bowden and Vansteelandt, 2011). We refer the in-
terestedreadertoRobins(1994)andMullahy(1997)
odds Y(1)=1X(1)>X(0),C
(3) { | }. for inference on the conditional relative risk
odds Y(0)=1X(1)>X(0),C
{ | } P(Y =1X,Z,C)
Inference for principal stratification causal odds ra- (5) | ,
P Y(0)=1X,Z,C
tios is also more rigid in the sense of having no { | }
flexible extensions to more general settings involv- andto van der Laan, HubbardandJewell (2007)for
ingcontinuousinstrumentsandexposures.Whiledi- inference on the so-called switch relative risk, which
is defined as (5) for subjects with values (X,Z,C)
chotomization of the instrument and/or exposure is
often employed in view of this, it not only implies for which P(Y =1X,Z,C) P Y(0)=1X,Z,C
| ≤ { | }
and as
a loss of information, but may also induce a viola-
tion of the exclusion restriction and may make the P(Y =0X,Z,C)
| ,
relevance of the principal stratum “X(1) >X(0)” P Y(0)=0X,Z,C
{ | }
become dubious (see Pearl, 2011, for further discus-
for all remaining subjects. The latter causal effect
sion of these issues).
parameter is more difficult to interpret, but has the
In view of the aforementioned limitations, our at-
advantage that models for the switch relative risk,
tention in this article will focus on causal effects
unlikemodels for(5), guarantee probabilities within
which are defined within identifiable subsets of the
the unit interval.
population. Special attention will be given to the
For policy making, the interest lies more usually
conditionalcausaloddsratio(Robins,2000;Vanstee-
in population-averaged or marginal effect measures
landtandGoetghebeur,2003;RobinsandRotnitzky,
(Greenland, 1987; Stock, 1988) such as
2004), which we define as
odds Y(x+1)=1
odds(Y =1X,Z,C) (6) { },
(4) | . odds Y(x)=1
odds Y(0)=1X,Z,C { }
{ | } where x is a user-specified reference level, or
Itexpressestheeffect ofsettingtheexposuretozero
within subgroups defined by the observed exposure odds Y(X +1)=1
{ } or
level X, instrumental variables Z and covariates C. odds Y(X)=1
(7) { }
In the special case where X is a dichotomous treat-
odds Y(1.1 X)=1
mentvariable,takingthevalue1fortreatmentand0 { × }.
odds Y(X)=1
for no treatment, (4) evaluated at X =1, that is, { }
Here, (6) evaluates the effect of changing the expo-
odds Y(1)=1X =1,Z,C
{ | } sure from level x to x+1 uniformly in the popula-
odds Y(0)=1X =1,Z,C
{ | } tion. It thus reflects the effect that would have been
is sometimes referred to as the treatment effect in estimated had an ideal randomized controlled trial
the treated who are observed to have IV level Z (i.e., with 100% compliance) in fact been possible,
(Hern´anandRobins,2006;Robins,VanderWeeleand randomizing subjects over exposure level x versus
Richardson,2006;Didelez,MengandSheehan,2010; x+1.Incontrast,theeffectmeasuresin(7)allowfor
Tan,2010). Conditional causal oddsratios would be natural variation in the exposure between subjects
of special interest if the goal of the study were to by expressing the effect of an absolute or relative
examine the impact of setting the exposure to zero increase in the observed exposure. This may ulti-
forthosewithagivenexposurelevelX,forexample, mately be of most interest in many observational
to examine the impact of preventing nosocomial in- studies, considering that many public health inter-
4 VANSTEELANDT,BOWDEN, BABANEZHADANDGOETGHEBEUR
ventions would target a change in exposure level 2006) which is commonly adopted in the principal
(e.g.,diet,BMI,physicalexercise,...),startingfrom stratification approach. In spite of the randomiza-
some natural, subject-specific exposure level X. tion assumption [cf. IV assumption (c)], it may be
We review estimation of the conditional causal violated because subjects with exposure level X are
odds ratio (4) in Section 2. By casting different de- not exchangeable over levels of the IV, so that they
velopments withinthesamecausalframeworkbased might in particular experience different effects. The
on counterfactuals, new insights into their intercon- additional assumption of a linear exposure effect is
nectionswillbedeveloped.Weproposenovelestima- only relevant for exposures that take on more than
tors of the marginal causal odds ratios given in (6) twolevels.Itmustbecautiouslyinterpretedbecause
and(7)inSection 3,aswellasforthecorresponding the conditional causal odds ratio (4) expresses ef-
effect measures expressed as risk differences or rela- fects for differently exposed subgroups which may
tive risks. Extensive simulation studies are reported not be exchangeable. Both these assumptions are
inSection4andanevaluation on3datasetsisgiven critical because they are empirically unverifiable.
in Section 5. VansteelandtandGoetghebeur(2005)assessthesen-
sitivity of the conditional causal odds ratio estima-
2. IV ESTIMATION OF THE CONDITIONAL tor to violation of the linearity assumption andnote
CAUSAL ODDS RATIO that,underviolationofthelinearityassumption,the
estimator can still yield a meaningful first order ap-
Identification of the conditional causal odds ra- proximation. In the remainder of this work, we will
tio (4) is studied in detail in Robins and Rotnitzky assume that model (8) is correctly specified.
(2004) and Vansteelandt and Goetghebeur (2005),
2.1 Approximate Estimation
who find that—as for other IV estimators (Hernan
and Robins, 2006)—parametric restrictions are re- ApproximateIVestimatorsoftheconditionalcau-
quiredinadditiontothestandardinstrumentalvari- sal odds ratio can be obtained by averaging over
ablesassumptions.Inparticular,nonlinearexposure the observed exposure values in model (8) using the
effects and modification of the exposure effect by following approximations:
the instrumental variable are not nonparametrically
E logitE(Y X,Z,C)Z,C
identified. We will therefore consider estimation of { | | }
(10)
the conditional causal odds ratio under so-called lo- logitE(Y Z,C),
gistic structural mean models (Robins, 2000; Van- ≈ |
E[logitE Y(0)X,Z,C Z,C]
steelandt and Goetghebeur, 2003; Robins and Rot-
{ | }|
(11)
nitzky, 2004), which impose parametric restrictions
logitE Y(0)Z,C .
on the conditional causal odds ratio (4). In partic- ≈ { | }
ular, these models postulate that the exposure ef- This together with the logistic structural mean mo-
fect is linear in the exposure on the conditional log del (8) implies
odds ratio scale, and independent of the instrumen-
logitE(Y Z,C)
tal variable, in the sense that |
(12) logitE Y(0)Z,C +m(C;ψ∗)E(X Z,C)
odds(Y =1X,Z,C) ≈ { | } |
(8) | =exp m(C;ψ∗)X , =logitE Y(0)C +m(C;ψ∗)E(X Z,C),
odds Y(0)=1X,Z,C { } { | } |
{ | }
upon noting that the combined IV assumptions (b)
wherem(C;ψ) is aknownfunction (e.g., ψ +ψ C),
0 1
and (c), conditional on C, imply Y(x) Z C for
smooth(i.e., withcontinuous first-orderderivatives) ⊥⊥ |
all x. It follows that approximate IV estimators of
in ψ, and ψ∗ is an unknown finite dimensional pa-
the conditional causal odds ratio can be obtained
rameter. In the absence of covariates, this gives rise
via the following two-stage approach:
to a relatively simple model of the form
1. Estimate the expected exposure in function of
odds(Y =1X,Z)
(9) | =exp(ψ∗X). the IV and covariates by fitting an appropriate
odds Y(0)=1X,Z
{ | } regression model. Let the predicted exposure be
Theassumptionthattheexposureeffectisnotmod- Xˆ Eˆ(X Z,C).
≡ |
ifiedbytheIVsubstitutesthemonotonicityassump- 2. RegresstheoutcomeoncovariatesC andonm(C;
tion[thatX(z) X(z′)ifz z′](HernanandRobins, ψ)Xˆ through standard logistic regression to ob-
≥ ≥
IV ESTIMATION OF CAUSALODDSRATIOS 5
tain an estimate of ψ∗. In the absence of co- causal effect because Y Z at the null hypothe-
⊥⊥
variates, this involves fitting a logistic regression sis so that the usual maximum likelihood estimator
model of the form of ψ in model (13) will then converge to 0 in proba-
(13) logitE(Y Z)=ω+ψXˆ. bility. Further, note that the standard IV estimator
| requires correct specification of the first stage re-
When,furthermore,theIVisdichotomous,itfol-
gression model for the expected exposure (Didelez,
lows from (12) that
Meng and Sheehan, 2010; Rassen et al., 2009; Hen-
odds(Y =1Z =1) neman, van der Laan and Hubbard, 2002). In spite
OR |
Y|Z ≡ odds(Y =1Z =0) ofitsapproximatenature,thestandardIVestimator
(14) | continues to bemuch usedin Mendelian randomiza-
exp(ψ∗)∆X|Z, tion studies because of its simplicity, because it can
≈
where ∆ E(X Z = 1) E(X Z = 0), so beusedinmeta-analysesofsummarystatistics,even
X|Z
that ψ∗ can b≡e estim|ated as l−ogOˆR | /∆ˆ . when information on ORY|Z and ∆X|Z is obtained
Y|Z X|Z
from different studies (Minelli et al., 2004; Smith
The estimator obtained using the above two-stage et al., 2005; Bowden et al., 2006), and because the
approach is referred to as the standard IV estima- underlying principle extends to case–control studies
torinPalmeretal.(2008),aWald-typeestimator in when the first stage regression is evaluated on the
Didelez, Meng and Sheehan (2010) and the 2-stage controls and the disease prevalence is low (Smith
logistic approach in Rassen et al. (2009). It is com- et al., 2005; Bowden and Vansteelandt, 2011). For
monly employed in the analysis of Mendelian ran- relative risk estimators, the resulting bias due to
domization studies (Thompson et al., 2003; Palmer basing the first stage regression on controls rather
et al., 2008), where it is typically viewed as an ap- than a random population sample amounts to the
proximate estimator of the conditional causal odds difference between the log relative risk and the log
ratio (1). Our alternative development shows that odds ratio between Y and Z, inflated by the recip-
it can also be viewed as an approximate estimator rocal of the exposure distortion ∆ (Bowden and
X|Z
of the conditional causal odds ratio (4). To gain in- Vansteelandt, 2011).
sight into the adequacy of the approximations (10) Thebias ofthestandardIVestimator can someti-
and(11),supposeforsimplicitythattherearenoco- mes be attenuated by including the first-stage resi-
variates,thattheexposurehasanormaldistribution dual R X Xˆ as an additional regressor to Xˆ in
with constant variance σx2 conditional on Z, that model(≡13).T−hisisknownasthecontrolfunctionsap-
logitE(Y X,Z) =β +β X +β Z and that m(C;
0 x z proach in econometrics (Smith and Blundell, 1986;
|
ψ) = ψ. Then it is easily shown, using results in
Rivers and Vuong, 1988) and has also been conside-
Zeger and Liang (1988), that
red in the biostatistical literature on noncompliance
logitE(Y Z) β β2σ2 +β β2σ2 E(X Z) adjustment (Nagelkerke et al., 2000) and Mendelian
| ≈ 0{ x x} x{ x x} |
randomization (Palmer etal., 2008).Acontrol func-
+β β2σ2 Z,
z{ x x} tion refers to a random variable conditioning on
logitE Y(0)Z β (β ψ∗)2σ2 which renders the exposure independent of the un-
{ | }≈ 0{ x− x}
measured variables that confound the association
+(β ψ∗) (β ψ∗)2σ2 E(X Z)
x− { x− x} | between exposure and outcome. Intuitively, the re-
+β (β ψ∗)2σ2 Z, gression residual R may apply as a control function
z{ x− x}
because it captures (part of) those confounders. In
where for any parameter β and variance compo-
nentσ2,wedefineβ σ2 =β(c2σ2+1)−1/2 with c= particular, let us summarize (without loss of gen-
16√3/15π. It can re{lati}vely easily be deduced from erality) all confounders of the exposure effect into
a scalar measurement U. Assume that the contri-
these expressions and the fact that E Y(0)Z =
{ | } butions of the instrument Z and confounder U are
E Y(0) that
{ } additive in the sense that X =h(Z)+U for some
ψ∗
logitE(Y Z) β′ + E(X Z), function h. Suppose for simplicity that there are no
| ≈ 0 c2β2σ2+1 | covariates and that the conditional mean E(X Z) is
x x
|
for some β0′,suggesting inpcreasingbias withincreas- known so that Xˆ =h(Z) (here we use that U ⊥⊥Z,
ing association between X and Y (given Z) and as implied by the IV assumptions). Then R=U so
with increasing residual variance in X (given Z). that a (correctly specified) logistic regression of Y
This is true except at the null hypothesis of no on X and R (or, equivalently, Xˆ and R) will yield
6 VANSTEELANDT,BOWDEN, BABANEZHADANDGOETGHEBEUR
aconsistentestimator oftheconditionalcausalodds (Imbens and Newey, 2009). In the next section we
ratio (1), which is here identical to (4) because U is review direct approaches to the estimation of the
completely determinedby X andZ.Moregenerally, conditional causal odds ratio (4) which do not rely
following the lines of Smith and Blundell (1986), on assumptions about the exposure distribution.
assume that X =h(Z)+V, U =β˜∗V +ε, where ε
1 2.2 Consistent Estimation
follows a standard logistic distribution, and that
Y(x)=1ifandonlyifβ˜∗+ψ∗x+U >0forsomeβ˜∗, Rememberthat,althoughY maywelldependonZ
0 0
β˜∗. Then it also follows that Y =1 if and only if (in the presence of an exposure effect), the IV as-
1
ε>−β˜0∗−ψ∗X−β˜1∗V, from which saunmdpGtoioentsghiembpeulyr (t2h0a0t3)Ym(0a)k⊥e⊥uZse|Cof.tVhiasntsoteoelbatnadint
logitE(Y X,V)=β˜∗+ψ∗X+β˜∗V. a consistent estimator of ψ∗ in model (8), which is
| 0 1
chosen to make this independence happen. Because
Upon substituting V with the estimated regression
residual R, one obtains an estimator exp(ψˆ) which this is not possible without making additional para-
metricmodelingassumptions(RobinsandRotnitzky,
consistently estimates the conditional causal odds
2004), they model the expected observed outcome,
ratio (1). In the Appendix we demonstrate that this
conditional on the exposure and IV, for example,
isalsoaconsistentestimatoroftheconditionalcausal
odds ratio (4) when the exposure is normally dis- logitP(Y =1X,Z,C)
tributed with constant variance, conditional on the (15) |
=β∗+β∗X +β∗Z+β∗XZ+β∗C,
instrument, butnot necessarily otherwise. Standard 0 1 2 3 4
error calculation for the standard and adjusted IV where β∗,β∗,β∗,β∗ and β∗ are unknown scalar pa-
0 1 2 3 4
estimators is also detailed in the Appendix. rameters. More generally, one may postulate that
Over recent years, semiparametric analogs to the (16) logitE(Y X,Z,C)=m(X,Z,C;β∗),
adjusted IV approach have been developed in the |
where m(X,Z,C;β) is a known function, smooth
econometrics literature to alleviate concerns about
in β, and β∗ is an unknown finite-dimensional pa-
model misspecification. Blundell and Powell (2004)
rameter.Anestimator βˆofβ∗ canbeobtainedusing
andRothe (2009), for instance, avoid parametric re-
standard methods (e.g., using maximum likelihood
strictionsontheconditionalexpectationsE(X Z,C)
| estimation). Combining the causal model (8) with
and E(Y X,Z,C) (and, in particular, on the dis-
| the so-called association model (16) yields a predic-
tribution of ε) by using kernel regression estima-
tion for the counterfactual outcome Y(0) for each
tors and semiparametric maximum likelihood esti-
subject which, for given ψ, equals
mation, respectively. Imbens and Newey (2009) al-
low for the contributions of the instrument Z and H(ψ,βˆ)=expit m(X,Z,C;βˆ) m(C;ψ)X ,
{ − }
confounder U on the exposure to be nonadditive by
where expit(a) exp(a)/ 1 + exp(a) . Because
extending the previous works to nonseparable ex- ≡ { }
E Y(0)Z,C) = E Y(0)C under the IV assump-
posure models of the form X =h(Z,U,C) for some tio{ns, t|he value o{f ψ∗ |can} now be chosen as the
function h. They show that the association between
value ψ which makes this mean independence hap-
exposureandoutcomeisunconfoundeduponadjust- pen, once Y(0) is replaced by H(ψ,βˆ). When there
ing for R =F (X Z,C) as a control function,
X|Z,C are no covariates and the instrument Z is dichoto-
|
whereF istheconditionalcumulativedistribu-
X|Z,C mous, taking the values 0 and 1, one thus chooses ψ
tion function of X, given Z and C. To avoid para-
such that
metric restrictions on the conditional expectations
H (ψ,βˆ)Z H (ψ,βˆ)(1 Z )
FX|Z,C(X Z,C) and E(Y X,Z,C), they base infer- (17) i i i = i i − i .
ence on lo|cal linear regres|sion estimators. P iZi P i(1−Zi)
Alimitationofallthesesemiparametricapproaches When also the exposure is dichotomous, then mo-
P P
is that, by avoiding assumptions on the distribution del (15) is guaranteed to hold and a closed-form es-
of ε, the causal parameter ψ∗ becomes difficult to timator is obtained, as given in the Appendix. In
interpret so that it may be exclusively of interest most cases, the solution to (17) gives a unique esti-
for the calculation of marginal causal odds ratios mator of the causal odds ratio, although multiple or
(see Section 3). A further limitation is that all fore- no solutions are sometimes obtained when precision
going approaches require the exposure to be con- is limited due to small sample size or the outcome
tinuously distributed (Rothe, 2009); some addition- mean beingclose to0or 1.Thisis illustrated in Fig-
ally require the IV to be continuously distributed ure 1, which displays the left- and right-hand side
IV ESTIMATION OF CAUSALODDSRATIOS 7
Fig. 1. Plot of the left- (solid) and right-hand side (dotted) of expression (17) as a function of ψ. Top: simulated data set
[Right: with β∗=0 in model (15)]; Bottom: data set analyzed in Section 5.1.
4
of (17) in function of ψ for 3 settings. The top 2 For general instruments, a consistent point esti-
panels are based on the same simulated data set. mator of ψ∗ can be found by solving unbiased esti-
They show that 2 or no solutions can be obtained mating equation
for the same data set, depending on whether the n
association model (16) includes an interaction be- 0= [d(Z ,C ) E d(Z ,C )C ]
i i i i i
− { | }
tween exposure and instrument (left panel) or not
i=1
(18) X
(right panel). The bottom panel corresponds to the
[H (ψ,βˆ) E H (ψ,βˆ)C ]
data analysis of Section 5.1, where a single solution i i i
· − { | }
was obtained. Our experience indicates that, when for ψ, where d(Z ,C ) is an arbitrary function of Z
i i i
2 solutions are obtained, one of them corresponds andC ,forexample,d(Z ,C )=Z (seeBowdenand
i i i i
to an effect size which is so large that it would be Vansteelandt, 2011, for choices that yield a semi-
deemed unrealistic [and correspondingly yield unre- parametricefficientestimatorofψ∗).Thisthusleads
alistically small or large values of E Y(0) ]. When to the following 2-stage approach:
{ }
no solutions are obtained, this can sometimes be re-
1. First fit the association model (16), for instance,
solved by choosing a less parsimonious association
usingmaximumlikelihoodestimation,andobtain
model (as in Figure 1, top), but must be seen as an
an estimator βˆ of β∗;
indication that information is very limited. In the
2. Next, solve equation (18) to obtain an estima-
simulation experiments of Section 4, a single solu-
tor ψˆ of ψ∗.
tion was always obtained, but convergence of the
root-finding algorithm (nlm in R) was sometimes CorrespondingR-code is available from the first au-
very dependent on the choice of an adequate start- thor’s website (users.ugent.be/˜svsteela). This app-
ing value. roach is extended in Tan (2010) to enable estima-
8 VANSTEELANDT,BOWDEN, BABANEZHADANDGOETGHEBEUR
tion of the treatment effect on the treated at the IV tion
level z ofexposurex,asdefinedin(2),thusavoiding
logitE Y(0)X,Z,C
conditioning on C. { | }
In the Appendix we show that when the associa- (19) logitE Y(0)X =0,Z,C
− { | }
tion model includes an additive term in d(Z ,C )
i i − =q(X,Z,C;η∗),
E d(Z ,C )C and is fitted using maximum likeli-
i i i
{ | }
hoodestimationasinstandardgeneralizedlinearmo- where q(X,Z,C;η) is a known function satisfying
delsoftware,thenitssolutionisrobusttomisspecifi- q(0,Z,C;η)=0, smooth in η, and η∗ is an unknown
cation of the association model (16) when ψ∗=0. finite-dimensionalparameter.Thatq(X,Z,C;η∗)en-
This means that a consistent estimator of ψ∗ = 0 codes the degree of selection bias can be seen be-
is obtained, even when all models are misspecified. causeq(X,Z,C;η∗)=0forallX impliesthatE Y(0)
{ |
In the absence of covariates and with d(Zi,Ci)=Zi X,Z,C =E Y(0)Z,C and thus implies that the
and E{d(Zi,Ci)|Ci} = nj=1Zj/n, this is satisfied associat}ion b{etwee|n exp}osure and outcome [more
as soon as the association model includes an inter- precisely, Y(0)] is unconfounded (conditional on Z
P
cept and main effect in Zi [as in model (15)]. The and C). Relying on a parametric model for the con-
proposed approach then yields a valid (Wald and ditional exposure distribution, f(X Z,C)=f(X Z,
score) test of thecausal nullhypothesis that ψ∗=0, C;α∗) (fitted using maximum likeli|hood inferen|ce,
even when both models (8) and (16) are misspeci-
for instance), their approach involves the following
fied.Thisproperty,whichwerefertoasa“local” ro-
iterative procedure. First, for each fixed ψ (start-
bustness property (Vansteelandt and Goetghebeur,
ing from an initial value ψ ), maximum likelihood
0
2003), also guarantees that estimators of the causal
estimators ηˆ(ψ) and ωˆ(ψ) are computed for the pa-
odds ratio will have small bias under model mis- rameters η∗ and ω∗ indexingtheimpliedassociation
specification when the true exposure effect is close
model
to, but not equal to, zero.
A drawback of the parameterization by Vanstee- P(Y =1X,Z,C;ψ,η∗,ω∗)
|
landtandGoetghebeur(2003)isthattheassociation (20) =expit m(C;ψ)X +q(X,Z,C;η∗)
model may be uncongenial with the causal model. {
Specifically, given the observed data law f(X,Z C) +v(Z,C;η∗,ω∗) ,
| }
andthelimitingvalueβ∗ ofβˆ,theremaybenovalue where v(Z,C;η∗,ω∗) logitE Y(0)X =0,Z,C is
of the causal parameter ψ for which E H(ψ,β∗)Z, ≡ { | }
the solution to the integral equation
{ |
C =E H(ψ,β∗)C over the entire support of Z
an}d C.{In the A|pp}endix, we show that this may logitE Y(0)C =t(C;ω∗)
{ | }
happen when parametric restrictions are imposed
on the main effect of the instrumental variable in (21) = expit q(X =x,Z,C;η∗)
{
the association model (16), along with its interac- Z
+v(Z,C;η∗,ω∗)
tion with covariates C, but not when that main ef-
}
fect is left unrestricted. It follows that no congenial- f(X =x Z,C;α∗)dx,
ity problems arise in the common situation of a di- · |
wheret(C;ω)isaknownfunctionofC,smoothinω,
chotomous instrument and no covariates, so long as
a main effect of the IV is included in the associa- and where ω∗ is an unknown finite-dimensional pa-
tion model. This continues to betruefor categorical rameter.Forthegivenestimators ηˆ(ψ)andωˆ(ψ),an
IVs with morethan 2levels whendummyregressors estimator of ψ is then obtained by solving a linear
areused for theinstrumentin theassociation model combinationoftheestimatingequations(18)andes-
and there are no covariates. For general IVs, one timating equations for the parameters indexing the
may consider generalized additive association mod- associationmodel(20).Boththesestepsarethenite-
elswhichleave themaineffectoftheIVunrestricted rated until convergence of the estimator. In the Ap-
(apart from smoothness restrictions). pendixwesuggestasomewhatsimplerstrategywhich,
Robins and Rotnitzky (2004) developed an alter- nonetheless, also involves solvingintegral equations.
native approach for estimation of ψ∗ in model (8), Alternatively, one could focus on the switch relative
which guarantees a congenial parameterization by risk of van der Laan, Hubbard and Jewell (2007),
avoidingdirectspecificationofanassociationmodel. introduced in Section 1, to avoid the uncongeniality
They parameterize instead the selection-bias func- problems associated with the odds ratio.
IV ESTIMATION OF CAUSALODDSRATIOS 9
An advantage of the approach of Robins and Rot- and using the previous identity, we obtain
nitzky (2004) is that it guarantees that E Y(0)Z, θ∗+θ∗α∗+φ∗α∗Z+θ∗C
C =E Y(0)C for all Z and C, although{only u|n- E(Y Z,C)=Φ 0 1 0 1 3 .
} { | } | 1+θ∗2σ2∗
der correct specification of the law f(X Z,C). Un- (cid:18) 1 (cid:19)
|
der the approach of Vansteelandt and Goetghebeur This suggests regressing thpe outcome on the instru-
(2003), this is only guaranteed under congenial pa- mental variable and covariate using the probit re-
rameterizations as suggested previously, butregard- gression model
l(ecsosrroefctwlyh)estpheercifiaedm.oAdeflurfothretrhaedvlaawntafg(eXi|sZt,hCa)t iist (25) Φ−1{E(Y|Z,C)}=λ∗0+λ∗1Z+λ∗2C
might possiblygive somewhat moreefficient estima- to obtain an estimate λˆ1 for theunknownregression
tors by fully exploiting the a priori knowledge that slope λ∗, and then estimating φ∗ as
1
E Y(0)Z,C =E Y(0)C toestimateunknownpa-
ra{meter|s [i.e}., v(Z{,C)] a|nd} by additionally relying (26) φˆ= λˆ1 1+θˆ12σˆ2.
on a model for the exposure distribution. A draw- q αˆ
1
back is that the approach is computationally de-
Wewillrefertothisestimatorasthe“Probit-Normal
manding,especiallyforcontinuousIVsand/orinthe
SMM estimator” throughout. It is related to the in-
presence of covariates, as it involves solving integral
strumentalvariables probit(Lee,1981)andthegen-
equations for each (Z,C) and this within each iter-
eralized two-stage simultaneous probit (Amemiya,
ation of the algorithm. In addition, standard error
1978), both of which instead infer effect estimates
calculations are more complex. A further drawback
conditionalontheunmeasuredconfounderU.When
is that consistent estimation (away from the null)
the outcome mean lies between 10% and 90%, the
requires correct specification of the conditional ex-
above estimator yields an approximate estimate of
posure distribution f(X Z,C). thecausaloddsratiothroughtheidentityexp(ψ∗)
|
The estimation procedure for logistic structural exp(φ∗/0.6071) (McCullagh and Nelder, 1989). Fo≈r
mean models simplifies when the logit link is re-
dichotomous exposures, related estimators can be
placed with the probit link and the exposure is as-
obtained via probit structural equation models that
sumed to be normally distributed conditional on
replacethelinearregressionmodelforX inassump-
i
theinstrumentalvariableandcovariates (withmean
tion 1 above, with a probit regression model (see,
α∗+α∗Z+α∗C andconstantstandarddeviationσ∗,
0 1 2 e.g., Rassen et al., 2009).
where α∗,α∗,σ∗ are unknown). For instance, com-
0 1
bining the probit structural mean model 3. IV ESTIMATION OF THE MARGINAL
Φ−1 E(Y X,Z,C) Φ−1 E(Y(0)X,Z,C) CAUSAL ODDS RATIO
(22) { | }− { | }
We will now turnattention to theidentification of
=φ∗X,
marginalcausaleffects.Underlinearstructuralmod-
where Φ−1 is the probit link and φ∗ is unknown, els,thesecoincidewithconditionalcausaleffectsun-
with the probit association model dertypical assumptions(HernanandRobins,2006).
Consider, for instance, the extended linear struc-
(23) Φ−1 E(Y X,Z,C) =θ∗+θ∗X+θ∗Z+θ∗C,
{ | } 0 1 2 3 tural mean model which imposes the restriction
whereθ∗,θ∗,θ∗ areunknown,andaveraging overthe
0 1 2 E Y Y(x)X,C,Z =m(C,x;ψ∗)(X x)
exposure,conditionalonZ andC(seetheAppendix), { − | } −
gives for each feasible exposure level x, where m(C,x;ψ)
is a known function (e.g., ψ +ψ C+ψ x), smooth
0 1 2
E Y(0)Z,C
{ | } in ψ, and ψ∗ an unknown finite dimensional param-
=Φ (θ∗+θ∗Z eter. Then it follows from the restriction
{ 0 2
(24)
+(θ∗ φ∗)(α∗+α∗Z+α∗C)+θ∗C) E Y m(C,x;ψ∗)(X x)C,Z
1− 0 1 2 3 { − − | }
=E Y m(C,x;ψ∗)(X x)C
( 1+(θ∗ φ∗)2σ2∗)−1 . { − − | }
· 1− } for each x, that
q
Because this does not dependon Z undertheIV as-
E Y m(C,x;ψ∗)X C,Z
sumptions, it follows that θ∗=(φ∗ θ∗)α∗. Averag- { − | }
ing over the exposure in the2 associ−atio1n m1odel (23) =E Y m(C,x;ψ∗)X C
{ − | }
10 VANSTEELANDT,BOWDEN, BABANEZHADANDGOETGHEBEUR
for each x, and thus that m(C,x;ψ∗) does not de- n
pˆ =n−1 expit m(X ,Z ,C ;βˆ)
pendon x. This then implies that the marginal cau- X+1 i i i
{
sal effect equals Xi=1
+m(C ;ψˆ ) .
E Y(x∗) Y(x)C =m(C,0;ψ∗)(x∗ x). i Xi+1 }
{ − | } −
Standard error calculations are reported in the Ap-
Unfortunately, this result does not extend to logis-
pendix. Using the above expressions, also estima-
tic structural mean models, so that the conditional
tors of the marginal risk difference P Y(x+1) =
causaloddsratio correspondingtoasingle reference {
1 P Y(x)=1 or relative risk P Y(x+1)=1 /
exposure level (e.g., 0) does not uniquely map into }− { } { }
P Y(x)=1 can straightforwardly be obtained.
the marginal causal odds ratio. { }
A drawback of this strategy, which we discuss in
Let us therefore assume that in addition to the
the Appendix, is that even when model (27) is con-
association model (16), the extended logistic struc-
genial with the association model (16) for x=0 (or
tural mean model holds, which we define by the re-
some other reference level), it need not be a well-
striction
specified model for all x. We conjecture that when
odds(Y =1X,Z,C)
this would happen, this may be partially detectable
|
odds Y(x)=1X,Z,C in the sense of yielding estimating equations with
(27) { | }
no solution, as the uncongeniality is then due to the
=exp m(C;ψ∗)(X x) ,
{ x − } nonexistence of a value of ψ∗ for some x so that
x
for each feasible exposurelevel x,wherem(C;ψ ) is E Y(x)Z,C =E Y(x)C for all (Z,C). As with
x
{ | } { | }
a known function (e.g., ψ +ψ C), smooth in ψ , other causal models that are not guaranteed to be
x0 x1 x
and ψ∗ an unknown finite-dimensional parameter. congenial(e.g., Petersen etal.,2007;Tan,2010)and
x
Themarginalcausaloddsratio(6)can nowbeiden- as confirmed in simulation studies in the next sec-
tified upon noting that tion, we believe this is unlikely to induce an im-
portant bias. The concern for bias is further allevi-
P Y(x)=1
{ } ated by the aforementioned local robustness prop-
=E[expit m(X,Z,C;β∗) m(C;ψ∗)(X x) ] erty, which continues to hold for extended logistic
{ − x − }
structural mean models.
and the marginal causal odds ratio [(7), left] upon
The idea of using conditional causal effect esti-
noting that
mates as plug-in estimates in inference for marginal
P Y(X+1)=1 effects has been advocated in the biostatistical and
{ }
epidemiologicalliterature(see,e.g.,Greenland,1987;
=E[expit m(X,Z,C;β∗)+m(C;ψ∗ ) ].
{ X+1 } TenHave etal., 2003)andiscommonly employed in
Aconsistentestimatorof(6)isthusobtainedbyfirst the econometrics literature (see, e.g., Blundell and
obtaining consistent estimators of β∗,ψ∗ and ψ∗ , Powell, 2004; Imbens and Newey, 2009), where re-
x x+1
using the strategy of the previous section, and then latedproposalshavebeenmadestartingfromasemi-
calculating pˆ (1 pˆ )/ pˆ (1 pˆ ) , where for parametric control functions approach. Alternative
x+1 x x x+1
− { − }
given x approaches involve assumingthat all confounders of
the exposure effect can be captured into a scalar
n
pˆ =n−1 expit m(X ,Z ,C ;βˆ) variate U, which has an additive effect on the out-
x i i i
{ come(Amemiya,1974;Foster,1997;Johnstonetal.,
i=1
X
2008; Rassen et al., 2009) in the sense that
m(C ;ψˆ )(X x) .
i x i
− − } (28) E(Y X,C,U)=expit(β∗+ψ˜∗X+β∗C)+U,
A consistent estimator of [(7), left] is obtained by | 0 1
first obtaining consistent estimators of β∗ and ψ∗ whereβ∗,β∗,ψ˜∗areunknownandwhereE(U C)=0;
x+1 0 1 |
foreachobservedvalueX forxusingthestrategyof note that E(U X,C)=0 when there is confounding.
i | 6
the previous section, and then calculating pˆ (1 Because,foreach x,Y(x) X U,C,model(28)im-
X+1 − ⊥⊥ |
pˆ )/ pˆ (1 pˆ ) , where plies the marginal structural model
X X X+1
{ − }
n E Y(x)C =E[E Y(x)X =x,C,U C]
pˆ =n−1 Y , { | } { | }|
X i =expit(β∗+ψ˜∗x+β∗C)
i=1 0 1
X