Open Journal of Statistics, 2013, 3, 7-15
http://dx.doi.org/10.4236/ojs.2013.34A002 Published Online August 2013 (http://www.scirp.org/journal/ojs)
A Simple Method of Measuring Vaccine Effects on
Infectiousness and Contagion
Yasutaka Chiba
Division of Biostatistics, Clinical Research Center, Kinki University School of Medicine, Osaka, Japan
Received June 26, 2013; revised July 26, 2013; accepted August 3, 2013
Copyright © 2013 Yasutaka Chiba. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
The vaccination of one person may prevent another from becoming infected, either because the vaccine may prevent the
first person from acquiring the infection and thereby reduce the probability of transmission to the second, or because, if
the first person is infected, the vaccine may impair the ability of the infectious agent to initiate new infections. The for-
mer mechanism is referred as a contagion effect and the latter is referred as an infectiousness effect. By applying a prin-
cipal stratification approach, the conditional infectiousness effect has been defined, but the contagion effect is not de-
fined using this approach. Recently, new definitions of unconditional infectiousness and contagion effects were pro-
vided by applying a mediation analysis approach. In addition, a simple relationship between conditional and uncondi-
tional infectiousness effects was found under a number of assumptions. These two infectiousness effects can be as-
sessed by very simple estimation and sensitivity analysis methods under the assumptions. Nevertheless, such simple
methods to assess the contagion effect have not been discussed. In this paper, we review the methods of assessing infec-
tiousness effects, and apply them to the inference of the contagion effect. The methods provided here are illustrated
with hypothetical vaccine trial data.
Keywords: Indirect Effect; Interference; Mediation Analysis; Potential Outcome; Principal Stratification
1. Introduction
Evaluating the effect of vaccination on reducing infec-
tiousness has important public health consequences [1].
Even if a vaccine does not provide strong protection
against an infection, it could substantially reduce the total
number of cases if transmission from an infected vacci-
nated person is reduced compared to that from a non-
vaccinated person. This is because the vaccine status of
one person may affect whether another person becomes
infected. This phenomenon is referred as “interference”
in the statistical literature [2], or the “indirect effect” in
the infectious disease context [3]. In the presence of such
interference or indirect effects, a further distinction is
drawn, as mentioned below.
Considering households consisting of two persons, one
(person 1) is randomized to receive a vaccine or a control,
and the other (person 2) receives nothing. The vaccina-
tion of person 1 may prevent person 2 from becoming
infected via the following two mechanisms: 1) the vac-
cine may prevent person 1 from acquiring the infection
and thereby reduce the probability of transmission to
person 2, or 2) if person 1 is infected irrespective of the
vaccine, the vaccine may impair the ability of the infec-
tious agent to initiate new infections; i.e., make the agent
less infectious. The first of these mechanisms is referred
as the “contagion effect” [4] and the second is referred as
the “infectiousness effect” [5].
To give a formal definition of these two effects, the
principal stratification and mediation analysis approaches
have been adapted. Recently, by applying the principal
stratification approach, the conditional infectiousness
effect was defined [3,6], but unfortunately the contagion
effect was not. No one may be able to define it using this
approach. More recently, unconditional infectiousness
and contagion effects were defined by applying the me-
diation analysis approach [4]. Furthermore, a simple re-
lationship between conditional and unconditional infec-
tiousness effects was found under a number of assump-
tions [7]. These two infectiousness effects can be as-
sessed using very simple statistical methods under the
assumptions. Nevertheless, such methods to assess the
contagion effect have not been discussed. In this paper,
we review the methods used to assess infectiousness ef-
fects, and apply them to the inference of the contagion
C
opyright © 2013 SciRes. OJS

Y. CHIBA
8
effect.
This paper is organized as follows. Section 2 presents
the concepts and definitions used throughout this paper.
Section 3 reviews the relationship between the condi-
tional and unconditional infectiousness effects, and ex-
presses the contagion effect in terms of principal stratifi-
cation. In Section 4, we describe a simple method of es-
timating these effects upon their identification, and pro-
vide a simple sensitivity analysis method of assessing
how inferences would change under violations of the
identification assumption in Section 5. The methods pre-
sented in Sections 4 and 5 are illustrated using hypo-
thetical randomized vaccine trial data in Section 6. Sec-
tion 7 concludes with a discussion.
2. Concepts and Definitions
The notation and fundamental assumptions used through-
out this paper are presented in Section 2.1. Section 2.2
presents the crude estimator of the infectiousness effect,
and indicates the problem with this estimator. Section 2.3
formalizes the conditional infectiousness effect by ap-
plying the principal stratification approach, and in Sec-
tion 2.4 the unconditional infectiousness and contagion
effects are formalized by applying the mediation analysis
approach.
2.1. Notation and Assumptions
We consider a setting in which there are N households
indexed by in which each household con-
sists of two persons indexed by j = 1, 2. Let Aij denote the
vaccine status of person j in household i, where Aij = 1 if
the person received the vaccination and Aij = 0 if the
person did not. Let Yij denote the infection status of per-
son j in household i, where Yij = 1 if the person was in-
fected and Yij = 0 if the person was not. Finally, let
iii im
denote a set of baseline covari-
ates for household i. Because person 1 is randomized and
person 2 receives nothing, Xi affects Yi1 and Yi2 but does
not affect Ai1 and Ai2. The relationship among these vari-
ables is represented by a directed acyclic graph (DAG)
[8,9] (Figure 1), in which A2 is not displayed because the
vaccine status of person 2 (A2) is identical among all
households (i.e., person 2 is never vaccinated) (A2 = 0).
1,, ,i

12
,,,X X
N



XX
We assume that the vaccine status of the persons in
one household does not affect the outcomes of those in
other households; this is sometimes referred to as an as-
sumption of partial interference [3,10]. We let
denote the potential outcome for person j in
household i if the two persons in household i had, possi-
bly contrary to fact, a vaccine status of . This
assumption of partial interference might be plausible if
the various households are sufficiently geographically
separated or do not interact. We further require the con-
sistency assumption, which means that the value of Yij
that would have been observed if Aij had been set to what
in fact it was is equal to the value of Yij that was ob-
served; i.e.,

12
,
ij ii
Yaa

12
,
ii
aa


12
,
ijij ii
YYAA [11]. In addition, by ran-
dom assignment to person 1, it is assumed that


12
,
ij ii
Yaa is independent of A
i1. This independency
can also be assumed conditional on Xi. Because person 2
is always unvaccinated, we simplify the notation as




11
:,0a
ij iij
Ya Yi
Using this notation, on the vaccine efficacy scale, the
indirect effect is formalized by
.






22
1Pr1 1Pr01YY
ii


(i.e., the effect on person 2 of person 1 being vaccinat-
ed) [3]. In contrast, the effect on person 1 of person 1
being vaccinated is called the direct effect, and is for-
malized by



  
11
1Pr11Pr01
ii
YY

.
In the current setting, due to random assignment to per-
son 1 and the consistency assumption, these direct and
indirect effects are identified by the sample proportions

 
11 11
1Pr10
ii ii
YA YA1Pr1

 ,

 
21 21
1 1Pr10
ii ii
YA YA1Pr

 ,
respectively. The indirect effect is classified into infec-
tiousness and contagion effects due to the existence of
two infection mechanisms, as noted in Section 1.
2.2. Crude Estimator of the Infectiousness Effect
The crude estimator of the infectiousness effect might be
taken as [12]:



211
211
11, 1
10, 1
iii
iii
YAY
YAY
Pr
1Pr





. (2.1)
This is a comparison of the infection proportions for
person 2 in the subgroup in which person 1 was vacci-
nated and infected versus that in which person 1 was
unvaccinated and infected. This is an appealing intuitive
method of capturing the extent to which the vaccine may
render those infected less contagious, which may in turn
prevent the second person from being infected (i.e., in-
fectiousness effect).
However, the measure is subject to selection bias. Al-
though the vaccine status of person 1, Ai1, is randomized,
conditioning on a variable that occurs after treatment (i.e.,
the infection status of person 1) breaks the randomization.
This can be indicated using the DAG shown in Figure 1.
Even if an arrow from X to A1 does not exist due to ran-
domization, conditioning for Y1 would induce non-iden-
tifiability due to the induced structural relationship of a
possible double arrow between A1 and X, as shown in
Figure 2 [13].
As a result, the subgroup with person 1, who was vac-

Y. CHIBA 9
A
1
Y
1
Y
2
X
Figure 1. A directed acyclic graph representing the rela-
tionship among A1, Y1, Y2, and X in the current context.
Y
1
A
1
Y
2
X
Figure 2. A graph after conditioning on Y1.
cinated and infected, may be quite different from that in
which person 1 is unvaccinated and infected. For exam-
ple, those in the vaccinated group who become infected
may be a less healthy subpopulation than those in the
unvaccinated group who become infected. If the persons
who are infected are less healthy even though they have
been vaccinated, they may also more likely to be conta-
gious and to pass on the disease. We are then computing
infection proportions for person 2 for subpopulations that
are quite different with respect to person 1.
2.3. Conditional Infectiousness Effect
By applying the principal stratification approach, con-
sider the following contrast [14]:






  
211
211
Pr1 1101
:1 Pr0 110 1
iii
C
iii
YYY
IYYY

  . (2.2)
This compares the infection status of person 2 if per-
son 1 was vaccinated, , versus that if person 1 was
unvaccinated, 2, but only in the subgroup of
households in which person 1 would have been infected
irrespective of whether person 1 was vaccinated; i.e.,
ii . Such a subgroup is sometimes re-
ferred to as a principal stratum [15]. Because we are con-
sidering only the subgroup of households for whom per-
son 1 would have been infected irrespective of whether
person 1 was vaccinated, person 2 is exposed to the in-
fection of person 1, and thus any effect of the vaccine
ought to occur through a change in the infectiousness.
Therefore, the conditional infectiousness effect is defined
by Equation (2.2), because it is conditioned on
ii . Moreover, unlike with the crude
comparison in Equation (2.1), we are now comparing the
infection proportions of person 2 for the same subpopu-
lation in Equation (2.2). We are no longer considering a
healthier or unhealthier subgroup for person 1.

21 i
Y

0
i
Y
01
01

11
1YY

11
1YY
Unfortunately, we do not know which households fall
into the subpopulation in which person 1 would have
been infected irrespective of whether he or she was vac-
cinated. This is because we can observe only the out-
come of person 1, either with or without the vaccine but
not under both scenarios. Because we do not know which
households fall into this subpopulation, we cannot com-
pute the conditional infectiousness effect in a straight-
forward manner. However, in Section 4, we will show
that the effect can be estimated simply under a number of
assumptions.
2.4. Unconditional Infectiousness and Contagion
Effect
Suppose that, in addition to potentially intervening to
vaccinate person 1, we could, at least hypothetically, in-
tervene to infect or not infect person 1. Then,


2121
,,
iiii
Yaay
would denote the infection status of per-
son 2 if we would set the vaccine status of person 1 and
person 2 to ai1 and ai2 and the infection status of person 1
to yi1. The assumption that person 2 is always unvacci-
nated allows a simplified notation:




,0,Ya y
211
,:
iii
Yay 21 1ii i . This potential outcome is
used to define unconditional infectiousness and conta-
gion effects [4].
Consider the following contrast:





 

21
21
Pr1, 11
:1 Pr0, 11
ii
U
ii
YY
IYY

 . (2.3)
This compares the potential infection status of person
2 if person 1 had been vaccinated versus unvaccinated
and person 1 had the infection status that would occur if
vaccinated. If Equation (2.3) is non-zero, this will be
because even when person 1 is vaccinated and infected,
the vaccine itself affects whether person 2 is infected by
person 1. In some ways, it is analogous to what is called
an infectiousness effect. However, this measure defined
by Equation (2.3) differs from the conditional infec-
tiousness effect defined by Equation (2.2) in that it is not
conditional on person 1 actually being infected.
Consider now the other contrast:
 


 

21
21
Pr0,11
:1 Pr0, 01
ii
ii
YY
CYY

 . (2.4)
The term




21
0, 1
ii
YY considers what the potential
infection status of person 2 is if person 1 is unvaccinated,
but we set the infection status of person 1 to the level it
would have been if person 1 was vaccinated. Equation
(2.4) compares this potential outcome to




21
0, 0
ii
YY ,
which is the potential infection status of person 2 if per-
son 1 is not vaccinated, and we set the infection status of
person 1 to the level it would be if person 1 was unvac-
cinated. For Equation (2.4) to be nonzero,


11 i
Y and

Y. CHIBA
10

10 i
Y have to differ; i.e., vaccination of person 1 would
have to affect the infection status of person 1, and that
change in infection for person 1 would have to change
the infection status for person 2, even if person 1 had
remained unvaccinated. Essentially, Equation (2.4) is
non-zero if the vaccine prevents infection in person 1,
and that in turn prevents person 2 from being infected.
Thus, the contagion effect is defined by Equation (2.4).
These definitions of the unconditional infectiousness
and contagion effects have the feature that we can decom-
pose an indirect effect into unconditional infectiousness
and contagion effects; i.e., from Equations (2.3) and (2.4),
 
 
 

 






 

 


 

 
21212 1
2
22121 21
Pr 1,11Pr1,11Pr0,11
Pr1 1
11 1
Pr0 1Pr0, 01Pr0,1 1Pr0, 01
11 1.
iiiii i
i
iiiii ii
UUU
YYYYY Y
Y
YYYYY YY
ICICIC



 



  
(2.5)
come infected from outside the household. This decomposition is analogous to what are referred
to as “natural direct and indirect effects” [16,17] in the
mediation analysis.
Assumption 2.


11
1
ii
YY

0 for all i.
Assumption 1 implies that person 2 cannot be infected
unless person 1 is infected. Assumption 2 implies






11
Pr1 1,000
ii
YY

 ; i.e., there is no household
in which person 1 would be infected if vaccinated, but
uninfected if unvaccinated.
3. Relationship between Conditional and
Unconditional Infectiousness Effects
Under these two assumptions,



211
Pr, 11
iii
YaY

can be expressed as follows [7]:
Here, we require the following two assumptions:
Assumption 1. Only person 1, not person 2, can be-
 

   

  

      


 

 



 

11
211211 1111
00
11
2111 11
00
2111 11
21111 11
Pr,11Pr,111,0 Pr(1,0
Pr,11,0Pr1,0
Pr,111 1,01Pr1 1,01
Pr,1101 Pr
iiiiii iiii
st
iiii ii
st
iiii ii
iiiii ii
YaYYaYYsY tYsY t
YasYsYtYsY t
YaYY YY
YaYaY YY


 









   
21111 1
11
Pr1101Pr11,
iiiiii
YaYYY A


where the third equality is because
 

 
21 11

,0 110,00YaYY t 


Pr ii by Assump-
tion 1 and by Assumption
2, the fourth is by Assumption 2, and the last is by ran-
dom assignment to person 1 and the consistency assump-
tion.
i i


11
Pr1 1,000
ii
YY
Using this equation and Equations (2.2) and (2.3), it is
readily confirmed that IU = IC under Assumptions 1 and 2.
Furthermore, substituting this equation into Equation
(2.4) derives the following form for the contagion effect:




 




211 11
21
Pr01101 Pr11
1,
Pr1 0
iiiii
ii
YYYYA
CYA

 
4. Estimation
where the denominator is by
 

  
21
22
Pr0, 01
Pr0 1Pr1.0
ii
ii
YY
YYA


Section 4.1 presents the inverse-probability-weighting
(IPW) estimators for the infectiousness and contagion
effects. Section 4.2 shows that the analysis for the IPW
can be implemented easily.
1i

Therefore, to assess the unconditional infectiousness and
contagion effects, we can apply the statistical methods
developed for the conditional infectiousness effect. Such
methods are described in Sections 4 and 5. We note that
this form for the contagion effect is not its definition un-
der the principal stratification approach.
4.1. Estimators
Under Assumption 2,
 




21 11
Pr110 1
ii ii
Ya YY
an be expressed as follows: c

Y. CHIBA 11












 


 
21 1121 121 11
211 1
Pr1 101Pr111Pr111,1
Pr11,1.
ii iiii iii ii
iii i
Ya YYYa YYa YA
YaAY
  

Specifically,
 

211
211
Pr1 11,1
Pr11, 1
iii
iii
YAY
YAY


by the consistency assumption. Thus, under Assumptions
1 and 2, the respective infectiousness and contagion ef-
fects can be expressed as:


 
211
211
Pr11, 1
1Pr0 11,1
iii
CU
iii
YAY
II YAY

 
, (4.1)



 

211 11
21
Pr011,1Pr11
1 Pr1 0
iiiii
ii
YAY YA
CYA
 
  . (4.2)
To derive estimators of



211
Pr0 11,1
iii
YAY,
we require the following assumption:
Assumption 3. is independent of Ai1, condi-
tional on Yi1 and Xi.

21ii
Ya
This assumption states that all baseline covariates that
affect the vaccine status of both persons 1 and 2 are
measured, and implies that, in Figure 2, all paths from A1
to Y2 are blocked except for the direct path A1 → Y2.
Under Assumption 3,
 
211
Pr01 1,1
iii
YAY
can be expressed as follows [18]:
 













2 112 1111
21111
211 11
211
11
Pr 011,1Pr 011,1,Pr1,1
Pr010, 1,Pr1,1
Pr1 0,1,Pr1,1
Pr1,0,1,1
Pr 1,
iiiiii iiii
x
iiii iii
x
iii iiii
x
ix iiiiix
x
ii
YA YYA YXxXxA Y
YAYXxXxAY
YAYXx XxAY
pYAYXxp
AY
 


 







,
1
where

11
11,
ixi ii
pPrA YXx

. Following the
theory of Hirano et al. [19], once pix has been modeled
and calculated, the last equation can be estimated by:

11
1
11
10,1 1
N
ix
ii
iix
p
2 i
I
AY Y
N



p
,
, (4.3)
where N11 is the number of households with
and
 
11
,1, 1
ii
AY 

.
I
denotes the indicator func-
tion with for households with
and

0, 1
1

0,


11
1
ii
Y
IA
IA
 
11
,0,
ii
AY 11 1 0
ii
Y

 for the oth-
ers. The value of pix is often predicted from a model for
the regression of Ai1 on Xi (e.g., logistic regression mod-
el) in the subgroup of households with Yi1 = 1.
We note that we can derive the other types of estima-
tors; i.e., the model-based standardization estimator [20],
which uses a model for the regression of Yi2 on Xi rather
than the regression of Ai1 on Xi, and the doubly robust
estimator [21], which uses both regression models. See
Chiba and Taguri [18] for details.
4.2. Implementation
Equation (4.3) implies that we can implement the analy-
sis by limiting the analysis set to households with Yi1 = 1.
(4.3) has the same form as those for the average causal
effect of an exposure on the outcome with the exposed
group as the target population in the setting of observa-
tional studies [20,22], where the exposure and outcome
correspond to Ai1 and Yi2, respectively. Therefore, we can
easily calculate the IPW estimate and the confidence
interval (CI) using a marginal structural model (MSM)
[23,24]. The regression parameters in this model are es-
timated by a weighted regression model with the form
Furthermore, except conditioning on Yi1 = 1, Equation


2011
exp
ii
YA  , where the weights are wi = 1 for
= 1 and

households with Ai11
iixix
wp p for
households with Ai1 = 0, where theated
by the robust variance, which provides a conservative CI
that is guaranteed to cover the true at least 95% of the
time in large samples [24]. The SAS code is given else-
where [7,18].
In this MSM
variance is estim
, 1
ˆ

corresponds to an estimator of






211
211
r
log Pr011,,1
iii
iii
Y
YAY
log P011,1AY


and 0
ˆ

corresponds to an estimator of




1
211
log Pr011,
iii
YAY

.
Therefore, an estimator of the infectiousness effect is:

Y. CHIBA
12
1
ˆ
ˆˆ
1e
CU
II

, (4.4)
and that of the contagion effect is:


0
ˆ1
1
ˆ
Pr 1
ˆ
0
i
i
Y
A

1
1
i
A
2
1e ˆ
Pr 1
i
CY





The CI of
. (4.5)
ˆˆ
CU
I
I is evaluated by

1
1expCI ofˆ

,
by and similarl is evaluated
y the CI of C
ˆ



1
11
1 1pCI oi
ii
A
Y

In other words, the variances of these estimtes are
used to obtain the CIs, where the delta method is used to
yield the varia
Here, we provide a simple sensitivity analysis method to
change under violation of
We set the sensitivity parameter as [18,25]:

01
ˆˆ
log Pr
ˆ
log Pr10
1ex f
.
i
Y
A
 



a
nces of estimates on a log-scale.
5. Sensitivity Analysis
assess how inference would
Assumption 3. The sensitivity analysis formula is pro-
vided in Section 5.1, and a plausible range of the sensi-
tivity parameter is provided in Section 5.2.
5.1. Sensitivity Analysis Formulas


 
211
Pr0 11,
iii
YAY
21 1
Pr 0 10,1
ii i
YA
Y
 
1. (5.1)
The sensitivity parameter α is the ratio
outcome that would have been observed if
unvaccinate
between the
person 1 was
d in comparing two different populations: the
population in the numerator is that in which person 1 was
vaccinated (Ai1 = 1), and the population in the denomi-
nator is that in which person 1 was unvaccinated (Ai1 = 0),
where the infection status is equal in these two popula-
tions (Yi1 = 1). The interpretation of α is then simply the
ratio of the expected outcomes under non-vaccination for
these two populations.
Using Equation (5.1), Equations (4.1) and (4.2) can be
expressed as follows:


211
Pr11, 1
1
1iii
CU
YAY
II

211
Pr 10, 1
iii
YAY





, (5.2)


11
11
Pr1 1
1Pr1 0
ii
ii
YA
CYA



 
, (5.3)
respectively, where Equation (5.3) can be derived be-
cause, by Assumption 1,




21
1
21111
0
211 11
Pr1 0
Pr1 0,Pr0
Pr 10,1Pr10.
ii
iiiii
y
iii ii
YA
YAYyYyA
YAY YA



 
 

Using Equations (5.2) and (5.3), a sensitivity analysis
can be conducted as follows. The sensitivity parameter α
is set by the investigator according to what is considered
plausible. The parameter can be varied over a range of
plausible values to examine how conclusions vary ac-
cording to differences in parameter values. The results of
the sensitivity analysis can be displayed graphically,
where the horizontal axis represents the sensitivity pa-
rameter and the vertical axis represents the true infec-
tiousness and contagion effects.
Generally, to obtain the CIs, the variances are calcu-
lated from Equations (5.2) and (5.3) with a fixed value of
α. However, this calculation yields narrower CIs than
those calculated from Equations (4.4) and (4.5). To avoid
this problem, we use the variances calculated from Equa-
tions (4.4) and (4.5) to obtain the CIs of the infectious-
ness and contagion effects estimated from Equations
(5.2) and (5.3).
5.2. Range of the Sensitivity Parameter
In some situations, it may be troublesome for investiga-
tors to determine a range of values of α to examine.
Therefore, we present a range of values that α can take;
i.e., bounds for α.
Initially, we apply the large sample bounds [26-28] for




   
211
211
Pr0 11,1
Pr0 110.1
iii
iii
YAY
YYY



under Assumption 2. The large sample bounds are calcu-
lated from the expected number if households with








1,01, 1YY 
11ii were assigned to the unvacci-
nated group (Ai1 = 0), as follows.

 
,0,1AY 
11ii are those with either Households with








1, 01,1YY 
 
11ii or



1, 00,1YY 
11ii . There-
fore, under Assumption 2, of households with




,0,1AY 
11ii , the expected number of households
with








1, 01,1YY 
11ii assigned to the unvaccinated
group (Ai1 = 0), N*, is calculated as:
 

 
 

 




  






 
11
1
111 11111
11111
111 11
11
Pr0,1PPr10 Pr0 Pr11
*Pr101Pr10,0 1Pr01
Pr10 Pr0 Pr11Pr0Pr11 .
Pr1 0
iii iiiii
iii ii
iii ii
iii
ii
NAYNYAAY
NYYY YY
NY AAY ANAYA
YA
 

 
 


r101Y Y 

Y. CHIBA 13
Using this number, the lower bound of
 
   
211
211
Pr0 11,1
Pr0 110 1
iii
iii
YAY
YYY

 
is

010 010
0*
*0 NNN N

 


1
max ,
**NN






,
and the upper bound is

011 011
*0
*1
min ,
**
NNN
N
NN
 




,
1





where

is the num-

*112
Pr, ,*
ayyi ii
NNAaYyYy

112
,,,
iii
ber of households with

,*
A
YY ay
ing bounds for α:
y. Some
algebra yields the follow
11 11
max0, 111min,,
PQ PQ


 
 
 

 
 
 

where


11
Pr11 P
ii
PYA d
1 1
r 10|
i i
YA
an

211
Pr10, 1
iii
QYAY
. We note that thr
bound is the same as or smaller than that deri
e uppe
ved by
Chiba and Taguri [7], which is 1/P under Assumptions 1
and 2.
Next, we consider the bounds for α under Assumption
e following assumption [25,29]:
Assumption 4.
2 and th
 


Pr 0 10,.1YAY
le insofar as the
poperson 1 was vaccinated is likely to
be less healthy (or the infection is more virulent) than
that in which person 1 was unvaccinated. Tpopu-
lation in which person 1 was infected despreceiving
the vaccination would be less healthy than the second
po
ran-
ve pneumococcal conjugate vaccine
umococcus serotype. The colonization
status with respect to the given serotype of the 1-year-old
child and the mother is also monitored. Because pneu-
mococcus is highly prevalent in young childrewho at-
tend day care, the mother is more likely to acquire the
pneumococcus from the child than through other trans-
211
Pr0 11,1
iii
YAY
21 1iii
This assumption is arguably reasonab
pulation in which
he first
ite
pulation in which person 1 was unvaccinated and in-
fected. Thus, under the scenario in which both persons
are unvaccinated, person 2 is more likely to be infected
in the first than in the second population.
Under Assumptions 2 and 4, it is obvious that α ≥ 1.
While the large sample bounds generally yield a wide
range, if Assumption 4 is plausible, it will yield a nar-
rower width than the large sample bounds only.
6. Illustration
To illustrate the methods presented in Sections 4 and 5,
we employ data from a hypothetical vaccine trial used
elsewhere [4]. Consider now a vaccine trial setting in
which 1-year-old children at a day-care center are
domized to recei
against a given pne
n
mission routes. Here, we assume that the second person
(the mother) can be infected only from the first (the
1-year-old child). The hypothetical data are given in Ta-
ble 1.
Under Assumptions 1-3, using a MSM, Equations
(4.4) and (4.5) yielded an IPW estimate of the infec-
tiousness effect of 0.44 (95% CI: 0.33, 0.53) and an IPW
estimate of the contagion effect of 0.54 (95% CI: 0.46,
0.61). The overall indirect effect was 1 − (79/1000)/
(305/1000) = 0.74 (95% CI: 0.67, 0.79), which equals the
decomposition of Equation (2.5): 0.44 + 0.54 − 0.44 ×
0.54 = 0.74.
To assess how inference would change under violation
of Assumption 3, we implemented the sensitivity analy-
sis presented in Section 5. Before the implementation, we
determined the bounds for α to determine the range to be
examined. The large sample bounds yielded bounds for α
of 0.04 ≤ α ≤ 1.80. By adding Assumption 4, this range
was narrowed to 1.00 ≤ α ≤ 1.80. For this range of α, we
implemented the sensitivity analysis using Equations
(5.2) and (5.3). The results are shown in Figure 3 for the
infectiousness effect and Figure 4 for the contagion ef-
fect.
For this range of α, the respective lower and upper
limits of the infectiousness effect were 0.43 (95% CI:
0.31, 0.53) and 0.68 (95% CI: 0.62, 0.74), and those of
the contagion effect were 0.18 (95% CI: 0.04, 0.30) and
0.55 (95% CI: 0.47, 0.61). The IPW estimates under As-
sumption 3 correspond to the estimates for α = 1.01 in
Figures 3 and 4.
Table 1. Numbers infected (Yi1, Yi2) from a hypothetical
randomized trial of pneumococcal conjugate vaccine with
2000 householdsa.
Yi1 = 0
Yi2 = 0
Yi1 = 1
Yi2 = 0
Yi1 = 1
Yi2 = 1 Total
Low SES
Ai1 = 0, Ai2 = 0 200 120 180 500
Ai1 = 1, Ai2 = 0 350 96 54 500
High SES
i1i2400 75 25 500
Ai1 = 0, Ai2 = 0 250 125 125 500
A = 1, A = 0
aPerson 1 (the 1-year-olas ed 1:1 to vaccine rol,
and person 2 (the mothet vavactatus [Ai1, Ai2]),
and half the households have either low or high socioeconomic status (SES).
d cild) w
r) was no
hran omiz
ccinated (
dor cont
cination s

Y. CHIBA
14
0.3
0.4
0.5
ious
0.6
ess E
0.7
0.8
1.11.3.5 1.1.8
ct nf
Alpha
Fiy analysis of fectiess effe
11.21.4 16 1.7
Infe fect
gure 3. Sensitivitthe inousnect; th
solid line indicates the infectiousness effect and broken lines
indicate 95% confidence intervals.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
11.1 1.21.31.41.5 1.61.71.8
Contagion Effect
Alpha
Figure 4. Sensitivity analysis of the contagion effect; the
solid line indicates the contagion effect and broken lines
indicate 95% confidence intervals.
7. Discussion
In this paper, by applying a simple relationship between
conditional and unconditional infectiousness effects,
have presented simple statistical methods to assess t
infectiousness and contagion effects. Although the meth-
ods for the infectiousness effects are a review of the past
literature, we have summarized them in the present
We have further applied them to infer the contion ef-
n be partially veri
data by examining whether
we
he
paper.
ag
fect.
The methods presented here are limited by the as-
sumptions. While Assumption 1 cafied
from the observed

21
Pr1 0
ii
YY0, Assumption 2 cannot be verified
he principal stratific
from the observed data. Although we believe that As-
sumption 2 will be plausible in many settings, whether
the assumption holds will depend on the nature of the
vaccine under study. Therefore, the assumption will not
be applicable to all vaccines [18]. In such situations, un-
fortunately, the methods presented here cannot be ap-
plied. Nevertheless, for unconditional infectiousness and
contagion effects, we can still apply the methods devel-
oped in the context of the mediation analysis approach
[30-33], and for the conditional infectiousness effect, we
can also apply the methods developed in the context of
tation approach [25-28]. However,
these methods have a weakness in that they are more
complex than those in this paper.
The framework used also made an assumption of par-
tial interference. This might be plausible if the various
households are sufficiently geographically separated or
do not interact. In certain settings, this assumption might
be plausible. Nevertheless, future work will attempt to
generalize the methods given here to allow for violations
of this assumption.
8. Acknowledgements
This work was supported partially by Grant-in-Aid for
Scientific Research (No. 23700344) from the Ministry of
Education, Culture, Sports, Science, and Technology of
Japan.
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