eAPPENDIX 1
P(CAP | vacc) is the risk of CAP among those vaccinated against influenza, and P(CAP | vacc) is
the risk of CAP among those not vaccinated against influenza. Then
RRCAP 
P(CAP | vacc)
P(CAP | vacc)
(1)
is the relative risk of CAP for being vaccinated against influenza versus not being vaccinated
against influenza. The effectiveness of influenza vaccine against CAP is expressed as
VECAP  (1  RRCAP )100 % .
We then express the risk of CAP among the vaccinated as
P(CAP | vacc) = P(CAP  vacc ) / P( vacc )


= P(CAP  flu  vacc)  P(CAP  flu  vacc) / P(vacc) =
P(CAP | flu  vacc)P( flu | vacc)P(vacc)  P(CAP | flu  vacc)P( flu | vacc)P(vacc)/ P(vacc)
So that
P(CAP | vacc) = P(CAP | flu  vacc) P( flu | vacc)  P(CAP | flu  vacc) P( flu | vacc)
Similarly we have the risk of CAP among the unvaccinated as
P(CAP | vacc)  P(CAP | flu  vacc) P( flu | vacc)  P(CAP | flu  vacc) P( flu | vacc)
We then redefine RRCAP as follows
RRCAP 
P(CAP | flu  vacc) P( flu | vacc)  P(CAP | flu  vacc) P( flu | vacc)
P(CAP | flu  vacc) P( flu | vacc)  P(CAP | flu  vacc) P( flu | vacc)
(2)
We now make a simplifying assumption that once it is known whether a person has influenza or
not, the risk of CAP becomes independent of vaccination status and becomes a function of
influenza disease status alone. Under this assumption we have
P(CAP | flu  vacc)  P(CAP | flu  vacc)  P(CAP | flu)
P(CAP | flu  vacc)  P(CAP | flu  vacc)  P(CAP | flu)
Substituting the above two expressions into equation 2 and using the following
P( flu | vacc)  1  P( flu | vacc)
P( flu | vacc)  1  P( flu | vacc)
We have
RRCAP 
P(CAP | flu)  P(CAP | flu)P( flu | vacc)  P(CAP | flu)
P(CAP | flu)  P(CAP | flu)P( flu | vacc)  P(CAP | flu)
(3)
Now define
RR flu 
P( flu | vacc)
P( flu | vacc)
(4)
as the relative risk of influenza disease for being vaccinated against influenza versus not being
vaccinated, with P ( flu | vacc) being the background rate of influenza infection among the
unvaccinated. Note that RRflu ≤ 1 and that VE flu  (1  RR flu )100% is the effectiveness of
influenza vaccine against influenza infection.
Substituting P( flu | vacc)  RR flu  P( flu | vacc) into equation 3 we have
RRCAP 
P(CAP | flu)  P(CAP | flu) RR  P( flu | vacc)  P(CAP | flu)
P(CAP | flu)  P(CAP | flu) P( flu | vacc)  P(CAP | flu)
flu
Now define
RRCAP| flu 
P(CAP | flu)
P(CAP | flu)
(6)
(5)
as the relative risk of CAP for having influenza disease versus not being having influenza. Note
that RRCAP| flu  1 . We have that
P(CAP | flu)  RRCAP| flu  P(CAP | flu)
P(CAP | flu)  P(CAP | flu)  P(CAP | flu)  (RR
CAP| flu
 1)
Substituting these into Equation (5) we have
RRCAP 
P(CAP | flu)( RRCAP| flu  1) RR flu  P( flu | vacc)  P(CAP | flu)
P(CAP | flu)( RRCAP| flu  1) P( flu | vacc)  P(CAP | flu)
And dividing both numerator and denominator by P(CAP | flu) we obtain
RRCAP 
RR flu  ( RRCAP| flu  1) P( flu | vacc)  1
( RRCAP| flu  1) P( flu | vacc)  1
(7)
eAPPENDIX 2
In addition to estimating the effectiveness of influenza vaccine against CAP in general, we may
wish to estimate the effectiveness of influenza vaccine specifically against that subset of CAP
with pre-disposing influenza infection, or with influenza co-infection. That is, we wish to
estimate
 
RRCAP
P(CAP  flu | vacc)
P(CAP  flu | vacc)
(1)
We note the following expressions
P(CAP  flu | vacc )  P(CAP  flu  vacc ) / P( vacc )
 P(CAP | flu  vacc ) P( flu | vacc ) P( vacc ) / P( vacc )
P(CAP  flu | vacc )  P(CAP | flu  vacc ) P( flu | vacc )
Similarly
P(CAP  flu | vacc)  P(CAP | flu  vacc) P( flu | vacc)
and therefore we have
 
RRCAP
P(CAP  flu | vacc) P(CAP | flu  vacc) P( flu | vacc)

P(CAP  flu | vacc) P(CAP | flu  vacc) P( flu | vacc)
(2)
Once it is known whether a person is infected with influenza or not, we assume the risk of CAP
then becomes independent of vaccination status and becomes a function on influenza infection
status alone. Under this assumption we have
P(CAP | flu  vacc)  P(CAP | flu  vacc)  P(CAP | flu)
and substituting into Equation (2) we have
 
RRCAP
P(CAP | flu ) P( flu | vacc)
P(CAP | flu ) P( flu | vacc)

P( flu | vacc)
P( flu | vacc)
Therefore under the assumption of conditional independence of CAP and vaccination status
given influenza infection status, the relative risk of CAP with influenza is the same as the
relative risk of influenza:
 
RRCAP
P(CAP  flu | vacc) P( flu | vacc)

 RR flu
P(CAP  flu | vacc) P( flu | vacc)
(3)
Similarly, the vaccine effectiveness against CAP with influenza is the same as the vaccine
effectiveness against influenza:
 )  (1  RR flu )  VE flu
VECAP flu  (1  RRCAP
eAPPENDIX 3
Equations 4 through 7 of this article and equation 7 of Appendix 1 require an estimate of
RRCAP|flu, the relative risk of CAP for being infected with influenza versus not being infected
with influenza. An estimate of RRCAP|flu would be difficult to obtain. However, P(flu|CAP), the
proportion of CAP that is associated with influenza may be more easily estimated from
observational studies of patients with CAP or surveillance systems. Using Bayes theorem we
derive an expression for RRCAP|flu that is a function of P(flu|CAP), the proportion of CAP that is
associated with influenza, and which may be conveniently substituted into equations that require
RRCAP|flu.
RRCAP| flu 
P(CAP | flu) P(CAP  flu) / P( flu) P( flu | CAP) P(CAP) / P( flu)


P(CAP | flu) P(CAP  flu) / P( flu) P( flu | CAP) P(CAP) / P( flu)
And therefore
RRCAP| flu 
P( flu | CAP) 1  P( flu)

1  P( flu | CAP) P( flu)
(1)
The risk of influenza in the general population is shown as P ( flu ) which can be expressed as a
weighted average of the influenza rates among the vaccinated and unvaccinated as follows:
P( flu )  P( flu | vacc) P(vacc)  P( flu | vacc) P(vacc)
 RR flu P( flu | vacc) P(vacc)  P( flu | vacc)1  P(vacc)
 P( flu | vacc)RR flu  P(vacc)  1  P(vacc)
P( flu )  P( flu | vacc)P(vacc)RR flu  1  1
Any of these expressions for P ( flu ) can be substituted into Equation 1 for RRCAP|flu.
Equation 1 is the odds of influenza among CAP cases divided by the odds of influenza among
the general population. The use of this exposure odds ratio to estimate a disease relative risk is an
essential feature of case-cohort or case-based studies (Pierce 1993; Knol et al. 2008).
eAPPENDIX REFERENCES
Knol MJ, Vandenbroucke JP, Scott P, Egger M. What do case-control studies estimate? Survey
of methods and assumptions in published case-control research. Am J Epidemiol.
2008;168(9):1073-1081.
Pierce N. What does the odds ratio estimate in a case-control study? Int J Epidemiol.
1993;22(6):1189-1192.