eAppendix
1. Estimation of measures of additive interaction
Consider model (1) i.e.
p
ln (1−p) = b0 + b1 X + b2 Y + b3 XY + b4 Z1 + ⋯ + bn+3 Zn
The combined effect of the arbitrary increment dx and dy of the continuous
determinants X and Y is:
ln (OR X:x0 →x0 +dx ,Y:y0 →y0 +dy /Zi ) = ln (
odds(X = x0 + dx , Y = y0 + dy / Zi = zi )
)
odds(X = x0 , Y = y0 / Zi = zi )
= ln[odds(X = x0 + dx , Y = y0 + dy / Zi = zi )] − ln[odds(X = x0 , Y = y0 / Zi = zi )]
= b0 + b1 (x0 + dx ) + b2 (y0 + dy ) + b3 (x0 + dx )(y0 + dy ) + b4 z1 + ⋯ + bn+3 zn
−(b0 + b1 (x0 ) + b2 (y0 ) + b3 (x0 )(y0 ) + b4 z1 + ⋯ + bn+3 zn )
= b1 (dx ) + b2 (dy ) + b3 (dx dy + x0 dy + y0 dx )
→ OR x0 →x0 +dx ,y0 →y0 +dy /Zi = eb1 (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
The effect of an arbitrary increment dx of X, for a specific value y0 of Y:
ln(OR X:x0 →x0 +dxY:,y0 →y0 /Zi ) = ln (
odds(X = x0 + dx , Y = y0 / Zi = zi )
)
odds(X = x0 , Y = y0 / Zi = zi )
= ln[odds(X = x0 + dx , Y = y0 / Zi = zi )] − ln[odds(X = x0 , Y = y0 / Zi = zi )]
= b0 + b1 (x0 + dx ) + b2 (y0 ) + b3 (x0 + dx )(y0 ) + b4 z1 + ⋯ + bn+3 zn
−(b0 + b1 (x0 ) + b2 (y0 ) + b3 (x0 )(y0 ) + b4 z1 + ⋯ + bn+3 zn )
= b1 (dx ) + b3 (dx )y0
→ OR x0 →x0 +dx,y0 →y0 /Zi = eb1 (dx )+b3 (dx )y0
The effect of arbitrary increment dy of Y, for a specific value x0 of X:
ln(OR X:x0 →x0 ,Y:y0 →y0+dy/Zi ) = ln (
odds(X = x0 , Y = y0 + dy / Zi = zi )
)
odds(X = x0 , Y = y0 / Zi = zi )
= ln[odds(X = x0 , Y = y0 + dy / Zi = zi )] − ln[odds(X = x0 , Y = y0 / Zi = zi )]
= b0 + b1 (x0 ) + b2 (y0 + dy ) + b3 (x0 )(y0 + dy ) + b4 z1 + ⋯ + bn+3 zn
−(b0 + b1 (x0 ) + b2 (y0 ) + b3 (x0 )(y0 ) + b4 z1 + ⋯ + bn+3 zn )
= b2 (dy ) + b3 x0 (dy )
→ OR x0 →x0 ,y0 →y0 +dy/Zi = eb2 (dy )+b3 x0 (dy )
By assuming that odds ratios (ORs) from (1) approximate relative risks (RR), it is
straightforward that
RERIx0 →x0 +dx ,y0 →y0 +dy = OR x0 →x0 +dx ,y0 →y0 +dy/Zi − OR x0 →x0 +dx,y0 →y0 /Zi
−OR x0 →x0 ,y0 →y0 +dy/Zi + 1
=e
b1 (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
APx0 →x0 +dx ,y0 →y0+dy =
− eb1 (dx )+b3 (dx )y0 − eb2 (dy )+b3 x0 (dy ) + 1
RERIx0 →x0 +dx,y0→y0+dy
ORx0 →x0 +dx ,y0→y0+dy/Zi
= 1 − e−b2 (dy )−b3 (x0 +dx )(dy ) − e−b1 (dx )−b3 (dx )(y0+dy ) + e−b1 (dx )−b2 (dy )−b3 (dx dy +x0 dy +y0 dx )
Sx0 →x0 +dx ,y0 →y0 +dy = OR
ORx0→x0 +dx ,y0→y0+dy/Zi −1
x0 →x0 +dx ,y0 →y0 /Zi +ORx0 →x0 ,y0 →y0 +dy/Zi −2
=
(d )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
eb1 x
−1
eb1(dx)+b3 (dx)y0 +eb2(dy)+b3 x0(dy) −2
Knol and colleagues1 estimated Rothman's indexes in case of continuous risk
factors; however, their formulae hold only for increases by 1 unit of the 2 risk factors
from a background level of exposure of x0 =0 and y0 =0.
2. Estimation of variances and confidence intervals for indexes of additive
interaction
The partial derivatives of RERI, AP and ln(S) (f, g and ln(h) respectively) with
respect to 𝑏𝑗 evaluated at 𝑏̂𝑗 , are shown below:
Relative excess risk due to interaction (RERI):
RERIx0 →x0 +dx ,y0 →y0 +dy = f(b1 , b2 , b3 / x0 , y0 , dx, dy) = eb1
(dx )+b2 (dy )+b3 (dx dy +x0 dy +y0dx )
−eb1 (dx )+b3 (dx )y0 − eb2 (dy )+b3 x0 (dy ) + 1
∂f
̂
̂
̂
̂
̂
̂
̂
̂
̂
̂
f1 = ∂b | b̃=b̃̂ = dx (eb1 (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx ) − eb1 (dx )+b3 (dx )y0 )
1
f2 =
∂f
| ̃=b̃̂
∂b2 b
= dy (eb1 (dx )+b2 (dy )+b3 (dx dy+x0 dy +y0 dx ) − eb2 (dy )+b3 x0 (dy ) )
∂f
̂
̂
, so
(S.1)
(S.2)
̂
f3 = ∂b | b̃=b̃̂ = (dx dy + x0 dy + y0 dx )eb1 (dx )+b2 (dy)+b3 (dx dy +x0 dy +y0 dx )
3
̂
̂
̂
̂
−((dx )y0 )eb1 (dx )+b3 (dx )y0 − (x0 (dy )) eb2 (dy )+b3 x0 (dy )
(S.3)
Proportion attributable to interaction (AP):
APx0 →x0 +dx ,y0→y0 +dy = g(b1 , b2 , b3 / x0 , y0 , dx, dy) = 1 − e−b2
(dy )−b3 (dx dy +x0 dy )
̂
−e−b1 (dx )−b3 (dx dy+y0 dx ) + e−b1 (dx )−b2 (dy )−b3 (dx dy +x0 dy +y0 dx )
∂g
̂
̂
̂
̂
̂
̂
̂
̂
̂
̂
,so
g1 = ∂b | b̃=b̃̂ = dx (e−b1 (dx )−b3 (dx dy +y0 dx ) − e−b1 (dx )−b2 (dy )−b3 (dx dy +x0 dy +y0 dx ) ) (S.4)
1
∂g
g 2 = ∂b | b̃=b̃̂ = dy (e−b2 (dy)−b3 (dx dy +x0 dy ) − e−b1 (dx )−b2 (dy )−b3 (dx dy +x0 dy +y0 dx ) ) (S.5)
2
g3 =
∂g
̂
̂
= (dx dy + x0 dy )e−b2 (dy )−b3 (dx dy+x0 dy )
|
∂b3 b̃ = b̂̃
̂
̂
+(dx dy + y0 dx )e−b1 (dx )−b3 (dx dy +y0 dx )
− (dx dy + x0 dy + y0 dx ) e
̂ (d )−b
̂ (d )−b
̂ (d d +x d +y d )
−b
1 x
2 y
3 x y
0 y
0 x
(S.6)
Synergy index (S): After converting S to ln(S), we have
ln(S)x0 →x0 +dx ,y0→y0 +dy = ln[h(b1 , b2 , b3 / x0 , y0 , dx, dy)] =
= ln (eb1 (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx ) − 1) − ln (eb1 (dx )+b3 (dx )y0 + eb2 (dy )+b3 x0 (dy ) − 2)
so,
ln(h1 ) =
= (dx ) (
∂ln(h)
|
=
∂b1 b̃ = b̂̃
b (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
e 1
b (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
e 1
−1
ln(h2 ) =
= (dy ) (
eb1 (dx)+b3 (dx)y0
e
b1 (dx )+b3 (dx )y0
)
(S.7)
)
(S.8)
+eb2 (dy)+b3 x0 (dy ) −2
∂ln(h)
|
=
∂b2 b̃ = b̂̃
b (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
e 1
b (dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
e 1
−1
ln(h3 ) =
−
∂ln(h)
| ̃=b̃̂
∂b3 b
=
−
eb2 (dy)+b3 x0 (dy)
e
b1 (dx )+b3 (dx )y0
+eb2 (dy)+b3 x0 (dy) −2
(d )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
(dx dy +x0 dy +y0 dx )eb1 x
(d )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
eb1 x
−1
−
b (d )+b3 x0 (dy )
(dx )y0 eb1 (dx )+b3 (dx )y0 +x0 (dy )e 2 y
eb1 (dx)+b3 (dx)y0 +eb2 (dy)+b3 x0 (dy) −2
(S.9)
Estimation of variances and confidence intervals for measures of additive interaction
between two continuous risk factors using logistic regression
For estimating confidence intervals for each one of the Rothmans indexes, we
calculated the necessary variance estimators, by applying the delta method. Let 𝑏̃̂ =
(𝑏̂1 , 𝑏̂2 , 𝑏̂3 ) denote the 3 estimated coefficients from model (1). Rothman’s indexes
estimated in (3), (5), (7) in the Research Letter can be seen as functions of 𝑏̂1 , 𝑏̂2 and
𝑏̂3 , with parameters the background level of exposure x0 on X and y0 on Y, and the
arbitrary increment dx of X and dy of Y, i.e:
RERIx0 →x0 +dx ,y0 →y0+dy = eb1
(dx )+b2 (dy )+b3 (dx dy +x0 dy +y0 dx )
− eb1 (dx )+b3 (dx )y0
−eb2 (dy )+b3 x0 (dy ) + 1 = f(b̂ 1 , b̂ 2 , b̂ 3 / x0 , y0 , dx, dy)
APx0 →x0 +dx ,y0→y0 +dy = 1 − e−b2
(dy )−b3 (dx dy +x0 dy )
− e−b1 (dx )−b3 (dx dy +y0 dx )
+e−b1 (dx )−b2 (dy )−b3 (dx dy +x0 dy+y0 dx ) = g(b̂ 1 , b̂ 2 , b̂ 3 / x0 , y0 , dx, dy)
Sx0 →x0 +dx ,y0→y0 +dy =
(S.10)
b (d )+b (d )+b (d d +x d +y d )
e 1 x 2 y 3 x y 0 y 0 x −1
eb1 (dx)+b3 (dx )y0 +eb2 (dy )+b3 x0 (dy ) −2
̂1 , b̂2 , b̂3 / x0 , y0 , dx, dy)
= h (b
(S.11)
(S.12)
The variances of (S.10) – (S.12) can then be estimated as follows:
̂ [RERI
̂ x →x +d ,y →y +d ] = Var
̂ [f (b̂̃ / x0 , y0 , dx , dy )] , so
Var
0
0
x 0
0
y
̂ [RERI
̂ x →x +d ,y →y +d ] = f12 σ
Var
̂12 + f22 σ
̂22 + f32 σ
̂23 + 2f1 f2 σ̂ 12 + 2f1 f3 σ̂ 13 + 2f2 f3 σ̂ 23
0
0
x 0
0
y
Where:
𝑓𝑗 is the partial derivative of f with respect to 𝑏𝑗 evaluated at 𝑏̂̃ , j=1,2,3, that is:
fj =
∂f
|
∂bj b̃ = b̃̂
̂ (b̂j , b̂k ) , k=2,3 and k>j.
̂ (b̂j ) and σ
σ2j = Var
̂
̂jk = Cov
Similarly for AP :
̂ x →x +d ,y →y +d ] = Var
̂ [AP
̂ [g (b̂̃ / x0 , y0 , dx , dy )] ,
Var
0
0
x 0
0
y
̂ x →x +d ,y →y +d ] = g12 σ
̂ [AP
Var
̂12 + g 22 σ
̂22 + g 23 σ
̂23 + 2g1 g 2 σ̂ 12 + 2g1 g 3 σ̂ 13 + 2g 2 g 3 σ̂ 23 ,
0
0
x 0
0
y
where
gj =
∂g
|
∂bj b̃ = b̃̂
For S, since it is a ratio, the coverage properties are better if a confidence interval for
ln(S) rather than for S is calculated2. Therefore, at first a confidence region for ln(S)
was formed and then the limits of this confidence region were exponentiated in order
to calculate a confidence interval for S. So, ln(S) and its variance can be written as:
ln [Ŝx0 →x0 +dx ,y0 →y0 +dy ] = ln (eb1 (dx )+b2 (dy)+b3 (dx dy +x0 dy +y0 dx ) − 1)
−ln (eb1 (dx )+b3 (dx )y0 + eb2 (dy )+b3 x0 (dy) − 2)
so
̂ [ln [Ŝx →x +d ,y →y +d ]] = Var
̂ [ln [h (b̂̃ / x0 , y0 , dx , dy )]] , that is
Var
0
0
x 0
0
y
̂ [ln [Ŝx →x +d ,y →y +d ]] = [ln(h1 )]2 σ
Var
̂12 + [ln(h2 )]2 σ
̂22 + [ln(h3 )]2 σ
̂23
0
0
x 0
0
y
+2ln(h1 )ln(h2 )σ̂ 12 + 2ln(h1 )ln(h3 )σ̂ 13 + 2ln(h2 )ln(h3 )σ̂ 23
where
ln(hj ) =
∂ln(h)
|
∂bj b̃ = b̂̃
So the 95% confidence intervals for RERI, AP and S can be calculated as follows:
̂ − 1.96√Var
̂ [RERI
̂ ], RERI
̂ + 1.96√Var
̂ [RERI
̂] )
For RERI: (RERI
,
̂ − 1.96√Var
̂ ], AP
̂ + 1.96√Var
̂] )
̂ [AP
̂ [AP
for AP: (AP
,
and for S: (e
̂ [ln(Ŝ)]
ln(Ŝ)−1.96√Var
,e
̂ [ln(Ŝ)]
ln(Ŝ)+1.96√Var
)
3. Illustrative dataset
We used data from the Greek EPIC cohort. EPIC is a European longitudinal study
designed to investigate the relationships between diet, lifestyle and environmental
factors and the incidence of cancer and other chronic diseases3. In our paradigm, our
sample consists of adult women, aged less than 45 years old of the Greek segment of
the EPIC study4.
The binary outcome we considered was whether a subject was hypertensive or not at
enrollment (cross sectional analysis). The information of the dichotomous outcome
was collected using as criteria if the participant’s diastolic or systolic blood pressure
was measured more than 90 mmHg or 140 mmHg respectively at baseline. We further
considered BMI (in kg/m2) and level of adherence in Mediterranean Diet (MD)2 at
recruitment as continuous and ordinal risk factors respectively. The variable related to
(non-) adherence to MD (NMD), has 3 levels; in the 1st level, we considered those
participants that their MD score is 6-9 (i.e. high adherence to MD), in the 2nd level,
those with MD score between 2-5 (i.e. moderate adherence to MD) and in the 3rd
level, those with MD score 0-1 (i.e. low adherence to MD). Moreover, we include in
the analysis, age (in years), height (in cm) and education (3 levels; categorically
modeled) as possible confounders. With respect to the variable related to education,
we considered in the 1st level those with elementary education, in the 2nd level those
with secondary education and in the 3rd level those who have, at least, a university
degree. Participants with missing values in any of the above variables were excluded
this analysis. Descriptive statistics of all variables included in the analysis are shown
in table S1.
We hypothesized that increasing BMI and increasing levels of NMD would result in
increasing prevalence of hypertension. In order to test our hypothesis, we ran a
logistic regression model with hypertension as the dependent variable, while BMI
(variable X) and increasing levels of NMD (variable Y) are considered the risk factors
of interest. Age (variable Z1), education (Z2 is the dummy variable corresponding to
the comparison of the 2nd level of education vs the 1st and Z3 is the dummy variable
for the comparison of the 3rd level of education vs the 1st) and height (variable Z4)
are considered as potential confounders. Let us consider the following model (S.13)
p
ln (1−p) = a0 + a1 X + a2 Y + a3 Z1 + a4 Z2 + a5 Z3 + a6 Z4
(S.13)
where p is the probability of the occurrence of hypertension and a0 − a6 the estimated
coefficients of the model. This is the model without the interaction between X and Y
in order to illustrate that BMI and NMD are risk factors for the occurrence of
hypertension. The results of the logistic regression model (S.13) are presented in
table S2. Both BMI and NMD are positively associated with higher occurrence of
hypertension although only BMI is deemed statistically significant. Subsequently we
estimated the measures for additive interaction between BMI and NMD on the
prevalence of hypertension whilst adjusting for the effects of the above potential
confounders. In order to estimate Rothman's measures for additive interaction, we ran
the following logistic regression model (S.14) which is (1) of the Research Letter
when n=4, i.e.
p
ln (1−p) = b0 + b1 X + b2 Y + b3 XY + b4 Z1 , +b5 Z2 + b6 Z3 + b7 Z4
(S.14)
The results of model (S.14) [or (1)] are presented in the Research Letter in Table 1.
Script. Stata code for estimating additive interaction between any variables X
and Y on binary outcome d.
*Programming code for additive interaction between variables X, Y on binary outcome d
************************************************************************************
* At the beginning, the user has to define the variables d, x, y and x_y (i.e. the
*product term x*y), and any potential confounder that should be included in the
*logistic regression model.
* Here we use 3 such variables, Z1, Z2, Z3
global
global
global
global
global
global
global
d
x
y
x_y
z1
z2
z3
"
"
"
"
"
"
"
hyper "
bmi "
nmd "
bmi_nmd "
age "
i.education "
height "
* In the Research Letter, for our illustrative example, the variables we used are:
*
1) d=hypertension (yes/no) --> hyper
*
2) X=Body Mass Index (BMI), calculated in kg/m2 --> bmi
*
3) Y=Non adherence to Mediterrenean diet NMD (ordered in three levels) --> nmd
*
4) X_Y=Product term of BMI and NMD --> bmi_nmd=bmi*nmd
*
5) Z1=age (continuous in years) --> age
*
6) Z2=education (categorically in 3 levels) -> i.education
*
7) Z3=height (continuous in cm)--> height
* The user has to state also the parameters x0, y0, dx and dy.
* In our example, in the first line of table 1, we have used x0=25, y0=1, dx=2, dy=1
global
global
global
global
x0
y0
dx
dy
"25"
"1"
"2"
"1"
logit $d $x $y $x_y $z1 $z2 $z3
*To estimate RERI:
nlcom(RERI:exp($dx*_b[$x]+$dy*_b[$y]+($dx*$dy+$dy*$x0+$dx*$y0)*_b[$x_y])exp($dx*_b[$x]+$dx*$y0*_b[$x_y])-exp($dy*_b[$y]+$dy*$x0*_b[$x_y])+1 )
*the output give the point estimate of RERI for x0, y0, dx, dy, its associated
*standard error using the delta method and the 95% CI
*To estimate AP:
nlcom ( AP:1-exp(-_b[$y]*$dy-_b[$x_y]*($dx*$dy+$x0*$dy))-exp(-_b[$x]*$dx_b[$x_y]*($dx*$dy+$y0*$dx))+exp(-_b[$x]*($dx)-_b[$y]*($dy)_b[$x_y]*($dx*$dy+$y0*$dx+$x0*$dy)) )
*the output give the point estimate of AP for x0, y0, dx, dy, its associated standard
*error using the delta method and the 95% CI
* To estimate S:
*Since S is a ratio, one has to estimate ln(S) (named as "ln_syn_index" in the nlcom
*command below) and the SE for ln(S) for x0, y0, dx, dy:
nlcom (ln_syn_index:ln(exp($dx*_b[$x]+$dy*_b[$y]+($dx*$dy+$dy*$x0+$dx*$y0)*_b[$x_y])1)-ln(exp($dx*_b[$x]+$dx*$y0*_b[$x_y])+exp($dy*_b[$y]+$dy*$x0*_b[$x_y])-2) ), post
*After obtaining _b[ln_syn_index] and _se[ln_syn_index] from the above command we
*estimate S and the associated 95% CI as follows:
***Estimate S for x0, y0, dx, dy
scalar Syn_index=exp(_b[ln_syn_index])
***Estimate 95% CI for S, for x0, y0, dx, dy
scalar Syn_index_low95 =exp(_b[ln_syn_index]-invnormal(0.975)*_se[ln_syn_index])
scalar Syn_index_high95=exp(_b[ln_syn_index]+invnormal(0.975)*_se[ln_syn_index])
****And present in the output these estimates for S in one line
mat define Synergy_index=(Syn_index, Syn_index_low95, Syn_index_high95)
mat rown
Synergy_index= Syn_index
mat coln
Synergy_index= S_index S_low95 S_high95
mat list
Synergy_index
****End of programming code**********
************************************************************************************
References
1. Knol MJ, van der Tweel I, Grobbee DE, Numans ME, Geerlings MI. Estimating
interaction on an additive scale between continuous determinants in a logistic
regression model. Int J Epidemiol 2007;36:1111-8.
2. Hosmer DW, Lemeshow S. Confidence interval estimation of interaction.
Epidemiology 1992;3:452-6.
3. Riboli E and Kaaks R. The EPIC Project: rationale and study design. European
Prospective Investigation into Cancer and Nutrition. Int J Epidemiol 1997;26
Suppl 1:S6-14.
4. Trichopoulou A, Costacou T, Bamia C, Trichopoulos D. Adherence to a
Mediterranean diet and survival in a Greek population. N Engl J Med
2003;348:2599-608
Table S1: Descriptive statistics of age, body mass index (BMI), height, hypertension,
non adherence in Mediterranean Diet (NMD) and education in 4805 women aged less
than 45 from the Greek-EPIC cohort
Continuous variables
Age (in years)
BMI (in kg/m2)
Height (in cm)
Mean
37.99
26.62
160.02
SD
4.61
5.00
6.03
Categorical variables
N
Hypertension
No
Yes
Non adherence to Mediterranean Diet (NMD)
1-->Med. Diet score 6-9
2-->Med. Diet score 2-5
3-->Med. Diet score 0-1
Education level
1-->Elementary
2-->Secondary
3-->University degree or higher
(%)
4336 (90.24%)
469
(9.76%)
1133 (23.58%)
3448 (71.76%)
224
(4.66%)
1407
1917
1481
(29.28%)
(39.90%)
(30.82%)
Table S2: Estimated coefficients from the logistic regression model (S.13) of the prevalence of hypertension in women aged less than 45 yrs old
Logistic regression model (S.13)
2
Body Mass Index (in kg/m )
Non adherence to Mediterranean Diet (3 levels - ordered)
Age (in years)
Education: (reference category - elementary school)
secondary
university
Height (in cm)
b
0.130
0.166
0.068
se
0.009
0.105
0.013
Odds Ratio (eb)
1.14
1.18
1.07
95% Confidence Interval for Odds Ratio
(1.12 - 1.16)
(0.96 - 1.45)
(1.04 - 1.10)
-0.541
-0.897
0.021
0.118
0.147
0.009
0.58
0.41
1.02
(0.46 - 0.73)
(0.31 - 0.54)
(1.0 - 1.04)