Introduction
Let \(Y\) be a binary
response, \(A\) a binary
exposure, and \(V\) a vector
of covariates.
DAG for the statistical model with the dashed
edge representing a potential interaction between exposure \(A\) and covariates \(V\).
In a common setting, the main interest lies in quantifying the
treatment effect, \(\nu\), of \(A\) on \(Y\) adjusting for the set of covariates,
and often a standard approach is to use a Generalized Linear Model
(GLM):
\[g\{ E(Y\mid A,V) \} = A\nu^tW +
\underset{\mathrm{nuisance}}{\mu^tZ}\]
with link function \(g\), and \(W = w(V)\), \(Z=
v(V)\) known vector functions of \(V\).
The canonical link (logit) leads to nice computational properties
(logistic regression) and parameters with an odds-ratio interpretation.
But ORs are not collapsible even under randomization. For
example
\[
E(Y\mid X) = E[ E(Y\mid X,Z) \mid X ] = E[\operatorname{expit}( \mu +
\alpha X + \beta Z ) \mid X]
\neq \operatorname{expit}[\mu + \alpha X + \beta E(Z\mid X)],
\]
When marginalizing we leave the class of logistic regression. This
non-collapsibility makes it hard to interpret odds-ratios and to compare
results from different studies
Relative risks (and risk differences) are
collapsible and generally considered easier to interpret than
odds-ratios. Richardson et
al (JASA, 2017) proposed a regression model for a binary exposures
which solves the computational problems and need for parameter
contraints that are associated with using for example binomial
regression with a log-link function (or identify link for the risk
difference) to obtain such parameter estimates. In the following we
consider the relative risk as the target parameter
\[
\mathrm{RR}(v) = \frac{P(Y=1\mid A=1, V=v)}{P(Y=1\mid A=0, V=v)}.
\]
Let \(p_a(V) = P(Y \mid A=a, V),
a\in\{0,1\}\), the idea is then to posit a linear model for \[ \theta(v) = \log \big(RR(v)\big) \],
i.e., \[\log \big(RR(v)\big) =
\alpha^Tv,\]
and a nuisance model for the odds-product
\[ \phi(v) =
\log\left(\frac{p_{0}(v)p_{1}(v)}{(1-p_{0}(v))(1-p_{1}(v))}\right)
\]
noting that these two parameters are variation independent
as illustrated by the below L’Abbé plot.
p0 <- seq(0,1,length.out=100)
p1 <- function(p0,op) 1/(1+(op*(1-p0)/p0)^-1)
plot(0, type="n", xlim=c(0,1), ylim=c(0,1),
xlab="P(Y=1|A=0)", ylab="P(Y=1|A=1)", main="Constant odds product")
for (op in exp(seq(-6,6,by=.25))) lines(p0,p1(p0,op), col="lightblue")
p0 <- seq(0,1,length.out=100)
p1 <- function(p0,rr) rr*p0
plot(0, type="n", xlim=c(0,1), ylim=c(0,1),
xlab="P(Y=1|A=0)", ylab="P(Y=1|A=1)", main="Constant relative risk")
for (rr in exp(seq(-3,3,by=.25))) lines(p0,p1(p0,rr), col="lightblue")
Similarly, a model can be constructed for the risk-difference on the
following scale
\[\theta(v) = \operatorname{arctanh}
\big(RD(v)\big).\]
Simulation
First the targeted package is loaded
library(targeted)
This automatically imports lava (CRAN) which we can
use to simulate from the Relative-Risk Odds-Product (RR-OP) model.
m <- lava::lvm(a ~ x,
lp.target ~ 1,
lp.nuisance ~ x+z)
m <- lava::binomial.rr(m, response="y", exposure="a", target.model="lp.target", nuisance.model="lp.nuisance")
The lvm call first defines the linear predictor for the
exposure to be of the form
\[\mathrm{LP}_A := \mu_A + \alpha
X\]
and the linear predictors for the /target parameter/ (relative risk)
and the /nuisance parameter/ (odds product) to be of the form
\[\mathrm{LP}_{RR} :=
\mu_{RR},\]
\[\mathrm{LP}_{OP} := \mu_{OP} + \beta_x X
+ \beta_z Z.\]
The covariates are by default assumed to be independent and standard
normal \(X, Z\sim\mathcal{N}(0,1)\),
but their distribution can easily be altered using the
lava::distribution method.
The binomial.rr function
args(lava::binomial.rr)
#> function (x, response, exposure, target.model, nuisance.model,
#> exposure.model = binomial.lvm(), ...)
#> NULL
then defines the link functions, i.e.,
\[\operatorname{logit}(E[A\mid X,Z]) =
\mu_A + \alpha X,\]
\[\operatorname{log}(E[Y\mid X,Z,
A=1]/E[Y\mid X, A=0]) = \mu_{RR},\]
\[\operatorname{log}\{p_1(X,Z)p_0(X,Z)/[(1-p_1(X,Z))(1-p_0(X,Z))]\}
=
\mu_{OP}+\beta_x X + \beta_z Z\]
with \(p_a(X,Z)=E(Y\mid
A=a,X,Z)\).
The risk-difference model with the RD parameter modeled on the \(\operatorname{arctanh}\) scale can be
defined similarly using the binomial.rd method
args(lava::binomial.rd)
#> function (x, response, exposure, target.model, nuisance.model,
#> exposure.model = binomial.lvm(), ...)
#> NULL
We can inspect the parameter names of the modeled
coef(m)
#> m1 m2 m3 p1 p2
#> "a" "lp.target" "lp.nuisance" "a~x" "lp.nuisance~x"
#> p3 p4
#> "lp.nuisance~z" "a~~a"
Here the intercepts of the model are simply given the same name as
the variables, such that \(\mu_A\)
becomes a, and the other regression coefficients are
labeled using the “~”-formula notation, e.g., \(\alpha\) becomes a~x.
Intercepts are by default set to zero and regression parameters set
to one in the simulation. Hence to simulate from the model with \((mu_A, \mu_{RR}, \mu_{OP}, \alpha, \beta_x,
\beta_z)^T =
(-1,1,-2,2,1,1)^T\), we define the parameter vector
p given by
p <- c('a'=-1, 'lp.target'=1, 'lp.nuisance'=-1, 'a~x'=2)
and then simulate from the model using the sim
method
d <- lava::sim(m, 1e4, p=p, seed=1)
head(d)
#> a x lp.target lp.nuisance z y
#> 1 0 -0.6264538 1 -2.4307854 -0.8043316 0
#> 2 0 0.1836433 1 -1.8728823 -1.0565257 0
#> 3 0 -0.8356286 1 -2.8710244 -1.0353958 0
#> 4 1 1.5952808 1 -0.5902796 -1.1855604 1
#> 5 0 0.3295078 1 -1.1709317 -0.5004395 1
#> 6 0 -0.8204684 1 -2.3454571 -0.5249887 0
Notice, that in this simulated data the target
parameter \(\mu_{RR}\) has
been set to lp.target = 1.
Estimation
MLE
We start by fitting the model using the maximum likelihood
estimator.
args(riskreg_mle)
#> function (y, a, x1, x2 = x1, weights = rep(1, length(y)), std.err = TRUE,
#> type = "rr", start = NULL, control = list(), ...)
#> NULL
The riskreg_mle function takes vectors/matrices as input
arguments with the response y, exposure a,
target parameter design matrix x1 (i.e., the matrix \(W\) at the start of this text), and the
nuisance model design matrix x2 (odds product).
We first consider the case of a correctly specified model, hence we
do not consider any interactions with the exposure for the odds product
and simply let x1 be a vector of ones, whereas the design
matrix for the log-odds-product depends on both \(X\) and \(Z\)
x1 <- model.matrix(~1, d)
x2 <- model.matrix(~x+z, d)
fit1 <- with(d, riskreg_mle(y, a, x1, x2, type="rr"))
fit1
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9512 0.03319 0.8862 1.0163 1.204e-180
#> odds-product:(Intercept) -1.0610 0.05199 -1.1629 -0.9591 1.377e-92
#> odds-product:x 1.0330 0.05944 0.9165 1.1495 1.230e-67
#> odds-product:z 1.0421 0.05285 0.9386 1.1457 1.523e-86
The parameters are presented in the same order as the columns of
x1and x2, hence the target parameter estimate
is in the first row
estimate(fit1, keep=1)
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9512 0.03336 0.8858 1.017 7.159e-179
DRE
We next fit the model using a double robust estimator (DRE) which
introduces a model for the exposure \(E(A=1\mid V)\) (propensity model). The
double-robustness stems from the fact that the this estimator remains
consistent in the union model where either the odds-product model or the
propensity model is correctly specified. With both models correctly
specified the estimator is efficient.
with(d, riskreg_fit(y, a, target=x1, nuisance=x2, propensity=x2, type="rr"))
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9372 0.0339 0.8708 1.004 3.004e-168
The usual /formula/-syntax can be applied using the
riskreg function. Here we illustrate the double-robustness
by using a wrong propensity model but a correct nuisance
paramter (odds-product) model:
riskreg(y~a, nuisance=~x+z, propensity=~z, data=d, type="rr")
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9511 0.03333 0.8857 1.016 4.547e-179
Or vice-versa
riskreg(y~a, nuisance=~z, propensity=~x+z, data=d, type="rr")
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9404 0.03727 0.8673 1.013 1.736e-140
whereas the MLE in this case yields a biased estimate of the relative
risk:
fit2 <- with(d, riskreg_mle(y, a, x1=model.matrix(~1,d), x2=model.matrix(~z, d)))
estimate(fit2, keep=1)
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 1.243 0.02778 1.189 1.298 0
Interactions
The more general model where \[\log RR(V)
= A \alpha^TV\] for a subset \(V\) of the covariates can be estimated
using the target argument:
fit <- riskreg(y~a, target=~x, nuisance=~x+z, data=d)
fit
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.95267 0.03365 0.88673 1.01862 2.361e-176
#> x -0.01078 0.03804 -0.08534 0.06378 7.769e-01
As expected we do not see any evidence of an effect of \(X\) on the relative risk with the 95%
confidence limits clearly overlapping zero.
Note, that when the propensity argument is omitted as
above, the same design matrix is used for both the odds-product model
and the propensity model.
Risk-difference
The syntax for fitting the risk-difference model is similar. To
illustrate this we simulate some new data from this model
m2 <- lava::binomial.rd(m, response="y", exposure="a", target.model="lp.target", nuisance.model="lp.nuisance")
d2 <- lava::sim(m2, 1e4, p=p)
And we can then fit the DRE with the syntax
riskreg(y~a, nuisance=~x+z, data=d2, type="rd")
#> Estimate Std.Err 2.5% 97.5% P-value
#> (Intercept) 0.9831 0.02052 0.9429 1.023 0
Influence-function
The DRE is a regular and asymptotic linear (RAL) estimator,
hence \[\sqrt{n}(\widehat{\alpha}_{\mathrm{DRE}} -
\alpha) = \frac{1}{\sqrt{n}}\sum_{i=1}^{n} \phi_{\mathrm{eff}}(Z_{i}) +
o_{p}(1)\] where \(Z_i = (Y_i, A_i,
V_i), i=1,\ldots,n\) are the i.i.d. observations and \(\phi_{\mathrm{eff}}\) is the influence
function.
The influence function can be extracted using the IC
method
head(IC(fit))
#> (Intercept) x
#> 1 0.6226459 -0.4585424
#> 2 1.2319959 0.7925974
#> 3 0.3941326 -0.5798067
#> 4 -0.8854891 2.9621437
#> 5 -6.9949137 -5.2133648
#> 6 0.5853934 -0.7571452