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Estimating a relative risk or risk
difference with a binary exposure
Klaus Kähler Holst
2024-02-22
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):
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
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.
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.
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
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 \(Z\)
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.
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:
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
These binaries (installable software) and packages are in development. They may not be fully stable and should be used with caution. We make no claims about them.
