2026-07-23
Source:vignettes/riskregression.qmd
Let Y be a binary response, A a binary exposure, and V a vector of covariates.
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).
First the targeted package is loaded
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(), ...)
NULLthen 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(), ...)
NULLWe 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
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 0Notice, that in this simulated data the target parameter \mu_{RR} has been set to lp.target = 1.
We start by fitting the model using the maximum likelihood estimator.
args(riskreg_mle)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-86The 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-179We 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-168The 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-179Or 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-140whereas 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 0The 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-01As 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.
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) 0.9899 0.0201 0.9505 1.029 0The 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
R version 4.6.1 (2026-06-24)
Platform: aarch64-apple-darwin25.5.0
Running under: macOS Tahoe 26.5.2
Matrix products: default
BLAS: /Users/klaus/.asdf/installs/r/4.6.1/lib/R/lib/libRblas.dylib
LAPACK: /Users/klaus/.asdf/installs/r/4.6.1/lib/R/lib/libRlapack.dylib; LAPACK version 3.12.1
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: Europe/Copenhagen
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] targeted_0.8
loaded via a namespace (and not attached):
[1] mets_1.3.12 cli_3.6.6 knitr_1.51
[4] rlang_1.3.0 xfun_0.60 otel_0.2.0
[7] jsonlite_2.0.0 future.apply_1.20.2 listenv_1.0.0
[10] lava_1.9.2 htmltools_0.5.9 rmarkdown_2.31
[13] grid_4.6.1 evaluate_1.0.5 fastmap_1.2.0
[16] numDeriv_2016.8-1.1 mvtnorm_1.4-2 yaml_2.3.12
[19] timereg_2.0.7 compiler_4.6.1 codetools_0.2-20
[22] Rcpp_1.1.2 future_1.70.0 lattice_0.22-9
[25] digest_0.6.39 R6_2.6.1 parallelly_1.48.0
[28] parallel_4.6.1 splines_4.6.1 Matrix_1.7-5
[31] RcppArmadillo_15.4.0-1 tools_4.6.1 globals_0.19.1
[34] survival_3.8-6