riskreg: Risk regression

In targeted: Targeted Inference

Risk regression

Description

Risk regression with binary exposure and nuisance model for the odds-product.

Let A be the binary exposure, V the set of covariates, and Y the binary response variable, and define p_a(v) = P(Y=1 \mid A=a, V=v), a\in\{0,1\}.

The target parameter is either the relative risk

\mathrm{RR}(v) = \frac{p_1(v)}{p_0(v)}

or the risk difference

\mathrm{RD}(v) = p_1(v)-p_0(v)

We assume a target parameter model given by either

\log\{RR(v)\} = \alpha^t v

or

\mathrm{arctanh}\{RD(v)\} = \alpha^t v

and similarly a working linear 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) = \beta^t v

.

A propensity model for E(A=1|V) is also fitted using a logistic regression working model

\mathrm{logit}\{E(A=1\mid V=v)\} = \gamma^t v.

If both the odds-product model and the propensity model are correct the estimator is efficient. Further, the estimator is consistent in the union model, i.e., the estimator is double-robust in the sense that only one of the two models needs to be correctly specified to get a consistent estimate.

Usage

riskreg(
  formula,
  nuisance = ~1,
  propensity = ~1,
  target = ~1,
  data,
  weights,
  type = "rr",
  optimal = TRUE,
  std.err = TRUE,
  start = NULL,
  mle = FALSE,
  ...
)

Arguments

formula	formula (see details below)
nuisance	nuisance model (formula)
propensity	propensity model (formula)
target	(optional) target model (formula)
data	data.frame
weights	optional weights
type	type of association measure (rd og rr)
optimal	If TRUE optimal weights are calculated
std.err	If TRUE standard errors are calculated
start	optional starting values
mle	Semi-parametric (double-robust) estimate or MLE (TRUE gives MLE)
...	additional arguments to unconstrained optimization routine (nlminb)

Details

The 'formula' argument should be given as response ~ exposure | target-formula | nuisance-formula or response ~ exposure | target | nuisance | propensity

E.g., riskreg(y ~ a | 1 | x+z | x+z, data=...)

Alternatively, the model can specifed using the target, nuisance and propensity arguments: riskreg(y ~ a, target=~1, nuisance=~x+z, ...)

The riskreg_fit function can be used with matrix inputs rather than formulas.

Value

An object of class 'riskreg.targeted' is returned. See targeted-class for more details about this class and its generic functions.

Author(s)

Klaus K. Holst

References

Richardson, T. S., Robins, J. M., & Wang, L. (2017). On modeling and estimation for the relative risk and risk difference. Journal of the American Statistical Association, 112(519), 1121–1130. http://dx.doi.org/10.1080/01621459.2016.1192546

Examples

m <- lvm(a[-2] ~ x,
         z ~ 1,
         lp.target[1] ~ 1,
         lp.nuisance[-1] ~ 2*x)
distribution(m,~a) <- binomial.lvm("logit")
m <- binomial.rr(m, "y","a","lp.target","lp.nuisance")
d <- sim(m,5e2,seed=1)

I <- model.matrix(~1, d)
X <- model.matrix(~1+x, d)
with(d, riskreg_mle(y, a, I, X, type="rr"))

with(d, riskreg_fit(y, a, nuisance=X, propensity=I, type="rr"))
riskreg(y ~ a | 1, nuisance=~x ,  data=d, type="rr")

## Model with same design matrix for nuisance and propensity model:
with(d, riskreg_fit(y, a, nuisance=X, type="rr"))

## a <- riskreg(y ~ a, target=~z, nuisance=~x,  propensity=~x, data=d, type="rr")
a <- riskreg(y ~ a | z, nuisance=~x,  propensity=~x, data=d, type="rr")
a
predict(a, d[1:5,])

riskreg(y ~ a, nuisance=~x,  data=d, type="rr", mle=TRUE)

targeted documentation built on May 29, 2024, 2:08 a.m.