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.

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)

## Value

An object of class 'riskreg.targeted' is returned. See targeted-class

## 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.

Klaus K. Holst

## Examples

m <- lvm(a[-2] ~ x,
z ~ 1,
lp.target ~ 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"))
#>                          Estimate Std.Err    2.5%   97.5%   P-value
#> (Intercept)                0.9453  0.1150  0.7198  1.1707 2.092e-16
#> odds-product:(Intercept)  -0.8068  0.3031 -1.4009 -0.2126 7.783e-03
#> odds-product:x             2.3296  0.3914  1.5624  3.0969 2.659e-09

with(d, riskreg_fit(y, a, nuisance=X, propensity=I, type="rr"))
#>             Estimate Std.Err   2.5% 97.5%   P-value
#> (Intercept)   0.9469   0.113 0.7254 1.168 5.418e-17
riskreg(y ~ a | 1, nuisance=~x ,  data=d, type="rr")
#>             Estimate Std.Err   2.5% 97.5%   P-value
#> (Intercept)   0.9469   0.113 0.7254 1.168 5.418e-17

## Model with same design matrix for nuisance and propensity model:
with(d, riskreg_fit(y, a, nuisance=X, type="rr"))
#>             Estimate Std.Err   2.5% 97.5%   P-value
#> (Intercept)   0.9402  0.1173 0.7104  1.17 1.082e-15

## 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
#>             Estimate Std.Err    2.5%     97.5%   P-value
#> (Intercept)   0.9797  0.1212  0.7421  1.217281 6.366e-16
#> z            -0.2218  0.1115 -0.4404 -0.003198 4.674e-02

riskreg(y ~ a, nuisance=~x,  data=d, type="rr", mle=TRUE)
#>                          Estimate Std.Err    2.5%   97.5%   P-value
#> (Intercept)                0.9453  0.1150  0.7198  1.1707 2.092e-16
#> odds-product:(Intercept)  -0.8068  0.3031 -1.4009 -0.2126 7.783e-03
#> odds-product:x             2.3296  0.3914  1.5624  3.0969 2.659e-09