# Instrumental variable estimation of the causal exposure effect in Cox proportional hazards (PH) models `ivcoxph` performs instrumental variable estimation of the causal exposure effect in Cox PH models with individual-level data. Below, $Z$, $X$, and $T$ are the instrument, the exposure, and the outcome, respectively. $L$ is a vector of covariates that we wish to control for in the analysis; these would typically be confounders for the instrument and the outcome. ```r ivcoxph(estmethod, X, fitZ.L=NULL, fitX.LZ=NULL, fitX.L=NULL, fitT.LX=NULL, fitT.LZX=NULL, data, formula=~1, ctrl=FALSE, clusterid=NULL, t=NULL, vcov.fit=TRUE, ...) ``` ## Arguments - `estmethod`: a string specifying the desired estimation method; either `"ts"` for two-stage estimation, or `"g"` for G-estimation. - `X`: a string specifying the name of the exposure $X$ in `data`. This is not needed if `fitX.LZ` is specified. - `fitZ.L`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. This is a fitted GLM for $E(Z|L)$. If there are no covariates, then `fitZ.L` may be specified as a model with an intercept only. This argument is not used when `estmethod="ts"`. - `fitX.LZ`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. This is a fitted GLM for $E(X|L,Z)$. - `fitX.L`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. This is a fitted GLM for $E(X|L)$. If there are no covariates, then `fitX.L` may be specified as a model with an intercept only. This argument is not used when `estmethod="ts"`. - `fitT.LX`: an object of class `"coxph"`, as returned by the `coxph` function in the `survival` package. This is a fitted Cox PH model for $\lambda(t|L,X)$. This argument is not used when `estmethod="g"`. - `fitT.LZX`: either an object of class `"coxph"` or an object of class `"survfit"`, as returned by the `coxph` function in the `survival` package. This is a fitted Cox PH model for $\lambda(t|L,Z,X)$ or a non-parametric model for $S(t|L,Z,X)$, respectively. This argument is not used when `estmethod="ts"`. - `data`: a data frame containing the variables in the model. The covariates, instrument, exposure and outcome can have arbitrary names, e.g. they don't need to be called `L`, `Z`, `X` and `T`. - `formula`: an object of class `"formula"`, with no left-hand side. This specifies the causal interaction terms $m(L)$; see `Details'. Defaults to `~1`, i.e. main effect only. This argument is not used when `estmethod="ts"`. - `ctrl`: logical. Should the control function $R=X-\hat{X}$ be used when re-fitting `fitT.LX`? This argument is not used when `estmethod="g"`. - `clusterid`: an optional string containing the name of a cluster identification variable when data are clustered. Specifying `clusterid` corrects the standard errors but does not affect the estimates. - `t`: a numeric scalar specifying the time point at which to solve the estimating equation when `estmethod="g"`; see `Details'. If not specified, then the estimating equation is solved at the optimal value of `t`, defined as the value that minimizes $trace\{var(\hat{\psi})\}$; see Martinussen et al (2017). This argument is not used when `estmethod="ts"`. - `vcov.fit`: logical. Should the variance-covariance matrix be computed? - `...`: optional arguments passed on to the `nleqslv` function, which is used to solve the estimating equations when `estmethod="g"`. See the help pages for `nleqslv`. This argument is not used when `estmethod="ts"`. ## Details `ivcoxph` estimates the parameter $\psi$ in the causal Cox PH model $$ \textrm{log}\{\lambda(t|L,Z,X)\}-\textrm{log}\{\lambda_0(t|L,Z,X)\}=m^T(L)X\psi. $$ Here, $\lambda_0(t|L,Z,X)$ is counterfactual hazard function, had the exposure been set to 0. The vector function $m(L)$ contains interaction terms between $L$ and $X$. If `estmethod="ts"`, then these are specified implicitly through the model `fitT.LX`. If `estmethod="g"`, then these are specified explicitly through the `formula` argument. If `estmethod="ts"`, then two-stage estimation of $\psi$ is performed. In this case, the model `fitX.LZ` is used to construct predictions $\hat{X}=\hat{E}(X|L,Z)$. These predictions are subsequently used to re-fit the model `fitT.LX`, with $X$ replaced with $\hat{X}$. The obtained coefficient(s) for $\hat{X}$ in the re-fitted model is the two-stage estimator of $\psi$. If `estmethod="g"`, then G-estimation of $\psi$ is performed. In this case, the estimator is obtained as the solution to the estimating equation $$ H(\psi)=\sum_{i=1}^n\hat{d}(L_i,Z_i)h_i(\psi;t)=0, $$ where $$ h_i(\psi;t)=\hat{S}(t|L_i,Z_i,X_i)^{\textrm{exp}\{-m^T(L_i)\psi X_i\}}. $$ The estimated function $\hat{d}(L,Z)$ is chosen so that the true function has conditional mean 0, given $L$; $E\{d(L,Z)|L\}=0$. The specific form of $\hat{d}(L,Z)$ is determined by the user-specified models. If `fitX.LZ` and `fitX.L` are specified, then $\hat{d}(L,Z)=m(L)\{\hat{E}(X|L,Z)-\hat{E}(X|L)\}$, where $\hat{E}(X|L,Z)$ and $\hat{E}(X|L)$ are obtained from `fitX.LZ` and `fitX.L`, respectively. If these are not specified, then $\hat{d}(L,Z)=m(L)\{Z-\hat{E}(Z|L)\}$, where $\hat{E}(Z|L)$ is obtained from `fitZ.L`, which then must be specified. The estimating equation is solved at the value of $t$ specified by the argument `t`. $\hat{S}(t|L_i,Z_i,X_i)$ is an estimate of $S(t|L_i,Z_i,X_i)$ obtained from the model `fitT.LZX`. ## Returns `ivcoxph` returns an object of class `"ivcoxph"`, which inherits from class `"ivmod"`. An object of class `"ivcoxph"` is a list containing - **call**: the matched call. - **input**: `input` is a list containing all input arguments - **est**: a vector containing the estimate of $\psi$. - **vcov**: the variance-covariance matrix for the estimate of $\psi$, obtained with the sandwich formula. - **estfunall**: a matrix of all subject-specific contributions to the estimating functions used in the estimation process. One row for each subject, one column for each parameter. If `estmethod="ts"`, then the first columns correspond to the parameters estimated by `fitX.LZ`, and the last columns correspond to the parameters estimated by the re-fitted model `fitT.LX`. If `estmethod="g"`, then the first columns correspond to $\psi$, and the remaining columns correspond the parameters estimated by `fitZ.L`, `fitX.LZ`, `fitX.L` and `fitT.LZX`, whichever were used in the estimation process. - **d.estfun**: the jacobian matrix of `colMeans(estfunall)`. - **converged**: logical. Was a solution found to the estimating equations? - **fitT.LX**: the re-fitted model `fitT.LX` used in the estimation process when `estmethod="ts"`. This element is NULL when `estmethod="g"`. - **t**: the value of `t` used in the estimation process. This element is NULL when `estmethod="ts"`. ## References Martinussen T., Sorensen D.D., Vansteelandt S. (2019). Instrumental variables estimation under a structural Cox model. **Biostatistics** 20 (1), 65-79. Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. **Epidemiologic Methods** 8 (1), 1-20. Tchetgen Tchetgen E.J., Walter S., Vansteelandt S., Martinussen T., Glymour M. (2015). Instrumental variable estimation in a survival context. **Epidemiology** 26 (3), 402-410. ## Author(s) Arvid Sjolander. ## Note `ivcoxph` allows for weights. However, these are defined implicitly through the input models. Thus, when models are used as input to `ivcoxph`, these models have to be fitted with the same weights. When `estmethod="g"` the weights are taken from `fitX.LZ`, if specified by the user. If `fitX.LZ` is not specified then the weights are taken from `fitZ.L`. Hence, if weights are used, then either `fitX.LZ` or `fitZ.L` must be specified. ## Examples ```r require(survival) set.seed(9) ##Note: the parameter values in the examples below are chosen to make ##Y0 independent of Z, which is necessary for Z to be a valid instrument. n <- 10000 psi0 <- 0.5 Z <- rbinom(n, 1, 0.5) X <- rbinom(n, 1, 0.7*Z+0.2*(1-Z)) m0 <- exp(0.8*X-0.41*Z) #T0 independent of Z at t=1 T <- rexp(n, rate=exp(psi0*X+log(m0))) C <- rexp(n, rate=exp(psi0*X+log(m0))) #50% censoring d <- as.numeric(T

ivcoxph function