# Instrumental variable estimation of the causal exposure effect in generalized linear models `ivglm` performs instrumental variable estimation of the causal exposure effect in generalized linear models with individual-level data. Below, $Z$, $X$, and $Y$ 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 ivglm(estmethod, X, Y, fitZ.L=NULL, fitX.LZ=NULL, fitX.L=NULL, fitY.LX=NULL, fitY.LZX=NULL, data, formula=~1, ctrl=FALSE, clusterid=NULL, link, 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. - `Y`: a string specifying the name of the outcome $Y$ in `data`. This is not needed if `fitY.LX` or `fitY.LZX` 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"`. - `fitY.LX`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. This is a fitted GLM for $E(Y|L,X)$. This argument is not used when `estmethod="g"`. - `fitY.LZX`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. This is a fitted GLM for $E(Y|L,Z,X)$. This argument is not used when `estmethod="ts"`. It is also not used when `estmethod="g"` and `link="identity"` or `link="log"`. - `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 `Y`. - `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 `fitY.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. - `link`: a string specifying the link function for the causal generalized linear model; see `Details'. Either `"identity"`, `"log"`, or `"logit"`. 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 `ivglm` estimates the parameter $\psi$ in the causal generalized linear model $$ \eta\{E(Y|L,Z,X)\}-\eta\{E(Y_0|L,Z,X)\}=m^T(L)X\psi. $$ Here, $E(Y_0|L,Z,X)$ is counterfactual mean of the outcome, had the exposure been set to 0. The link function $\eta$ is either the identity, log or logit link, as specified by the `link` argument. The vector function $m(L)$ contains interaction terms between $L$ and $X$. If `estmethod="ts"`, then these are specified implicitly through the model `fitY.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 `fitY.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 $$ H(\psi)=\sum_{i=1}^n\hat{d}(L_i,Z_i)h_i(\psi)=0. $$ The function $h_i(\psi)$ is defined as $$ h_i(\psi)=Y_i-m^T(L_i)\psi X_i $$ when `link="identity"`, $$ h_i(\psi)=Y_i\textrm{exp}\{-m^T(L_i)\psi X_i\} $$ when `link="log"`, and $$ h_i(\psi)=\textrm{expit}[\textrm{logit}\{\hat{E}(Y|L_i,Z_i,X_i)\}-m^T(L_i)\psi X_i] $$ when `link="logit"`. In the latter, $\hat{E}(Y|L_i,Z_i,X_i)$ is an estimate of $E(Y|L_i,Z_i,X_i)$ obtained from the model `fitY.LZX`. 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. ## Returns `ivglm` returns an object of class `"ivglm"`, which inherits from class `"ivmod"`. An object of class `"ivglm"` 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 `fitY.LX`. If `estmethod="g"`, then the first columns correspond to $\psi$, and the remaining columns correspond to the parameters estimated by `fitZ.L`, `fitX.LZ`, `fitX.L` and `fitY.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? - **fitY.LX**: the re-fitted model `fitY.LX` used in the estimation process when `estmethod="ts"`. This element is NULL when `estmethod="g"`. ## References Bowden J., Vansteelandt S. (2011). Mendelian randomization analysis of case-control data using structural mean models. **Statistics in Medicine** 30 (6), 678-694. Sjolander A., Martinussen T. (2019). Instrumental variable estimation with the R package ivtools. **Epidemiologic Methods** 8 (1), 1-20. Vansteelandt S., Bowden J., Babanezhad M., Goetghebeur E. (2011). On instrumental variables estimation of causal odds ratios. **Statistical Science** 26 (3), 403-422. ## Author(s) Arvid Sjolander. ## Note `ivglm` allows for weights. However, these are defined implicitly through the input models. Thus, when models are used as input to `ivglm`, 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 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 <- 1000 psi0 <- 0.5 psi1 <- 0.2 ##---Example 1: linear model and interaction between X and L--- L <- rnorm(n) Z <- rnorm(n, mean=L) X <- rnorm(n, mean=Z) m0 <- X-Z+L Y <- rnorm(n, mean=psi0*X+psi1*X*L+m0) data <- data.frame(L, Z, X, Y) #two-stage estimation fitX.LZ <- glm(formula=X~Z, data=data) fitY.LX <- glm(formula=Y~X+L+X*L, data=data) fitIV <- ivglm(estmethod="ts", fitX.LZ=fitX.LZ, fitY.LX=fitY.LX, data=data, ctrl=TRUE) summary(fitIV) #G-estimation with model for Z fitZ.L <- glm(formula=Z~L, data=data) fitIV <- ivglm(estmethod="g", X="X", Y="Y", fitZ.L=fitZ.L, data=data, formula=~L, link="identity") summary(fitIV) #G-estimation with model for X fitX.LZ <- glm(formula=X~L+Z, data=data) fitX.L <- glm(formula=X~L, data=data) fitIV <- ivglm(estmethod="g", Y="Y", fitX.LZ=fitX.LZ, fitX.L=fitX.L, data=data, formula=~L, link="identity") summary(fitIV) ##---Example 2: logistic model and no covariates--- Z <- rbinom(n, 1, 0.5) X <- rbinom(n, 1, 0.7*Z+0.2*(1-Z)) m0 <- plogis(1+0.8*X-0.39*Z) Y <- rbinom(n, 1, plogis(psi0*X+log(m0/(1-m0)))) data <- data.frame(Z, X, Y) #two-stage estimation fitX.LZ <- glm(formula=X~Z, family="binomial", data=data) fitY.LX <- glm(formula=Y~X, family="binomial", data=data) fitIV <- ivglm(estmethod="ts", fitX.LZ=fitX.LZ, fitY.LX=fitY.LX, data=data, ctrl=TRUE) summary(fitIV) #G-estimation with model for Z fitZ.L <- glm(formula=Z~1, data=data) fitY.LZX <- glm(formula=Y~X+Z+X*Z, family="binomial", data=data) fitIV <- ivglm(estmethod="g", X="X", fitZ.L=fitZ.L, fitY.LZX=fitY.LZX, data=data, link="logit") summary(fitIV) ```

ivglm function