# Regression standardization in generalized linear models `stdGlm` performs regression standardization in generalized linear models, at specified values of the exposure, over the sample covariate distribution. Let $Y$, $X$, and $Z$ be the outcome, the exposure, and a vector of covariates, respectively. `stdGlm` uses a fitted generalized linear model to estimate the standardized mean $\theta(x)=E\{E(Y|X=x,Z)\}$, where $x$ is a specific value of $X$, and the outer expectation is over the marginal distribution of $Z$. ```r stdGlm(fit, data, X, x, clusterid, case.control = FALSE, subsetnew) ``` ## Arguments - `fit`: an object of class `"glm"`, as returned by the `glm` function in the `stats` package. If arguments `weights` and/or `subset` are used when fitting the model, then the same weights and subset are used in `stdGlm`. - `data`: a data frame containing the variables in the model. This should be the same data frame as was used to fit the model in `fit`. - `X`: a string containing the name of the exposure variable $X$ in `data`. - `x`: an optional vector containing the specific values of $X$ at which to estimate the standardized mean. If $X$ is binary (0/1) or a factor, then `x` defaults to all values of $X$. If $X$ is numeric, then `x` defaults to the mean of $X$. If `x` is set to `NA`, then $X$ is not altered. This produces an estimate of the marginal mean $E(Y)=E\{E(Y|X,Z)\}$. - `clusterid`: an optional string containing the name of a cluster identification variable when data are clustered. - `case.control`: logical. Do data come from a case-control study? Defaults to FALSE. - `subsetnew`: an optional logical statement specifying a subset of observations to be used in the standardization. This set is assumed to be a subset of the subset (if any) that was used to fit the regression model. ## Details `stdGlm` assumes that a generalized linear model $$ \eta\{E(Y|X,Z)\}=h(X,Z;\beta) $$ has been fitted. The maximum likelihood estimate of $\beta$ is used to obtain estimates of the mean $E(Y|X=x,Z)$: $$ \hat{E}(Y|X=x,Z)=\eta^{-1}\{h(X=x,Z;\hat{\beta})\}. $$ For each $x$ in the `x` argument, these estimates are averaged across all subjects (i.e. all observed values of $Z$) to produce estimates $$ \hat{\theta}(x)=\sum_{i=1}^n \hat{E}(Y|X=x,Z_i)/n, $$ where $Z_i$ is the value of $Z$ for subject $i$, $i=1,...,n$. The variance for $\hat{\theta}(x)$ is obtained by the sandwich formula. ## Returns An object of class `"stdGlm"` is a list containing - **call**: the matched call. - **input**: `input` is a list containing all input arguments. - **est**: a vector with length equal to `length(x)`, where element `j` is equal to $\hat{\theta}$(`x[j]`). - **vcov**: a square matrix with `length(x)` rows, where the element on row `i` and column `j` is the (estimated) covariance of $\hat{\theta}$(`x[i]`) and $\hat{\theta}$(`x[j]`). ## References Rothman K.J., Greenland S., Lash T.L. (2008). **Modern Epidemiology**, 3rd edition. Lippincott, Williams and Wilkins. Sjolander A. (2016). Regression standardization with the R-package stdReg. **European Journal of Epidemiology** 31 (6), 563-574. Sjolander A. (2016). Estimation of causal effect measures with the R-package stdReg. **European Journal of Epidemiology** 33 (9), 847-858. ## Author(s) Arvid Sjolander. ## Note The variance calculation performed by `stdGlm` does not condition on the observed covariates $\bar{Z}=(Z_1,...,Z_n)$. To see how this matters, note that $$ var\{\hat{\theta}(x)\}=E[var\{\hat{\theta}(x)|\bar{Z}\}]+var[E\{\hat{\theta}(x)|\bar{Z}\}]. $$ The usual parameter $\beta$ in a generalized linear model does not depend on $\bar{Z}$. Thus, $E(\hat{\beta}|\bar{Z})$ is independent of $\bar{Z}$ as well (since $E(\hat{\beta}|\bar{Z})=\beta$), so that the term $var[E\{\hat{\beta}|\bar{Z}\}]$ in the corresponding variance decomposition for $var(\hat{\beta})$ becomes equal to 0. However, $\theta(x)$ depends on $\bar{Z}$ through the average over the sample distribution for $Z$, and thus the term $var[E\{\hat{\theta}(x)|\bar{Z}\}]$ is not 0, unless one conditions on $\bar{Z}$. ## Examples ```r ##Example 1: continuous outcome n <- 1000 Z <- rnorm(n) X <- rnorm(n, mean=Z) Y <- rnorm(n, mean=X+Z+0.1*X^2) dd <- data.frame(Z, X, Y) fit <- glm(formula=Y~X+Z+I(X^2), data=dd) fit.std <- stdGlm(fit=fit, data=dd, X="X", x=seq(-3,3,0.5)) print(summary(fit.std)) plot(fit.std) ##Example 2: binary outcome n <- 1000 Z <- rnorm(n) X <- rnorm(n, mean=Z) Y <- rbinom(n, 1, prob=(1+exp(X+Z))^(-1)) dd <- data.frame(Z, X, Y) fit <- glm(formula=Y~X+Z+X*Z, family="binomial", data=dd) fit.std <- stdGlm(fit=fit, data=dd, X="X", x=seq(-3,3,0.5)) print(summary(fit.std)) plot(fit.std) ```

stdGlm function