For generalized linear models, let \(x_0\) and \(x_1\) be fixed constants. Let \(\psi =g\{\theta (x_0),\theta (x_1)\}\) be a function of \(\theta (x_0)\) and \(\theta (x_1)\), e.g. \(\theta (x_1)-\theta (x_0)\). Define \(\nu =\{\beta ,\theta (x_0),\theta (x_1),\psi \}\). The estimator \(\hat{\nu }=[\hat{\beta },\hat{\theta }(x_0),\hat{\theta }(x_1),g\{\hat{\theta } (x_0),\hat{\theta } (x_1)\}]\) is an M-estimator [5] that solves the estimating equation
$$\begin{aligned} \sum _{i=1}^nU_{\nu ,i}(\nu )=\sum _{i=1}^n \left[ \begin{array}{l} U_{\beta ,i}(\beta )\\ U_{\theta (x_0),i}\{\beta ,\theta (x_0)\}\\ U_{\theta (x_1),i}\{\beta ,\theta (x_1)\}\\ U_{\psi ,i}\{\theta (x_0),\theta (x_1),\psi \} \end{array}\right] =0, \end{aligned}$$
where \(U_{\beta ,i}(\beta )\) is the contribution to the maximum likelihood score function from subject i, \(U_{\theta (x),i}\{\beta ,\theta (x)\}=\eta ^{-1}\{h(X=x,Z_i;\beta )\}-\theta (x)\) for \(x=x_1\) and \(x=x_0\), and \(U_{\psi ,i}\{\theta (x_0),\theta (x_1),\psi \}=g\{\theta (x_0),\theta (x_1)\}-\psi\).
For Cox regression models, let \(x_0\), \(x_1\) and t be fixed constants. Let \(\psi =g\{\theta (t,x_0),\theta (t,x_1)\}\) be a function of \(\theta (t,x_0)\) and \(\theta (t,x_1)\), e.g. \(\theta (t,x_1)-\theta (t,x_0)\). Define \(\nu =\{\beta ,{\varLambda }_0(t),\theta (t,x_0),\theta (t,x_1),\psi \}\). The estimator \(\hat{\nu }=[\hat{\beta },\hat{{\varLambda }}_0(t),\hat{\theta }(t,x_0),\hat{\theta }(t,x_1),g\{\hat{\theta } (t,x_0),\hat{\theta } (t,x_1)\}]\) is an M-estimator [5] that solves the estimating equation
$$\begin{aligned} \sum _{i=1}^nU_{\nu ,i}(\nu )=\sum _{i=1}^n\left[ \begin{array}{l}U_{\beta ,i}(\beta )\\ U_{{\varLambda }_0(t),i}\{\beta ,{\varLambda }_0(t)\}\\ U_{\theta (t,x_0),i}\{\beta ,{\varLambda }_0(t),\theta (t,x_0)\}\\ U_{\theta (t,x_1),i}\{\beta ,{\varLambda }_0(t),\theta (t,x_1)\}\\ U_{\psi ,i}\{\theta (t,x_0),\theta (t,x_1),\psi \} \end{array}\right] =0, \end{aligned}$$
where \(U_{\beta ,i}(\beta )\) is the contribution to the Cox partial likelihood score function from subject i, \(U_{{\varLambda }_0(t),i}\{\beta ,{\varLambda }_0(t)\}\) is the contribution to the estimating function for Breslow’s estimator of the cumulative baseline hazard from subject i, \(U_{\theta (t,x),i}\{\beta ,{\varLambda }_0(t),\theta (t,x)\}=\text {exp}[-{\varLambda }_0(t)\text {exp}\{h(X=x,Z_i;\beta )\}]-\theta (t,x)\) for \(x=x_1\) and \(x=x_0\), and \(U_{\psi ,i}\{\theta (t,x_0),\theta (t,x_1),\psi \}=g\{\theta (t,x_0),\theta (t,x_1)\}-\psi\).
For both generalized linear models and Cox regression models it now follows from standard theory for M-estimators [5] that \(n^{1/2}(\hat{\nu }-\nu )\) is asymptotically normal with mean 0 and variance given by the ‘sandwich formula’
$$\begin{aligned} {\varSigma }=E^{\prime}\left\{ \frac{\partial U_{\nu ,i}(\nu )}{\partial \nu }\right\} ^{-1}\text {var}\{U_{\nu ,i}(\nu )\}E\left\{ \frac{\partial U_{\nu ,i}(\nu )}{\partial \nu }\right\} ^{-1}. \end{aligned}$$
(5)
A consistent estimate of the variance of \(\hat{\nu }\) is obtained by replacing \(\nu\) in (5) with \(\hat{\nu }\), and the population moments in (5) by their sample counterparts.
The sandwich formula assumes that \(U_{\nu ,i}(\nu )\) and \(U_{\nu ,i^{\prime}}(\nu )\) are independent, for \(i\ne i^{\prime}\). When data are clustered, as in the example in ‘Standardization with generalized linear models’ section, we may define \(U_{\nu ,i}(\nu )=\sum _{j=1}^{n_i}U_{\nu ,ij}(\nu )\), where \(U_{\nu ,ij}(\nu )\) is the contribution to the estimating equation from subject j within cluster i, and \(n_i\) is the total number of subjects in cluster i. Provided that the clusters are independent we thus have that \(U_{\nu ,i}(\nu )\) and \(U_{\nu ,i^{\prime}}(\nu )\) are independent as well, for \(i\ne i^{\prime}\), so that the sandwich formula still applies.
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