Model-based inference on average causal effect in observational clustered data

Abstract

We study causal inference using the framework of potential outcomes in clustered data settings where observational units are clustered in naturally occurring groups (e.g. patients within hospitals). To incorporate the correlated nature of the data, we employ mixed-effects models and a sandwich estimator to make inferences on the average causal effect (ACE). Our methods apply the concept of potential outcomes from the Rubin Causal Model (Holland in J Am Stat Assoc 81(396):945–960, 1986), and extend Schafer and Kang’s methods of estimating the variance of the ACE (Schafer and Kang in Psychol Methods 13(4):279–313, 2008). Particularly, we develop two model-based approaches to estimate the ACE and its variance under a dual-modeling strategy which adjusts for the confounding effect through inverse probability weighting. These two approaches use linear mixed-effects models for the estimation of potential outcomes, but differ in how clustering is handled in the treatment assignment model. We present a summary of our comprehensive simulation study assessing the repetitive sampling properties of the two approaches in a pseudo-random simulation environment. Finally, we report our findings from an application to study the ACE of inadequate prenatal care on birth weight among low-income women in New York State.

Appendix: Formulas for ACE variance estimation in clustered data

Some of the equations for ACE estimate and its variance estimate have been presented in the paper. Here we show the formulas for the computation behind the equations. In method 1, residuals from linear mixed-effects model are adjusted by inverse propensity scores that are estimated by logistic regression. The OLS estimates from linear mixed-effects model (Demidenko 2004) are:

$$\begin{aligned} {\hat{\beta }}_0&=\left( x_{ij0}^Tx_{ij0}\right) ^{-1}x_{ij0}^T\bar{y}_{i0},\\ {\hat{\beta }}_1&=\left( x_{ij1}^Tx_{ij1}\right) ^{-1}x_{ij1}^T\bar{y}_{i1}, \\ {\hat{\alpha }}_0&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(\bar{y}_{ij0}-\bar{x}_{i0}{\hat{\beta }}_0),\\ {\hat{\alpha }}_1&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(\bar{y}_{ij1}-\bar{x}_{i1}{\hat{\beta }}_1),\\ {\hat{\mu }}_0&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(x_{ij}\beta _0), \\ {\hat{\mu }}_1&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(x_{ij}\beta _1),\\ {\hat{\epsilon }}_1&=\frac{\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}{\hat{\pi }}_{ij}^{-1}(y_{ij}-x_{ij}{\hat{\beta }}_1-{\hat{\alpha }}_{i1})}{\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1}},\\ {\hat{\epsilon }}_0&=\frac{\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}(y_{ij}-x_{ij}{\hat{\beta }}_0-{\hat{\alpha }}_{i0})}{\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}} \end{aligned}$$

The matrix A is a 9 by 9 lower triangle matrix:

$$\begin{aligned} \begin{bmatrix} A_{11}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad A_{22}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{33}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad A_{42}&\quad 0&\quad A_{44}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{53}&\quad 0&\quad A_{55}&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad A_{62}&\quad 0&\quad 0&\quad 0&\quad A_{66}&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{73}&\quad 0&\quad 0&\quad 0&\quad A_{77}&\quad 0&\quad 0\\ A_{81}&\quad A_{82}&\quad 0&\quad 0&\quad 0&\quad A_{86}&\quad 0&\quad A_{88}&\quad 0\\ A_{91}&\quad 0&\quad A_{93}&\quad 0&\quad 0&\quad 0&\quad A_{97}&\quad 0&\quad A_{99} \end{bmatrix} \end{aligned}$$

where

$$\begin{aligned} \hat{A}_{11}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}{\hat{\pi }}_{ij}(1-\hat{\pi }_{ij})z_{ij}z_{ij}^T,\quad \hat{A}_{22}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})x_{ij}^Tx_{ij},\\ \hat{A}_{33}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}x_{ij}^Tx_{ij},\quad \hat{A}_{42}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}x_{ij}, \quad \hat{A}_{44}=-1, \quad \hat{A}_{53}=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}x_{ij}, \\ \hat{A}_{55}&=-1, \quad \hat{A}_{62}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\bar{x}_{i0},\quad \hat{A}_{66}=-1,\quad \hat{A}_{73}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\bar{x}_{i1},\quad \hat{A}_{77}=-1,\\ \hat{A}_{81}&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij}){\hat{\pi }}_{ij}(1-\hat{\pi }_{ij})^{-1}(y_{ij}-x_{ij}{\hat{\beta }}_0-\alpha _0-\epsilon _0)z_{ij}^T,\\ \hat{A}_{91}&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1}(1-{\hat{\pi }}_{ij})(y_{ij}-x_{ij}{\hat{\beta }}_1-\alpha _1-\epsilon _1)z_{ij}^T,\\ \hat{A}_{82}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}x_{ij},\quad \hat{A}_{93}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1}x_{ij},\\ \hat{A}_{86}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1},\quad \hat{A}_{97}=t_{ij}{\hat{\pi }}_{ij}^{-1},\\ \hat{A}_{88}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1},\quad \hat{A}_{99}=t_{ij}{\hat{\pi }}_{ij}^{-1} \end{aligned}$$

ACE can be estimated by Eq. (8) and its variance can be estimated by Eq. (7).

In method 2, residuals are adjusted by inverse propensity scores that are estimated by logistic mixed-effects model. The re-written equation ${\hat{{\textit{ACE}}}}=a^T\theta$ has a=(0,0,0,0,-1,1,-1,1,-1,1)$^T$ and ${\hat{\theta }}=({\hat{\gamma }}^T,{\hat{\sigma }}^2,{\hat{\beta }}_0^T,{\hat{\beta }}_1^T,{\hat{\mu }}_0^T,{\hat{\mu }}_1^T,{\hat{\alpha }}_0^T,{\hat{\alpha }}_1^T,{\hat{\epsilon }}_0^T,{\hat{\epsilon }}_1^T)$. The maximum likelihood estimates for the linear mixed-effects model are the same as in method 2. ${\hat{\gamma }}^T$ and ${\hat{\sigma }}^2$are estimated from logistic mixed-effects model: $P_{t_{ij}=1}(z_{ij};\gamma ,\zeta _i)=\pi _{ij}$ and $P_{t_{ij}=0}(z_{ij};\gamma ,\zeta _i)=1-\pi _{ij}$, where $\pi _{ij}=\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}$ and $\zeta \sim N(0,\sigma )^2)$. The log-likelihood for treatment group is: $l(\gamma ,\sigma ^2)=-\frac{N}{2}ln(2\pi \sigma ^2)+\gamma M+\sum _{i-1}^{m} ln\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta$, where $M=\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}z_{ij}, h_i(\gamma ,\zeta _i)=k_i\zeta _i-\frac{\zeta _i^2}{2\sigma ^2}-\sum \limits _{j=1}^{n_i}ln(1+e^{z_{ij}^T\gamma +\zeta _i}),K_i=\sum _{j=1}^{n_i}t_{ij}$. The maximum likelihood estimates can be obtained from:

$$\begin{aligned} \frac{\partial l(\gamma ,\sigma ^2)}{\partial \gamma }=M-\sum _{i=1}^m\frac{I_{i3}}{I_{i1}}, \frac{\partial l(\gamma ,\sigma ^2)}{\partial \sigma ^2}=-\frac{N}{2\sigma ^2}+\frac{1}{2\sigma ^4}\sum _{i=1}^m\frac{I_{i2}}{I_{i1}} \end{aligned}$$

(10)

where $I_{i1}=\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta ,I_{i2}=\int _{-\infty }^{\infty }\zeta ^2e^{h_i(\gamma ,\zeta )}d\zeta ,I_{i3}=\int _{-\infty }^{\infty }\sum _{j=1}^{n_i}Z_{ij}\frac{e^{h_i(\gamma ,\zeta )}}{1+e^{h_i(\gamma ,\zeta )}}e^{h_i(\gamma ,\zeta )}]d\zeta$

Matrix A is a 10 by 10 lower triangle matrix as:

$$\begin{aligned} \begin{bmatrix} A_{11}&\quad A_{12}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ A_{21}&\quad A_{22}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{33}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad A_{44}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{53}&\quad 0&\quad A_{55}&\quad 0&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad A_{64}&\quad 0&\quad A_{66}&\quad 0&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad A_{73}&\quad 0&\quad 0&\quad 0&\quad A_{77}&\quad 0&\quad 0&\quad 0\\ 0&\quad 0&\quad 0&\quad A_{84}&\quad 0&\quad 0&\quad 0&\quad A_{88}&\quad 0&\quad 0\\ A_{91}&\quad A_{92}&\quad A_{93}&\quad 0&\quad 0&\quad 0&\quad A_{97}&\quad 0&\quad A_{99}&\quad 0\\ A_{101}&\quad A_{102}&\quad 0&\quad A_{104}&\quad 0&\quad 0&\quad 0&\quad A_{108}&\quad 0&\quad A_{1010}\\ \end{bmatrix} \end{aligned}$$

where

$$\begin{aligned} \hat{A}_{11}\,&=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\left\{ -\frac{1}{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta }\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}z_{ij}^T\frac{e^{z_{ij}^T\gamma +\zeta _i}}{(1+e^{z_{ij}^T\gamma +\zeta _i})^2}e^{h_i(\gamma ,\zeta )}d\zeta \right. \\&\quad \left. + \left( \frac{\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij} \frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta }\right) ^2 \right\} ,\\ \hat{A}_{21}\,&=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\frac{1}{2\sigma ^4} \left\{ -\frac{\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}\zeta ^2d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta } \right. \\&\quad \left. +\frac{\int _{-\infty }^{\infty }\zeta ^2e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta } \right\} ,\\ \hat{A}_{91}&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}(1-\hat{\pi }_{ij})^{-1}(y_{ij}-x_{ij}{\hat{\beta }}_1-\alpha _1-\epsilon _1)z_{ij}^T,\\ \hat{A}_{101}&=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1}(1-{\hat{\pi }}_{ij})(y_{ij}-x_{ij}{\hat{\beta }}_1-\alpha _1-\epsilon _1)z_{ij}^T,\\ \hat{A}_{12}\,&=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\left\{ -\frac{\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}\frac{\zeta ^2}{2\sigma ^4}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta }, \right. \\&\quad \left. +\frac{\int _{-\infty }^{\infty }\frac{\zeta ^2}{2\sigma ^4}e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta } \right\} ,\\ \hat{A}_{22}\,&=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\left\{ \frac{1}{2\sigma ^4}-\frac{1}{\sigma ^6}\frac{\int _{-\infty }^{\infty }\zeta ^2e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta }+\frac{1}{2\sigma ^4} \left( \frac{\int _{-\infty }^{\infty }\frac{\zeta ^4}{2\sigma ^4}e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta } \right. \right. \\&\quad \left. \left. -\frac{\int _{-\infty }^{\infty }\zeta ^2e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }\frac{\zeta ^2}{2\sigma ^4}e^{h_i(\gamma ,\zeta )}d\zeta }{\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta \int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta } \right) \right\} ,\\ \hat{A}_{92}\,&=\,0, \quad \hat{A}_{102}\,=\,0, \quad \hat{A}_{33}\,=\,-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})x_{ij}^Tx_{ij}, \quad \hat{A}_{53}\,=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\ x_{ij}, \\ \hat{A}_{73}\,&=\,\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\ \bar{x}_{i0}, \quad \hat{A}_{93}\,=\,-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}x_{ij},\\ \hat{A}_{44}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}x_{ij}^Tx_{ij}, \quad \hat{A}_{64}=\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}x_{ij},\\ \hat{A}_{84}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}\bar{x}_{i1}, \quad \hat{A}_{104}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1}x_{ij}, \quad \hat{A}_{55}=-1, \quad \hat{A}_{66}=-1, \quad \hat{A}_{77}=-1,\\ \hat{A}_{97}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}, \quad \hat{A}_{88}=-1, \quad \hat{A}_{108}=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1},\\ \hat{A}_{99}&=-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}(1-t_{ij})(1-{\hat{\pi }}_{ij})^{-1}, \quad \hat{A}_{1010} =-\frac{1}{N}\sum \limits _{i}^{m}\sum \limits _{j}^{n_i}t_{ij}{\hat{\pi }}_{ij}^{-1} \end{aligned}$$

The 8 integrals in the elements of matrix A can be approximated by the method of Gauss-Hermite quadrature for integrals as described by Demidenko (2004).

$$\begin{aligned} f_i(1)&=\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta , \quad f_i(2)=\int _{-\infty }^{\infty }\zeta ^2e^{h_i(\gamma ,\zeta )}d\zeta , \quad f_i(3)=\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}d\zeta ,\\ f_i(4)&=\int _{-\infty }^{\infty }\frac{\zeta ^2}{2\sigma ^4}e^{h_i(\gamma ,\zeta )}d\zeta , \quad f_i(5)=\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}z_{ij}^T\frac{e^{z_{ij}^T\gamma +\zeta _i}}{(1+e^{z_{ij}^T\gamma +\zeta _i})^2}e^{h_i(\gamma ,\zeta )}d\zeta , \quad \\ f_i(6)&=\int _{-\infty }^{\infty }\frac{\zeta ^4}{2\sigma ^4}e^{h_i(\gamma ,\zeta )}d\zeta , \quad f_i(7)=\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}\frac{\zeta ^2}{2\sigma ^4}d\zeta ,\\ f_i(8)&=\int _{-\infty }^{\infty }\sum \limits _{j=1}^{n_i}z_{ij}\frac{e^{z_{ij}^T\gamma +\zeta _i}}{1+e^{z_{ij}^T\gamma +\zeta _i}}e^{h_i(\gamma ,\zeta )}\zeta ^2d\zeta \end{aligned}$$

where $k_i= \sum \limits _{j=1}^{n_i}T_{ij}$ and $h_i(\gamma ,\zeta )=k_i\zeta -\frac{\zeta ^2}{2\sigma ^2}- \sum \limits _{j=1}^{n_i}ln(1+e^{\gamma z_{ij}+\zeta })$.

Better precision can be achieved when the approximation is around the point of maximum of the integrand. For example, to approximate $f_i(1)=\int _{-\infty }^{\infty }e^{h_i(\gamma ,\zeta )}d\zeta$, we need to find $\zeta$ at the maximum value of $e^{h_i(\gamma ,\zeta )}d\zeta$, which is the minimum of $h_i(\gamma ,\zeta )=-k_i\zeta +\frac{\zeta ^2}{2\sigma ^2}+\sum \limits _{j=1}^{n_i}ln(1+e^{\gamma z_{ij}+\zeta })$. Given $\frac{dh_i}{d\zeta }=-k_i+\frac{\zeta }{\sigma ^2}+e^{\zeta }\sum \limits _{j=1}^{n_i}\frac{B_j}{1+B_je^\zeta }$ and $\frac{d^2h_i}{d\zeta ^2}=\frac{1}{\sigma ^2}+e^\zeta \sum \limits _{j=1}^{n_i}\frac{B_j}{(1+B_je^\zeta )^2}$, where $B_j=e^{\gamma z_{ij}}$, the second derivative is positive. Therefore, the function has a unique minimum. Newton-Raphson algorithm can be used to solve $\frac{dh_i}{d\zeta }=0$.

$$\begin{aligned} \zeta _{s+1}=\zeta _{s}-\left( \frac{dh_i}{d\zeta }\right) \left( \frac{d^2h_i}{d\zeta ^2}\right) ^{-1} \end{aligned}$$

(11)

Starting from zero, if $\zeta _{max}$ is the limiting point of the iterations, then $f_i(1)$ can be approximated by

$$\begin{aligned} \int _{-\infty }^{\infty }e^{h(\zeta )}d\zeta \approx \sqrt{2}{\hat{\nu }}_h\sum \limits _{k=1}^{K}w_kexp[\zeta _k^2+h(\zeta _{max}+\sqrt{2}{\hat{\nu }}_h\zeta _k)] \end{aligned}$$

(12)

where ${\hat{\nu }}_h=(-\frac{d^2h_i}{d\zeta ^2}|_{\zeta =\zeta _{max}})^{-\frac{1}{2}}$. All the integrals can be approximated by this algorithm. However, if the integral is not a unimodal function, the integral needs to be split into two intervals $(-\infty ,0)$ and $(0,\infty )$ for the approximation. Similarly, ACE can be estimated by Eq. (8) and its variance can be estimated by Eq. (7).