2. Potential Outcomes and Causal Estimands

To elaborate the relaxed SUTVA (Rubin Reference Rubin1986, Hong & Raudenbush Reference Hong and Raudenbush2006, Reference Hong and Raudenbush2008), we step back and reintroduce some notations. Suppose there is binary treatment $T_i=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_i=1$$\end{document} if student i is instructed by a newly graduated teacher prepared by intern-teaching fieldwork experience, and $T_i=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_i=0$$\end{document} if this student is instructed by a teacher with student-teaching experience. There is also a school assignment indicator $S_i=k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i=k$$\end{document} if student i is observed to have been assigned to school k.

Student’s learning outcome depends on their school assignments, but student-school assignment is typically far from random. To move forward without modeling the student-school assignment mechanism, we assume students are first assigned to schools and then, treatments are assigned to students within schools (the intact schools assumption; Hong & Raudenbush Reference Hong and Raudenbush2006, Reference Hong and Raudenbush2008), and fix our interest in the event $(T_i=t\mid S_i=k)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(T_i=t\mid S_i=k)$$\end{document} that occurs when student i who has been assigned to school k is assigned to treatment $t\in \{0,1\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \{0,1\}$$\end{document} . This event will be denoted by $T_{\mathrm{ik}}=t$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}=t$$\end{document} in the rest of the article for notational simplicity. Although the generalization of our causal inference is now restricted to the observed student-school allocation, the resultant estimates have practical value since students would typically attend schools in their neighborhood areas, not any school in the study population.

Then, we adopt a weaker form of the SUTVA to reduce the number of potential outcomes for each student. At the elementary level, the same teacher and students typically stay in the same classroom for all classes throughout the year. Hence, it seems reasonable to assume all students in the same classroom receive the same treatment and there is no interference between classrooms. Given $S_i=k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i=k$$\end{document} , student i’s has two potential outcomes, defined as $Y_{\mathrm{ik}}\left(t\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{ik}(t)$$\end{document} , $t\in \{0,1\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \{0,1\}$$\end{document} .

The difference between student i’s two potential outcomes given $S_i=k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i=k$$\end{document} , $Y_{\mathrm{ik}}\left(1\right)-Y_{\mathrm{ik}}\left(0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{ik}(1)-Y_{ik}(0)$$\end{document} is the student-specific causal effect of our interest. Let ${\Delta }_k=E[Y_{\mathrm{ik}}\left(1\right)-Y_{\mathrm{ik}}\left(0\right)\mid S_i=k]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _k=E[Y_{ik}(1)-Y_{ik}(0)\mid S_i=k]$$\end{document} denote the average treatment effect of all students who has been assigned to school k. Then, our causal estimand can be expressed as $\Delta =E\left({\omega }_k{\Delta }_k\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta =E(\omega _k \Delta _k)$$\end{document} , the weighted average of ${\Delta }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _k$$\end{document} ’s across all k’s. If we aim to generalize $\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} to a population of schools, each school should be weighted equally and ${\omega }_k\equiv 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _k\equiv 1$$\end{document} for all k’s. Suppose we are interested in generalizing $\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} to a population of students, ${\Delta }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _k$$\end{document} will be weighted in proportion to the number of students in school k, e.g. ${\omega }_k=\frac{n_{k}K}{N}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _k=\frac{n_k K}{N}$$\end{document} where $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} , K, and N are, respectively, the number of observed students in school k, the number of observed schools, and the total number of observed students across all k’s, assuming all schools and students in each school have equal probability to be observed in the sample.

3. Theoretical Model for the Potential Outcomes

Hierarchical linear models (HLM), also known as multilevel models or linear mixed effect models, is commonly used to accommodate the clustered structure of educational outcomes (Raudenbush & Bryk Reference Raudenbush and Bryk2002; Goldstein Reference Goldstein2011). To take into account the important role schools play in student learning without overcomplicating the exposition of the proposed methodology, we consider a simple two-level HLM—random coefficient model—for the potential outcomes of students i who has been assigned to school k:

(1) $\begin{array}{c}\begin{array}{ccc}{Y}_{\mathrm{ik}}\left(1\right)& =& {\beta }_{k1}+{ϵ}_{\mathrm{ik}}\left(1\right)\\ Y_{\mathrm{ik}}\left(0\right)& =& {\beta }_{k0}+{ϵ}_{\mathrm{ik}}\left(0\right)\end{array},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ccc} Y_{ik}(1) &{}=&{} \beta _{k1}+\epsilon _{ik}(1) \\ Y_{ik}(0) &{}=&{} \beta _{k0}+\epsilon _{ik}(0) \end{array}, \end{aligned}$$\end{document}

where ${ϵ}_{\mathrm{ik}}\left(t\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{ik}(t)$$\end{document} is the random error assumed independently and identically distributed as $N(0,{\sigma }_{ϵ}^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N(0,\sigma _{\epsilon }^2)$$\end{document} for $t\in \{0,1\}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in \{0,1\}$$\end{document} , and ${\beta }_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k1}$$\end{document} is the school k’s average intern-teaching outcome and ${\beta }_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _{k0}$$\end{document} the school k’s average student-teaching outcome that vary among schools as a function of the school random effects $b_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k1}$$\end{document} and $b_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k0}$$\end{document} :

(2) $\begin{array}{c}\begin{array}{ccc}{\beta }_{k1}& =& {\beta }_1+b_{k1}\\ {\beta }_{k0}& =& {\beta }_0+b_{k0}\end{array},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \begin{array}{ccc} \beta _{k1} &{}=&{} \beta _1+b_{k1} \\ \beta _{k0} &{}=&{} \beta _0+b_{k0} \end{array}, \end{aligned}$$\end{document}

where $b_k=(b_{k1},b_{k0})\sim N(0,\Omega )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {b}}_k=(b_{k1},b_{k0})\sim N(0,\Omega )$$\end{document} with $\Omega =\left(\begin{array}{cc}{\sigma }_1^2& \rho {\sigma }_1{\sigma }_0\\ \rho {\sigma }_1{\sigma }_0& {\sigma }_0^2\end{array}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\left( \begin{array}{cc} \sigma _1^2 &{} \rho \sigma _1\sigma _0 \\ \rho \sigma _1\sigma _0 &{} \sigma _0^2 \end{array}\right) $$\end{document} , and ${\beta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1$$\end{document} and ${\beta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0$$\end{document} are, respectively, the population average intern-teaching outcome and the population average student-teaching outcome. The difference of school k’s averages $({\beta }_{k1}-{\beta }_{k0})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta _{k1}-\beta _{k0})$$\end{document} corresponds to the ${\Delta }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _k$$\end{document} defined in Sect. 2, and the difference of population averages $({\beta }_1-{\beta }_0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta _1-\beta _0)$$\end{document} corresponds to our causal estimand $\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} with ${\omega }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _k$$\end{document} incorporated in the estimation stage, as shown in the latter sections. Although not the focus of this article, this model also supplies the following estimands: ${\sigma }_1^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _1^2$$\end{document} the variance of the average intern-teaching outcome across schools, ${\sigma }_0^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0^2$$\end{document} the variance of the average student-teaching outcome across schools, and $-1<\rho <1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-1<\rho <1$$\end{document} the correlation between average intern-teaching outcome and student-teaching outcome across schools.

4. Model for the Observed Data

The fundamental problem in estimating $({\beta }_1-{\beta }_0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta _1-\beta _0)$$\end{document} , or equivalently $\Delta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} , is the fact that we only observe one of the two potential outcomes for each student. The observed outcome for student i in school k can be written as a function of the observed $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} , $Y_{\mathrm{ik}}=T_{\mathrm{ik}}{Y}_{\mathrm{ik}}\left(1\right)+(1-T_{\mathrm{ik}})Y_{\mathrm{ik}}\left(0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{ik}= T_{ik}Y_{ik}(1)+(1-T_{ik})Y_{ik}(0)$$\end{document} , which results in,

(3) $\begin{array}{c}{Y}_{\mathrm{ik}}=T_{\mathrm{ik}}({\beta }_1+b_{k1})+(1-T_{\mathrm{ik}})({\beta }_0+b_{k0})+e_{\mathrm{ik}}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y_{ik}=T_{ik}(\beta _1+b_{k1})+(1-T_{ik})(\beta _0+b_{k0})+e_{ik} \end{aligned}$$\end{document}

where $e_{\mathrm{ik}}=T_{\mathrm{ik}}{ϵ}_{\mathrm{ik}}\left(1\right)+(1-T_{\mathrm{ik}}){ϵ}_{\mathrm{ik}}\left(0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_{ik}=T_{ik}\epsilon _{ik}(1)+(1-T_{ik})\epsilon _{ik}(0)$$\end{document} . This model also has the form of a random coefficient model, but the conventional maximum likelihood estimation (Raudenbush & Bryk Reference Raudenbush and Bryk2002; West et al. Reference West, Welch and Galecki2014; Bates et al. Reference Bates, Maechler, Bolker and Walker2015) does not yield consistent estimates of ${\beta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1$$\end{document} and ${\beta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0$$\end{document} unless $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} is independent of ${ϵ}_{\mathrm{ik}}\left(1\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{ik}(1)$$\end{document} , ${ϵ}_{\mathrm{ik}}\left(0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{ik}(0)$$\end{document} , $b_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k1}$$\end{document} and $b_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k0}$$\end{document} for all i’s and k’s, i.e., the treatment assignments are completely randomized (Ebbes et al. Reference Ebbes, Bckenholt and Wedel2004; Wooldridge Reference Wooldridge2010). In our observational study, we impose the following two assumptions to proceed:

(Ignorability)

Random treatment assignment within each school, or equivalently,

(4) $\begin{array}{c}{Y}_{\mathrm{ik}}\left(1\right),Y_{\mathrm{ik}}\left(0\right)\perp T_{\mathrm{ik}}\mid b_k,\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y_{ik}(1),Y_{ik}(0)\perp T_{ik}\mid {\mathbf {b}}_k, \end{aligned}$$\end{document}

since $b_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {b}}_k$$\end{document} is controlled, although not directly observed, once the school identity is given. In other words, $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} might be correlated with $b_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {b}}_k$$\end{document} , but is independent of ${ϵ}_{\mathrm{ik}}\left(1\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{ik}(1)$$\end{document} and ${ϵ}_{\mathrm{ik}}\left(0\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{ik}(0)$$\end{document} .

(Positivity)

Define the probability of treatment as $Pr(T_{\mathrm{ik}}=1\mid b_k)={\pi }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Pr(T_{ik}=1\mid {\mathbf {b}}_k)=\pi _k$$\end{document} for $i=1,\dots ,n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i=1,\ldots ,n_k$$\end{document} in school k, then,

(5) $\begin{array}{c}0<{\pi }_k<1\text{for}\text{all}{k}^{\prime }\text{s.}\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0<\pi _k<1 \text{ for } \text{ all } k'\text{ s. }\end{aligned}$$\end{document}

Since treatment assignment is random within each school, ${\pi }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _k$$\end{document} can be consistently estimated by the proportion of the sample assigned to $T_{\mathrm{ik}}=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}=1$$\end{document} in school k (Arpino & Mealli Reference Arpino and Mealli2011; Li et al. Reference Li, Zaslavsky and Landrum2013; Raudenbush 2014; Raudenbush & Schwartz 2016):

(6) $\begin{array}{c}{\hat{\pi }}_k=\frac{n_{k1}}{n_k},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\hat{\pi }}_k=\frac{n_{k1}}{n_k}, \end{aligned}$$\end{document}

where $n_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}$$\end{document} is the number of intern-teaching observations in school k. When $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} , ${\hat{\pi }}_k=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\pi }}_k=0$$\end{document} , and ${\hat{\pi }}_k=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\pi }}_k=1$$\end{document} if $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} , causing the so-called practical violation of the positivity violation and problematic IPTW estimates, as shown in the next section.

5. IPTW Estimating Function Under Practical Positivity

The IPTW method, proposed by Robins et al. (Reference Robins, Hernan and Brumback2000) in single-level settings, has been integrated into a broad class of HLM to study causal effects in multilevel settings (Hong & Raudenbush Reference Hong and Raudenbush2008). Similar to the single-level setting, each observation is weighted in proportion to the inverse probability of its assigned treatment to create a pseudo-sample that approximates a sample collected under randomization. Specifically, Hong & Raudenbush (Reference Hong and Raudenbush2008) showed that given the value of the variance components, like the unweighted complete-data score function from randomized treatment assignments, the weighted complete-data score function also has expectation zero. Therefore, equating the weighted complete-data score function to zero and jointly solving for fixed effects and random effects yields consistent causal estimates. In our example, the complete data for student i in school k include $(Y_{\mathrm{ik}},T_{\mathrm{ik}},b_k)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Y_{ik}, T_{ik}, {\mathbf {b}}_k)$$\end{document} where $b_k=(b_{k1},b_{k0})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {b}}_k=(b_{k1}, b_{k0})$$\end{document} . Given $\Omega$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and ${\sigma }_{ϵ}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\epsilon }^2$$\end{document} , the weighted complete-data score functions for $\theta =({\beta }_1,{\beta }_0,\dots ,b_{k1},b_{k0},\dots )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta =(\beta _1,\beta _0,\ldots ,b_{k1}, b_{k0},\ldots )$$\end{document} can be written as (Hong & Raudenbush Reference Hong and Raudenbush2008; Bates 2014),

where $(h(T_{\mathrm{ik}};\theta )=-\frac{1}{{\sigma }_{ϵ}^2}\frac{d}{d\theta }{e}_{\mathrm{ik}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(h(T_{ik};\theta )=-\frac{1}{\sigma _{\epsilon }^2}\frac{d}{d\theta }e_{ik})$$\end{document} , $(v_k=\frac{{\omega }_{k}N}{n_{k}K})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v_k=\frac{\omega _k N}{n_k K})$$\end{document} with ${\omega }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _k$$\end{document} as specified in Sect. 2, and $w_{\mathrm{ik}}=T_{\mathrm{ik}}\left(\frac{c}{{\hat{\pi }}_k}\right)+(1-T_{\mathrm{ik}})\left(\frac{1-c}{1-{\hat{\pi }}_k}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ik}=T_{ik}\left( \frac{c}{{\hat{\pi }}_k}\right) +(1-T_{ik})\left( \frac{1-c}{1-{\hat{\pi }}_k}\right) $$\end{document} with a constant c chosen to normalize the weights such that ${\sum }_{k=1}^K{v}_k\left({\sum }_{i=1}^{n_k},w_{\mathrm{ik}}\right)=N$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{k=1}^K v_k\left( \sum _{i=1}^{n_k}w_{ik}\right) =N$$\end{document} .

Proof

When $0<n_{k1}<n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<n_{k1}<n_k$$\end{document} for all k’s, we have $(2+2K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2 + 2K)$$\end{document} score functions in (7) associated with the observed data. Equating them to zero results in $(2+2K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2 + 2K)$$\end{document} estimating equations. Then, the consistency of the resultant estimates follows by showing that the weighted complete-data score function in (7) has expectation zero (see Appendix A). $$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

However, when $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} or $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} for some k’s, the number of score functions in (7) associated with the observed data reduces to $(2+K+\tilde{K})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+K+{\tilde{K}})$$\end{document} , where $\tilde{K}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{K}}$$\end{document} is the number of schools that have variations in the observed values of $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} . This is because in (7.1), ${\sum }_{i=1}^{n_k}\frac{w_{\mathrm{ik}}}{{\sigma }_{ϵ}^2}{T}_{\mathrm{ik}}{e}_{\mathrm{ik}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^{n_k}\frac{w_{ik}}{\sigma _{\epsilon }^2}T_{ik}e_{ik}=0$$\end{document} when $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} , and in (7.2), ${\sum }_{i=1}^{n_k}\frac{w_{\mathrm{ik}}}{{\sigma }_{ϵ}^2}(1-T_{\mathrm{ik}})e_{\mathrm{ik}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^{n_k}\frac{w_{ik}}{\sigma _{\epsilon }^2}(1-T_{ik})e_{ik}=0$$\end{document} when $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} . Equating them to zero results in a system of $(2+K+\tilde{K})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+K+{\tilde{K}})$$\end{document} estimating equations as follows,

$\begin{array}{cc}& \sum _{k=1}^K{v}_k\sum _{i=1}^{n_k}\left(I_{(0<n_{1k}<n_k)},\left(w_{\mathrm{ik}},h,(T_{\mathrm{ik}};\tilde{\theta }),e_{\mathrm{ik}},-,\frac{1}{n_k(1-{\rho }^2)},\left(\frac{d}{d\tilde{\theta }},b_k\right),\left(\begin{array}{c}\frac{b_{k1}}{{\sigma }_1^2}-\frac{\rho b_{k0}}{{\sigma }_1{\sigma }_0}\\ \frac{b_{k0}}{{\sigma }_0^2}-\frac{\rho b_{k1}}{{\sigma }_1{\sigma }_0}\end{array}\right)\right)\right)\\ & +I_{(n_{k1}=n_k)}\left(w_{\mathrm{ik}},h,(T_{\mathrm{ik}};\tilde{\theta }),e_{\mathrm{ik}},-,\frac{1}{n_k(1-{\rho }^2)},\left(\frac{d}{d\tilde{\theta }},b_{k1}\right),\left(\frac{b_{k1}}{{\sigma }_1^2},-,\frac{\rho b_{k0}}{{\sigma }_1{\sigma }_0}\right)\right)\\ & +I_{(n_{k1}=0)}\left(\left(w_{\mathrm{ik}},h,(T_{\mathrm{ik}};\tilde{\theta }),e_{\mathrm{ik}},-,\frac{1}{n_k(1-{\rho }^2)},\left(\frac{d}{d\tilde{\theta }},b_{k0}\right),\left(\frac{b_{k0}}{{\sigma }_0^2},-,\frac{\rho b_{k1}}{{\sigma }_1{\sigma }_0}\right)\right)\right)=0\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\sum _{k=1}^K v_k\sum _{i=1}^{n_k} \left\{ I_{(0<n_{1k}<n_k)} \left[ w_{ik}h(T_{ik};\tilde{\theta })e_{ik}-\frac{1}{n_k(1-\rho ^2)}\left( \frac{d}{d\tilde{\theta }}{\mathbf {b}}_k\right) \left( \begin{array}{l} \frac{b_{k1}}{\sigma _1^2}-\frac{\rho b_{k0}}{\sigma _1\sigma _0} \\ \frac{b_{k0}}{\sigma _0^2}-\frac{\rho b_{k1}}{\sigma _1\sigma _0} \end{array}\right) \right] \right. \\&\quad +I_{(n_{k1}=n_k)} \left[ w_{ik}h(T_{ik};\tilde{\theta })e_{ik}-\frac{1}{n_k(1-\rho ^2)} \left( \frac{d}{d\tilde{\theta }}b_{k1}\right) \left( \frac{b_{k1}}{\sigma _1^2}-\frac{\rho b_{k0}}{\sigma _1\sigma _0}\right) \right] \\&\quad + I_{(n_{k1}=0)} \left. \left[ w_{ik}h(T_{ik};\tilde{\theta })e_{ik}-\frac{1}{n_k(1-\rho ^2)} \left( \frac{d}{d\tilde{\theta }}b_{k0}\right) \left( \frac{b_{k0}}{\sigma _0^2}-\frac{\rho b_{k1}}{\sigma _1\sigma _0}\right) \right] \right\} =0 \end{aligned}$$\end{document}

where $\tilde{\theta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{\theta }$$\end{document} is a length $(2+K+\tilde{K})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+K+{\tilde{K}})$$\end{document} vector that includes all elements in $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} , except for $b_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k0}$$\end{document} if $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} and $b_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k1}$$\end{document} if $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} . The left hand side of these estimating equations does not have expectation zero, because $E(b_{k1}\mid n_{k1})\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(b_{k1}\mid n_{k1})\ne 0$$\end{document} and $E(b_{k0}\mid n_{k1})\ne 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(b_{k0}\mid n_{k1})\ne 0$$\end{document} , causing bias in the resultant estimates.

If theoretical positivity holds, practical positivity is less likely to be violated as sample size increases in $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} , i.e., $n_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}$$\end{document} is unlikely to be 0 or $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} , as $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} approaches infinity. But in finite samples, $n_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}$$\end{document} can equal 0 or $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} by chance. In the next section, we propose to augment the weighted score function to correct the bias that occurs in such situations.

6. Augmented IPTW Estimating Function when Positivity is Practically Violated

When $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} or $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} for some k’s, we consider the following augmented weighted complete-data score function for $\theta$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} :

where $(Q(1,k)=\hat{E}[Y_{\mathrm{ik}}\left(1\right)\mid S_i=k]-({\beta }_1+b_{k1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q(1,k)={\hat{E}}[Y_{ik}(1)\mid S_i=k]-(\beta _1+b_{k1})$$\end{document} is the difference between an estimate of the school-specific potential outcome derived from the observed data and their true expected value based on the model assumption in (1) and (2). Similarly, $(Q(0,k)=\hat{E}[Y_{\mathrm{ik}}\left(0\right)\mid S_i=k]-({\beta }_0+b_{k0})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(Q(0,k)={\hat{E}}[Y_{ik}(0)\mid S_i=k]-(\beta _0+b_{k0})$$\end{document} . Note that (8) differs from (7) only in (8.1) and (8.2), and (8) becomes (7) when $0<n_{k1}<n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<n_{k1}<n_k$$\end{document} for all k’s.

Proof

As seen in (8.3) and (8.4), all of the $(2+2K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2 + 2K)$$\end{document} score functions in (8) are associated with the observed data, whether or not $0<n_{k1}<n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<n_{k1}<n_k$$\end{document} for all k’s. Equating them to zero results in $(2+2K)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2 + 2K)$$\end{document} estimating equations. The resultant estimates are consistent if the augmented weighted complete-data score function in (8) can be shown to converge in expectation to zero:

$\begin{array}{cc}& E\left(\sum _{k=1}^K,v_k,\sum _{i=1}^{n_k},S_{\mathrm{ik}}^{\mathrm{aw}}\right)\cong 0\text{, which follows from,}\\ & E\left(S_{\mathrm{ik}}^{\mathrm{aw}}\right)=E\left(I_{(0<n_{k1}<n_k)}\right)E[w_{\mathrm{ik}}h(T_{\mathrm{ik}};\theta )e_{\mathrm{ik}}\mid b_k]\\ & +I_{(n_{k1}=n_k)}\left(\frac{c(n_k+1)}{n_k},h,(1;\theta ),{ϵ}_{\mathrm{ik}},\left(1\right),+,\frac{(1-c)(n_k+1)}{n_k},h,(0;\theta ),Q,(0,k)\right)\\ & +I_{(n_{k1}=0)}\left(\frac{(1-c)(n_k+1)}{n_k},h,(0;\theta ),{ϵ}_{\mathrm{ik}},\left(0\right),+,\frac{c(n_k+1)}{n_k},h,(1;\theta ),Q,(1,k)\right)\\ & \left(-,\frac{1}{n_k},\left(\frac{d}{d\theta },b_k\right),{\Omega }^{-1},b_k^{\prime }\right)\\ & =E\left(I_{(0<n_{k1}<n_k)},\left(\frac{ch(1;\theta ){ϵ}_{\mathrm{ik}}\left(1\right)}{{\hat{\pi }}_k},{\pi }_k,+,\frac{(1-c)h(0;\theta ){ϵ}_{\mathrm{ik}}\left(0\right)}{1-{\hat{\pi }}_k},(1-{\pi }_k)\right)\right)\\ & +I_{(n_{k1}=n_k)}\frac{c(n_k+1)}{n_k}h(1;\theta )E\left[{ϵ}_{\mathrm{ik}}\left(1\right)\right]+\frac{(1-c)(n_k+1)}{n_k}h(0;\theta )E\left(I_{(n_{k1}=n_k)},Q,(0,k)\right)\\ & +I_{(n_{k1}=0)}\frac{(1-c)(n_k+1)}{n_k}h(0;\theta )E\left[{ϵ}_{\mathrm{ik}}\left(0\right)\right]+\frac{c(n_k+1)}{n_k}h(1;\theta )E\left(I_{(n_{k1}=0)},Q,(1,k)\right)\\ & -\frac{1}{n_k}\left(\frac{d}{d\theta },b_k\right){\Omega }^{-1}E\left(b_k^{\prime }\right)\\ & =I_{(0<n_{k1}<n_k)}ch(1;\theta )E\left[{ϵ}_{\mathrm{ik}}\left(1\right)\right]+I_{(0<n_{k1}<n_k)}(1-c)h(0;\theta )E\left[{ϵ}_{\mathrm{ik}}\left(0\right)\right]\\ & +0+\frac{(1-c)(n_k+1)}{n_k}h(0;\theta )E\left(I_{(n_{k1}=n_k)},E,\left(Q,(0,k),\mid ,n_{k1}\right)\right)\\ & +0+\frac{c(n_k+1)}{n_k}h(1;\theta )E\left(I_{(n_{k1}=0)},E,\left(Q,(1,k),\mid ,n_{k1}\right)\right)-0\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&E\left( \sum _{k=1}^K v_k\sum _{i=1}^{n_k}S_{ik}^{aw}\right) \cong 0\text {, which follows from,}\\&E(S_{ik}^{aw})=E\left\{ I_{(0<n_{k1}<n_k)}\right. E[w_{ik}h(T_{ik};\theta )e_{ik}\mid {\mathbf {b}}_k] \nonumber \\&\qquad +\,I_{(n_{k1}=n_k)} \left[ \frac{c(n_k+1)}{n_k}h(1;\theta )\epsilon _{ik}(1)+\frac{(1-c)(n_k+1)}{n_k} h(0;\theta )Q(0,k)\right] \nonumber \\&\qquad +\,I_{(n_{k1}=0)} \left[ \frac{(1-c)(n_k+1)}{n_k}h(0;\theta )\epsilon _{ik}(0)+\frac{c(n_k+1)}{n_k} h(1;\theta )Q(1,k)\right] \nonumber \\&\qquad \left. -\frac{1}{n_k}\left( \frac{d}{d\theta }{\mathbf {b}}_k\right) \Omega ^{-1}{\mathbf {b}}_k^\prime \right\} \\&\quad = E\left\{ I_{(0<n_{k1}<n_k)}\left[ \frac{ch(1;\theta )\epsilon _{ik}(1)}{{\hat{\pi }}_k}\pi _k +\frac{(1-c)h(0;\theta )\epsilon _{ik}(0)}{1-{\hat{\pi }}_k}(1-\pi _k)\right] \right\} \nonumber \\&\qquad +\,I_{(n_{k1}=n_k)}\frac{c(n_k+1)}{n_k}h(1;\theta )E[\epsilon _{ik}(1)]+\frac{(1-c)(n_k+1)}{n_k} h(0;\theta )E\left[ I_{(n_{k1}=n_k)}Q(0,k)\right] \nonumber \\&\qquad +\,I_{(n_{k1}=0)}\frac{(1-c)(n_k+1)}{n_k}h(0;\theta )E[\epsilon _{ik}(0)]+\frac{c(n_k+1)}{n_k} h(1;\theta )E\left[ I_{(n_{k1}=0)}Q(1,k)\right] \nonumber \\&\qquad -\frac{1}{n_k}\left( \frac{d}{d\theta }{\mathbf {b}}_k\right) \Omega ^{-1}E({\mathbf {b}}_k^\prime ) \nonumber \\&\quad = I_{(0<n_{k1}<n_k)}ch(1;\theta )E[\epsilon _{ik}(1)]+I_{(0<n_{k1}<n_k)}(1-c)h(0;\theta )E[\epsilon _{ik}(0)] \nonumber \\&\qquad +\,0+\frac{(1-c)(n_k+1)}{n_k}h(0;\theta )E\left\{ I_{(n_{k1}=n_k)}E\left[ Q(0,k)\mid n_{k1}\right] \right\} \nonumber \\&\qquad +0+\,\frac{c(n_k+1)}{n_k}h(1;\theta )E\left\{ I_{(n_{k1}=0)}E\left[ Q(1,k)\mid n_{k1}\right] \right\} -0 \end{aligned}$$\end{document}

Therefore, $E\left(S_{\mathrm{ik}}^{\mathrm{aw}}\right)\cong 0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E\left( S_{ik}^{aw}\right) \cong 0$$\end{document} , if $E[Q(1,k)\mid n_{k1}]=E\left[Q(1,k)\right]=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Q(1,k)\mid n_{k1}]=E[Q(1,k)]=0$$\end{document} and $E[Q(0,k)\mid n_{k1}]=E\left[Q(0,k)\right]=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Q(0,k)\mid n_{k1}]=E[Q(0,k)]=0$$\end{document} in large samples. $$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square $$\end{document}

The values of the variance components $\Omega$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and ${\sigma }_{ϵ}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\epsilon }^2$$\end{document} are usually unknown and need to be estimated. Following Hong & Raudenbush (Reference Hong and Raudenbush2008), we adopt a maximum pseudo-likelihood approach and make use of existing software program for implementation, with further details provided in Appendix B. In brief, an augmented data $A={(A_1^{\prime },\dots ,A_K^{\prime })}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}=(A_1^\prime ,\ldots ,A_K^\prime )^\prime $$\end{document} is created that includes, for every schools k,

(9) $\begin{array}{c}{A}_k=\left(\begin{array}{ccc}& (Y_{1k},& T_{1k})\\ & ⋮& \\ & (Y_{n_{k}k},& T_{n_{k}k})\\ \text{if}{n}_{k1}=0,& (Y_{(n_k+1)k}=\hat{E}[Y_{\mathrm{ik}}\left(1\right)\mid S_i=k],& T_{(n_k+1)k}=1)\\ \text{if}{n}_{k1}=n_k,& (Y_{(n_k+1)k}=\hat{E}[Y_{\mathrm{ik}}\left(0\right)\mid S_i=k],& T_{(n_k+1)k}=0)\end{array}\right),\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A_k=\left[ \begin{array}{l@{\quad }c@{\quad }r} &{} (Y_{1k}, &{} T_{1k}) \\ &{} \vdots &{} \\ &{} (Y_{n_kk}, &{} T_{n_kk}) \\ \text {if } n_{k1}=0, &{} (Y_{(n_k+1)k}={\hat{E}}[Y_{ik}(1)\mid S_i=k], &{} T_{(n_k+1)k}=1 ) \\ \text {if } n_{k1}=n_k, &{} (Y_{(n_k+1)k}={\hat{E}}[Y_{ik}(0)\mid S_i=k], &{} T_{(n_k+1)k}=0) \end{array}\right] , \end{aligned}$$\end{document}

having $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} rows if $0<n_{k1}<n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<n_{k1}<n_k$$\end{document} , and $n_k+1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k+1$$\end{document} rows if $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} or $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} . Then, the estimates of ${\beta }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1$$\end{document} , ${\beta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _0$$\end{document} , $\Omega$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} and ${\sigma }_{ϵ}^2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\epsilon }^2$$\end{document} that maximize the likelihood function corresponding to the augmented weighted complete-data score function in (8) can be obtained by first calculating ${\hat{\pi }}_k^a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\pi }}_k^a$$\end{document} based on (6) as if $A_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k$$\end{document} is observed in school k, and then feeding $A$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {A}}$$\end{document} into the standard HLM estimation procedure with $w_{\mathrm{ik}}^a=T_{\mathrm{ik}}\left(\frac{c}{{\hat{\pi }}_k^a}\right)+(1-T_{\mathrm{ik}})\left(\frac{1-c}{1-{\hat{\pi }}_k^a}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ik}^a=T_{ik}\left( \frac{c}{{\hat{\pi }}_k^a}\right) +(1-T_{ik})\left( \frac{1-c}{1-{\hat{\pi }}_k^a}\right) $$\end{document} assigned as the weights. We call this the AIPTW estimator in the rest of the article.

7. Estimating the School-Specific Potential Outcomes

Estimating $E[Y_{\mathrm{ik}}\left(1\right)\mid S_i=k]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Y_{ik}(1)\mid S_i=k]$$\end{document} for school k whose $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} and $E[Y_{\mathrm{ik}}\left(0\right)\mid S_i=k]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Y_{ik}(0)\mid S_i=k]$$\end{document} when $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} is challenging because information regarding the unobserved $b_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k1}$$\end{document} and $b_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k0}$$\end{document} is limited for these schools. In a random intercept model, including the school-specific average $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} as an additional covariate in the model (Kim & Frees Reference Kim and Frees2006; Bafumi & Gelman 2006; Raudenbush Reference Raudenbush2009) has been used to obtained consistent fixed-effect estimates when $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} is not independent of the random intercepts. In that spirit, we re-parameterize model (3) as follows:

(10) $\begin{array}{c}{Y}_{\mathrm{ik}}=({\ddot{\beta }}_1+{\ddot{b}}_{k1}+{\gamma }_1{\overline{T}}_k)T_{\mathrm{ik}}+({\ddot{\beta }}_0+{\ddot{b}}_{k0}+{\gamma }_0{\overline{T}}_k)(1-T_{\mathrm{ik}})+e_{\mathrm{ik}},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y_{ik} = (\ddot{\beta }_1+\ddot{b}_{k1}+\gamma _1{\bar{T}}_k)T_{ik} + (\ddot{\beta }_0+\ddot{b}_{k0}+\gamma _0{\bar{T}}_k)(1-T_{ik}) + e_{ik}, \end{aligned}$$\end{document}

where ${\overline{T}}_k=\frac{{\sum }_i{T}_{\mathrm{ik}}}{n_k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{T}}_k=\frac{\sum _i T_{ik}}{n_k}$$\end{document} , ${\ddot{b}}_{k1}=b_{k1}-{\gamma }_1({\overline{T}}_k-\overline{\overline{T}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k1}=b_{k1}-\gamma _1({\bar{T}}_k-\bar{{\bar{T}}})$$\end{document} with $\overline{\overline{T}}=\frac{{\sum }_k{\overline{T}}_k}{K}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar{{\bar{T}}}=\frac{\sum _k {\bar{T}}_k}{K}$$\end{document} to ensure $E\left({\ddot{b}}_{k1}\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\ddot{b}_{k1})=0$$\end{document} , ${\ddot{b}}_{k0}=b_{k0}-{\gamma }_0({\overline{T}}_k-\overline{\overline{T}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k0}=b_{k0}-\gamma _0({\bar{T}}_k-\bar{{\bar{T}}})$$\end{document} so that $E\left({\ddot{b}}_{k0}\right)=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\ddot{b}_{k0})=0$$\end{document} , ${\ddot{\beta }}_1={\beta }_1-{\gamma }_1\overline{\overline{T}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\beta }_1=\beta _1-\gamma _1\bar{{\bar{T}}}$$\end{document} and ${\ddot{\beta }}_0={\beta }_0-{\gamma }_0\overline{\overline{T}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\beta }_0=\beta _0-\gamma _0\bar{{\bar{T}}}$$\end{document} . It can be shown that ${\ddot{b}}_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k1}$$\end{document} and ${\ddot{b}}_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k0}$$\end{document} are close to independent of $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} , in large K (see Appendix C). Therefore, standard maximum likelihood estimation can be used to obtain consistent estimates of ${\ddot{\beta }}_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\beta }_1$$\end{document} , ${\ddot{\beta }}_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{\beta }_0$$\end{document} and ${\gamma }_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _1$$\end{document} and ${\gamma }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _0$$\end{document} .

In the standard maximum likelihood estimation, random effect estimates shrink toward their marginal expectation, zero, when school has little or no relevant observations. Specifically, when $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} , ${\overline{T}}_k=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{T}}_k=0$$\end{document} and ${\hat{\ddot{b}}}_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\ddot{b}}_{k1}=0$$\end{document} , resulting in school k’s estimated potential outcome $\hat{E}[Y_{\mathrm{ik}}\left(1\right)\mid S_i=k]={\hat{\ddot{\beta }}}_1+{\hat{\ddot{b}}}_{k1}+{\hat{\gamma }}_1{\overline{T}}_k={\hat{\ddot{\beta }}}_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{E}}[Y_{ik}(1)\mid S_i=k]=\hat{\ddot{\beta }}_1+\hat{\ddot{b}}_{k1}+\hat{\gamma }_1{\bar{T}}_k=\hat{\ddot{\beta }}_1$$\end{document} , and $Q(1,k)={\hat{\ddot{\beta }}}_1-({\ddot{\beta }}_1+{\ddot{b}}_{k1})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(1,k)=\hat{\ddot{\beta }}_1-(\ddot{\beta }_1+\ddot{b}_{k1})$$\end{document} . Similarly, when $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} , ${\overline{T}}_k=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{T}}_k=1$$\end{document} and ${\hat{\ddot{b}}}_{k0}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\ddot{b}}_{k0}=0$$\end{document} , resulting in $Q(0,k)={\hat{\ddot{\beta }}}_0+{\hat{\gamma }}_0-({\ddot{\beta }}_0+{\ddot{b}}_{k0}+{\gamma }_0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q(0,k)=\hat{\ddot{\beta }}_0+{\hat{\gamma }}_0-(\ddot{\beta }_0+\ddot{b}_{k0}+\gamma _0)$$\end{document} . Since ${\hat{\ddot{\beta }}}_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\ddot{\beta }}_1$$\end{document} is consistent, $E\left[Q(1,k)\right]=E[{\hat{\ddot{\beta }}}_1-({\ddot{\beta }}_1+{\ddot{b}}_{k1})]$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Q(1,k)]=E[\hat{\ddot{\beta }}_1-(\ddot{\beta }_1+\ddot{b}_{k1})]$$\end{document} approaches $E\left({\ddot{b}}_{k1}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\ddot{b}_{k1})$$\end{document} and has expectation zero, as sample size increases. Similarly, E[Q(0, k)] approaches $E\left({\ddot{b}}_{k0}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(\ddot{b}_{k0})$$\end{document} and has expectation zero.

Furthermore, since ${\ddot{b}}_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k1}$$\end{document} and ${\ddot{b}}_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{b}_{k0}$$\end{document} are close to independent of $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} in large K, $E[Q(1,k)\mid n_{k1}]=E({\ddot{b}}_{k1}\mid n_{k1})=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Q(1,k)\mid n_{k1}]=E(\ddot{b}_{k1}\mid n_{k1})=0$$\end{document} and $E[Q(0,k)\mid n_{k1}]=E({\ddot{b}}_{k0}\mid n_{k1})=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E[Q(0,k)\mid n_{k1}]=E(\ddot{b}_{k0}\mid n_{k1})=0$$\end{document} , as sample size increases.

We call the model in (10) a school-average-T-corrected model, denoted by “SATC” in the rest of the article. To improve efficiency, we also consider a simplified version, called reduced SATC (RSATC), with one parameter less than SATC:

$\begin{array}{c}{Y}_{\mathrm{ik}}=({\dot{\beta }}_1+{\dot{b}}_{k1})T_{\mathrm{ik}}+({\dot{\beta }}_0+{\dot{b}}_{k0})(1-T_{\mathrm{ik}})+\gamma {\overline{T}}_k+e_{\mathrm{ik}},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Y_{ik}=({\dot{\beta }}_1+{\dot{b}}_{k1})T_{ik}+({\dot{\beta }}_0+{\dot{b}}_{k0})(1-T_{ik})+\gamma {\bar{T}}_k+e_{ik}, \end{aligned}$$\end{document}

where ${\dot{b}}_{k1}=b_{k1}-\gamma ({\overline{T}}_k-\overline{\overline{T}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{b}}_{k1}=b_{k1}-\gamma ({\bar{T}}_k-\bar{{\bar{T}}})$$\end{document} , ${\dot{b}}_{k0}=b_{k0}-\gamma ({\overline{T}}_k-\overline{\overline{T}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{b}}_{k0}=b_{k0}-\gamma ({\bar{T}}_k-\bar{{\bar{T}}})$$\end{document} , ${\dot{\beta }}_1={\beta }_1-\gamma \overline{\overline{T}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\beta }}_1=\beta _1-\gamma \bar{{\bar{T}}}$$\end{document} , ${\dot{\beta }}_0={\beta }_0-\gamma \overline{\overline{T}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\beta }}_0=\beta _0-\gamma \bar{{\bar{T}}}$$\end{document} and ${\dot{\beta }}_1-{\dot{\beta }}_0={\beta }_1-{\beta }_0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\dot{\beta }}_1-{\dot{\beta }}_0=\beta _1-\beta _0$$\end{document} . SATC reduces to RSATC when $cov(b_{k1},T_{\mathrm{ik}})=cov(b_{k0},T_{\mathrm{ik}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov(b_{k1},T_{ik})=cov(b_{k0},T_{ik})$$\end{document} . Therefore, RSATC is expected to be correct and more efficient when $cov(b_{k1},T_{\mathrm{ik}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov(b_{k1},T_{ik})$$\end{document} and $cov(b_{k0},T_{\mathrm{ik}})$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$cov(b_{k0},T_{ik})$$\end{document} are close enough. We will compare the performance of AIPTW based on SATC and RSATC using simulated data in the next section.

8. Simulation

We conducted a simulation study to explore the moderate sample size performance of the AIPTW when SATC or RSATC are used in estimating Q(1, k) and Q(0, k), denoted by AIPTW-SATC and AIPTW-RSATC, respectively, and to compare their performance with the IPTW using the original sample (denoted by IPTW-orig), and the IPTW using the trimmed sample (denoted by IPTW-trim). Two simulation settings were chosen which mimicked the real data example described in Sect. 9, and 1000 replicated data sets were generated for each setting using the random coefficient model specified in (1) and (2). In the first setting, we generated $K=150$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=150$$\end{document} clusters and within each cluster $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} observations where $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} follows a discrete uniform distribution ranging from 1 to 19 such that 26% of the schools have no more than 5 observations. The binary treatment indicator $T_{\mathrm{ik}}=1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}=1$$\end{document} if $g\left(b_k\right)>0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\mathbf {b}}_k)>0$$\end{document} and $T_{\mathrm{ik}}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}=0$$\end{document} if $g\left(b_k\right)<0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\mathbf {b}}_k)<0$$\end{document} where $g\left(b_k\right)=c_1+c_2{b}_{k0}+c_3{b}_{k1}+c_4{\zeta }_k+{\xi }_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g({\mathbf {b}}_k)=c_1+c_2b_{k0}+c_3b_{k1}+c_4\zeta _k+\xi _{ik}$$\end{document} with both ${\zeta }_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\zeta _k$$\end{document} and ${\xi }_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi _{ik}$$\end{document} generated from a standard normal distribution representing other unknown school-level and student-level factors in the treatment assignment mechanism; constants $c_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1$$\end{document} , $c_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2$$\end{document} , $c_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_3$$\end{document} and $c_4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_4$$\end{document} were chosen to have the correlation coefficient between $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} and $b_{k0}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k0}$$\end{document} : $r_0=0.4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0=0.4$$\end{document} , the correlation coefficient between $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} and $b_{k1}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_{k1}$$\end{document} : $r_1=0.4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_1=0.4$$\end{document} , the overall probability of treatment: $p=0.3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.3$$\end{document} , and 26% or 80% of the schools have practical positivity violations, i.e., $n_{k1}=0$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=0$$\end{document} or $n_{k1}=n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{k1}=n_k$$\end{document} in these schools. Then, the outcome $Y_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Y_{ik}$$\end{document} was generated based on Model (1) and (2) with $\beta =({\beta }_0,{\beta }_1)=(35,40)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =(\beta _0,\beta _1)=(35,40)$$\end{document} , ${\sigma }_0={\sigma }_1=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0=\sigma _1=8$$\end{document} , $\rho =0.8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho =0.8$$\end{document} and ${\sigma }_{\epsilon }=45$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\varepsilon }=45$$\end{document} . In the second setting, $K=200$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K=200$$\end{document} , $n_k$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_k$$\end{document} follows a discrete uniform distribution ranging from 1 to 49 such that 10% of the schools have no more than 5 observations, $\beta =(12,15)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =(12,15)$$\end{document} , ${\sigma }_0={\sigma }_1=8$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _0=\sigma _1=8$$\end{document} , and ${\sigma }_{ϵ}=35$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _{\epsilon }=35$$\end{document} . And for $T_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{ik}$$\end{document} , $c_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_1$$\end{document} , $c_2$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_2$$\end{document} , $c_3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_3$$\end{document} and $c_4$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_4$$\end{document} were chosen to have various combinations of $(r_0,r_1,\rho )$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(r_0,r_1,\rho )$$\end{document} as detailed below, $p=0.3$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=0.3$$\end{document} , and 80% of the schools have practical positivity violations.

We focus on obtaining an estimate for $({\beta }_1-{\beta }_0)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\beta _1-\beta _0)$$\end{document} to be generalized to a population of students. In other words, we have ${\omega }_k=\frac{n_{k}K}{N}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _k=\frac{n_k K}{N}$$\end{document} , or equivalently, $v_k\equiv 1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$v_k\equiv 1$$\end{document} . For each data set, we obtain $\hat{\beta }=({\hat{\beta }}_0,{\hat{\beta }}_1)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}=({\hat{\beta }}_0,{\hat{\beta }}_1)$$\end{document} directly by feeding the original sample, the trimmed sample, the SATC augmented data and the RSATC augmented data into the R function lmer in the lme4 package (Bates et al. Reference Bates, Maechler, Bolker and Walker2015) with corresponding $w_{\mathrm{ik}}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ik}$$\end{document} , or $w_{\mathrm{ik}}^a$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w_{ik}^a$$\end{document} for the augmented data, assigned in its weights argument. For ${\hat{\pi }}_k=\frac{n_{k1}}{n_k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\pi }}_k=\frac{n_{k1}}{n_k}$$\end{document} , we choose $c=\frac{N_1}{N}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=\frac{N_1}{N}$$\end{document} to normalize the weights where $N_1$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_1$$\end{document} is the total number of the intern-teaching observations because they help to neutralize the impact of observations with extremely small or extremely large $\frac{n_{k1}}{n_k}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{n_{k1}}{n_k}$$\end{document} . For the standard error of $\hat{\beta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}$$\end{document} in IPTW-orig and IPTW-trim, we calculated the square root of the following robust estimator (Hong & Raudenbush Reference Hong and Raudenbush2008) using $({\hat{\sigma }}_0^2,{\hat{\sigma }}_1^2,\hat{\rho },{\hat{\sigma }}_{ϵ}^2)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\hat{\sigma }}_0^2, {\hat{\sigma }}_1^2, {\hat{\rho }}, {\hat{\sigma }}_\epsilon ^2)$$\end{document} returned from the lmer function,

$\begin{array}{c}cov\left({\hat{\beta }}_{\mathrm{IPTW}}\right)={\left(X^{\prime }\hat{W}X\right)}^{-1}{X}^{\prime }\hat{W}(Y-X\hat{\beta }){(Y-X\hat{\beta })}^{\prime }\hat{W}X{\left(X^{\prime }\hat{W}X\right)}^{-1},\end{array}$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} cov({\hat{\beta }}_{IPTW})=({\mathbf {X}}^\prime \hat{{\mathbf {W}}}{\mathbf {X}})^{-1}{\mathbf {X}}^\prime \hat{{\mathbf {W}}}({\mathbf {Y}}-{\mathbf {X}}{\hat{\beta }})({\mathbf {Y}}-{\mathbf {X}}{\hat{\beta }})^\prime \hat{{\mathbf {W}}}{\mathbf {X}}({\mathbf {X}}^\prime \hat{{\mathbf {W}}}{\mathbf {X}})^{-1}, \end{aligned}$$\end{document}

where $X^{\prime }=\left(\begin{array}{ccc}{T}_1^{\prime }& \dots & T_K^{\prime }\\ {(1-T_1)}^{\prime }& \dots & {(1-T_K)}^{\prime }\end{array}\right)$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {X}}^\prime =\left( \begin{array}{ccc} {\mathbf {T}}^\prime _1 &{} \ldots &{} {\mathbf {T}}^\prime _K \\ ({\mathbf {1}}-{\mathbf {T}}_1)^\prime &{} \ldots &{} ({\mathbf {1}}-{\mathbf {T}}_K)^\prime \end{array}\right) $$\end{document} with $T_k={(T_{1k},T_{2k},\dots ,T_{n_{k}k})}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {T}}_k =(T_{1k},T_{2k},\ldots ,T_{n_kk})^\prime $$\end{document} , ${\hat{W}}^{-1}=\text{diag}\{{\hat{\sigma }}_0^2(1-T_k){(1-T_k)}^{\prime }+{\hat{\sigma }}_1^2{T}_k{T}_k^{\prime }+\hat{\rho }(1-T_k)T_k^{\prime }+\hat{\rho }{T}_k{(1-T_k)}^{\prime }+{\hat{\sigma }}_{ϵ}^2{W}_k^{-1}{\}}_{k=1}^K$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{{\mathbf {W}}}^{-1} = \text {diag} \Big \{{\hat{\sigma }}_0^2 ({\mathbf {1}}-{\mathbf {T}}_k) ({\mathbf {1}}-{\mathbf {T}}_k)^\prime + {\hat{\sigma }}_1^2 {\mathbf {T}}_k {\mathbf {T}}_k^\prime + {\hat{\rho }} ({\mathbf {1}}-{\mathbf {T}}_k) {\mathbf {T}}_k^\prime + {\hat{\rho }} {\mathbf {T}}_k({\mathbf {1}}-{\mathbf {T}}_k)^\prime + {\hat{\sigma }}_\epsilon ^2 {\mathbf {W}}_k^{-1}\Big \}_{k=1}^K$$\end{document} with $W_k={(w_{1k},w_{2k},\dots ,w_{n_{k}k})}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {W}}_k=(w_{1k},w_{2k},\ldots ,w_{n_kk})^\prime $$\end{document} , and $Y={(Y_1^{\prime },Y_2^{\prime },\dots ,Y_K^{\prime })}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {Y}}=({\mathbf {Y}}_1^\prime ,{\mathbf {Y}}_2^\prime ,\ldots ,{\mathbf {Y}}_K^\prime )^\prime $$\end{document} with $Y_k={(Y_{1k},Y_{2k},\dots ,Y_{n_{k}k})}^{\prime }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {Y}}_k=({\mathbf {Y}}_{1k},{\mathbf {Y}}_{2k},\ldots ,{\mathbf {Y}}_{n_kk})^\prime $$\end{document} . To estimate the standard error of $\hat{\beta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}$$\end{document} in AIPTW-SATC and AIPTW-RSATC, we employed the bootstrap procedure by resampling the clusters with replacement 30 times (Field & Welsh Reference Field and Welsh2007) and then calculated the sample standard deviation of the 30 AIPTW $\hat{\beta }$ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\hat{\beta }}$$\end{document} ’s from these bootstrap samples. Readers can find in the supplementary materials, the program code in R with a generic function AIPTW-HLM that can be used to obtain the IPTW-orig, IPTW-trim, AIPTW-SATC and AIPTW-RSATC estimates, and the sample code to generate the simulated data and obtain the simulation results for one of the settings.