Table 2.

Estimands of Interest: Conditional Average Treatment Effects (CATEs)

Analysis	Teachers reporting fixed mindsets	Teachers reporting growth mindsets
Assuming no confounding of the moderator	CATES = Fixed = $\begin{array}{l}\mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Mindset},S_j=\mathrm{Fixed}\right]-\\ \mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Control},S_j=\mathrm{Fixed}\right]\end{array}$	CATES = Growth = $\begin{array}{l}\mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Mindset},S_j=\mathrm{Growth}\right]-\\ \mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Control},S_j=\mathrm{Growth}\right]\end{array}$
$\mathbb{E}$ Adjusting for potential confounding (primary estimand of interest)	CATES = Fixed, C = c = $\begin{array}{l}\mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Mindset},S_j=\mathrm{Fixed},C_j=c\right]-\\ \mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Control},S_j=\mathrm{Fixed},C_j=c\right]\end{array}$	CATES = Growth, C = c = $\begin{array}{l}\mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Mindset},S_j=\mathrm{Growth},C_j=c\right]-\\ \mathbb{E}\left[Y_{ij}|T_{ij}=\mathrm{Control},S_j=\mathrm{Growth},C_j=c\right]\end{array}$

Note: CATE refers to the treatment effect within a subgroup; i indexes students, j indexes teachers. Y = math grades, T (for treatment) = treatment status, S = teacher mindset, C (for confounds) = vector of teacher-mindset confounds, c = population average for potential teacher or school confounds. See proofs and justifications in the article by Yamamoto et al. (2019).