. Author manuscript; available in PMC: 2021 Jul 1.

Published in final edited form as: Stat Methods Med Res. 2019 Oct 1;29(7):1891–1912. doi: 10.1177/0962280219877520

Table 2:

Correlation estimators for selected working correlation structures, assuming constant within-person variance over time. $d \in D$ is an embedded DTR, $W_{i}^{(d)}$ is shorthand for W^(d)(A_1,i, R_i,A_2,i) and ${\hat{e}}_{i, t}^{(d)} (\hat{θ})$ is the estimated residual $Y_{i, t} - μ_{t}^{(d)} (X_{i}; \hat{θ})$ .

Cor. structure	$Cor (Y_{t_{j}}^{(d)}, Y_{t_{k}}^{(d)})$	Estimator
AR(1)	${\begin{array}{l} 1 & t_{j} = t_{k} \\ {(ρ^{(d)})}^{\| j - k \|} & t_{j} \neq t_{k} \end{array}$	${\hat{ρ}}^{(d)} = \frac{\sum_{i = 1}^{n} W_{i}^{(d)} \sum_{m = 1}^{T - 1} {\hat{e}}_{i, t_{m}}^{(d)} (\hat{θ}) {\hat{e}}_{i, t_{m + 1}}^{(d)} (\hat{θ})}{{({\hat{σ}}^{(d)})}^{2} \cdot n \cdot (T - 1)}$
Exchangeable	${\begin{array}{l} 1 & t_{j} = t_{k} \\ ρ^{(d)} & t_{j} \neq t_{k} \end{array}$	${\hat{ρ}}^{(d)} = \frac{\sum_{i = 1}^{n} W_{i}^{(d)} \sum_{l < m} {\hat{e}}_{i, t_{l}}^{(d)} (\hat{θ}) {\hat{e}}_{i, t m}^{(d)} (\hat{θ})}{{({\hat{σ}}^{(d)})}^{2} \cdot n \cdot T (T - 1) / 2}$
Unstructured	${\begin{matrix} 1 & t_{j} = t_{k} \\ ρ_{t_{j}, t_{k}}^{(d)} & t_{j} \neq t_{k} \end{matrix}$	${\hat{ρ}}_{t_{j}, t_{k}}^{(d)} = \frac{\sum_{i = 1}^{n} W_{i}^{(d)} {\hat{e}}_{i, t_{j}}^{(d)} (\hat{θ}) {\hat{e}}_{i, t_{k}}^{(d)} (\hat{θ})}{{({\hat{σ}}^{(d)})}^{2} \cdot n}$