Table 1.

Time-specific balance metrics for confounding of a time-varying exposure A(t) by time-varying covariates C(t) in the presence of censoring S(t)

Diagnostic 1: Time-varying confounding i.e. C(t − k) across levels of A(t) (for all t ∈ {0, …, t} and chosen k ∈ {0, …, t} where 0 ≤ k ≤ t′)
Balance metrica	Definitions
E[C(t − k)|A(t) = a′,Ā(t − 1),S̄(t) = 0] − E[C(t − k)|A(t) = a″,Ā(t − 1),S(t) = 0]	N/A
Diagnostic 2: Exposure-covariate feedback i.e. C(t) across levels of A(t − k) (for all t ∈ {0, …, t} and all k ∈ {1, …, t} where 1 ≤ k ≤ t′)
Balance metrica
a. by inverse probability weighting	Weight Was(t−k) = Wa(t−k) × Ws(t)
E[Wa′s(t−k) × I(A(t−k) = a′)×C(t)|Ā(t − k − 1), S̄(t) = 0] − E[Wa″s(t−k) × C(t)|Ā(t − k − 1), S̄(t) = 0]	$W_{a(t-k)}=\frac{P[A(t-k)=a\mid \overline{A}(t-k-1),\overline{S}(t-k)=0]}{P[A(t-k)=a\mid \overline{A}(t-k-1),\overline{C}(t-k),\overline{S}(t-k)=0]}$ $W_{s(t)}={\prod }_{k=0}^{k=t}\frac{P[S(t-k)=0\mid \overline{S}(t-k-1)=0]}{P[S(t-k)=0\mid \overline{S}(t-k-1)=0,\overline{A}(t-k-1),\overline{C}(t-k-1)]}$
b. by propensity score stratificationb	Propensity score ea(t−k) and weight Ws(t)
E[Ws(t − k) × I(A(t − k) = a′ × C(t)|ea′(t − k),S̄(t) = 0] − E[Ws(t − k) × I(A(t − k) = a″ × C(t)|ea′(t − k),S̄(t) = 0]	ea(t −k) = P[A(t − k) = a′|Ā(t − k − 1), S̄(t − k) = 0 $W_{s(t)}={\prod }_{k=0}^{k=t}\frac{P[S(t-k)=0\mid \overline{S}(t-k-1)=0]}{P[S(t-k)=0\mid \overline{S}(t-k-1)=0,\overline{A}(t-k-1),\overline{C}(t-k-1)]}$
Diagnostic 3: Residual time-varying confounding i.e. C(t − k) across levels of A(t) (for all t ∈ {0, …, t} and chosen k ∈ {0, …, t} where 0 ≤ k ≤ t′)
Balance metrica	Weight Wa(t)
E[Wa′(t) × I(A(t) = a′) × C(t − k)|Ā(t − 1),S̄(t) = 0] − E[Wa″(t) × I(A(t) = a″) × C(t − k)|Ā(t − 1), S̄(t) = 0]	$W_{a(t)}={\prod }_{k=0}^{k=t}\frac{P[A(t-k)=a\mid \overline{A}(t-k-1),\overline{S}(t-k)=0]}{P[A(t-k)=a\mid \overline{A}(t-k-1),\overline{C}(t-k),\overline{S}(t-k)=0]}$

^aThese balance metrics are on the mean difference scale. They can be reported on the standardized mean difference scale by dividing by the (unweighted) pooled standard deviation conditional on exposure history (for Diagnostics 1, 2a and 3) or propensity-score strata (for Diagnostic 2b). Doing so places metrics for binary and continuous covariates on the same scale.

^bFor categorical exposures one needs to jointly condition on the predicted probabilities for each non-referent exposure level