Table 2.

Summary balance metrics averaged over person-time by standardization

Time-specific metrics for Diagnostics 1, 2a, and 3 have the general forma,b: $$Q_{H(\tau ),\tau ,k}=E[W(\tau )\times I(A(\tau )=a^{\prime })\times C({\tau }_c)\mid H(\tau ),\tau ,k]-E[W(\tau )\times I(A(\tau )=a^{″})\times C({\tau }_c)\mid H(\tau ),\tau ,k]$$ Where k is the distance between the covariate time τc and the exposure time τ i.e. k = |τ − τc|. H(τ) is the exposure history at time τ i.e. A(τ − 1); W(τ) is equal to one or an inverse probability weight, and follows the exposure time τ except for Diagnostic 2 under censoring where it follows covariate time τc. Each diagnostic QH(τ),τ,k is estimated from weighted person-timec: $$N_{H(\tau ),\tau ,k}=\sum W(\tau )\times I(A(\tau )=a^{\prime })\mid H(\tau ),\tau ,k+\sum W(\tau )\times I(A(\tau )=a^{″})\mid H(\tau ),\tau ,k$$
Summary metric	Example Interpretation for Diagnostic 1
Average over exposure history H(τ)c
${\overline{Q}}_{\tau ,k}={\displaystyle \sum _{H(\tau )}}(Q_{H(\tau ),\tau ,k}\times N_{H(\tau ),\tau ,k})/{\displaystyle \sum _{H(\tau )}}{N}_{H(\tau ),\tau ,k}$	The average balance for exposure A assessed at time 2 and covariate C assessed at time 1 is…
Average over exposure history H(τ) and time τ
${\overline{Q}}_k={\displaystyle \sum _{H(\tau ),\tau }}(Q_{H(\tau ),\tau ,k}\times N_{H(\tau ),\tau ,k})/{\displaystyle \sum _{H(\tau ),\tau }}{N}_{H(\tau ),\tau ,k}$	The average balance for exposure A at any given time τ and covariate C assessed at time τ − 1 is…
Average over exposure history H(τ), time τ, and distance kd
$\overline{Q}={\displaystyle \sum _{H(\tau ),\tau ,k}}(Q_{H(\tau ),\tau ,k}\times N_{H(\tau ),\tau ,k})/{\displaystyle \sum _{H(\tau ),\tau ,k}}{N}_{H(\tau ),\tau ,k}$	The average balance for exposure A assessed at any given time and covariate C assessed at any prior time is…

^aFor Diagnostic 2b replace exposure history H(τ) with propensity score strata e(τ). The standardization over person-time defined by e(τ) gives the average balance for an exposure assessed at a specific time τ and a covariate assessed at a specific time τ_c

^bWe omitted censoring here to simplify the expressions but this can be incorporated as in Table 1 or eAppendix Table 1

^cThe time-specific metrics are implicitly indexed by their non-referent value a′. When there are many such values, one can first average metrics over the non-referent person-time distribution within levels of exposure history H(τ), time τ, distance k. The person-time contributing to this metric is the sum over all exposure levels. This metric can then be averaged over exposure history, time, and distance.

^dOne can first define disjoint ranges of distance p, and then average over distance k for each range p. An example interpretation for Diagnostic 1 would be “the average balance for exposure A assessed at time τ and covariate C assessed anywhere between time τ − 1 and τ − 10.”