Outline 19.1 The causal effect of time-varying treatments 19.2 Treatment strategies 19.3 Sequentially randomized experiments 19.4 Sequential exchangeability 19.5 Identifiability under some but not all treatment strategies 19.6 Time-varying confounding and time-varying confounders 3
前回まで ▣ Time-fixed treatment variable A ▣ 1: treated, 0: untreated ▣ Outcome variable Y ▣ measured 60 months later 5 ⇨ Average causal effect of A on the outcome Y [!"#] − [!"$]
今回から ▣ Time-varying treatment variable Ak ▣ フォロ⑲アップ期間中、値が変化する 6 例) 5-year follow-up study of individuals with HIV ▣ k = 0, 1, 2 …. K with K = 59 ▣ Ak = 1: received antiretroviral therapy in month k 0: otherwise § 原則：研究開始前は誰も治療を受けていない。(A-1 = 0 for all individuals)
Treatment の種類 ▣ Treatment strategy § A rule to assign treatment at each time k of follow-up 1. Static or non-dynamic treatment strategy § Strategies 2 for which treatment does not depend on covariates. 2. Dynamic treatment strategy § Strategies in which the treatment ak at time k depends on the evolution of an individual’s time varying covariate(s) 2 k 10
Dynamic treatment strategy ▣ At time 0, all individuals have a high CD4 cell count (L0 = 0) ▣ Do not treat while Lk = 0, start treatment when Lk = 1 and treat continuously after that □ 2 = (a0 , a1 , … aK ) というように書くことができない 12 例) 2 k : CD4 cell count measured at month k in all individuals. □ 1 = low CD4 cell count (a bad prognosis) □ 0 = otherwise
HIV Study 18 ▣ Lk = CD4 cell count at time k ▣ Uk = immune system at time k ▣ Y = health status ▣ immune system に対するダメ⑲ジが大きいほど、CD4 cell count は低くなり、健康状態が悪化する。
Figure 3 – HIV Study 22 ▣ SRE ▣ Ak に治療するかは前月までの治療歴＋E k + H Uk によって決まる ▣ 測定できない変数によってランダム化 の確率を算出することはできない。 ▣ SREは、測定できないUk から治療変数 Ak に直接→がない場合のみcausal diagram で表すことができる。
今回から ▣ Causal inference with time-varying treatments requires adjusting for the time varying covariates 2 k to achieve conditional exchangeability at each time point. 27 Sequential conditional exchangeability
SCE in observational studies 30 ▣ The mean of the counterfactual outcome E[!] under all strategies is identified. ▣ No mean of the counterfactual outcome E[!] is identified.
Other causal diagrams – observational studies 31 ▣ HIV Study: an indicator for a scheduled clinic visit at time 0 that was not recorded in our database. ▣ The mean counterfactual outcome is identified under any static strategy; however, it is not identified under any dynamic strategy.
SCE in Figure 19.5 34 ▣ HIV Study ▣ HIV Study ▣ HIV Study … this path is blocked. … both hold for any static strategy. = static sequential exchangeability for % &
Static sequential exchangeability (SSE) 35 ▣ Static sequential exchangeability for " # is weaker than sequential exchangeability !. ▣ Static sequential exchangeability is sufficient to identify the mean counterfactual outcome under any static strategy = E .
SSE in observational studies (Figure 19.6) 36 ▣ Static sequential exchangeability also holds in Figure 19.6. ▣ In any observational study represented by Figure 19.6, we can identify the mean counterfactual outcome under any static strategy.
SSE in observational studies (Figure 19.11) 37 ▣ Neither sequential exchangeability for ! nor static sequential exchangeability for ! hold. ▣ In observational study represented by Figure 19.11, we cannot use the data to validly estimate causal effects involving any strategies.
SCE under dynamic regimes? (Figure 19.5) 38 ▣ $ = 0 for everyone ▣ %(% !) = 1 when % ! = 1 , %(% !) = 0 when % ! = 0 ▣ $ = 0 for everyone ▣ We can identify the mean counterfactual outcome under all strategy . … both hold for any strategy
SCE under dynamic regimes? (Figure 19.6) 39 ▣ find that does not hold because of the open path below. ▣ We cannot identify the mean counterfactual outcome for any strategy .
All the measured covariates sufficient to ensure sequential exchangeability? ▣ We need to adjust for confounders of the effect of A1 on Y. ▣ Block all open back door paths between A1 and Y. 42
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