Time-varying treatment(Causal inference: What if, Chapter 19)

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Chapter 19. TIME-VARYING TREATMENTS

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Part III Causal inference from complex longitudinal data 2

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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

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The Causal Effect of Time-varying Treatments 1.

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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 [!"#] − [!"$]

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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)

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記号の意味 ▣ ̅ = (A0 , A1 , … Ak ) § time 0 から time k までの治療歴 ▣ ̅ K = ̅ § 研究開始から研究終了までの治療歴 7

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HIV Study ▣ ̅ = (A0 = 1, A1 = 1, … A59 = 1) = 2 1 § 研究開始から終了まで治療を受けた参加者 ▣ ̅ = (A0 = 0, A1 = 0, … A59 = 0) = 2 0 § 研究開始から終了まで治療を受けなかった参加者 8 v 多くの人はフォロ⑲アップ期間中に治療を受けた月があったり、受けなかった月があったりする。

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Treatment Strategies 2.

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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

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Static treatment strategy 例) 11 [!"#] − [!"$] ⇨ Average causal effect of A on the outcome Y ▣ “Always treat” § 2 = (1, 1, … 1) = 2 1 ▣ “Never treat” § 2 = (0, 0, … 0) = 2 0

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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

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平均因果効果の表し方 13 例) HIV Study [ ! "] − [ ! "#] q strategy 2 (“always treat”) vs. strategy 2 ’ (“never treat”) q strategy 2 (“always treat”) vs. strategy (“treat only after CD4 cell count is low”) [ % !"& #] − [!'"& $] [ % !"& #] −

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平均因果効果の表し方 (continued) 14 q strategy 2 (“always treat”) vs. strategy (“treat only after CD4 cell count is low”) [ % !"& #] − ⇨ time-varying treatment の平均因果効果の表し方は一つではない。 v = any static or dynamic strategy v $%! " がよく使われる。($, ! ")

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Sequentially Randomized Experiments 3.

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Sequentially randomized experiments (SRE) とは？ ▣ An experiment in which treatment is randomly assigned to each individual at each time k 16

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各変数の定義 17 § Lk : the set of measured variables at k § Uk : the set of unmeasured variables at k § common causes of at least two other variables

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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。

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Figure 1 – HIV Study 19 1. 前月に治療を受けなかった人 (Ak-1 = 0) § 0.5 の確率で治療を施す 2. 前月に治療を受けた人 (Ak-1 = 1) § 1 の確率で治療を施す ▣ SRE ▣ Ak に治療するかは前月までの治療歴によって決まる

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Figure 1 – HIV Study (continued) 20 ▣ Static treatment strategy の平均因果効果 ▣ SRE ̅ = & ] ▣ Dynamic treatment strategy の平均因果効果 g-methods を使わなければ算出できない

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Figure 2 – HIV Study 21 1. 前月に治療を受けず、CD4 cell countが高い人 (Ak-1 = 0, Lk = 1) § 0.4 の確率で治療を施す 2. 前月に治療を受けず、CD4 cell countが低い人 (Ak-1 = 1, Lk = 0) § 0.8 の確率で治療を施す 3. CD4 cell countの値に関わらず、前月に治療を受けた人 (Ak-1 = 1) § 0.5 の確率で治療を施す ▣ SRE ▣ Ak に治療するかは前月までの治療歴＋E k によって決まる

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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。

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Observational Studies? 23 ▣ Ak に治療するかは outcome predictors (prognostic factors) によって決まる

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Observational Studies – HIV Study 24 ▣ CD4 cell count (Lk )が低い ⇨ 治療が施される ▣ Ak に治療するかは ̅ k-1 ＋E k によって決まる ▣ CD4 cell count (Lk )が低い ⇨ 治療が施される

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Sequential Exchangeability 4.

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前回まで ▣ Valid causal inferences about time-fixed treatments typically require conditional exchangeability. 26 ! ⊥ |

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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

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Sequential conditional exchangeability (SCE) 28 ( ⊥ ̅ | ̅ )*#"( ̅ ,!"#, % -!"% , > ) for all strategies and k = 0,1…K

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どんな場合にSCEが成立するの？ 29 ▣ Sequential exchangeability for $ holds in; □ sequentially randomized experiments □ observational studies ■ 治療を受ける確率が ̅ k-1 ＋E k によって決まる場合

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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.

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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.

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Other identifiability conditions 32 ▣ Sequential conditional exchangeability ▣ Positivity ▣ Consistency v ３つの条件が成立した場合、 ̅ k-1 とE k を調整することで、the mean counterfactual outcome E[!] を確認することができる。 • g-formula (standardization) • IP weighting • g-estimation

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Identifiability under some but not all treatment strategies 5.

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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 % &

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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 .

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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.

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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.

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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

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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 .

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Summary ▣ In Figure 19.5, sequential exchangeability for $ holds. ▣ In Figure 19.6, only the weaker condition for static strategies holds. 40

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Time-varying confounding and time-varying confounders 6.

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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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43 Thank you!