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

38e2af7f8bdad4f2087ab3d42b627e33?s=47 Shuntaro Sato
November 25, 2020

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

Keywords: 因果推論, Time-fixed treatment(時間固定),Time-varying treatment(時間変動),Sequentially exchangeability, Static sequential exchangeability, Dynamic sequential exchangeability, SWIG

38e2af7f8bdad4f2087ab3d42b627e33?s=128

Shuntaro Sato

November 25, 2020
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  1. Chapter 19. TIME-VARYING TREATMENTS

  2. Part III Causal inference from complex longitudinal data 2

  3. 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
  4. The Causal Effect of Time-varying Treatments 1.

  5. 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 [!"#] − [!"$]
  6. 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)
  7. 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
  8. 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。
  9. Treatment Strategies 2.

  10. 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
  11. 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
  12. 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
  13. 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”) [ % !"& #] − [!'"& $] [ % !"& #] −
  14. 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 $%! " がよく使われる。($, ! ")
  15. Sequentially Randomized Experiments 3.

  16. Sequentially randomized experiments (SRE) とは? ▣ An experiment in which

    treatment is randomly assigned to each individual at each time k 16
  17. 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
  18. 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。
  19. 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
  20. 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
  21. 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
  22. 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。
  23. Observational Studies? 23 ▣ Ak に治療するかは outcome predictors (prognostic factors)

    によって決まる
  24. 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
  25. Sequential Exchangeability 4.

  26. 前回まで ▣ Valid causal inferences about time-fixed treatments typically require

    conditional exchangeability. 26 ! ⊥ |
  27. 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
  28. Sequential conditional exchangeability (SCE) 28 ( ⊥ ̅ | ̅

    )*#"( ̅ ,!"#, % -!"% , > ) for all strategies and k = 0,1…K
  29. 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
  30. 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.
  31. 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.
  32. 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
  33. Identifiability under some but not all treatment strategies 5.

  34. 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 % &
  35. 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 .
  36. 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.
  37. 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.
  38. 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
  39. 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 .
  40. Summary ▣ In Figure 19.5, sequential exchangeability for $ holds.

    ▣ In Figure 19.6, only the weaker condition for static strategies holds. 40
  41. Time-varying confounding and time-varying confounders 6.

  42. 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
  43. 43 Thank you!