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

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

Shuntaro Sato

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

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  2. Part III
    Causal inference from
    complex longitudinal data
    2

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  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

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

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  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
    [!"#] − [!"$]

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  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)

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  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

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  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。

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  9. Treatment Strategies
    2.

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  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

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  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

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  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

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  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”)
    [ %
    !"&
    #] − [!'"&
    $]
    [ %
    !"&
    #] −

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  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 $%!
    " がよく使われる。($, !
    ")

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  15. Sequentially
    Randomized
    Experiments
    3.

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  16. 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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  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

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  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。

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  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

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  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

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  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

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  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。

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  23. 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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  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

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  25. Sequential
    Exchangeability
    4.

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

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  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

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  28. Sequential conditional exchangeability (SCE)
    28
    ( ⊥ ̅

    | ̅
    )*#"( ̅
    ,!"#,
    %
    -!"%
    , >
    )
    for all strategies and k = 0,1…K

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  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

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  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.

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  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.

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  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

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  33. Identifiability under some but
    not all treatment strategies
    5.

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  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 %
    &

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  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
    .

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  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.

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  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.

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  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

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  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 .

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  40. 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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  41. Time-varying confounding
    and time-varying confounders
    6.

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  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

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  43. 43
    Thank you!

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