Shuntaro Sato
November 25, 2020
2.1k

# 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

## Transcript

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!