## Slide 1

### Slide 1 text

Chapter 19. TIME-VARYING TREATMENTS

## Slide 2

### Slide 2 text

Part III Causal inference from complex longitudinal data 2

## Slide 3

### Slide 3 text

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

## Slide 4

### Slide 4 text

The Causal Effect of Time-varying Treatments 1.

## Slide 8

### Slide 8 text

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。

## Slide 9

### Slide 9 text

Treatment Strategies 2.

## Slide 10

### Slide 10 text

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

## Slide 11

### Slide 11 text

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

## Slide 12

### Slide 12 text

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

## Slide 15

### Slide 15 text

Sequentially Randomized Experiments 3.

## Slide 16

### Slide 16 text

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

## Slide 18

### Slide 18 text

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。

## Slide 19

### Slide 19 text

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

## Slide 20

### Slide 20 text

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

## Slide 21

### Slide 21 text

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

## Slide 22

### Slide 22 text

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。

## Slide 23

### Slide 23 text

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

## Slide 24

### Slide 24 text

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

## Slide 25

### Slide 25 text

Sequential Exchangeability 4.

## Slide 28

### Slide 28 text

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

## Slide 29

### Slide 29 text

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

## Slide 30

### Slide 30 text

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.

## Slide 31

### Slide 31 text

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.

## Slide 32

### Slide 32 text

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

## Slide 33

### Slide 33 text

Identifiability under some but not all treatment strategies 5.

## Slide 34

### Slide 34 text

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 % &

## Slide 35

### Slide 35 text

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 .

## Slide 36

### Slide 36 text

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.

## Slide 37

### Slide 37 text

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.

## Slide 38

### Slide 38 text

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

## Slide 39

### Slide 39 text

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 .

## Slide 40

### Slide 40 text

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

## Slide 41

### Slide 41 text

Time-varying confounding and time-varying confounders 6.

## Slide 42

### Slide 42 text

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 Thank you!