Defining General, Hypothetical Interventions

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Have you ever begun reading a paper in the methodological causal inference literature and encountered the phrase “assume the treatment or exposure is a binary…”? (Most papers we read assume this!!) While assuming exposure variables are binary can simplify the definition of causal effects, many exposures of interest in reality are not binary.

Instead, we will work in situations where \(A\) is a binary, categorical, multivariate, or continuous variable!

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In the previous section, we defined \(\dd(a_t, h_t, \epsilon_t)\) as a function that maps \(A_t\), \(H_t\), and potentially a randomizer \(\epsilon_t\) to a new value of \(A_t\). Our focus henceforth is estimating the causal effect of an intervention, characterized by \(\dd\) on the outcome \(Y\), through the causal parameter

\[ \theta = \E[Y^{\bar A^{\dd}}]\text{,} \]

where \(Y^{\bar A^{\dd}}\) is the counterfactual outcome in a world, where possibly contrary to fact, each entry of \(\bar{A} = (A_1, \ldots, A_\tau)\) was modified according to the function \(\dd\).

• When \(Y\) is continuous, \(\theta\) is the mean population value of \(Y\) under intervention \(\dd\).

• When \(Y\) is dichotomous, \(\theta\) is the population proportion of event \(Y\) under intervention \(\dd\).

• When \(Y\) is the indicator of an event by end of the study, \(\theta\) is defined as the cumulative incidence of \(Y\) under intervention \(\dd\).

But what is this function \(\dd\), how can it be defined, and how does using this function to define interventions solve the problem? Let’s start from simple to more complex examples of functions \(\dd\).

Static Interventions

Let \(A\) denote a binary vector, such as receiving a medication, and define \(\dd(a_t, h_t, \epsilon) = 1\). This intervention characterizes a hypothetical world where all members of the population receive treatment.

Static intervention

An intervention is static if the function \(\dd_t\) always returns the same value regardless of the input.

Example

Let’s say we were interested in the effect of randomizing patients with opioid use disorder to injection naltrexone (\(A=1\)) vs. sublingual buprenorphine (\(A=0\)). We would contrast the counterfactual outcomes in a hypothetical world in which all units were treated with injection naltrexone \(\dd_1=1\) versus a hypothetical world in which all units were treated with buprenorphine \(\dd_0=0\). This gives us the well-known average (comparative) treatment effect (ATE). \[\E[Y^{\bar A^{\dd_1}} - Y^{\bar A^{\dd_0}}] = \E[Y^{A=1} - Y^{A=0}] \].

Dynamic Treatment Regime

Let \(A_t\) denote a binary vector, such as receiving a medication, and \(H_t\) a numeric vector, such as a measure of discomfort. For a given value of \(\delta\), define \[ \dd(a_t, h_t, \epsilon) = \begin{cases} 1 &\text{ if } h_t > \delta \\ 0 &\text{ otherwise.} \end{cases} \]

Dynamic treatment regime

Interventions where the output of the function \(\dd\) depends only on the covariates \(H_t\) are referred to as being dynamic.

Example

Rudolph, Williams, et al. (2022) examined a Buprenorphine (BUP-NX) dosing strategy among a population of patients who were taking BUP-NX for opioid use disorder. Under the hypothetical intervention, patients who reported opioid use during the week prior to a physicians exam received a BUP-NX dose increase while patients who did not report prior-week opioid use maintained the same dose. Let \(A_t\) be a binary indicator for BUP-NX dose increase at week \(t\) compared to week \(t-1\) and \(X_t\) be an indicator for opioid use at week \(t\). Then,

\[ \dd(a_t, h_t, \epsilon) = \begin{cases} 1 \text{ if } x_{t-1} = 1\\ 0 \text{ otherwise} \end{cases} \]

Modified Treatment Policies

While much attention is given to static and dynamic interventions, their use is often accompanied by a few key problems.

1. Defining causal effects in terms of hypothetical interventions where treatment is applied to all units may be inconceivable. For example, we may be interested to know if reducing surgery time reduces surgical complications. However, it’s inconceivable to set all surgeries to a given duration, even if this duration depends on patient covariates.

2. Defining causal effects in terms of hypothetical interventions where treatment is applied to all units may induce positivity violations.

A solution to these problems is to instead define causal effects using modified treatment policies (MTP).

Modified treatment policy

An intervention characterized by a hypothetical world where the natural value of treatment is modified is called a modified treatment policy.

Additive and multiplicative shift MTP

Let \(A_t\) denote a numeric vector. Assume that \(A_t\) has support in the data such that \(P(A_t \leq u(h_t) \mid H_t = h_t) = 1\). For an analyst-defined value of \(\delta\), define \[ \dd(a_t, h_t, \epsilon) = \begin{cases} a_t + \delta &\text{ if } a_t + \delta \leq u(h_t) \\ a_t &\text{ otherwise.} \end{cases} \]

Under this intervention, the natural value of exposure at time \(t\) is increased by the analyst-defined value \(\delta\), whenever such an increase is feasible. This MTP is referred to as an additive shift MTP.

Example

Dı́az et al. (2023) estimated the effect of increasing P/F ratio (a measure of hypoxemia) by 50 units on survival among those patients with acute respiratory failure (a P/F ratio < 300).

\[ \dd_t(a_t, h_t, \epsilon) = \begin{cases} a_t + 50 &\text{ if } a_t \leq 300 \\ a_t &\text{ otherwise} \end{cases} \]

We can similarly define a multiplicative shift MTP as

\[ \dd(a_t, h_t, \epsilon) = \begin{cases} a_t \times \delta &\text{ if } a_t \times \delta \leq u(h_t) \\ a_t &\text{ otherwise}. \end{cases} \]

Example

Nugent and Balzer (2023) evaluated the association between county-level measures of mobility and incident COVID-19 cases in the United States in the Summer and Fall of 2022. They considered both hypothetical additive and multiplicative MTPs; for example, they defined a multiplicative MTP where a measure for the density of mobile devices visiting commercial locations was decreased by 25%:

\[ \dd(a_t, h_t, \epsilon) = a_t \times 0.75. \]

Randomized Interventions

Let \(A\) denote a binary vector, \(\epsilon \sim U(0, 1)\), and \(\epsilon\) be an analyst-defined value between 0 and 1. We may then define randomized interventions. For example, imagine we are interested in a hypothetical world where half of all smokers quit smoking. This intervention would be defined as

\[ \dd(a_t, \epsilon_t) = \begin{cases} 0 &\text{ if } \epsilon_t < 0.5 \text{ and } a_t = 1 \\ a_t &\text{ otherwise} \end{cases}. \]

Incremental Propensity Score Interventions Based on the Risk Ratio

Let \(A\) denote a binary variable, \(\epsilon \sim U(0, 1)\), and \(\delta\) be an analyst-defined risk ratio limited to be between \(0\) and \(1\). In addition, define \(P(A_t = a_t\mid H_t)= \g(a_t \mid H_t)\).

If we were interested in an intervention that decreased the likelihood of receiving treatment, define

\[ \dd_t(a_t, h_t, \epsilon_t) = \begin{cases} a_t &\text{ if } \epsilon_t < \delta \\ 0 &\text{ otherwise} \end{cases}. \] In this case, we have \(\g^\dd(a_t \mid H_t) = a_t \delta \g_t(1 \mid H_t) + (1 - a_t) (1 - \delta \g_t(1\mid H_t))\), which leads to a risk ratio of \(\g_t^\dd(1 \mid H_t)/\g_t(1\mid H_t) = \delta\) for comparing the propensity score post- vs pre-intervention.

Conversely, if we were interested in an intervention that increased the likelihood of receiving treatment, define

\[ \dd_t(a_t, h_t, \epsilon_t) = \begin{cases} a_t &\text{ if } \epsilon_t < \delta \\ 1 &\text{ otherwise.} \end{cases} \]

Now \(\g_t^\dd(a_t \mid H_t) = a_t (1 - \delta \g_t(0\mid H_t)) + (1 - a_t) \delta \g_t(0 \mid H_t)\), which implies a risk ratio \(\g_t^\dd(0\mid H_t)/\g(0\mid H_t) = \delta\).

Incremental propensity score intervention

An intervention where the conditional probability of treatment is shifted. (As an aside, interventions where the shift is in the odds ratio scale where previously proposed, but the effects of odds-ratio shifts should not be estimated with the lmtp package, we will discuss this further.

Interventions where the shift is in the odds ratio scale were previously proposed, but the effects of odds-ratio shifts should not be estimated with lmtp, we will discuss this more later.

Example

Using electronic health record data, Wen, Marcus, and Young (2023) estimated the effect of increasing the proportion of PrEP uptake on bacterial STI among cis-gender males being tested for STIs and that do not have HIV. Let \(A_t\) be a binary indicator for PrEP initiation at week \(t\), and \(L_t\) be a binary indicator for any STI testing and being HIV-free at week \(t\). They defined a “medium” successful PrEP uptake intervention as

\[ \dd(a_t, h_t, \epsilon) = \begin{cases} a_t &\text{ if } l_t = 1 \text{ and } \epsilon_t < 0.85 \\ 1 &\text{ otherwise}. \end{cases} \]

Identification of the causal parameter

Recall that the fundamental problem of causal inference is that we can’t observe the alternative worlds which we use to define causal effects. If we can’t observe counterfactual variables, then how can we learn a causal effect? Under a set of certain assumptions, we can identify a causal parameter from observed data. These assumptions are called identification assumptions.

Identification Assumptions

1. Positivity. If \((a_t, h_t) \in \text{supp}\{A_t, H_t\}\) then \(\dd(a_t, h_t) \in \text{supp}\{A_t, H_t\}\) for \(t \in \{1, ..., \tau\}\).

   If there is a unit with observed treatment value \(a_t\) and covariates \(h_t\), there must also be a unit with treatment value \(\dd(a_t, h_t)\) and covariates \(h_t\).

2. No unmeasured confounders. All the common causes of \(A_t\) and \((L_s, A_s, Y)\) are measured and contained in \(H_t\) for all \(s \in \{t+1, ..., \tau\}\).

   For all times \(t\), the history \(H_t\) contains sufficient variables to adjust for confounding of \(A_t\) and any subsequent variables, including future treatment.

Question

When might these assumptions be violated?

Assuming the above, \(\theta\) is identified from the observed data with:

Theorem

Set \(\m_{\tau+1} = Y\) For \(t = \tau, ..., 1\), recursively define

\[ \m_t: (a_t, h_t) \rightarrow \E[\m_{t + 1}(A^{\dd}_{t+1}, H_{t + 1}) \mid A_t = a_t, H_t = h_t], \]

and define \(\theta = E[\m_1(A^{\dd}_1, L_1)]\).

As an example, consider the following data where \(\tau = 2\).

We can compute the identification formula in the following steps:

1. Set \(\m_3(A_3^\dd, H_3) = Y\)

2. Compute the regression of \(\m_3(A_3^\dd, H_3)\) on \((A_2, H_2)\). This gives a predictive function, call that predictive function \(\m_2(A_2,H_2)\).

3. Use the predictive function to compute what would have occurred if the intervention had been implemented at time \(t=2\), i.e., compute \(\m_2(A_2^\dd,H_2)\).

4. Compute the regression of \(\m_2(A_2^\dd,H_2)\) on \((A_1, H_1)\). This gives a predictive function, call that predictive function \(\m_1(A_1,H_1)\).

5. Use the predictive function to compute what would have occurred if the intervention had been implemented at time \(t=1\), i.e., compute \(\m_1(A_1^\dd,H_1)\).

6. Compute the mean of \(\m_1(A_1^\dd,H_1)\). This mean is equal to \(\theta\).

References

Dı́az, Iván, Katherine L Hoffman, and Nima S Hejazi. 2024. “Causal Survival Analysis Under Competing Risks Using Longitudinal Modified Treatment Policies.” Lifetime Data Analysis 30 (1): 213–36.

Dı́az, Iván, Nicholas Williams, Katherine L Hoffman, and Edward J Schenck. 2023. “Nonparametric Causal Effects Based on Longitudinal Modified Treatment Policies.” Journal of the American Statistical Association 118 (542): 846–57.

Haneuse, Sebastian, and Andrea Rotnitzky. 2013. “Estimation of the Effect of Interventions That Modify the Received Treatment.” Statistics in Medicine 32 (30): 5260–77.

Kennedy, Edward H. 2019. “Nonparametric Causal Effects Based on Incremental Propensity Score Interventions.” Journal of the American Statistical Association 114 (526): 645–56.

Nugent, Joshua R, and Laura B Balzer. 2023. “A Demonstration of Modified Treatment Policies to Evaluate Shifts in Mobility and COVID-19 Case Rates in US Counties.” American Journal of Epidemiology 192 (5): 762–71.

Rudolph, Kara E, Catherine Gimbrone, Ellicott C Matthay, Ivan Diaz, Corey S Davis, Katherine Keyes, and Magdalena Cerdá. 2022. “When Effects Cannot Be Estimated: Redefining Estimands to Understand the Effects of Naloxone Access Laws.” Epidemiology 33 (5): 689–98.

Rudolph, Kara E, Nicholas T Williams, Alicia T Singham Goodwin, Matisyahu Shulman, Marc Fishman, Iván Dı́az, Sean Luo, John Rotrosen, and Edward V Nunes. 2022. “Buprenorphine & Methadone Dosing Strategies to Reduce Risk of Relapse in the Treatment of Opioid Use Disorder.” Drug and Alcohol Dependence 239: 109609.

Wen, Lan, Julia L Marcus, and Jessica G Young. 2023. “Intervention Treatment Distributions That Depend on the Observed Treatment Process and Model Double Robustness in Causal Survival Analysis.” Statistical Methods in Medical Research 32 (3): 509–23.