Technical Details : identification and estimation

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causalTransportR relies on a framework for the missing data problem in generalization and transportation that is outlined in gory detail in this document.

Setup

Data Preliminaries

\(a \in \Sett{A} := \SetB{1, \dots, K}\) treatment
\(X \in \R^p\) covariates
\(Z \in \R^q\) surrogate (short-term) outcomes
\(Y \in \R\) outcome
\(S \in \SetB{0, 1}\) selection indicator: takes on value 1 in the experimental sample, 0 otherwise.

We conjecture the existence of potential outcomes \(Y^a\) and \(Z^a\) for each value of treatment \(a\) for the long-term and surrogate outcomes respectively.

The selection indicator \(S\) defines two subpopulations.

\(S = 1\) : The trial sample. We observe \(\SetB{A_i, X_i, Y_i}_{i = 1}^n\) for these observations. Call these individuals \(\Sett{S}_1\).
\(S = 0\) : The extrapolation sample. There are two data availability cases for this subsample:
- Individual covariates \(X_i\) observed for each observation in the trial sample. Call these \(\Sett{S}_0\). The treatment \(a\) may or may not be observed, and the outcome \(Y\) is never observed.
- Aggregate summary statistics : \(\Ol{X}\) sample averages and higher moments observed for extrapolation sample. No treatment or outcome information is available.

The union of these subpopulations is called the overall sample \(\Sett{S}_0 \cup \Sett{S}_1\).

With the above data, our estimators will rely on a handful of nuisance functions (sample analogues \(\wh{\alpha}\) fit using machine learning estimators with cross-fitting)

\(\mu^a(x) = \Exp{Y | A = a , X = x}\): Outcome model. Typically only estimable in \(\Sett{S}_1\).
\(\pi^a(x) = \Prob{A = a | X = x , S = 1}\): Treatment propensity model (henceforth Propensity score). Typically known in an RCT (i.e. the \(\Sett{S}_1\) sample).
\(\rho(x) = \Prob{S = 1 | X = x}\) : Selection propensity model (henceforth Selection score). Typically unknown.
\(\nu(a, z, x) = \Exp{Y | A = a , Z = z, X = x}\): Augmented outcome model which includes surrogate predictors.

Estimands

We may be interested in a few estimands. We write marginal means as \(\phi\) and causal contrasts as \(\tau\).

Baseline

\[ \phi^1_a = \Exp{Y^a | S = 1} \]

\[ \tau^1(a, a') = \Exp{\tau^1_{a, a'}(X) | S = 1} = \Exp{Y^a - Y^{a'} | S = 1} \]

This is identified under standard assumptions since the treatment is randomized in the experimental sample.

Generalisation

\[ \phi_a = \Exp{Y^a} \]

\[ \tau(a, a') = \Exp{ \tau^0_{a, a'}(X)} = \Exp{Y^a - Y^{a'} } \]

This is the treatment effect in the overall population \(\Sett{S} := \Sett{S}_1 \cup \Sett{S}_0\), which encompasses the trial sample and the extrapolation sample.

Transportation

\[ \phi^0_a = \Exp{Y^a | S = 0} \]

\[ \tau^0(a, a') = \Exp{\tau^0_{a, a'}(X) | S = 0} = \Exp{Y^a - Y^{a'} | S = 0} \]

Estimating this is challenging because we don’t observed \(Y\) for anyone in the \(\Sett{S}_0\) sample, so we must extrapolate our estimates of \(\tau(\cdot)\) from the \(\Sett{S}_1\) sample. All potential outcomes are missing.

Identification Assumptions

Consistency / SUTVA : \(Y_i = \Indic(A_i = a) Y^a_i\). This rules out interference.
Unconfoundedness in experiment: \(Y^0, \dots, Y^a \indep A | X = x, S = 1\). This is guaranteed by randomization.
Overlap
1. Propensity overlap in trial: \(\Prob{A = a | X = x , S = 1}\)
2. Selection overlap: \(\Prob{S = 1 | X = x) > 0}\)
Ignorability of trial participation / Missingness at Random (MAR)
1. \(Y^0, \dots, Y^a \indep S | X = x\). This requires that trial participation is as good as random given covariates \(X\). Note that this is weaker than Missingness Completely at Random (MCAR), which would require unconditional independence between potential outcomes and selection and obviate the need for any adjustment. This can be weakened to
2. Marginal model / Heterogeneous effect stability: This requires ‘outcome model stability’ (\(\phi_a(X | S = 0) = \phi_a(X | S = 1)\)), i.e. expected potential outcomes as a function of covariates are identical across trial and extrapolation populations. This also implies that causal contrasts are stable across trial and extrapolation populations. \(\Exp{\tau_{a, a'}(X)| S = 1} = \Exp{\tau_{a, a'}(X)| S = 0}\)
3. Mean exchangeability: \(\Exp{Y^a | X = x, S = 1, A = a} = \Exp{Y^a | X = x, A = a} = \Exp{Y^a | X = x}\).

Under A1-A4, marginal means \(\phi_a, \phi^0_a\), and causal contrasts \(\tau(a, a')\) and \(\tau^0(a, a')\) in the overall and target sample respectively are point identified. A4 is the most demanding condition and must be evaluated with care for each use case. Under weak violations of A4, partial identification is still possible.

Estimators

We write estimators for generic counterfactual means \(\phi\). Each estimator characterised by its influence function \(\psi(z)\), which solves the moment condition \(\Exp{\psi(\cdot)} = 0\), where \(\psi(z)\) is typically of form \(\text{Estimator} - \tau\) such that it obeys

\[ \sqrt{n} (\wh{\phi} - \phi) = \frac{1}{n} \sumin \psi(z) / \sqrt{n} + O_p(1) \]

This makes it a Regular and Asymptotically Normal (RAL) and allows us to construct valid confidence intervals. For each influence function, we omit the \(-\tau\) piece for clarity.

To illustrate the estimation library, we generate some synthetic experimental data and illustrate the use of the ateGT function for each use case.

# simulate RCT
treatprob <- 0.5
# workhorse
dgp <- function(n = 10000, p = 10, treat.prob = treatprob,
                # bounds of X
                Xbounds = c(-1, 1),
                # nonlinear heterogeneity
                tauF = function(x) 1 / exp(-x[3]),
                # nonlinear y0
                y0F = function(x) pmax(x[1] + x[2], 0) + sin(x[5]) * pmax(x[7], 0.5),
                # nonlinear selection
                selF = function(x) x[1] - 3 * x[3] + pmax(x[4], 0)) {
    X <<- matrix(runif(n * p, Xbounds[1], Xbounds[2]), n, p)
    a <<- rbinom(n, 1, treat.prob)
    # generate outcomes using supplied functions
    TAU <<- apply(X, 1, tauF)
    Y0 <- apply(X, 1, y0F)
    selscore <- apply(X, 1, selF)
    # outcome
    y <<- (a * TAU + Y0 + rnorm(n))
    # selection
    s <<- rbinom(n, 1, plogis(selscore)) |> as.logical()
    # set outcomes for s = 0 as missing
    y[s == 0] <<- NA
    cat("True effect\n")
    cat(mean(TAU))
}
# this function populates the global namespace using <<- ; not recommended for real use
dgp()

## True effect
## 1.16714

The true generalization effect is 1.1671402. The true transportation effect is 1.585578. They are different because the selection score includes \(X_3\), which also moderates the treatment effect.

The naive difference in means is severely downward biased.

mean(y[s == 1 & a == 1]) - mean(y[s == 1 & a == 0])

## [1] 0.8135869

Using the doubly-robust AIPW estimator does not improve things much.

aipw(y[s == 1], a[s == 1], X[s == 1, ], noi = F)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.3630470 0.02128147 0.3213353 0.4047587    0
## 2      E{Y(1)} 1.1918666 0.02194327 1.1488578 1.2348754    0
## 3 E{Y(1)-Y(0)} 0.8288196 0.02867180 0.7726228 0.8850163    0

Outcome Modelling (OM)

If high quality estimation of conditional response surfaces \(\Exp{Y | A = a, S = 1}\) is feasible, and the outcome model truly is stable across treatment and selection subsamples (A4b), the counterfactual means and causal contrasts in both generalization and transportation cases are identified by simple extrapolation over the relevant samples.

generalisation

\[ \Exp{Y^a} = \Expt{\Sett{S}_0 \cup \Sett{S}_1}{\mu^a(X)} \]

where the expectation is taken with respect to the overall sample (i.e. marginalizing over \(S=0\) and \(S=1\) subsamples). Its sample analogue is

\[ \wh{\phi}_a = \frac{1}{\abs{\Sett{S}_0 \cup \Sett{S}_1}} \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\mu}^{a}(X_i) \]

ateGT(y, a, X, s, target = "generalize", estimator = "OM", noi = F)$res

##      parameter       est          se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.3156708 0.005133740 0.3056087 0.3257329    0
## 2      E{Y(1)} 1.4340213 0.007440637 1.4194376 1.4486049    0
## 3 E{Y(1)-Y(0)} 1.1183505 0.005334528 1.1078948 1.1288062    0

transportation

\[ \Exp{Y^a} = \Expt{\Sett{S}_0}{\mu^a(X)} \]

with sample analogue

\[ \wh{\phi}^0_a = \frac{1}{\abs{\Sett{S}_0}} \sum_{i \in \Sett{S}_0} \wh{\mu}^{a}(X_i) \]

ateGT(y, a, X, s, target = "transport", estimator = "OM", noi = F)$res

##      parameter       est          se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.2600017 0.007354422 0.2455871 0.2744164    0
## 2      E{Y(1)} 1.7208079 0.009964147 1.7012782 1.7403376    0
## 3 E{Y(1)-Y(0)} 1.4608062 0.006205822 1.4486428 1.4729696    0

Reweighting

Another reasonable strategy is to reweight the observations in \(\Sett{S}_1\) to resemble the target sample (either the overall sample \(\Sett{S}_0 \cup \Sett{S}_1\) or the extrapolation sample \(\Sett{S}_0\)). Different target samples imply different weights.

Generic estimators for the two targets are

\[ \Exp{Y^a} = \omega_i \frac{\Indic{A = a}}{\pi^a(X)} Y \]

\[ \Exp{Y^a | S = 0} = \omega_i^0 \frac{\Indic{A = a}}{\pi^a(X)} Y \]

Inverse Selection Weights (ISW)

Inverse selection scores are an obvious candidate for a first set weights.

generalisation

\[ \Exp{Y^a} = \frac{S}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} Y \]

The inverse selection weights \(1/\rho(X)\) downweight observations that are over-represented in the experimental sample relative to the overall sample, and vice versa.

Sample analogue:

\[ \wh{\phi}_a = \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \frac{S_i}{\wh{\rho}(X_i)} \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} Y_i \]

ateGT(y, a, X, s, target = "generalize", estimator = "ISW", noi = F)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.3253488 0.03124566 0.2641073 0.3865903    0
## 2      E{Y(1)} 1.4652592 0.06120733 1.3452928 1.5852256    0
## 3 E{Y(1)-Y(0)} 1.1399104 0.06941169 1.0038634 1.2759573    0

transportation

\[ \Exp{Y^a | S = 0} = \frac{S(1 - \rho(X))}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} Y \]

The inverse selection weights \((1-\rho(X))/\rho(X)\) downweight observations that are over-represented in the experimental sample relative to the extrapolation sample, and vice versa.

Sample analogue:

\[ \wh{\phi}^1_a = \frac{1}{\abs{\Sett{S}_0}} \sum_{i \in \Sett{S}_1} \frac{S_i(1 - \wh{\rho}(X_i))}{\wh{\rho}(X_i)} \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} Y_i \]

ateGT(y, a, X, s, target = "transport", estimator = "ISW", noi = F)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.2738876 0.03650136 0.2023449 0.3454303    0
## 2      E{Y(1)} 1.8460260 0.08032468 1.6885897 2.0034624    0
## 3 E{Y(1)-Y(0)} 1.5721384 0.08849108 1.3986959 1.7455810    0

Augmented ISW (AISW)

We may want to combine the ISW and OM approaches into a more robust approach that works well when either \(\rho, \pi\) or \(\mu\) is correctly specified

generalisation

\[ \Exp{Y^a} = \mu(a, X) + \frac{1}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) \]

This estimator ‘augments’ the outcome model with an weighted average of residuals (\(Y - \mu(\cdot)\)).

Sample analogue:

\[ \wh{\phi}_a = \frac{1}{\abs{\Sett{S}_0 \cup \Sett{S}_1}} \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\mu}^{a}(X_i) + \frac{S_i}{\wh{\rho}(X_i)} \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} (Y_i - \wh{\mu}^a(X_i) ) \]

ateGT(y, a, X, s, target = "generalize", estimator = "AISW", noi = F)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.3026286 0.02816369 0.2474278 0.3578295    0
## 2      E{Y(1)} 1.4963845 0.02742900 1.4426237 1.5501454    0
## 3 E{Y(1)-Y(0)} 1.1937559 0.03866555 1.1179714 1.2695404    0

transportation

\[ \Exp{Y^a | S = 0} = \mu(a, X) + \frac{1}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) \]

Sample analogue:

\[ \wh{\phi}^1_a = \frac{1}{\abs{\Sett{S}_0 }} \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} (1 - S_i) \wh{\mu}^{a}(X_i) + \frac{S_i(1 - \wh{\rho}(X_i))}{\wh{\rho}(X_i)} \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} (Y_i - \wh{\mu}^a(X_i) ) \]

ateGT(y, a, X, s, target = "transport", estimator = "AISW")

Calibration Weighting (CW)

Estimating a propensity score and taking its inverse can have sub-optimal behaviour if certain observations have very small values, or if the propensity model is misspecified. Alternatively, we may seek to directly solve for balancing weights that guarantee balance

\[ \sum_{i \in \Sett{S}_0} \omega_i c_{ij}(X_ij) \approx \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} c_{ij}(X_ij) \]

in specified moment functions \(c_{ij}\) (typically identity functions, which implies means and higher moments) between the experimental sample and target sample (either the overall or extrapolation sample). These weights are ‘automatic’ in the sense that they do not require a choice of parametric model for the outcome or propensity model, and have been shown to exhibit excellent performance in finite samples.

A popular calibration weight is the entropy weight, which involves solving the following (convex) program

\[\begin{align*} \max_{\mathbf{w}} H(w) &= - \sum_{i : i \in \mathcal{S}_1 } w_i \log w_i \\ \text{Balance constraints:} & \sum_{i : i \in \mathcal{S}_1} w_i c_{ri}(\mathbf{X}_i) = m_r(X_j) \text{ with } r \in 1, \dots, R \\ \text{Proper weights:} & \sum_{i : i \in \mathcal{S}_1} w_i = 1 \; \; \text{and } w_i \geq 0 \; \forall \; \{i: i \in \mathcal{S}_1 \} \end{align*}\]

This convex program has \(R\) balance constraints that seek to equate functions \(c_{ri}(X_i)\) in a source sample \(\mathcal{S}_1\) to a target value \(m_r\) (which may be different for the overall or extrapolation sample). Solving the dual of the above problem is computationally very fast and scales well.

Entropy balancing is also double-robust: it is consistent when the outcome model is linear OR if the selection model is log-linear.

generalisation

Set \(m_r(X_j)\) to be the average covariate values in the overall sample. Solve the ebal program and obtain weights \(\wh{\omega}_i\)

\[ \wh{\phi}_a = \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\omega}_i \frac{\Indic{A = a}}{\wh{\pi}^a(X)} Y \]

ateGT(y, a, X, s, target = "generalize", estimator = "CW")$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.4742417 0.03478469 0.4060638 0.5424197    0
## 2      E{Y(1)} 1.8476831 0.05210291 1.7455614 1.9498048    0
## 3 E{Y(1)-Y(0)} 1.3734414 0.06517506 1.2456983 1.5011845    0

transportation

Set \(m_r(X_j)\) to be the average covariate values in the extrapolation sample. Solve the ebal program and obtain weights \(\wh{\omega}_i^1\)

\[ \wh{\phi}_a = \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\omega}_i^1 \frac{\Indic{A = a}}{\wh{\pi}^a(X)} Y \]

ateGT(y, a, X, s, target = "transport", estimator = "CW")$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.4629304 0.04965513 0.3656064 0.5602545    0
## 2      E{Y(1)} 2.3293385 0.09136245 2.1502681 2.5084089    0
## 3 E{Y(1)-Y(0)} 1.8664081 0.10587890 1.6588854 2.0739307    0

Augmented Calibration weighting (ACW)

As with AISW, we may want to combine the ICW and OM approaches into a more robust approach that works well when either weights or \(\mu\) is correctly specified

generalisation

\[ \Exp{Y^a} = \mu(a, X) + \omega \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) \]

This estimator ‘augments’ the outcome model with an weighted average of residuals (\(Y - \mu(\cdot)\)).

Sample analogue:

\[ \wh{\phi}_a = \frac{1}{\abs{\Sett{S}_0 \cup \Sett{S}_1}} \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\mu}^{a}(X_i) + \wh{\omega}_i \frac{\Indic{A = a}}{\wh{\pi}^a(X)} (Y - \wh{\mu}^a(X) ) \]

ateGT(y, a, X, s, target = "generalize", estimator = "ACW")$res

##      parameter      est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.312567 0.01732877 0.2786026 0.3465314    0
## 2      E{Y(1)} 1.442708 0.01776030 1.4078982 1.4775186    0
## 3 E{Y(1)-Y(0)} 1.130141 0.02374645 1.0835983 1.1766844    0

transportation

\[ \Exp{Y^a | S = 0} = \mu(a, X) + \omega^1 \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) \]

Sample analogue:

\[ \wh{\phi}^1_a = \frac{1}{\abs{\Sett{S}_0 }} \sum_{i \in \Sett{S}_0} \wh{\mu}^{a}(X_i) + \wh{\omega}_i^1 \frac{\Indic{A = a}}{\wh{\pi}^a(X)} (Y - \wh{\mu}^a(X) ) \]

ateGT(y, a, X, s, target = "transport", estimator = "ACW")$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 0.2318273 0.03578722 0.1616844 0.3019703    0
## 2      E{Y(1)} 1.8084866 0.03650755 1.7369318 1.8800414    0
## 3 E{Y(1)-Y(0)} 1.5766593 0.05009439 1.4784743 1.6748443    0

Surrogate Models

When surrogate outcomes (outcomes that are measured between baseline randomisation and endline), they may be incorporated into the analysis and produce precision gains (since outcomes tend to be correlated over time).

The additional assumption required for consistency here is

Cohort \(S\) is Missing at Random (MAR): \(S \indep Y^a | (X , A = a, Z^a)\)

Augmented Inverse Selection Weighting with Surrogates (AISWS)

Surrogates can be incorporated into the AISW estimator with an additional debiasing piece

generalisation

\[ \Exp{Y^a} = \mu(a, X) + \frac{1}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) + \frac{\Indic{A = a}}{\pi^a(X)} (\nu^a(Z, X) - \mu^a(X)) \]

This estimator ‘augments’ the outcome model with two weighted averages of residuals \(Y - \mu(\cdot)\) and \(\nu(\cdot) - \mu(\cdot)\).

Sample analogue:

transportation

\[ \Exp{Y^a} = \mu(a, X) + \frac{1 - \rho(X)}{\rho(X)} \frac{\Indic{A = a}}{\pi^a(X)} (Y - \mu^a(X) ) + \frac{\Indic{A = a}}{\pi^a(X)} (\nu^a(Z, X) - \mu^a(X)) \]

Sample analogue:

\[ \wh{\phi}_a = \frac{1}{\abs{\Sett{S}_0 \cup \Sett{S}_1}} \sum_{i \in \Sett{S}_0 \cup \Sett{S}_1} \wh{\mu}^{a}(X_i) + \frac{S_i(1- \wh{\rho}(X_i))}{\wh{\rho}(X_i)} \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} (Y_i - \wh{\mu}^a(X_i) ) + \frac{\Indic{A = a}}{\wh{\pi}^a(X_i)} (\wh{\nu}^a(X_i, Z_i) - \wh{\mu}^a(X_i)) \]

Estimation Function for aggregate data for target: `ateCAL`

We simulate some data and withold individual level data from the generalization / target sample.

# generate RCT with 10k obs
n <- 10000
p <- 10
treat.prob <- 0.5 # random assignment
a <- rbinom(n, 1, treat.prob)
X <- matrix(rnorm(n * p, 0, 2), n, p)
TAU <- 1 / (1 + exp(-X[, 3]))
# step fn in outcome model
y0 <- X[, 1] + X[, 2] + X[, 5] + pmax(X[, 7], 2)
y1 <- a * TAU + y0 + rnorm(n)
y <- (1 - a) * y0 + a * y1
# selection model misspecified
selscore <- X[, 1] - 0.5 * X[, 3] + pmax(X[, 4], 0)
s <- rbinom(n, 1, plogis(selscore)) |> as.logical()
# now, don't feed the algorithm a, s, X for the s = 0 group

Calibration and Augmented calibration estimators can be used for generalization and transportation by setting the target moments to the appropriate sample.

generalisation

Truth: 0.4998786

target_moments <- colMeans2(X)
ateCAL(y[s == 1], a[s == 1], X[s == 1, ],
    treatProb = treat.prob,
    targetMoments = target_moments,
    estimator = "ACW", noi = F
)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 3.0407581 0.04451131 2.9535160 3.1280003    0
## 2      E{Y(1)} 3.4773413 0.05296667 3.3735267 3.5811560    0
## 3 E{Y(1)-Y(0)} 0.4365832 0.03225989 0.3733538 0.4998126    0

ateCAL(y[s == 1], a[s == 1], X[s == 1, ],
    treatProb = treat.prob,
    targetMoments = target_moments,
    estimator = "CW", noi = F
)$res

##      parameter       est        se    ci.ll     ci.ul  pval
## 1      E{Y(0)} 3.8525818 0.1011644 3.654300 4.0508640 0.000
## 2      E{Y(1)} 4.2450770 0.1072732 4.034821 4.4553325 0.000
## 3 E{Y(1)-Y(0)} 0.3924952 0.1650848 0.068929 0.7160614 0.017

transportation

Truth: 0.5918173

target_moments <- colMeans2(X[s == 0, ])
ateCAL(y[s == 1], a[s == 1], X[s == 1, ],
    treatProb = treat.prob,
    targetMoments = target_moments,
    estimator = "ACW", noi = F
)$res

##      parameter       est         se     ci.ll     ci.ul pval
## 1      E{Y(0)} 3.0407845 0.04625990 2.9501151 3.1314539    0
## 2      E{Y(1)} 3.4697880 0.06309539 3.3461210 3.5934550    0
## 3 E{Y(1)-Y(0)} 0.4290035 0.04857097 0.3338044 0.5242026    0

ateCAL(y[s == 1], a[s == 1], X[s == 1, ],
    treatProb = treat.prob,
    targetMoments = target_moments,
    estimator = "CW", noi = F
)$res

##      parameter       est        se     ci.ll     ci.ul  pval
## 1      E{Y(0)} 3.2393090 0.1395562 2.9657789 3.5128392 0.000
## 2      E{Y(1)} 3.7852859 0.1481834 3.4948464 4.0757255 0.000
## 3 E{Y(1)-Y(0)} 0.5459769 0.2134626 0.1275903 0.9643635 0.011

Apoorva Lal

2022-10-23

Setup

Data Preliminaries

Estimands

Baseline

Generalisation

Transportation

Identification Assumptions

Estimators

Outcome Modelling (OM)

generalisation

transportation

Reweighting

Inverse Selection Weights (ISW)

generalisation

transportation

Augmented ISW (AISW)

generalisation

transportation

Calibration Weighting (CW)

generalisation

transportation

Augmented Calibration weighting (ACW)

generalisation

transportation

Surrogate Models

Augmented Inverse Selection Weighting with Surrogates (AISWS)

generalisation

transportation

Estimation Function for aggregate data for target: `ateCAL`

generalisation

transportation

Technical Details : identification and estimation

Apoorva Lal

2022-10-23

Setup

Data Preliminaries

Estimands

Baseline

Generalisation

Transportation

Identification Assumptions

Estimators

Outcome Modelling (OM)

generalisation

transportation

Reweighting

Inverse Selection Weights (ISW)

generalisation

transportation

Augmented ISW (AISW)

generalisation

transportation

Calibration Weighting (CW)

generalisation

transportation

Augmented Calibration weighting (ACW)

generalisation

transportation

Surrogate Models

Augmented Inverse Selection Weighting with Surrogates (AISWS)

generalisation

transportation

Estimation Function for aggregate data for target: ateCAL

generalisation

transportation

Estimation Function for aggregate data for target: `ateCAL`