Generating Pseudo Population

Usage

Input parameters:

w A data.frame of observed continues exposure, including id and w columns. c A data frame or matrix of observed baseline covariates, also includes id column,
ci_appr The causal inference approach. Options are “matching” and “weighting”.
dist_measure Distance measuring function.
scale specified scale parameter to control the relative weight that is attributed to the distance measures of the exposure versus the GPS estimates
delta_n specified caliper parameter on the exposure
covar_bl_method specified covariate balance method
covar_bl_trs specified covariate balance threshold
max_attempt maximum number of attempt to satisfy covariate balance

Technical Details for Matching

The matching algorithm aims to match an observed unit \(j\) to each \(j'\) at each exposure level \(w^{(l)}\).

1. We specify delta_n (\(\delta_n\)), a caliper for any exposure level \(w\), which constitutes equally sized bins, i.e., \([w-\delta_n, w+\delta_n]\). Based on the caliper delta_n , we define a predetermined set of \(L\) exposure levels \(\{w^{(1)}=\min(w)+ \delta_n,w^{(2)}=\min(w)+3 \delta_n,...,w^{(L)} = \min(w)+(2L-1) \delta_n\}\), where \(L = \lfloor \frac{\max(w)-\min(w)}{2\delta_n} + \frac{1}{2} \rfloor\). Each exposure level \(w^{(l)}\) is the midpoint of equally sized bins, \([w^{(l)}-\delta_n, w^{(l)}+\delta_n]\).

2. We implement a nested-loop algorithm, with \(l\) in \(1,2,\ldots, L\) as the outer-loop, and \(j'\) in \(1 ,\ldots,N\) as the inner-loop. The algorithm outputs the final product of our design stage, i.e., a matched set with \(N\times L\) units.
   for \(l = 1,2,\ldots, L\) do
     Choose one exposure level of interest \(w^{(l)} \in \{w^{(1)}, w^{(2)}, ..., w^{(L)}\}\).
     for \(j' = 1 ,\ldots,N\) do
     2.1 Evaluate the GPS \(\hat{e}(w^{(l)}, \mathbf{c}_{j'})\) (for short \(e^{(l)}_{j'}\)) at \(w^{(l)}\) based on the fitted GPS model in Step 1 for each unit \(j'\) having observed covariates \(\mathbf{c}_{j'}\).
     2.2 Implement the matching to find an observed unit – denoted by \(j\) – that matched with \(j'\) with respect to both the exposure \(w_{j}\approx w^{(l)}\) and the estimated GPS \(\hat{e}(w_j, \mathbf{c}_{j}) \approx e^{(l)}_{j'}\) (under a standardized Euclidean transformation). More specifically, we find a \(j\) as \[ j_{{gps}}(e^{(l)}_{j'},w^{(l)})=\text{arg} \ \underset{j: w_j \in [w^{(l)}-\delta_n,w^{(l)}+\delta_n]}{\text{min}} \ \mid\mid( \lambda \hat{e}^{*}(w_j,\mathbf{c}_j), (1-\lambda)w^{*}_j) -(\lambda e_{j'}^{(l)*}, (1-\lambda) w^{(l)*})\mid\mid, \] where dist_measure (\(||.||\)) is a pre-specified two-dimensional metric, scale (\(\lambda\)) is the scale parameter assigning weights to the corresponding two dimensions (i.e., the GPS and exposure), and \(\delta\) is the caliper defined in Step 2 allowing that only the unit \(j\) with an observed exposure \(w_j \in [w^{(l)}-\delta,w^{(l)}+\delta]\) can get matched.
     2.3 Impute \(Y_{j'}(w^{(l)})\) as: \(\hat{Y}_{j'}(w^{(l)})=Y^{obs}_{j_{{gps}}(e^{(l)}_{j'},w^{(l)})}\).
     end for

   end for

3. After implementing the matching algorithm, we construct the matched set with \(N\times L\) units by combining all \(\hat{Y}_{j'}(w^{(l)})\) for \(j'=1,\ldots,N\) and for all \(w^{(l)} \in \{w^{(1)},w^{(2)},...,w^{(L)}\}\).

Technical Details for Covariate Balance

We introduce the absolute correlation measure (covar_bl_method = “absolute”) to assess covariate balance for continuous exposures . The absolute correlation between the exposure and each pre-exposure covariate is a global measure and can inform whether the whole matched set is balanced. The measures above build upon the work by (Austin 2019) who examine covariate balance conditions with continuous exposures. We adapt them into the proposed matching framework.

In a balanced pseudo population dataset, the correlations between the exposure and pre-exposure covariates should close to zero, that is \(E [\mathbf{c}_{i}^{*} w_{i}^{*} ] \approx \mathbf{0}.\) We calculate the absolute correlation in the pseudo population dataset as
\[\begin{align*} \big\lvert \sum_{i=1}^{N\times L} \mathbf{c}_{i}^{*} w_{i}^{*} \big\lvert \end{align*}\]

The average absolute correlations are defined as the average of absolute correlations among all covariates. Average absolute correlation: \[\begin{align*} \overline{\big\lvert \sum_{i=1}^{N\times L} \mathbf{c}_{i}^{*} w_{i}^{*} \big\lvert} < \boldsymbol{\epsilon}_1. \end{align*}\] We specify a pre-specified threshold covar_bl_trs (\(\boldsymbol{\epsilon}_1\)), for example 0.1, on average absolute correlation as the threshold for covariate balance in the pseudo population dataset.