# Generating Pseudo Population

Pseudo population dataset is computed based on user-defined causal inference approaches (e.g., matching or weighting). A covariate balance test is performed on the pseudo population dataset. Users can specify covariate balance criteria and activate an adaptive approach and number of attempts to search for a target pseudo population dataset that meets the covariate balance criteria.

## Usage

Input parameters:

`Y` a vector of observed outcome
`w` a vector of observed continues exposure
`c` data frame or matrix of observed baseline covariates
`ci_appr` The causal inference approach. Options are “matching”, “weighting”, and “adjusting”.
`matching_fun` specified matching 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 `matching_fun` (\(||.||\)) 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.

## References

Austin, Peter C. 2019. “Assessing Covariate Balance When Using the Generalized Propensity Score with Quantitative or Continuous Exposures.” Statistical Methods in Medical Research 28 (5): 1365–77.