Methods for Causal Inference with Multiple Continuous Exposures using Multivariate Generalized Propensity Score, mvGPS

class: title-slide

# Methods for Estimating Causal Effects for Multivariate Continuous Exposures

## Justin Williams

### 2020-11-05

---

class: center, middle, inverse
# Introduction
---
# What If...

.pull-left[
.small[
- Q: "*What if* the batter had swung?"

- "What-if" questions require us to think causally and imagine unobserved potential outcomes

- Inability to observe all potential outcomes
    - Fundamental problem of causal inference (Holland, 1986)
    - Third rung of ladder of causation (Pearl and Mackenzie, 2018)
]
]
.pull-right[
<video width="560" height="315" controls>
    <source src="https://sporty-clips.mlb.com/8b0a7a89-c0af-40a7-aa57-032817d03177.mp4" type="video/mp4">
</video>
]
---
# Background

- Gold standard for answering causal questions is the randomized clinical trial
    - Balance on known and unknown confounders
- However, using the gold standard RCT may not be possible in real world settings
    - Unrealistic, e.g., surgery vs. placebo
    - Unethical, e.g., forcing patients to smoke

- How to estimate causal effects without randomization?
    - Exposure may be systematically related to outcome

- Utilize propensity score method (Rosenbaum and Rubin, 1983)

> .small[.dodger-blue[Propensity score] = Probability of receiving exposure given set of confounders]

`\begin{align}
\Pr(D=1\mid C)
\end{align}`

---
# Brief History of Propensity Score

**Estimation**
1. Originally proposed by Rosenbaum and Rubin (1983)
    - Handled binary exposure only, i.e., exposed vs. control
1. Methods extended to categorical, or multiple, treatments by Imbens (2000)
    - Term "generalized propensity score" (GPS) is introduced
1. Adaptions for continuous treatments introduced by Hirano and Imbens (2004); Imai and Van Dyk (2004)
    - Use Gaussian densities for estimating probability of exposure level

**Bias Removal**
1. Covariate adjustment
1. Subclassification/stratification
1. Matching
1. Inverse probability of treatment weighting (IPTW)

???
- Throughout we use GPS to refer to continuous rather than categorical treatment
---
# Propensity Score with Continuous Exposure
.small[
- Popular applications
    - *Economics*: difference in employment outcomes due to length of time in job
    training program (Flores, Flores-Lagunes, Gonzalez, and Neumann, 2012; Kluve, Schneider, Uhlendorff, and Zhao, 2012)
    - *Health outcomes*: childhood obesity and duration of breastfeeding (Jiang and Foster, 2013)
    - *Education policy*: transfer rate and number of credits taken (Doyle, 2011)
- Methodology development
    - SuperLearner (Kreif, Grieve, Díaz, and Harrison, 2015)
    - Kernel density estimation (Flores, Flores-Lagunes, Gonzalez, and Neumann, 2012)
    - Gradient boosting (Zhu, Coffman, and Ghosh, 2015)
    - Covariate balancing generalized propensity score (CBGPS) (Fong, Hazlett, and Imai, 2018)
    - Entropy balancing (Vegetabile, Griffin, Coffman, Cefalu, and McCaffrey, 2020; Tübbicke, 2020)
]
---
# Multivariate Continuous Exposures

- All methodological development has focused on single univariate continuous exposure
    - Multiple exposures noted only briefly in Imai and Van Dyk (2004)
- Need for methods to handle multivariate exposures
    - Identify potential combination therapies from observational studies
        - COVID-19 treatments (Sanders, Monogue, Jodlowski, and Cutrell, 2020)
    - Estimate combined effects of childhood obesity intervention programs in the community
- Challenges:
    - Define multivariate densities for generalized propensity score
    - Multiple sets of confounders
    - Correlation of exposures
    - Assessing high dimensional balance
    - Properly defining estimable multivariate region for inference
---
class: center, middle, inverse
# Motivating Example
---
# Project Goal
- High obesity rates of low-income 2-5 year olds in Los Angeles County (PHFE WIC, 2010)
    - 20% classified as obese during period from 2003-2009

- Estimate causal effects of non-randomly assigned obesity interventions on WIC preschool-aged child obesity rates
    - Early Childhood Obesity Systems Science (ECOSyS) study funded by NIH
[R01 HD072296](https://grantome.com/grant/NIH/R01-HD072296-01A1)

- Intervention Period: 2010-2016

- Exposure:
    - WIC intervention programs

- Outcome:
    - Change in obesity prevalence from post- to pre-intervention measured at census tracts

<div class="wic-logo-right"></div>
???
- In 2018 WIC served approximately half of all children under age 5 in Los Angeles County
---

# Outcome Data

.pull-left[
.small[
- Source:
    - WIC administrative data
        - Part of the [WIC Data Mining Project](https://lawicdata.org)
    - Pulled from 2007-2016
- Quantification:
    - Obesity prevalence at census tracts
    - Include only census tracts with at least 30 WIC children for all years
        - Total of 1079 included
    - Pre-intervention=2007-2009
    - Post-intervention=2012-2016
        - Allow for lagged treatment effect
    `$$Y=\bar{p}_{post}-\bar{p}_{pre}$$`
]
]
.pull-right[

<img src="times_ct_wic.png" width="95%" style="display: block; margin: auto;" />
]
---

# Exposure Data

.pull-left[
.small[
- Source:
    - Interviews with WIC staff
    - Total of 32 programs identified across 7 WIC agencies in LAC
    - Targeted 8 regions during interview process
- Quantification:
    1. Categorize intervention strategies used by program
    2. Construct adapted "intervention dose index"
    3. Identify clinics that implemented intervention
    4. Map dose from clinics to census tracts
]
]
.pull-right[
<img src="all_lac_wic_site.png" width="90%" style="display: block; margin: auto;" />
]
---
# Intervention Strategies

- Classified interventions into 9 potential strategies
    - Grouped as **micro** (targeted individuals) or **macro** (general population)

<table>
 <thead>
  <tr>
   <th style="text-align:right;font-weight: bold;color: #f3f6f6 !important;background-color: #4f86a7 !important;"> # </th>
   <th style="text-align:left;font-weight: bold;color: #f3f6f6 !important;background-color: #4f86a7 !important;"> Name </th>
   <th style="text-align:left;font-weight: bold;color: #f3f6f6 !important;background-color: #4f86a7 !important;"> Group </th>
   <th style="text-align:left;font-weight: bold;color: #f3f6f6 !important;background-color: #4f86a7 !important;"> n (%) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #f3f6f6 !important;"> 1 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Government Policies </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Macro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> 1 (3%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #f3f6f6 !important;"> 2 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Institutional Polices </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Macro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> 4 (12%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #f3f6f6 !important;"> 3 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Infrastructure Investments </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Macro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> 3 (9%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #f3f6f6 !important;"> 4 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Business Practices </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> Macro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #f3f6f6 !important;"> 4 (12%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #D3D7DE !important;"> 5 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Group Education </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Micro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> 21 (66%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #D3D7DE !important;"> 6 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Counseling </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Micro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> 14 (44%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #D3D7DE !important;"> 7 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Health Communication </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Micro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> 17 (53%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #D3D7DE !important;"> 8 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Home Visitation </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Micro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> 8 (25%) </td>
  </tr>
  <tr>
   <td style="text-align:right;color: #40514e !important;background-color: #D3D7DE !important;"> 9 </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Screening and Referral </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> Micro </td>
   <td style="text-align:left;color: #40514e !important;background-color: #D3D7DE !important;"> 14 (44%) </td>
  </tr>
</tbody>
</table>
???
- Micro strategies were more frequently used than macro
- Program could use multiple strategies
---
# Intervention Dose Index

.pull-left[
.footnotesize[
- Developed to quantify community exposure (Wang, Crespi, Jiang, Nobari, Roper-Fingerhut, Rauzon, Robles, Blocklin, Davoudi, Kuo, MacLeod, Seto, Whaley, and Prelip, 2018)
`\begin{align}
    IDI = RS \times FS \times SS
\end{align}`
- `$RS\in[0,1]$` = reach score:
    - probability of receiving the intervention program
- `$FS\in[0,1]$` = fidelity score:
    - degree to which program was followed during implementation
- `$SS\in[1,9]$` = strength score:
    - effectiveness of strategy as rated by subject matter experts

- Estimated for each strategy and then summed by group, macro and micro
]
]
.pull-right[
<br><br><br>
<div class="figure" style="text-align: center">
<img src="dose_constr_overview_v2.png" alt="Adapted IDI construction example" width="95%" />
<p class="caption">Adapted IDI construction example</p>
</div>

]
???
- We assume FS=1 for all programs and omit SS in our adapted version as we want to estimate this from the data

---
# Catchment Areas
.panelset[
.panel[.panel-name[Description]
- Inherent geographic misalignment
    - Exposure data on intervention programs at **clinic locations**
    - Outcome data on obesity rates at **census tracts**
- Construct mapping using catchment areas
- Catchment area construction:
    1. Based on client participation patterns of attendance at each clinic
    1. Form a circle encompassing X% of clients who attend the clinic
        - Macro strategies: 80%
        - Micro strategies: 50%
    1. Census tracts that overlap catchment area receive program IDI
]
.panel[.panel-name[Comparison]
.pull-left[
***Micro***

<img src="catchment_area/lac_wic_buffer_map_avg_thres_50_v2_2.jpeg" width="80%" style="display: block; margin: auto;" />
]
.pull-right[
***Macro***

<img src="catchment_area/lac_wic_buffer_map_avg_thres_80_v2_2.jpeg" width="80%" style="display: block; margin: auto;" />
]
]
]
???
- Key feature is that macro catchment areas are larger than micro reflecting belief that they have broader impact than micro
- Outlined we also see the 8 regions of interest where interviews were targeted
---
# Exposure Dose
<img src="joint_distribution/macro_micro_joint_dose_full.png" width="65%" style="display: block; margin: auto;" />
???
- In red is the trimmed convex hull region at the 95th quantile
- Each point represents a census tract
---
class: center, middle, inverse

# Multivariate Generalized Propensity Score
## mvGPS

---
# Notation

- `$Y$`: outcome of interest
- `$\mathbf{D}$`: multivariate exposure of dimension `$m$`
    - In simulation study and application we fix `$m=2$`, i.e., bivariate exposure
- `$\mathcal{C}=\{\mathbf{C}_{1}, \dots, \mathbf{C}_{m}\}$`: set of random variables of
length `$m$`
    - Each `$\mathbf{C}_{j}$` for `$j=1,\dots,m$`, is a `$p_{j}$` vector of baseline
    confounders associated with the `$j^{th}$` exposure and outcome
- `$Y_{i}(\mathbf{d})$`: potential outcome of `$i^{th}$` subject assigned exposure
vector `$\mathbf{d}=(d_{1},\dots,d_{m})$`
- `$\mu(\mathbf{d})=\mathbb{E}[Y(\mathbf{d})]$`: average dose-response function
- `$n$`: total number of units
- `$C_{ijk}$`: `$k^{th}$` confounder of the `$j^{th}$` exposure for the `$i^{th}$` individual

- Observed data
    - `$(Y_{i}, D_{i1},\dots,D_{im}, C_{i11}, \dots, C_{i1p_{1}}, \dots, C_{im1}, \dots, C_{imp_{m}})$`
---
# Identifying Assumptions
.panelset[
.panel[.panel-name[Overview]

- Three key assumptions for valid causal quantities

1. Weak ignorability
1. Positivity
1. Stable-unit treatment value (SUTVA)

]
.panel[.panel-name[Weak ignorability]

- Exposure is conditionally independent of the potential outcomes given the
appropriate set of confounders.

`\begin{align}
Y_{i}(\mathbf{d})\perp \!\!\! \perp\mathbf{D}_{i}\mid \mathcal{C} \quad \forall \quad \mathbf{d}\in\mathcal{D}.
\end{align}`

- Replace `$\mathcal{C}$` with `$f_{\mathbf{D}\mid\mathcal{C}}$`
    - Especially with high dimensional set and/or continuous confounders

- Difficult to rationalize
    - Requires perfect knowledge *and* collection of confounders
    - We assume it holds, i.e., no unmeasured confounding

- "Weak" refers to conditional independence for *each* level of exposure not *joint* distribution of exposures

]
.panel[.panel-name[Positivity]
.pull-left[
- All units have the potential to receive a particular level of exposure given
any value of the confounders

`\begin{align}
0<f_{\mathbf{D}\mid\mathcal{C}}(\mathbf{D}=\mathbf{d}\mid \mathcal{C})<1  \quad \forall \quad \mathbf{d}\in\mathcal{D}
\end{align}`

- Grid or **convex hull**

- **Trimmed** or untrimmed
]
.pull-right[

<img src="sim_images/chull_sim.jpeg" width="80%" style="display: block; margin: auto;" />
- `$\text{Cov}(D_{1}, D_{2})=0.5$`
- .indian-red[Convex hull] and .dodger-blue[grid]
]
]
.panel[.panel-name[SUTVA]
- The potential outcome of each unit does not depend on the exposure that other
units receive *and* that there exists only one version of each exposure

- Assumes there is no interference between units
    - No herd immunity

- No errors in defining the potential outcomes caused by **multiple versions** of
the exposure

- `$Y_{i} = Y_{i}(\mathbf{d})$`
]
]
---
# mvGPS

- Conditional probability of receiving multivariate exposure given set of confounders

`\begin{align}
f(\mathbf{D}|\mathbf{C}_{1},\dots,\mathbf{C}_{m})
\end{align}`

- For bias removal construct inverse probability of treatment weights

`\begin{align}
w=\frac{f(\mathbf{D})}{f(\mathbf{D}|\mathbf{C}_{1},\dots,\mathbf{C}_{m})},
\end{align}`

- Propose multivariate normal models
`\begin{align}
\mathbf{D}\sim \text{N}_{m}(\boldsymbol{\mu}, \boldsymbol{\Sigma}) \quad \mathbf{D}\mid \mathbf{C}_{1},\dots,\mathbf{C}_{m}\sim \text{N}_{m}\Bigg(\begin{bmatrix}\boldsymbol{\beta}_{1}^{T}\mathbf{C}_{1}\\ \vdots \\ \boldsymbol{\beta}_{m}^{T}\mathbf{C}_{m}\end{bmatrix}, \boldsymbol{\Omega}\Bigg),
\end{align}`
    - Tractable, well-behaved asymptotics, full univariate conditionals
    - Estimate parameters using ordinary least squares
---
# Properties of mvGPS weights

- Returns consistent estimate of average dose-response function
    - `$\mathbb{E}[wY\mid\mathbf{D}]=\mathbb{E}[Y(\mathbf{d})]$`

- Covariance between each exposure, `$D_{j}$` and confounder, `$C_{jk}$`, is zero
    - `$\mathbb{E}[w(D_{j}-\mu_{D_j})(C_{jk}-\mu_{C_{jk}})]=0$`

- Weights are normalized and maintain marginal moments
    - `$\mathbb{E}[w]=1$`
    - `$\mathbb{E}[w D_{j}]=\mathbb{E}[D_{j}]$`
    - `$\mathbb{E}[w\mathbf{C}_{j}]=\mathbb{E}[\mathbf{C}_{j}]$`
---
class: center, middle, inverse

# Simulation Study
---
# Design
- Sample size : `$n=200$`

- Repetitions: `$B=1000$`

- Compare mvGPS to other univariate alternative methods
    - Covariate balancing generalized propensity score, CBGPS, (Fong, Hazlett, and Imai, 2018)
    - Entropy balancing (Tübbicke, 2020)
    - Generalized linear propensity score, PS

- Simulate bivariate normal exposure
    - marginal correlation
    - overlap of confounding sets

- Effect of trimming weights (Kang and Schafer, 2007)
    - None vs. 99th quantile vs 95th quantile

???
- Univariate methods may perform well when exposures are highly correlated and have the same confounders
    - Perform poorly with weakly correlated exposures and/or separate sets of confounders
---
# Scenarios
.pull-left[
.small[
- M1: No common confounding
    - Each exposure has separate set of confounders
    - `$\mathbf{C}_{1}$` and `$\mathbf{C}_{2}$`

- M2: Partially common confounding
    - Each exposure have some unique confounders but also share set of common confounders
    - `$\mathbf{C}_{1}$`, `$\mathbf{C}_{2}$`, and `$\mathbf{C}_{12}$`

- M3: Common confounding
    - Both exposures share exact same set of common confounders
    - `$\mathbf{C}_{12}$`
]

]
.pull-right[

<img src="DAGS/sim_dags_c_only.jpeg" width="65%" style="display: block; margin: auto;" />
]
---
# Balance Results

.panelset[
.panel[.panel-name[Setup]

- Three balance metrics were assessed
    1. Average exposure-covariate correlation
    1. Maximum exposure-covariate correlation
    1. Effective sample size, `$(\Sigma_i w_i)^{2}/\Sigma_i w_i^2$`
]
.panel[.panel-name[Avg Correlation]

<img src="sim_images/avg_corr_plot.jpeg" width="57%" style="display: block; margin: auto;" />
]
.panel[.panel-name[Max Correlation]

<img src="sim_images/max_corr_plot.jpeg" width="57%" style="display: block; margin: auto;" />
]
.panel[.panel-name[ESS]

<img src="sim_images/ess_plot.jpeg" width="57%" style="display: block; margin: auto;" />
]
]
???
- Average summarizes total balance on all confounders
- Max describes the most imbalanced confounder
- ESS is equivalent to the relative power for inference

---
# Dose Response Results
.panelset[
.panel[.panel-name[Setup]

- Two performance metrics were assessed
    1. Absolute total bias
    1. Root mean squared error (RMSE)
]
.panel[.panel-name[Bias]

<img src="sim_images/bias_plot.jpeg" width="57%" style="display: block; margin: auto;" />
]
.panel[.panel-name[RMSE]

<img src="sim_images/hull_rmse_plot.jpeg" width="57%" style="display: block; margin: auto;" />
]
]
---

# Takeaways

- Multivariate methods are critical for achieving proper balance across multiple exposures
    - Protects against any single confounder being strongly imbalanced
    - Slight penalty of lower average balance in certain scenarios
- Univariate methods achieved best balance with high overlap of confounders
    - Despite balance, high total bias of treatment effects
- Using multivariate weights decreased ESS resulting in lower power and higher RMSE
    - Weight trimming reduced these effects, `$q=0.99$`

---
class: center, middle, inverse
# Application Results
## Estimating Causal Effect of WIC Intervention
---
# Potential Confounders
.small[
- Three data sources used: American Community Survey, WIC Admin, and NETS
- All values for propensity equation are average of three years prior to intervention, i.e., 2007-2009

American Community Survey
1. % Spanish Speaking
1. Median Household Income
1. % with at least High School Diploma
1. % of population less than 5 years old

WIC Admin
1. Obesity Prevalence
1. Overweight Prevalence

NETS (National Establishment Time-Series)
1. Healthy Food Outlets per sq. mile
1. Unhealthy Food Outlets per sq. mile
]
---
# Covariate Balance Total

---
# Covariate Balance Summary

| **Max Abs. Corr.** | **Avg. Abs. Corr.** | **ESS** |   **Method**    |
|:------------------:|:-------------------:|:-------:|:---------------:|
|        0.12        |        0.04         |   637   |      mvGPS      |
|        0.18        |        0.08         |   580   |  CBGPS (Macro)  |
|        0.22        |        0.07         |   659   |   PS (Macro)    |
|        0.25        |        0.08         |   541   | Entropy (Macro) |
|        0.35        |        0.12         |   779   | Entropy (Micro) |
|        0.37        |        0.14         |   828   |   PS (Micro)    |
|        0.49        |        0.16         |   679   |  CBGPS (Micro)  |
|        0.41        |        0.19         |  1079   |   Unweighted    |

---
# Dose-Response Surface

---
# Takeaways

**Balance**

- Significant imbalance prior to weighting
    - mvGPS had greater reduction in imbalance than univariate methods

- ESS for all methods are reduced but power is still reasonably high

**Dose-Response**

- *Unweighted* estimate was monotonic plane where effectiveness increased as you increased either exposure dose

- *mvGPS* estimate was a saddle with optimal effectiveness using high micro and low macro
    - Showed that high levels of both exposures was less effective
---
# Exploring Tail Behavior

---
# Discussion

- Achieved balance across multiple simultaneous exposures
- Adapted for confounding sets and/or correlation of exposures
- Visualized and interpreted effect of two continuous interventions with dose-response surface
- Defined high dimensional positivity space with convex hull
- [mvGPS](https://cran.r-project.org/package=mvGPS) R package available now on CRAN

### Limitations
- Multivariate normal parametric distribution
- Investigation of higher dimensional exposure
- Time-varying multivariate exposure
- Multiple versions possible for macro and micro doses
- Geographic interference (SUTVA violations)

---
# References I
.tiny[
Doyle, W. R. (2011). "Effect of increased academic momentum on transfer
rates: an application of the generalized propensity score". In:
_Economics of Education Review_ 30.1, pp. 191-200.

Flores, C. A., A. Flores-Lagunes, A. Gonzalez, and T. C. Neumann
(2012). "Estimating the effects of length of exposure to instruction in
a training program: The case of job corps". In: _Review of Economics
and Statistics_ 94.1, pp. 153-171.

Fong, C., C. Hazlett, and K. Imai (2018). "Covariate balancing
propensity score for a continuous treatment: application to the
efficacy of political advertisements". In: _Annals of Applied
Statistics_ 12.1, pp. 156-177.

Hirano, K. and G. W. Imbens (2004). "The propensity score with
continuous treatments". In: _Applied Bayesian Modeling and Causal
Inference from Incomplete-Data Perspectives_. Ed. by A. Gelman and X.
Meng. Hoboken, N.J.: John Wiley & Sons, pp. 73-84.

Holland, P. W. (1986). "Statistics and Causal Inference". In: _Journal
of the American Statistical Association_ 81.396, pp. 945-960.

Imai, K. and D. A. Van Dyk (2004). "Causal inference with general
treatment regimes: Generalizing the propensity score". In: _Journal of
the American Statistical Association_ 99.467, pp. 854-866.

Imbens, G. W. (2000). "The role of the propensity score in estimating
dose-response functions". In: _Biometrika_ 87.3, pp. 706-710.

Jiang, M. and E. M. Foster (2013). "Duration of breastfeeding and
childhood obesity: a generalized propensity score approach". In:
_Health Services Research_ 48.2.1, pp. 628-651.

Kang, J. D. and J. L. Schafer (2007). "Demystifying double robustness:
A comparison of alternative strategies for estimating a population mean
from incomplete data". In: _Statistical Science_ 22.4, pp. 523-539.

Kluve, J., H. Schneider, A. Uhlendorff, and Z. Zhao (2012). "Evaluating
continuous training programmes by using the generalized propensity
score". In: _Journal of the Royal Statistical Society: Series A_ 175.2,
pp. 587-617.

Kreif, N., R. Grieve, I. Díaz, and D. Harrison (2015). "Evaluation of
the effect of a continuous treatment: A machine learning approach with
an application to treatment for traumatic brain injury". In: _Health
Economics_ 24.9, pp. 1213-1228.
]
---
# References II
.tiny[
Pearl, J. and D. Mackenzie (2018). _The Book of Why: The New Science of
Cause and Effect_. New York: Basic Books.

PHFE WIC (2010). _WIC data 2003-2009: A report on low income families
with young children in Los Angeles County_. Technical Report. Special
Supplemental Nutrition Assistance Program for Women, Infants and
Children.

Rosenbaum, P. R. and D. B. Rubin (1983). "The central role of the
propensity score in observational studies for causal effects". In:
_Biometrika_ 70.1, pp. 41-55.

Sanders, J. M., M. L. Monogue, T. Z. Jodlowski, and J. B. Cutrell
(2020). "Pharmacologic Treatments for Coronavirus Disease 2019
(COVID-19): A Review". In: _JAMA_ 323.18, pp. 1824-1836.

Tübbicke, S. (2020). "Entropy Balancing for Continuous Treatments". In:
_arXiv preprint arXiv:2001.06281_. arXiv: 2001.06281 [econ.EM].

Vegetabile, B. G., B. A. Griffin, D. L. Coffman, M. Cefalu, et al.
(2020). "Nonparametric Estimation of Population Average Dose-Response
Curves using Entropy Balancing Weights for Continuous Exposures". In:
_arXiv preprint arXiv:2003.02938_. arXiv: 2003.02938 [stat.ME].

Wang, M. C., C. M. Crespi, L. H. Jiang, T. Nobari, et al. (2018).
"Developing an index of dose of exposure to early childhood obesity
community interventions". In: _Preventive Medicine_ 111, pp. 135-141.

Zhu, Y., D. L. Coffman, and D. Ghosh (2015). "A boosting algorithm for
estimating generalized propensity scores with continuous treatments".
In: _Journal of Causal Inference_ 3.1, pp. 25-40.
]
---
class: inverse

# Thank You!

.pull-right[
<br><br><br>

<a href="mailto:williazo@ucla.edu">
.white[<i class="fas  fa-envelope "></i> williazo@ucla.edu]
</a>

<a href="http://github.com/williazo">
.white[<i class="fab  fa-github "></i> williazo]
</a>

<a href="https://orcid.org/0000-0002-5045-2764">
.white[<i class="ai  ai-orcid "></i> ORCID]
</a>

<a href="https://arxiv.org/search/?query=0000-0002-5045-2764&searchtype=orcid&source=header">
.white[<i class="ai  ai-arxiv "></i> arXiv]
</a>

]

<div class="ucla-seal-left"></div>
<div class="fsph-logo-right"></div>
---

# Dose Refinements

1. Radius truncation

1. Micro exposure inverse distance weighting

1. Overlapping catchment area adjustment

---

# Dose Refinement Example