Methods for Estimating Causal Effects for Multivariate Continuous ExposuresJustin Williams2020-11-05
1/37

Introduction2/37

What If...

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)

3/37

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)

Propensity score = Probability of receiving exposure given set of confounders

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

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Brief History of Propensity Score

Estimation

Originally proposed by Rosenbaum and Rubin (1983)
- Handled binary exposure only, i.e., exposed vs. control
Methods extended to categorical, or multiple, treatments by Imbens (2000)
- Term "generalized propensity score" (GPS) is introduced
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

Covariate adjustment
Subclassification/stratification
Matching
Inverse probability of treatment weighting (IPTW)

5/37

Throughout we use GPS to refer to continuous rather than categorical treatment

Propensity Score with Continuous ExposurePopular applicationsEconomics: 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 developmentSuperLearner (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)


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Multivariate Continuous ExposuresAll methodological development has focused on single univariate continuous exposureMultiple exposures noted only briefly in Imai and Van Dyk (2004)

Need for methods to handle multivariate exposuresIdentify potential combination therapies from observational studiesCOVID-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

7/37

Motivating Example8/37

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

Intervention Period: 2010-2016
Exposure:
- WIC intervention programs
Outcome:
- Change in obesity prevalence from post- to pre-intervention measured at census tracts

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In 2018 WIC served approximately half of all children under age 5 in Los Angeles County

Outcome Data

Source:
- WIC administrative data
  - Part of the WIC Data Mining Project
- 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 $Y=\bar{p}_{post}-\bar{p}_{pre}$

10/37

Exposure Data

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

11/37

Intervention StrategiesClassified interventions into 9 potential strategiesGrouped as micro (targeted individuals) or macro (general population)


 
    # 
    Name 
    Group 
    n (%) 
  


    1 
    Government Policies 
    Macro 
    1 (3%) 
  

    2 
    Institutional Polices 
    Macro 
    4 (12%) 
  

    3 
    Infrastructure Investments 
    Macro 
    3 (9%) 
  

    4 
    Business Practices 
    Macro 
    4 (12%) 
  

    5 
    Group Education 
    Micro 
    21 (66%) 
  

    6 
    Counseling 
    Micro 
    14 (44%) 
  

    7 
    Health Communication 
    Micro 
    17 (53%) 
  

    8 
    Home Visitation 
    Micro 
    8 (25%) 
  

    9 
    Screening and Referral 
    Micro 
    14 (44%) 
  

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#	Name	Group	n (%)
1	Government Policies	Macro	1 (3%)
2	Institutional Polices	Macro	4 (12%)
3	Infrastructure Investments	Macro	3 (9%)
4	Business Practices	Macro	4 (12%)
5	Group Education	Micro	21 (66%)
6	Counseling	Micro	14 (44%)
7	Health Communication	Micro	17 (53%)
8	Home Visitation	Micro	8 (25%)
9	Screening and Referral	Micro	14 (44%)

Micro strategies were more frequently used than macro
Program could use multiple strategies

Intervention Dose Index

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

Adapted IDI construction example

13/37

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

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
2. Form a circle encompassing X% of clients who attend the clinic
  - Macro strategies: 80%
  - Micro strategies: 50%
3. Census tracts that overlap catchment area receive program IDI

Micro

Macro

14/37

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

15/37

In red is the trimmed convex hull region at the 95th quantile
Each point represents a census tract

Multivariate Generalized Propensity ScoremvGPS16/37

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
- $(Y_{i}, D_{i1},\dots,D_{im}, C_{i11}, \dots, C_{i1p_{1}}, \dots, C_{im1}, \dots, C_{imp_{m}})$

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Identifying Assumptions

Three key assumptions for valid causal quantities

Weak ignorability
Positivity
Stable-unit treatment value (SUTVA)

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

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

$\text{Cov}(D_{1}, D_{2})=0.5$
Convex hull and grid

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
$Y_{i} = Y_{i}(\mathbf{d})$

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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

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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
- $\mathbb{E}[w]=1$
- $\mathbb{E}[w D_{j}]=\mathbb{E}[D_{j}]$
- $\mathbb{E}[w\mathbf{C}_{j}]=\mathbb{E}[\mathbf{C}_{j}]$

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Simulation Study21/37

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

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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

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
- $\mathbf{C}_{12}$

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Balance Results

Three balance metrics were assessed
1. Average exposure-covariate correlation
2. Maximum exposure-covariate correlation
3. Effective sample size, $(\Sigma_i w_i)^{2}/\Sigma_i w_i^2$

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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

Two performance metrics were assessed
1. Absolute total bias
2. Root mean squared error (RMSE)

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TakeawaysMultivariate methods are critical for achieving proper balance across multiple exposuresProtects 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 confoundersDespite balance, high total bias of treatment effects

Using multivariate weights decreased ESS resulting in lower power and higher RMSEWeight trimming reduced these effects, q=0.99

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Application ResultsEstimating Causal Effect of WIC Intervention27/37

Potential Confounders

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

% Spanish Speaking
Median Household Income
% with at least High School Diploma
% of population less than 5 years old

WIC Admin

Obesity Prevalence
Overweight Prevalence

NETS (National Establishment Time-Series)

Healthy Food Outlets per sq. mile
Unhealthy Food Outlets per sq. mile

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Covariate Balance Total 29/37

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

30/37

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 31/37

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

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Exploring Tail Behavior
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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 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)

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References I

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.

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References II

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.

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Thank You!

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Radius truncation
Micro exposure inverse distance weighting
Overlapping catchment area adjustment

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Methods for Estimating Causal Effects for Multivariate Continuous ExposuresJustin Williams2020-11-05
1/37

Introduction2/37

What If...

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)

3/37

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)

Propensity score = Probability of receiving exposure given set of confounders

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

4/37

Brief History of Propensity Score

Estimation

Originally proposed by Rosenbaum and Rubin (1983)
- Handled binary exposure only, i.e., exposed vs. control
Methods extended to categorical, or multiple, treatments by Imbens (2000)
- Term "generalized propensity score" (GPS) is introduced
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

Covariate adjustment
Subclassification/stratification
Matching
Inverse probability of treatment weighting (IPTW)

5/37

Throughout we use GPS to refer to continuous rather than categorical treatment

Propensity Score with Continuous ExposurePopular applicationsEconomics: 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 developmentSuperLearner (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)


6/37

Multivariate Continuous ExposuresAll methodological development has focused on single univariate continuous exposureMultiple exposures noted only briefly in Imai and Van Dyk (2004)

Need for methods to handle multivariate exposuresIdentify potential combination therapies from observational studiesCOVID-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

7/37

Motivating Example8/37

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

Intervention Period: 2010-2016
Exposure:
- WIC intervention programs
Outcome:
- Change in obesity prevalence from post- to pre-intervention measured at census tracts

9/37

In 2018 WIC served approximately half of all children under age 5 in Los Angeles County

Outcome Data

Source:
- WIC administrative data
  - Part of the WIC Data Mining Project
- 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 $Y=\bar{p}_{post}-\bar{p}_{pre}$

10/37

Exposure Data

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

11/37

Intervention StrategiesClassified interventions into 9 potential strategiesGrouped as micro (targeted individuals) or macro (general population)


 
    # 
    Name 
    Group 
    n (%) 
  


    1 
    Government Policies 
    Macro 
    1 (3%) 
  

    2 
    Institutional Polices 
    Macro 
    4 (12%) 
  

    3 
    Infrastructure Investments 
    Macro 
    3 (9%) 
  

    4 
    Business Practices 
    Macro 
    4 (12%) 
  

    5 
    Group Education 
    Micro 
    21 (66%) 
  

    6 
    Counseling 
    Micro 
    14 (44%) 
  

    7 
    Health Communication 
    Micro 
    17 (53%) 
  

    8 
    Home Visitation 
    Micro 
    8 (25%) 
  

    9 
    Screening and Referral 
    Micro 
    14 (44%) 
  

12/37

#	Name	Group	n (%)
1	Government Policies	Macro	1 (3%)
2	Institutional Polices	Macro	4 (12%)
3	Infrastructure Investments	Macro	3 (9%)
4	Business Practices	Macro	4 (12%)
5	Group Education	Micro	21 (66%)
6	Counseling	Micro	14 (44%)
7	Health Communication	Micro	17 (53%)
8	Home Visitation	Micro	8 (25%)
9	Screening and Referral	Micro	14 (44%)

Micro strategies were more frequently used than macro
Program could use multiple strategies

Intervention Dose Index

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

Adapted IDI construction example

13/37

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

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
2. Form a circle encompassing X% of clients who attend the clinic
  - Macro strategies: 80%
  - Micro strategies: 50%
3. Census tracts that overlap catchment area receive program IDI

Micro

Macro

14/37

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

15/37

In red is the trimmed convex hull region at the 95th quantile
Each point represents a census tract

Multivariate Generalized Propensity ScoremvGPS16/37

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
- $(Y_{i}, D_{i1},\dots,D_{im}, C_{i11}, \dots, C_{i1p_{1}}, \dots, C_{im1}, \dots, C_{imp_{m}})$

17/37

Identifying Assumptions

Three key assumptions for valid causal quantities

Weak ignorability
Positivity
Stable-unit treatment value (SUTVA)

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

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

$\text{Cov}(D_{1}, D_{2})=0.5$
Convex hull and grid

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
$Y_{i} = Y_{i}(\mathbf{d})$

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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

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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
- $\mathbb{E}[w]=1$
- $\mathbb{E}[w D_{j}]=\mathbb{E}[D_{j}]$
- $\mathbb{E}[w\mathbf{C}_{j}]=\mathbb{E}[\mathbf{C}_{j}]$

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Simulation Study21/37

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

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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

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
- $\mathbf{C}_{12}$

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Balance Results

Three balance metrics were assessed
1. Average exposure-covariate correlation
2. Maximum exposure-covariate correlation
3. Effective sample size, $(\Sigma_i w_i)^{2}/\Sigma_i w_i^2$

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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

Two performance metrics were assessed
1. Absolute total bias
2. Root mean squared error (RMSE)

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TakeawaysMultivariate methods are critical for achieving proper balance across multiple exposuresProtects 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 confoundersDespite balance, high total bias of treatment effects

Using multivariate weights decreased ESS resulting in lower power and higher RMSEWeight trimming reduced these effects, q=0.99

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Application ResultsEstimating Causal Effect of WIC Intervention27/37

Potential Confounders

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

% Spanish Speaking
Median Household Income
% with at least High School Diploma
% of population less than 5 years old

WIC Admin

Obesity Prevalence
Overweight Prevalence

NETS (National Establishment Time-Series)

Healthy Food Outlets per sq. mile
Unhealthy Food Outlets per sq. mile

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Covariate Balance Total 29/37

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

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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 31/37

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

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Exploring Tail Behavior
33/37

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 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)

34/37

References I

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.

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.

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References II

Pearl, J. and D. Mackenzie (2018). The Book of Why: The New Science of Cause and Effect. New York: Basic Books.

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].

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.

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Thank You!

37/37

Radius truncation
Micro exposure inverse distance weighting
Overlapping catchment area adjustment

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Methods for Estimating Causal Effects for Multivariate Continuous Exposures

Justin Williams

2020-11-05

Introduction

What If...

Background

Brief History of Propensity Score

Propensity Score with Continuous Exposure

Multivariate Continuous Exposures

Motivating Example

Project Goal

Outcome Data

Exposure Data

Intervention Strategies

Intervention Dose Index

Catchment Areas

Exposure Dose

Multivariate Generalized Propensity Score

mvGPS

Notation

Identifying Assumptions

mvGPS

Properties of mvGPS weights

Simulation Study

Design

Scenarios

Balance Results

Dose Response Results

Takeaways

Application Results

Estimating Causal Effect of WIC Intervention

Potential Confounders

Covariate Balance Total

Covariate Balance Summary

Dose-Response Surface

Takeaways

Exploring Tail Behavior

Discussion

Limitations

References I

References II

Thank You!

Dose Refinements

Dose Refinement Example

Introduction

Help

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

Methods for Estimating Causal Effects for Multivariate Continuous Exposures

Justin Williams

2020-11-05

Introduction

What If...

Background

Brief History of Propensity Score

Propensity Score with Continuous Exposure

Multivariate Continuous Exposures

Motivating Example

Project Goal

Outcome Data

Exposure Data

Intervention Strategies

Intervention Dose Index

Catchment Areas

Exposure Dose

Multivariate Generalized Propensity Score

mvGPS

Notation

Identifying Assumptions

mvGPS

Properties of mvGPS weights

Simulation Study

Design

Scenarios

Balance Results

Dose Response Results

Takeaways

Application Results

Estimating Causal Effect of WIC Intervention

Potential Confounders

Covariate Balance Total