Delivery Facility Choice Model

Background

The care that a birthing person receives during labor and delivery depends on where the delivery took place, as not all facilities have the same access to resources and personnel. Healthcare providers trained in emergency obstetric and newborn care (EmONC) are crucial for reducing maternal and neonatal deaths, especially in high-burden settings.

Seven essential obstetric services, known as “signal functions,” have been designated as fundamental to basic emergency obstetric and newborn care (BEmONC): parenteral antibiotic administration; parenteral anticonvulsant administration; parenteral uterotonic administration; manual removal of retained products (manual vacuum aspiration); assisted vaginal delivery; manual placental removal; and newborn resuscitation. [UNICEF_2009] Comprehensive emergency obstetric and newborn care (CEmONC) encompasses all BEmONC services plus surgical capability (e.g., for c-sections) and blood transfusion capacity. These critical services determine a health facility’s capability to manage obstetric and newborn emergencies. [UNICEF_2009] Due to data constraints and for simplicity, in our model we assume that all hospitals are CEmONC facilities and all other delivery facilities (not including home births) are BEmONC facilities.

To capture the complex relationship between choice of delivery facility (home birth vs a BEmONC facility vs a CEmONC facility), the belief about gestational age (believed pre-term vs believed full term), and the related factors of antenatal care (ANC), and low birth weight and short gestation (LBWSG) risk exposure, we will include two novel affordances in our simulation: (1) correlated propensities for ANC, in-facility delivery (IFD), and LBWSG category; and (2) causal conditional probabilities for in-facility delivery that differ based on the believed preterm status when labor begins.

As a simplification, we will model the choice of delivery location in two steps: First, the birthing person decides whether to deliver at home or go to a facility, depending on the believed preterm status at the start of labor. Then, among simulants delivering in-facility, we will randomly assign simulants to BEmONC or CEmONC facilities, independent of other choices that have been made. Additionally, we do not currently model facility transfers, and we think of the delivery facility as the final location of the delivery if there were transfers between facilities.

Coming up with values for the needed correlations and causal probabilities for facility choice that are consistent with GBD and external evidence is detailed at the end of this document. But before we get to that complexity, let’s start with how we will use these correlations and causal probabilities in the simulation.

Note that the calibration procedure, and hence the values we’re using here (i.e., the correlations and the values of \(\Pr[\text{IFD status} \mid \operatorname{do}(\text{believed preterm status})]\)) depend critically on the implementations of other pieces of the model that are described elsewhere (most notably, choice of ultrasound type given ANC status, and deriving an estimated gestational age from the true gestational age – see the AI Ultrasound Module for more details).

Causal model

The following causal diagram shows the simulant attributes needed for choosing each simulant’s delivery facility, and the causal relationships between them that we will simulate:

Causal graph showing the variables affecting birth facility

Legend

Nodes

black and white oval:

dichotomous variable

green oval:

polytomous variable

orange oval:

continuous variable

blue-grey rectangle:

propensity, \(u \sim \operatorname{Uniform}([0,1])\)

Edges

dashed line:

correlation

black arrow:

probabilistic causal relationship

purple arrow:

deterministic causal relationship

blue-grey arrow:

input a propensity to simulate randomness

Note that the only exogenous variables in the model are the propensities, and the simulant attributes in all the ovals are endogenous, being completely determined once the propensities are specified.

The causal model calibration uses observed data and an optimization procedure to find consistent values for the three correlations between the propensities \(u_\text{ANC}\), \(u_\text{IFD}\), and \(u_\text{cat}\), and the causal probabilities \(\Pr[\text{IFD status} \mid \operatorname{do}(P')]\) for the arrow from believed preterm status to in-facility delivery status. The sections below record the values of these correlations and causal probabilities and detail how to use them in the Vivarium simulation to assign the final birth facility node, F.

Related modules and submodels

Instructions for assigning the variables in the causal model are spread out across the pregnancy component modules and the later sections in this document:

Location of documentation for causal model variables

Documentation sections	Variables
Initial attributes module Correlated propensities (below)	ANC propensity (\(u_\text{ANC}\)) IFD propensity (\(u_\text{IFD}\)) LBWSG category propensity (\(u_\text{cat}\))
Pregnancy module LBWSG risk exposure model Special ordering of the categories for categorical variables (below)	Sex of infant (S) LBWSG exposure (cat, BW, GA) Low birth weight status (LBW) Preterm status (P)
ANC module Special ordering of the categories for categorical variables (below)	ANC attendance
AI ultrasound module	Ultrasound type (U) Error in gestational age estimation (E) Estimated gestational age (GA’) Believed preterm status (P’)
Facility choice module Causal conditional probabilities for in-facility delivery (below) Special ordering of the categories for categorical variables (below)	In-facility delivery status (IFD status) Birth facility (F)

Assumptions and limitations

• The causal model was designed to capture the effect of expanded coverage of AI ultrasound on choice of delivery facility, so only the variables deemed important for this effect were included. If in the future we want to intervene on variables besides the ultrasound (U) node (for example, expand ANC coverage), we would likely need to add more nodes and/or edges to the model.

• Moving to a higher level care facility during the intrapartum period is common (referred up once labor begins if there is an issue) and the ability to do this is often a result of available transport, distance to clinics, etc. We currently do not include this level of detail and instead have simulants remain at a single facility for the whole intrapartum period. In the future, we may devise a strategy to model facility transfers, which may necessitate some changes to the facility choice model.

• The timing of a standard ultrasound affects its accuracy in determining gestational age (ultrasounds in the first trimester are more accurate than ultrasounds in later pregnancy). However, the facility choice model currently uses a dichotomous variable for ANC (“no ANC” vs. “some ANC”), so we are unable to model the timing of the ultrasound, instead defining a single category “standard ultrasound” that uses the average measurement error for ultrasounds taken at any point during pregnancy. In Wave II, we are planning to add more detail to the timing of ANC visits, which should allow us to more accurately model the uncertainty in GA estimation with standard ultrasounds, using the data in this paper.

• The diagram posits a causal relationship of gestational age (GA) on the error (E) in estimating the gestational age. Specifically, we have some empirical data from GF that shows that, in the absence of an accurate ultrasound, larger gestational ages are more likely to be underestimated, while smaller gestational ages are more likely to be overestimated. E.g., if the true GA is 42 when you go into labor, you are more likely to think that the GA is 40 than to think it is 44, since very few pregnancies last 44 weeks. This effect would correspond to having the mean of the distribution of E depend on the value of GA, but for simplicity we do not model this effect, instead assuming that the mean error is 0 regardless of GA. Thus, in our current modeling strategy, the arrow from GA to E is a “no-op” relationship, and E depends only on the ultrasound type. The impact on our results of omitting this effect will likely be small since the effect is more pronounced at the extremes of the GA distribution and not as pronounced near the preterm cutoff of 37 weeks.

• The causal model includes birth weight (BW) and low birth weight status (LBW), but these are not currently used in the causal model optimization due to lack of data.

Correlated propensities

This section describes how we will model an “intrinsic correlation” of ANC, home delivery, and LBWSG (see also the Initial attributes module). In short, we will use a Gaussian copula to model this, which has three parameters capturing the correlation between each pair of the three propensities.

The motivation for these correlations is as follows: we hypothesize that there are important “common causes” that are not shown explicitly in the diagram above. For example, having a home delivery and having no ANC visits might both be influenced by rurality — if all health services are offered far away, it is logical that people will be able to access them less. Similarly, it is likely that there are social exclusion factors causing both exposure to LBWSG risk and lack of access to ANC and in-facility birth. In a simulation model where we have not included scenarios that change these common-cause factors, we do not have to model their effects explicitly. For our purposes, it is sufficient to capture the correlations between ANC, in-facility birth, and LBWSG risk exposure.

In Vivarium, we use values selected uniformly at random from the interval [0,1], which we call propensities, to keep attributes like LBWSG and ANC calibrated at the population level while reducing variance between scenarios at the simulant level. This makes it straightforward to represent the correlation in our factors by generating correlated propensities. The statsmodels.distributions.copula.api.GaussianCopula implementation can make them:

>>> from statsmodels.distributions.copula.api import GaussianCopula
>>> # Input is a correlation matrix
>>> copula = GaussianCopula([[1.,   .63, .2],
...                          [.63, 1.,   .2],
...                          [.2,  .2,   1.]])
>>> # Each row contains 3 correlated propensities
>>> copula.rvs(10_000)
array([[0.29526683, 0.46781445, 0.43541525],
       [0.99146813, 0.94380918, 0.85479776],
       [0.46910608, 0.02300572, 0.49231122],
       ...,
       [0.01671794, 0.05403445, 0.0198954 ],
       [0.17063032, 0.27517952, 0.1050379 ],
       [0.66795735, 0.8360376 , 0.83390585]])

The argument of the GaussianCopula constructor is a correlation matrix, whose \((i,j)^\text{th}\) entry specifies the correlation between variable \(i\) and variable \(j\) (note that this implies that the matrix is symmetric with 1’s on the diagonal, and furthermore is positive semidefinite). The three “intrinsic correlations” are the values in the upper right (or lower left) triangle.

We may eventually specify draw-level estimates of each model parameter, but for now we will specify a single set of consistent parameters for each location, representing our best estimate or “mean draw” of the parameters.

Propensity correlations for mean draw

Factor A	Factor B	Ethiopia	Nigeria	Pakistan	Notes
ANC propensity \(u_\text{ANC}\)	IFD propensity \(u_\text{IFD}\)	0.63	0.41	0.35	Correlation found from causal model optimization after the other two correlations were fixed
ANC propensity \(u_\text{ANC}\)	LBWSG category propensity \(u_\text{cat}\)	0.2	0.2	0.2	Chosen arbitrarily as a plausible value
IFD propensity \(u_\text{IFD}\)	LBWSG category propensity \(u_\text{cat}\)	0.2	0.2	0.2	Chosen arbitrarily as a plausible value

The above correlations were computed in the facility_choice_optimization_3_countries notebook in the MNCNH Portfolio research repository.

Note

The causal model has 5 independent unknown parameters (3 correlations and 2 causal probabilities), but we have insufficient data to solve for all of them. Consequently, we fix two of the correlations and run the optimization to find the other three parameters (the third correlation and the two causal probabilities). Eventually we will want to run sensitivity analyses where we change the values of the fixed correlations (currently set to 0.2 in the table above), which requires updating the other three parameters to consistent values based on the results of the causal model optimization.

One way to do this would be to specify the two fixed correlations in model_spec.yaml and use a branches file to run parallel sims with different values, but this would require the simulation to call the optimization code, which takes 10-15 minutes to run. Alternatively, we could precompute several sets of consistent parameters, and then different scenarios would only have to specify which set of values to use.

Special ordering of the categories for categorical variables

Our method of inducing correlations using a Gaussian copula is equivalent to specifying the polychoric correlation between ordinal variables, and it relies on having a known ordering of each variable’s values. We will follow the convention of ordering the categories of all categorical variables from “highest risk” to “lowest risk” (GBD often follows this convention for risk factors), so that larger propensities are generally “better” for the simulant.

We use an ordering of the LBWSG categories that we hypothesize will make them have large polychoric correlation with the ANC and IFD propensities. Our chosen ordering also facilitates convergence of the causal model optimization, whose objective function involves the conditional probability of preterm status given facility choice. Specifically, we order the LBWSG categories first by preterm status (preterm < term), then from highest average RR to lowest average RR in the early neonatal age group (averaged across all draws), separately for each sex.

Important

• All preterm categories (< 37 weeks) are ordered before all term categories (37+ weeks)

• The ordering is sex-specific (the ordering is different for males and females)

• Within each preterm status (preterm or term), LBWSG categories are ordered in decreasing order by (sex-specific) average relative risk across draws

• The ordering is based on the RRs for the early neonatal age group since we’re interested in the risk right after birth

This ordering must be used when initializing the LBWSG category from its (correlated) propensity \(u_\text{cat}\), following the strategy described on the LBWSG risk exposure page.

We will also order the ANC and IFD propensities from highest to lowest risk:

ANC attendance categories

no ANC < ANC in later pregnancy < ANC in 1st trimester < ANC in 1st trimester and later pregnancy

IFD status categories

home birth < in-facility birth

These orderings must be used when initializing simulants’ ANC status and IFD status from the corresponding (correlated) propensities \(u_\text{ANC}\) and \(u_\text{IFD}\). See the Antenatal care attendance module for more details on assigning ANC status; see the Causal conditional probabilities for in-facility delivery section below for an explicit description of how to assign IFD status.

Note

The facility choice causal optimization model has not yet been updated to make use of all four ANC attendance categories or the corresponding additional detail for ultrasound timing. Accordingly, the AI-ultrasound module currently groups the last three ANC categories together, effectively making ANC attendance a dichotomous variable with categories ordered “no ANC” < “some ANC”.

In a future version of the model, we plan to use the more detailed ANC attendance information to determine whether simulants get a standard ultrasound in the 1st trimester or in later pregnancy, which affects the accuracy of GA estimation. Making these changes will require updating the facility choice causal optimization code and the final outputs used in the Vivarium simulation.

To be more explicit about how the ordered categories and propensities work in code: If the categories are ordered from highest risk to lowest risk as \(c_1, \dotsc, c_n\), divide the unit interval \([0,1]\) into \(n\) subintervals \(I_1, \dotsc, I_n\) ordered from left to right, such that the length of \(I_j\) is \(\Pr(c_j)\). Then a uniform propensity \(u \in [0,1]\) corresponds to category \(c_j\) precisely when \(u \in I_j\). This correspondence specifies how each ordinal variable should be initialized from its corresponding propensity. [[A picture would probably help, should we add one here?]]

Causal conditional probabilities for in-facility delivery

In addition to correlation, we posit that a belief about preterm status is influential in the decision to have a home delivery (see the Facility choice module). We will model this as a causal conditional probability of home delivery given a belief about preterm status. Although deriving consistent values for these probabilities is complex, and described in the final section of this page, using the causal conditional probabilities is simple: Simply select in-facility delivery with probability \(\text{Pr}[\text{in-facility}\mid \operatorname{do}(\text{believed preterm})]\) or \(\text{Pr}[\text{in-facility}\mid \operatorname{do}(\text{believed term})]\) for the corresponding cases, using the correlated IFD propensity and category ordering defined in the previous section.

Causal conditional probabilities of in-facility delivery for mean draw

Causal probability	Ethiopia	Nigeria	Pakistan
\(\text{Pr}[\text{at-home}\mid \operatorname{do}(\text{believed preterm})]\)	0.38	0.38	0.17
\(\text{Pr}[\text{in-facility}\mid \operatorname{do}(\text{believed preterm})]\)	1 - 0.38	1 - 0.38	1 - 0.17
\(\text{Pr}[\text{at-home}\mid \operatorname{do}(\text{believed term})]\)	0.55	0.51	0.26
\(\text{Pr}[\text{in-facility}\mid \operatorname{do}(\text{believed term})]\)	1 - 0.55	1 - 0.51	1 - 0.26

More explicitly, given the simulant’s believed preterm status (either “believed preterm” or “believed term”) and their IFD propensity, \(u_\text{IFD}\), the simulant’s IFD status is given by the following function \(f_\text{IFD}\):

\[\begin{split}\begin{aligned} \text{IFD status} &= f_\text{IFD}(\text{believed preterm status},\ u_\text{IFD}) \\ &= \begin{cases} \text{at-home}, & \text{if}\quad u_\text{IFD} < \text{Pr}[\text{at-home} \mid \operatorname{do}(\text{believed preterm status})] \\ \text{in-facility}, & \text{otherwise}. \end{cases} \end{aligned}\end{split}\]

Note that, as described in the previous section, smaller values of \(u_\text{IFD}\) correspond with home delivery, while larger values of \(u_\text{IFD}\) correspond with in-facility delivery. This ordering is important for the model to calibrate using the specified propensity correlations. The function \(f_\text{IFD}\) is one of the structural equations defining the causal model drawn above.

The above causal probabilities were computed in the facility_choice_optimization_3_countries notebook in the MNCNH Portfolio research repository.

Note

The above probabilities represent the causal effect of a simulant’s believed preterm status on their choice of home delivery or in-facility delivery. These will be different from the population’s observed conditional probabilities of IFD status given the believed preterm status, because of the correlations of \(u_\text{IFD}\) with \(u_\text{ANC}\) and \(u_\text{cat}\).

Choosing BEmONC vs. CEmONC

For simulants whose IFD status is “in-facility,” we assign CEmONC facility delivery using location-specific probabilities provided by the Health Systems team. These estimates represent the proportion of in-facility deliveries occurring in hospitals, which we are using as a proxy for CEmONC facilities. Since all in-facility deliveries occur in either BEmONC or CEmONC facilities, the probability of delivering in a BEmONC facility equals the complement of the CEmONC probability (i.e., 1 - P(CEmONC)). The decision of whether a simulant who gives birth in-facility delivers in a BEmONC or CEmONC facility should be independent from other choices in the model.

We have copied the HS team estimates to our J drive as-is. Before use in the simulation, we subset to our modeled locations and the latest year (2024) and retain only the draw columns.

import pandas as pd
hosp_any = pd.read_csv('/snfs1/Project/simulation_science/mnch_grant/MNCNH portfolio/hosp_any_st-gpr_results_weighted_aggregates_2025-06-06.csv')
location_ids = [165,179,214] # Pakistan, Ethiopia, Nigeria (modeled locations in the MNCNH portfolio simulation)
  # improvement: include some function to get location IDs for the locations used in the simulation
  # location_ids = get_location_ids(metadata.LOCATIONS)

hosp_ifd_proportion = hosp_any.loc[
  (hosp_any.location_id.isin(location_ids)) &
  (hosp_any.year_id == 2024) # Use most recent year available
  ].drop(columns=['mean', 'lower', 'upper'])

This data is specific to a given location ID and has 100 draws. To add the required 500 draws to the artifact for the MNCNH simulation for GBD 2021, duplicate the data five times such that draw 0 has the same value as draw 100, 200, 300, 400, etc. For GBD 2023, duplicate the data 2.5 times such that draw 0 has the same value as draw 100 and 200 and that draw 100 has the same value as draw 200 (data for draws 0-49 will be used three times as data for draws 50-99 will be used twice).

Once BEmONC or CEmONC has been chosen for all in-facility deliveries, use this choice in conjunction with the IFD status to assign one of the three values “home”, “BEmONC”, or “CEmONC” as the final birth facility (F) of each simulant.

Note

Before switching to using the HS team data, we used microdata-based estimates of the proportion of in-facility deliveries occurring in CEmONC facilities in Pakistan from BMGF. These estimates are not alarmingly different from the HS team estimates: 34% from the BMGF data vs. ~27% from the HS team data.

Overall delivery setting rates

While these values will not be used as direct inputs in assigning a delivery setting to simulants in the simulation, the population-level delivery setting rates will still be relevant in calculating PAFs for interventions that vary by delivery setting as well as for verification and validation. Therefore, the following parameters should be included in the artifact:

Delivery setting rate parameters to be included in the artifact

Parameter	Definition	Value	Use
bemonc_facility_fraction	Proportion of births that occur in facility settings (inclusive of both BEmONC and CEmONC facilities) that occur in BEmONC facilities	Defined in the Choosing BEmONC vs. CEmONC section	Directly used in assigning a delivery facility in the facility choice model
in_facility_delivery_proportion	Proportion of all births that occur in facility settings (including both BEmONC and CEmONC)	mean_value of GBD covariate 51 (do NOT include any parameter uncertainty in this parameter as only the mean_value was used as an input to the delivery facility model calibration)	Used in the calculation of the following parameters
p_home	Proportion of all births that occur at home	1 - in_facility_delivery_proportion	Used in calculating total population intervention coverage as a weighted average across delivery settings for intervention with coverage that varies by delivery facility at baseline and for V&V
p_bemonc	Proportion of all births that occur in BEmONC facilities	in_facility_delivery_proportion * bemonc_facility_fraction	Used in calculating total population intervention coverage as a weighted average across delivery settings for intervention with coverage that varies by delivery facility at baseline and for V&V
p_cemonc	Proportion of all births that occur in CEmONC facilities	in_facility_delivery_proportion * (1 - bemonc_facility_fraction)	Used in calculating total population intervention coverage as a weighted average across delivery settings for intervention with coverage that varies by delivery facility at baseline and for V&V

Challenge of calibrating the model

We have developed a nonlinear optimization model to find a consistent set of parameters for the Gaussian copula and the causal conditional probabilities. It will be described in detail here.

Code for running the causal optimization model can be found in the /facility_choice folder in the MNCNH Portfolio research repo. The original writeup describing the idea behind the optimization is on Sharepoint.

Todo

Add more details about how the calibration works.

Range of propensity and probabilities that are consistent with existing data

An important result of this optimization was to determine that the system is underdetermined. With the existing data we have available, there are a range of consistent values for the propensity and probability parameters. This section explores the tradeoffs between the parameters, to guide us in setting appropriate values.

It might be easier to think about “probability gaps”, meaning the difference between the conditional probabilities conditioned on believed full term and believed preterm than to think about the absolute magnitude of these probabilities.

References

[UNICEF_2009] (1,2)

UNICEF. (2009). Monitoring emergency obstetric care: a handbook. https://www.who.int/publications/i/item/9789241547734