. Author manuscript; available in PMC: 2021 Sep 3.

Published in final edited form as: J Expo Sci Environ Epidemiol. 2021 Mar 3;31(4):683–698. doi: 10.1038/s41370-021-00309-5

Table 1.

Approaches to estimating the effect of intervention on exposure¹

#	Approach	Data	Objective	Equation and Terms
1	Across Study Arms	All Post-Intervention	estimates the difference in study arm mean 48-hour exposure as compared to mean in Control study arm (β₁)	$log (y_{i j}) = β_{0 i j} + β_{1} StudyAr m_{j} + ε_{i j}$ y_ij = mean 48-hour air pollution exposure (either CO or PM_2.5) for participant i in study arm j, StudyArm_j = participant’s assigned study arm in two dummy variables for improved biomass and LPG study arms (ref: Control), ε_ij= error term for participant i in study arm j.
2	Before and After	All Data, study arm by study arm	estimates the difference in mean 48-hour exposure as compared to the baseline period (β₁)	$log (y_{i k}) = β_{0 i k} + β_{1} PostInterventio n_{k} + ε_{i k}$ y_ik = mean 48-hour CO exposure for participant i in intervention period k, PostIntervention_k = dummy variable for exposure estimate being before or after the intervention (ref: Pre-Intervention), and ε_ik= error term for participant i in intervention period k.
3	Difference-in-Differences (primary specification)	All Data	estimates the difference in mean 48-hour exposure observed in the post-intervention period from the baseline period as compared to the same difference occurring in the Control arm (β₃)	$log (y_{i j k}) = β_{0 i j k} + β_{1} StudyAr m_{j} + β_{2} PostInterventio n_{k} + β_{3} (StudyAr m_{j} \times PostInterventio n_{k}) + ε_{i j k}$ y_ijk = mean 48-hour CO exposure for participant i in study arm j in intervention period k, StudyArm_j = participants assigned cluster in two dummy variables for improved biomass and LPG study arms (ref: Control), PostIntervention_k = dummy variable for exposure estimate being before or after the intervention (ref: Pre-Intervention), StudyArm_j × PostIntervention_k = dummy variables for interaction terms between StudyArm_j and PostIntervention_k dummy variables (ref: Control and Pre-Intervention), and ε_ijk= error term for participant i in study arm j and intervention period k.
4	Session-specific Difference-in-Differences	All Data	estimates change in sessions compared to change from baseline in Control study arm (β₃)	$log (y_{i j l}) = β_{0 i j l} + β_{1} StudyAr m_{j} + β_{2} MonitoringSessio n_{l} + β_{3} (StudyAr m_{j} \times MonitoringSessio n_{l}) + ε_{i j l}$ y_ijl = mean 48-hour CO exposure for participant i in study arm j during monitoring session l, StudyArm_j = participants assigned cluster in two dummy variables for improved biomass and LPG study arms (ref: Control), MonitoringSession_l = dummy variable for monitoring session of exposure estimate, e.g., Session 1, Session 2, Session 3 (ref: Session 1), StudyArm_j × MonitoringSession_l = dummy variables for interaction terms between StudyArm_j and each MonitoringSession_l dummy variable (ref: Control and Session 1), and ε_ijl= error term for participant i in study arm j and monitoring session l.

This table describes the approach we took to estimate the effect of clean cooking interventions on personal air pollution exposure and is intended to illustrate the dependent and independent variable specifications, highlighting the coefficients of interest. However, all analyses are conducted utilizing generalized estimating equations, a non-parametric estimation framework that estimates population averaged effects. Standard errors in the GEEs account for multiple observations per participant and the village-clustered nature of the intervention deployment.