Table 4 Details on the estimation of effect sizes from the main analysis of the institutional deliveries outcome

The segmented regression model is: Y = β₀ + (β₁ x G) + (β₂ x time) + (β₃ x G x time) + ε. Y = outcome; time = time in quarters, with the intervention starting at time = 0; G is an indicator variable with a value of 0 (during baseline, when time < 0) or 1 (after intervention scale-up, when time ≥ 0), and ε is an error term. β₀ = predicted outcome value just before intervention scale-up, β₁ = level effect size (Table 3, column 2), β₂ = outcome slope during baseline, and β₃ = slope effect size (Table 3, column 3).

For institutional deliveries (main analysis), the model parameter estimates are: β₀ = 493.3, β₁ = 60.6, β₂ = 3.6, and β₃ = 8.7. Model-based estimates of the outcome are shown as diamonds at several key time points in Fig. 2a.

• The level effect (60.6 deliveries per 100,000 population) is the difference in the outcome immediately after intervention scale-up (553.9) minus the outcome at the very end of the baseline period (493.3).

• The slope effect (8.7 deliveries per 100,000 population per quarter) is the difference in the slope after intervention scale-up (β₂ + β₃, or 12.3) minus the baseline slope (β₂, or 3.6).

• The “combined” effect size for the intervention arm (Table 3, column 4) is the arithmetic difference between: a) the relative change in the outcome from the end of baseline (493.3 deliveries per 100,000 population) to the mid-point of the follow-up period (time = 4.5 quarters) based on modeled estimates of observed outcome values (609.0), which is a 23.5% increase—i.e., (609.0–493.3)/493.3; and b) the relative change in the outcome from the end of baseline to the follow-up mid-point based on the counterfactual, which is the baseline slope extended into the follow-up period (509.4), which is a 3.3% increase—i.e., (509.4–493.3)/492.3. Thus, the combined effect size is 23.5–3.3%, or a 20.2 percentage-point difference.

• The 2-armed combined effect size (Table 3, column 5) is the combined effect size for the intervention arm (20.2 percentage points) minus the combined effect size for the comparison arm (8.1 percentage points [not shown in Table 3]), or 12.1 percentage points.