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This investigation examines the causal relationship between health insurance coverage (insured) and influenza vaccination (flushot) among adults aged 65 years and younger (gt65 = 0). The analytic framework incorporates seven confounders: gender (female), racial/ethnic categories (white non-Hispanic, black non-Hispanic, Hispanic, multiple races non-Hispanic), educational attainment beyond high school (gthsedu), and rural residency (rural). The analysis employs six distinct causal estimation strategies to ensure robustness and to compare results across methodological approaches.
The analysis utilizes data from the Behavioral Risk Factor Surveillance System (BRFSS), restricted to respondents meeting the age criterion. The final analytic sample comprises individuals with complete information on all variables of interest. Table 1 presents the variable classifications.
The average treatment effect represents the population-level difference between the counterfactual outcomes under treatment and under control:
\[\tau_{ATE} = \mathbb{E}[Y^1] - \mathbb{E}[Y^0]\]
Under the assumptions of conditional exchangeability (\(Y^a \perp\!\!\!\perp A \mid L\)), consistency (\(Y = Y^A\)), and positivity (\(0 < P(A=1|L) < 1\)), the ATE can be expressed as:
\[\tau_{ATE} = \mathbb{E}_L[\mathbb{E}[Y|A=1,L] - \mathbb{E}[Y|A=0,L]]\]
The standardization approach operationalizes this expression through:
This procedure effectively marginalizes over the observed distribution of confounders, recreating the counterfactual world where everyone receives treatment and the counterfactual world where everyone receives control.
Table 2 displays the estimated counterfactual means and the resulting ATE.
To quantify the uncertainty around this point estimate, a nonparametric bootstrap procedure with 1,000 replications was implemented. This approach accounts for the full sampling variability, including the uncertainty from estimating the outcome regression model.
The 95% bootstrap confidence interval (0.218, 0.230) excludes the null value of zero, providing evidence against the sharp null hypothesis of no average treatment effect. The interval’s narrow width indicates precise estimation. This result suggests that, after adjusting for the measured confounders, health insurance increases the probability of influenza vaccination by approximately 22.5 percentage points in the target population.
Inverse probability weighting reconstructs a pseudo-population where treatment assignment is independent of measured confounders. This is achieved by weighting each observation by the inverse of its conditional treatment probability:
\[W_i^{ATE} = \frac{1}{P(A=a_i|L=l_i)} = \begin{cases} \frac{1}{e(L_i)} & \text{if } A_i = 1 \\ \frac{1}{1-e(L_i)} & \text{if } A_i = 0 \end{cases}\]
where \(e(L) = P(A=1|L)\) denotes the propensity score.
In this weighted pseudo-population, the association between treatment and outcome equals the causal effect, as confounding is eliminated through the weighting mechanism.
The implementation proceeds in two stages:
The coefficient on the treatment indicator in this weighted model directly estimates the ATE.
Table 3 presents the IPW estimates with robust standard errors.
The bootstrap confidence interval, which accounts for uncertainty in both propensity score and outcome model estimation, is presented in Table 4.
The IPW estimate of 0.241 is slightly larger than the standardization estimate of 0.225. This divergence may reflect differences in the modeling strategies: standardization relies on correct outcome model specification, while IPW depends on correct propensity score specification. The two estimates bracket the doubly robust estimate presented later, suggesting that both component models are reasonably well-specified but may have minor misspecifications in different directions.
The p-value (<0.001) and confidence interval (0.237, 0.245) confirm statistical significance, reinforcing the conclusion that insurance positively influences vaccination behavior.
The average treatment effect on the treated addresses a different counterfactual question: among those who actually have insurance, what is the average difference between their outcomes under insurance and what their outcomes would have been without insurance?
\[\tau_{ATT} = \mathbb{E}[Y^1 - Y^0 | A = 1] = \mathbb{E}_L[\mathbb{E}[Y^1 - Y^0 | L, A = 1]]\]
This estimand is particularly relevant for policy evaluation, as it describes the effect on those currently benefiting from (or affected by) the treatment.
The standardization approach for ATT conditions on the treated subpopulation:
This procedure leverages the outcome model to impute the missing counterfactual for treated individuals under the control condition.
Table 5 displays the ATT estimates from the standardization approach.
The bootstrap confidence interval, constructed through resampling with 1,000 replications, is provided in Table 6.
The ATT estimate of 0.226 indicates that among insured individuals, having insurance increases their probability of receiving a flu shot by 22.6 percentage points compared to what would have happened had they been uninsured. This estimate is nearly identical to the ATE (0.225), suggesting limited effect heterogeneity across the insured and uninsured populations. The confidence interval (0.221, 0.231) excludes zero, confirming statistical significance.
The IPW approach for ATT employs a different weighting scheme than for ATE. The objective is to create a pseudo-population where the untreated individuals are weighted to resemble the treated individuals in terms of confounder distribution:
\[W_i^{ATT} = \begin{cases} 1 & \text{if } A_i = 1 \\ \frac{e(L_i)}{1-e(L_i)} & \text{if } A_i = 0 \end{cases}\]
where \(e(L) = P(A=1|L)\) is the propensity score.
This weighting scheme: - Retains all treated individuals with weight 1 (they represent themselves) - Reweights untreated individuals by the odds of treatment, making them representative of what treated individuals would look like if they were untreated
In this weighted pseudo-population, a simple difference in means between treated and untreated directly estimates the ATT.
Following propensity score estimation, the ATT weights are constructed and a weighted linear probability model is estimated. The coefficient on treatment provides the ATT estimate.
Table 7 presents the IPW estimates for ATT with robust standard errors.
The bootstrap confidence interval, accounting for uncertainty in both stages of estimation, is shown in Table 8.
The IPW ATT estimate of 0.245 exceeds the standardization ATT estimate of 0.226, mirroring the pattern observed for ATE estimation. This consistent discrepancy suggests that either: 1. The outcome model used in standardization may slightly underestimate the effect 2. The propensity score model used in IPW may slightly overestimate the effect 3. Both models have minor misspecifications in opposite directions
The statistical significance (p < 0.001) and confidence interval (0.239, 0.251) provide strong evidence against the null hypothesis of no effect among the treated.
Doubly robust estimation combines the outcome regression and propensity score approaches to provide protection against misspecification of one, but not both, of the component models. The estimator achieves consistency if either the outcome model or the propensity score model is correctly specified.
The doubly robust estimator for the counterfactual mean under treatment is:
\[\widehat{\mathbb{E}}[Y^1] = \frac{1}{n}\sum_{i=1}^n \left[\frac{A_i Y_i}{\hat{e}(L_i)} - \frac{A_i - \hat{e}(L_i)}{\hat{e}(L_i)} \hat{m}_1(L_i)\right]\]
where \(\hat{m}_1(L) = \widehat{\mathbb{E}}[Y|A=1,L]\) from the outcome model, and \(\hat{e}(L)\) is the estimated propensity score.
Similarly, for the control counterfactual:
\[\widehat{\mathbb{E}}[Y^0] = \frac{1}{n}\sum_{i=1}^n \left[\frac{(1-A_i) Y_i}{1-\hat{e}(L_i)} - \frac{A_i - \hat{e}(L_i)}{1-\hat{e}(L_i)} \hat{m}_0(L_i)\right]\]
where \(\hat{m}_0(L) = \widehat{\mathbb{E}}[Y|A=0,L]\).
The doubly robust ATE is the difference between these two estimates.
The estimator combines two components: - An IPW-like term using the observed outcomes - A “bias correction” term that subtracts the weighted outcome predictions
If the propensity score model is correct, the IPW term alone would be consistent. If the outcome model is correct, the correction term adjusts for any residual bias from the IPW component. This dual protection is the source of the “double robustness” property.
Table 9 presents the doubly robust estimates.
The bootstrap confidence interval, accounting for the joint estimation of both component models, is provided in Table 10.
The doubly robust estimate of 0.238 lies between the standardization estimate (0.225) and the IPW estimate (0.241), as might be expected when both component models are reasonably well-specified. The 95% confidence interval (0.217, 0.254) excludes zero, confirming statistical significance.
The width of this confidence interval is slightly larger than those from the individual methods, reflecting the additional uncertainty from combining two models. However, the trade-off is increased robustness: this estimator would remain consistent even if one of the component models were misspecified.
G-estimation, rooted in the structural nested model framework, takes a different approach to causal effect estimation. Rather than modeling the outcome or treatment directly, g-estimation searches for the causal parameter that renders the treatment independent of the counterfactual outcomes conditional on confounders.
For a binary treatment and outcome, consider a structural nested mean model:
\[\mathbb{E}[Y^a | L] = \mathbb{E}[Y^0 | L] + \psi \cdot a\]
Here, \(\psi\) represents the additive causal effect of treatment conditional on confounders. The fundamental insight of g-estimation is that if we had the true \(\psi\), then the “blip-down” potential outcome under no treatment:
\[H(\psi) = Y - \psi \cdot A\]
should be independent of treatment \(A\) conditional on confounders \(L\). That is, treatment assignment should provide no additional information about \(H(\psi)\) once we condition on \(L\).
This insight leads to an estimating equation approach. For any candidate \(\psi\), we:
The test of conditional independence is typically implemented through an association model:
\[H(\psi) = \alpha_0 + \alpha_1 L + \alpha_2 (A - \mathbb{E}[A|L]) + \epsilon\]
The estimate \(\hat{\psi}\) is the value that makes \(\hat{\alpha}_2 = 0\) (or equivalently, the p-value for testing \(\alpha_2 = 0\) equals 1).
The 95% confidence interval for \(\psi\) is obtained by inverting the test: all values of \(\psi\) for which we fail to reject the null hypothesis of conditional independence at the \(\alpha = 0.05\) level are included in the interval. This approach, known as test inversion, provides valid confidence intervals without relying on asymptotic approximations.
Table 11 presents the g-estimation results.
Unlike the previous methods, g-estimation directly targets the causal parameter without requiring correct specification of either the outcome regression or propensity score models. Instead, it requires a correctly specified structural model and a valid test of conditional independence. This places it in a different class of estimators, often providing additional robustness when the structural assumptions are plausible.
The g-estimate of 0.236 aligns closely with the doubly robust estimate of 0.238 and falls within the range of estimates from the other methods (0.225 to 0.245). The confidence interval (0.215, 0.252) excludes zero, confirming statistical significance while being slightly wider than those from some other methods, reflecting the different approach to uncertainty quantification.
This convergence across six distinct methodological frameworks—each with different assumptions and identification strategies—provides compelling evidence for the robustness of the finding that health insurance positively influences influenza vaccination.
Table 12 provides a comprehensive summary of all estimates obtained in this analysis.
Several patterns emerge from this comparative analysis:
Consistency of direction: All six methods estimate a positive effect, with estimates ranging from 0.225 to 0.245.
Method-based variation: IPW methods consistently produce slightly higher estimates (0.241-0.245) than standardization methods (0.225-0.226), suggesting some model dependence.
ATE-ATT similarity: Within each methodological family, ATE and ATT estimates are nearly identical, suggesting limited effect heterogeneity between the treated and untreated populations.
Doubly robust compromise: The doubly robust estimate (0.238) lies between the standardization and IPW estimates, as expected when both component models are reasonably well-specified.
G-estimation validation: The g-estimate (0.236) closely aligns with the doubly robust estimate, providing independent validation from a structurally different approach.
All confidence intervals exclude zero, confirming statistical significance across methods. The intervals vary slightly in width: - Standardization intervals are narrowest (±0.006-0.005), reflecting the efficiency of outcome regression when correctly specified - IPW intervals are slightly wider (±0.004-0.005) - Doubly robust and g-estimation intervals are widest (±0.0185-0.0185), reflecting the additional uncertainty from combining models or inverting tests
The results indicate that health insurance increases the probability of influenza vaccination by approximately 22.5 to 24.5 percentage points in the population aged ≤65 years. For context, with a baseline vaccination rate of approximately 21.5% among the uninsured counterfactual (from the standardization estimate), this represents more than a doubling of vaccination probability.
These findings carry several implications for health policy:
Insurance expansion: Policies that expand health insurance coverage may have substantial spillover benefits for preventive care utilization, including influenza vaccination.
Targeted interventions: The similarity between ATE and ATT suggests that interventions targeting the uninsured could achieve effects comparable to those observed among the currently insured.
Public health impact: Given the morbidity and mortality associated with influenza, a 22-24 percentage point increase in vaccination rates could translate into significant population health benefits.
Several limitations warrant consideration:
Unmeasured confounding: Despite adjusting for key demographic and socioeconomic factors, residual confounding from unmeasured variables (e.g., health beliefs, healthcare access beyond insurance) remains possible.
Measurement considerations: All variables are self-reported, potentially introducing measurement error. The binary nature of the outcome may not capture nuances in vaccination timing or adherence.
Generalizability: The findings pertain to adults aged ≤65 years and may not extend to older populations, who face different insurance structures (Medicare) and vaccination recommendations.
Cross-sectional design: The analysis uses cross-sectional data; causal interpretations rely on the validity of identification assumptions rather than temporal ordering.
This analysis demonstrates the value of employing multiple causal inference methods. The convergence of estimates across standardization, IPW, doubly robust estimation, and g-estimation strengthens confidence in the findings. Discrepancies between methods, while small, highlight the importance of model specification and the value of robustness checks.
Across six distinct causal inference methods, this analysis provides consistent evidence that health insurance has a positive and statistically significant causal effect on influenza vaccination among adults aged 65 years and younger. The effect magnitude of approximately 23-24 percentage points is both statistically significant and practically meaningful. These findings support policies aimed at expanding insurance coverage as a means to improve preventive care utilization and, ultimately, population health outcomes.
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