Technical Summary: Dual System Estimation using Mixed Effects Loglinear Models

1. Problem Statement

Dual System Estimation (DSE) is a standard tool in official statistics for estimating population size by linking two data sources and estimating the number of units missed by both. The method relies on the assumption of independence between the two lists and homogeneous inclusion probabilities. To satisfy the homogeneity assumption, DSE is often stratified by variables such as region.

The standard approach employs fixed effects loglinear models, estimating parameters for every level of the stratifying variable. This approach faces significant challenges:

• Parameter Proliferation: When the number of stratification levels (e.g., regions) is large, the number of parameters becomes unmanageable.
• Small Sample Bias: In scenarios with many strata and small sample sizes within each stratum, fixed effects models (and the standard Lincoln-Petersen estimator) produce biased estimates.
• Infinite Estimates: In cases where the intersection of both lists is empty within a specific stratum ($n_{11}=0$), the fixed effects estimator yields infinite population estimates.

While the Chapman estimator is often used to mitigate small-sample bias in fixed effects models, it does not address the inefficiency of estimating many parameters independently. This paper investigates the use of mixed effects loglinear models (also known as multilevel or hierarchical models) as an alternative. These models replace fixed parameters for stratifying variables with random effects drawn from a distribution, allowing for "borrowing strength" across groups to improve estimation stability.

2. Methodology

2.1. Fixed Effects Loglinear Models

The authors define the baseline using Poisson regression with a log link.

• Without Stratification: A saturated model for two lists ($A$ and $B$) includes an intercept and interaction terms. Since the cell for units missed by both ($A=0, B=0$) is unobserved, the model assumes independence, reducing the model to a conditional independence form.
• With Stratification: When a stratifying variable $R$ (e.g., region) is introduced, the standard approach fits a conditional independence model $[AR][BR]$ for each level $l$ of $R$. This results in a separate set of parameters for every region, leading to a maximal model with zero degrees of freedom.

2.2. Mixed Effects Loglinear Models

The core proposal is to replace the fixed effects associated with the stratifying variable ($R$) and its interactions with random effects.

• Model Structure: The loglinear model is re-parameterized to include a random intercept ($u_{0l}$) and random slopes ($u_{1il}, u_{2jl}$) for the inclusion probabilities of lists $A$ and $B$ across regions.
  $$\log \mu_{ijl} = (\lambda + u_{0l}) + \lambda^A_i + \lambda^B_j + u_{1il} + u_{2jl}$$
  Here, $u_{0l} \sim N(0, \sigma^2_{u0})$ captures between-region variability in population size, while the slope terms capture variability in capture probabilities.
• Shrinkage Estimation: The random effect estimators act as shrinkage estimators (Best Unbiased Predictors). Group-level estimates are "shrunk" toward the overall mean, weighted by the reliability of the group's data (sample size) and the distance from the global mean. This reduces variance and mitigates small-sample bias without requiring infinite estimates.
• Implementation: Models are fitted using the lme4 package in R, utilizing the bobyqa optimizer to handle convergence issues common in small cell counts.

2.3. Bias Reduction and Extensions

• Chapman Estimator: For fixed effects models, the authors apply the Chapman estimator (adding 1 to the $n_{11}$ cell) to reduce small-sample bias. This adjustment is not applied to mixed effects models, as the shrinkage mechanism serves a similar bias-reduction purpose.
• Multiple System Estimation (MSE): The methodology is extended to three lists (Triple System Estimation). The authors formulate a mixed effects model for three-way tables, replacing fixed interaction terms with random effects to estimate the unobserved cell ($A=0, B=0, C=0$).

3. Simulation Study Design

The authors conducted a comprehensive simulation study to evaluate the performance of mixed effects models against fixed effects models (with and without the Chapman correction).

• Factors Varied:
  • Population Size ($N$): Ranging from 500 to 300,000.
  • Number of Regions: 5 to 30.
  • Capture Probabilities: High ($\pi_A=0.8, \pi_B=0.7$) and Low ($\pi_A=0.4, \pi_B=0.2$) scenarios.
  • Random Effect Distributions: Normal (assumed), Lognormal, and Pareto (to test robustness against normality violations).
• Metrics: Performance was assessed using Mean Absolute Relative Bias (MARb), Coefficient of Variation (CV), and Mean Squared Error (MSE).

4. Key Results

• Performance Comparison: The mixed effects loglinear model consistently outperformed or performed similarly to the fixed effects model with the Chapman estimator.
  • Bias: The mixed effects model showed lower MARb, particularly in scenarios with small population sizes and a large number of regions (where cell counts are sparse).
  • Variance: The mixed effects model demonstrated lower variability (CV) compared to fixed effects, especially when sample sizes were small.
• Robustness to Distributional Assumptions: The results indicated that the mixed effects model is robust to violations of the normality assumption. Estimates derived from lognormal and Pareto distributed random effects yielded MARb and CV comparable to those where the normality assumption held.
• Small Sample Behavior: In scenarios where fixed effects models without Chapman correction produced infinite estimates (due to zero counts in the intersection), the mixed effects model provided stable, finite estimates.
• Magnitude of Improvement: While the mixed effects model performed better, the authors note that the improvement in Mean Squared Error over the fixed effects model with the Chapman estimator was generally small.

5. Significance and Claims

The paper claims that mixed effects loglinear models offer a viable and robust alternative to traditional fixed effects approaches in Dual System Estimation, particularly in official statistics contexts involving:

1. Many Strata: Situations where the number of regions is large relative to the total population.
2. Small Sample Sizes: Scenarios where individual strata have sparse data, leading to instability in fixed effects estimates.
3. Low Capture Probabilities: Contexts where inclusion probabilities are low, increasing the risk of empty cells.

The authors conclude that while the Chapman estimator remains a strong performer, mixed effects models provide a "bias-robust approach" that borrows information across all groups. This approach naturally handles the trade-off between bias and variance, reducing the risk of infinite estimates and improving overall precision in difficult estimation scenarios. The methodology is successfully extended to multiple system estimation, suggesting broad applicability beyond the dual system case.

The paper maintains a modest tone, acknowledging that while the mixed effects model outperforms the fixed effects model, the gains in Mean Squared Error are often marginal in the tested scenarios. However, the stability and robustness of the mixed effects approach in small-sample, high-strata environments are highlighted as its primary advantages.