A Unified Framework for Understanding Model-Assisted, Model-Based, Hybrid, and Conditional Estimators in Finite Population Sampling

Contents
Introduction: Paradigms of Inference in Finite Population Sampling
The Fundamental Dichotomy: Design-Based vs. Model-Based Inference
The Design-Based (Randomization) Approach
The Model-Based (Prediction) Approach
The Role of the Superpopulation Model
Defining Key Estimator Properties: A Tale of Two Unbiasedness Concepts
The Model-Assisted Approach: Leveraging Models for Design-Based Efficiency
Core Philosophy: A Design-Based Heart with a Model-Based Brain
The Generalized Regression (GREG) Estimator: The Workhorse of Model-Assisted Estimation
Formulation and Components
The Calibration Perspective
Key Properties of GREG
Modern Extensions: Incorporating Machine Learning
Variance Estimation for GREG
The Model-Based Approach: Prediction-Theoretic Inference
Core Philosophy: The Model is the Message
The Best Linear Unbiased Predictor (BLUP)
The Achilles' Heel: Sensitivity to Model Misspecification
Strategies for Robustness in the Model-Based Framework
Empirical BLUP (EBLUP)
Hybrid Estimators: A Synthesis of Methodologies and Data Sources
Hybrid Inferential Philosophies
Hybrid Data Sources: Composite Estimation
Application 1: Repeated Surveys with Rotating Panels
Application 2: Combining Probability and Non-Probability Samples
Application 3: Combining Administrative Data and Survey Data
Hybrid Sampling Designs: Model-Aided Sampling (MAS)
The Conditional Approach: Inference for the Realized Sample
The Rationale for Conditioning
What to Condition On?
Conditional Bias and Conditional Variance
Constructing Conditional Estimators
Comparative Analysis and Recommendations for Practice
A Unified Comparative Framework
Choosing the Right Tool for the Job: A Practitioner's Guide
The Future: A Convergence of Paradigms?
Works cited

Introduction: Paradigms of Inference in Finite Population Sampling

The theory and practice of survey sampling are characterized by a rich and often contentious history, revolving around two fundamentally different philosophies of statistical inference: the design-based approach and the model-based approach. This foundational dichotomy, which solidified into a distinct scientific conflict around the 1970s, informs the development, properties, and application of virtually every estimator used in the field.1 Understanding this divide is not merely an academic exercise; it is a prerequisite for comprehending the distinct logic, strengths, and weaknesses of the various classes of estimators. The choice between these paradigms reflects a core disagreement about the source of randomness in a survey and, consequently, the basis upon which valid inferences about a finite population can be made. This report provides a comprehensive and detailed examination of four major classes of estimators that have emerged from this landscape: model-assisted, model-based, hybrid, and conditional estimators. It seeks to delineate their theoretical underpinnings, compare their statistical properties, and provide a clear framework for their practical application.

The Fundamental Dichotomy: Design-Based vs. Model-Based Inference

The two primary paradigms of inference from a sample to a finite population are distinguished by their assumptions about the population values and the mechanism that generates statistical randomness.

The Design-Based (Randomization) Approach

The design-based, or randomization, approach is the classical framework for finite population inference, championed in the seminal work of Neyman. Its core principle is that the values of the variable of interest for all units in the finite population, denoted by the vector (Y1​,Y2​,…,YN​), are considered a set of fixed, unknown constants.3 The sole source of randomness in this framework is the act of selecting a sample,
s, from the population, U. This selection is governed by a known probability sampling design, p(s), which assigns a known probability of selection to every possible subset of the population.1
Under this paradigm, the statistical properties of an estimator, such as its bias and variance, are evaluated with respect to this sampling distribution. That is, they are defined as averages over all possible samples that could have been drawn under the specified design.1 The randomization principle asserts that this known, man-made probability distribution is the only valid basis for making statistical inferences.8
The canonical design-based estimator is the Horvitz-Thompson (HT) estimator of a population total, Y=∑i∈U​Yi​, which is given by:

Y^HT​=i∈s∑​πi​Yi​​

where πi​ is the known probability that unit i is included in the sample.6 The HT estimator is celebrated for being design-unbiased for the population total, provided all inclusion probabilities are non-zero.
The primary strength of the design-based approach is its robustness. Because it makes no assumptions about the distribution of the Y values themselves, its validity does not depend on a statistical model for the data.6 This makes it a safe and dependable choice for large-scale, multi-purpose surveys where specifying a correct model for every variable of interest would be an impossible task. However, this robustness can come at the cost of efficiency. If the sampling design is not well-aligned with the underlying structure of the population, the resulting estimators can have large variances, sometimes dramatically so.6

The Model-Based (Prediction) Approach

In direct contrast, the model-based, or prediction, approach treats the finite population values (Y1​,…,YN​) not as fixed constants, but as a single realization of a set of random variables.3 These random variables are assumed to be generated from a hypothetical, infinite "superpopulation," which is described by a statistical model, denoted by
ξ.5
In this framework, the source of randomness is not the sampling process but the superpopulation model that generated the population values in the first place.5 Once a sample is drawn, its composition is considered fixed or ancillary. Inference is therefore
conditional on the specific sample that was realized.16 The sampling design itself is considered irrelevant to the inference process, a position strongly supported by the likelihood principle, which states that all information from the data about model parameters is contained in the likelihood function, and the likelihood function for a given sample is independent of the sampling design used to obtain it.2
The goal of the model-based approach is not to estimate a fixed total but to predict the values of the non-sampled units, Yi​ for i∈U−s, using the assumed model.13 The cornerstone of this approach is the Best Linear Unbiased Predictor (BLUP), which is designed to be optimal under the assumed model.8
The great appeal of the model-based approach is its potential for high efficiency. If the model is correctly specified, model-based estimators can achieve a given level of precision with a much smaller sample size than their design-based counterparts.9 The major drawback, and the primary point of criticism from design-based proponents, is its critical dependence on the model. If the assumed superpopulation model is mis‑specified, model-based estimators can be severely biased.8

The Role of the Superpopulation Model

The concept of the superpopulation model is the central theoretical construct that separates the two main inferential paradigms. It is not merely a tool but the very foundation of the model-based worldview, representing a complete reframing of the statistical problem. A superpopulation model, ξ, posits a stochastic process that generates the finite population values. A common formulation is a regression model:

Yi​=μ(xi​)+ϵi​

where xi​ is a vector of auxiliary variables for unit i, μ(⋅) is a function describing the relationship between the auxiliary variables and the response variable Yi​, and ϵi​ is a random error term, typically with Eξ​(ϵi​)=0.9
The philosophical divide hinges on the interpretation of this model.
●	In the model-based view, the model is treated as a representation of reality. The goal of inference is to estimate the parameters of this model (e.g., regression coefficients) or to predict outcomes based on it.18 The finite population is just one draw from this process, and interest often lies in the underlying process itself, an "analytic" use of survey data.19
●	In the design-based view, the model is considered a useful fiction. It is a "what if" scenario, a hypothetical construct used to guide the choice of a sampling design and estimator, and to evaluate their potential performance.1 The ultimate justification for inference, however, remains tied to the randomization distribution, not the model. This is the philosophy that gives rise to model-assisted estimation.

Defining Key Estimator Properties: A Tale of Two Unbiasedness Concepts

The differing views on the source of randomness lead to distinct definitions for the fundamental properties of estimators, most notably unbiasedness.
●	Design-Unbiasedness: An estimator θ^ of a fixed finite population parameter θ (e.g., the population mean Yˉ) is said to be design-unbiased if its expected value, taken over the probability distribution p(s) of all possible samples, is equal to θ. Formally:

Ep​(θ^)=s∈S∑​p(s)θ^s​=θ

where θ^s​ is the estimate from a specific sample s.7 This is an unconditional property, meaning it averages the performance of the estimator over all samples, including those that might be "good" and those that might be "bad" in retrospect.
●	Model-Unbiasedness: An estimator θ^ is model-unbiased (or prediction-unbiased) if, for a given sample s, its expected value under the superpopulation model ξ is equal to the expected value of the population parameter θ (which is itself a random variable under the model). Formally:

Eξ​(θ^−θ)=0

This property is conditional on the realized sample and is entirely dependent on the assumed model being correct.11
●	Asymptotic Properties: For many complex estimators, exact design-unbiasedness is not achievable. Instead, a weaker but highly valuable property is sought: design-consistency or asymptotic design-unbiasedness. An estimator is design-consistent if it converges in probability to the true population value as the sample size increases.21 This property is central to the model-assisted framework. It is important to distinguish this from exact unbiasedness, which holds for any sample size, whereas asymptotic unbiasedness is a large-sample property.23
●	Mean Squared Error (MSE): Ultimately, estimators are often compared based on their Mean Squared Error (MSE), which combines both variance and bias:

MSE(θ^)=Var(θ^)+2

The choice between a slightly biased estimator with a small variance and an unbiased estimator with a large variance is a recurring theme in survey sampling and a key motivation for moving beyond simple estimators.24

The Model-Assisted Approach: Leveraging Models for Design-Based Efficiency

The model-assisted approach represents a pragmatic and powerful synthesis of the two major inferential paradigms. It has become the dominant framework for estimation in national statistical offices and other large-scale survey organizations. It seeks to harness the efficiency gains offered by statistical models while retaining the robustness and theoretical protection of the design-based framework.

Core Philosophy: A Design-Based Heart with a Model-Based Brain

The foundational principle of model-assisted estimation is that it remains firmly rooted in the design-based world.6 The finite population values are treated as fixed, and the source of randomness for inference is the probability sampling design,
p(s).3 Statistical properties like bias and variance are evaluated with respect to this design distribution.
The role of the superpopulation model is purely instrumental; it is used to "assist" in the construction of an estimator that is more efficient (i.e., has lower design variance) than basic design-based estimators like the Horvitz-Thompson estimator.3 The model helps to predict the survey outcomes using auxiliary information, and these predictions are then incorporated into a design-based formula.9

The Generalized Regression (GREG) Estimator: The Workhorse of Model-Assisted Estimation

The quintessential model-assisted estimator is the Generalized Regression (GREG) estimator. Its structure elegantly embodies the compromise between the two paradigms, explicitly separating the task of model-based prediction from the task of design-based validation.

Formulation and Components

The GREG estimator for a population total, Y, is derived from the logic of the difference estimator. It predicts a value y^​i​ for every unit in the population using a "working model" and then uses the sample to estimate and correct for the aggregate prediction error. Its general form is:

Y^GREG​=i∈U∑​y^​i​+i∈s∑​πi​yi​−y^​i​​

where y^​i​=μ^​(xi​) are the predicted values based on auxiliary information xi​ for all units in the population U, and the predictions for the sampled units s are subtracted and re-added after being weighted by the inverse of their inclusion probabilities, πi​.9
The estimator consists of two distinct components:
1.	The Synthetic Component: The first term, ∑i∈U​y^​i​, is the sum of the model predictions over the entire population. This is effectively a model-based estimate of the total and is often called the "synthetic estimator".26 This component is what "borrows strength" from the auxiliary information to improve efficiency.
2.	The Bias-Correction Term: The second term, ∑i∈s​πi​yi​−y^​i​​, is the design-weighted sum of the sample prediction residuals. This term is a design-unbiased estimator of the total prediction error, ∑i∈U​(yi​−y^​i​). It acts as a correction that protects the GREG estimator from bias if the working model is mis‑specified. It is this term that anchors the GREG firmly in the design-based framework.9
Typically, the working model is a linear regression model, μ(xi​)=xiT​β, where the vector of coefficients β is estimated from the sample data using weighted least squares:

where ci​ are known constants related to the model error variance, often set to 1.9

The Calibration Perspective

An alternative and powerful way to view the GREG estimator is through the lens of calibration.6 A calibration estimator modifies the initial design weights
di​=1/πi​ to a new set of weights wi​ such that two conditions are met:
1.	The new weights wi​ are as "close" as possible to the original design weights di​ according to some distance function.
2.	The new weights satisfy a set of calibration constraints, which force the weighted sample totals of the auxiliary variables to equal their known population totals: ∑i∈s​wi​xi​=∑i∈U​xi​=X.
When a chi-square distance function is used, the resulting calibration estimator is identical to the linear GREG estimator.28 This perspective is intuitively appealing: it adjusts the weights to make the sample "look like" the population with respect to the known auxiliary variables.

Key Properties of GREG

The widespread adoption of the GREG estimator stems from its combination of two highly desirable properties: robustness and potential for efficiency.
●	Asymptotic Design-Unbiasedness: The GREG is not, in general, exactly design-unbiased for a finite sample because the estimated coefficient β^​ is itself a random variable that depends on the sample. However, it is asymptotically design-unbiased, or design-consistent.9 This means that as the sample size increases, the design bias of the estimator approaches zero. Crucially, this property holds
irrespective of whether the working model is correct or not.9 This celebrated robustness against model misspecification is the GREG's primary safeguard.
●	Efficiency: The design variance of the GREG estimator is approximately the design variance of the HT estimator of the population total of the model residuals, Ei​=Yi​−xiT​B, where B is the population regression coefficient. If the working model has predictive power, the residuals will have smaller variability than the original Yi​ values. Consequently, the GREG estimator will have a smaller design variance than the HT estimator.9 The gain in efficiency is directly proportional to the strength of the relationship between the auxiliary variables and the variable of interest, as captured by the model.8
It is critical to recognize that the "robustness" of GREG refers to its property of being asymptotically unbiased, not to its efficiency. A poorly specified model—one that does not capture the relationship between y and x well—will still yield a nearly unbiased GREG estimator, but its variance may be large, potentially even larger than that of the simple HT estimator.9 Therefore, model selection and diagnostics remain essential steps in the model-assisted approach, but their goal is to maximize efficiency, not to correct for bias.

Modern Extensions: Incorporating Machine Learning

The model-assisted framework is not static; its "plug-and-play" nature for prediction methods is a major strength that ensures its continued relevance in an era of rapid advances in data science. The "working model" used to generate the predictions y^​i​ is not restricted to simple linear regression. The GREG recipe can accommodate a vast array of modern, flexible prediction techniques, including:
●	Non-parametric regression and splines.25
●	Data-driven transformations like fractional polynomials.9
●	Penalized regression methods like the LASSO, which are useful for model selection when there is a large number of potential auxiliary variables.31
●	Other machine learning algorithms such as random forests, boosting, and support vector machines.22
By using these powerful techniques, practitioners can generate more accurate predictions, which in turn leads to model-assisted estimators with substantially lower variance. The fundamental structure of the GREG, with its bias-correction term, remains intact, continuing to provide the essential protection against model misspecification and overfitting.22 This adaptability allows survey statisticians to leverage cutting-edge predictive analytics while still operating within the safe harbor of design-based inference.

Variance Estimation for GREG

Estimating the variance of the GREG estimator is a non-trivial task that has generated considerable literature. The standard approach is to substitute the estimated sample residuals, ei​=yi​−y^​i​, into a standard design-based variance formula for an HT estimator.29 For a sampling design with negligible sampling fraction, a common variance estimator is:

where πij​ is the joint inclusion probability of units i and j, and gi​ are "g-weights" related to the calibration adjustment.33
A key theoretical and practical issue is that this standard approach typically ignores the variability introduced by having to estimate the model parameters β^​ from the sample. While this component of variance is often asymptotically negligible, ignoring it in finite samples can lead to a negative bias in the variance estimator (i.e., underestimation of the true uncertainty).25
To address this, more robust variance estimators have been developed. These methods often involve adjusting the squared residuals by factors analogous to the "leverages" in standard regression diagnostics, which gives them better properties from both a design-based and model-based perspective.34 Replication methods, particularly the delete-one jackknife, are also highly effective and popular in practice. The jackknife variance estimator implicitly incorporates these leverage-style adjustments and has been shown to perform well in simulations, making it a good dual-purpose choice for GREG variance estimation.34

The Model-Based Approach: Prediction-Theoretic Inference

The model-based approach represents a complete departure from the design-based philosophy. Here, the statistical model is not an assistant but the absolute foundation of inference. This paradigm offers the promise of maximum efficiency but at the cost of a strong and potentially unverifiable reliance on the correctness of its assumptions. It can be characterized as a "high-risk, high-reward" strategy.

Core Philosophy: The Model is the Message

In the model-based framework, all inferential properties are derived from the assumed superpopulation model, ξ. The sampling design, p(s), is considered irrelevant for inference once the sample data are available, a principle known as ignorability of the design.2 This follows from the likelihood principle, which posits that two experiments yielding proportional likelihood functions should lead to identical inferences.2 Since the likelihood function for the model parameters depends on the sample data but not on how the sample was chosen, the design is deemed non-informative for inference about those parameters.
Consequently, all statistical properties are evaluated conditionally on the specific sample, s, that has been drawn.16 The primary objective is to use the model, fit to the sample data, to construct the best possible predictor for the unobserved values in the population.

The Best Linear Unbiased Predictor (BLUP)

The central tool in the model-based arsenal is the Best Linear Unbiased Predictor (BLUP). As its name implies, it is an estimator defined by three properties under the assumed model ξ 8:
1.	Linear: It is a linear combination of the sample values of the response variable, yi​.
2.	Unbiased: It is model-unbiased, meaning Eξ​(Y^BLUP​−Y)=0.
3.	Best: It has the minimum model-based Mean Squared Error (MSE), which for a model-unbiased predictor is equivalent to minimum model variance, among all linear model-unbiased predictors.
The general form of the BLUP for a population total is the sum of the observed values in the sample plus the sum of the model-based predictions for the non-sampled units 8:
Y^BLUP​=i∈s∑​yi​+i∈U−s∑​y^​i​

where y^​i​ are the predictions for the non-sampled units derived from the model fit to the sample data. For a simple linear regression model through the origin, Yi​=βxi​+ϵi​ with Vξ​(ϵi​)=σ2v(xi​), the BLUP is the familiar ratio estimator under specific conditions.
The great appeal of the BLUP is its theoretical optimality. If the assumed model ξ is indeed the true data-generating process, the BLUP is, by definition, the most efficient linear predictor possible.8

The Achilles' Heel: Sensitivity to Model Misspecification

The optimality of the BLUP, and indeed the validity of the entire model-based approach, is critically contingent on the correct specification of the superpopulation model.8 This includes the functional form of the relationship (e.g., linear vs. non-linear) and the variance structure of the error terms. If the model is mis‑specified, the BLUP loses its property of unbiasedness and can be severely biased, leading to erroneous inferences.8 This lack of robustness is the most significant criticism of the model-based framework and the primary reason for the enduring appeal of the design-based approach.36

Strategies for Robustness in the Model-Based Framework

Recognizing this vulnerability, model-based theorists have developed strategies to provide some protection against model failure. This endeavor creates a fascinating paradox, as it often involves incorporating design-based ideas to protect a model-based estimator.
●	Balanced Sampling: The most important of these defenses is the concept of balanced sampling, proposed by Royall and Herson.8 A sample is said to be balanced on an auxiliary variable
x if the sample mean of x equals the known population mean of x (i.e., xˉs​=XˉU​).8 The power of this idea is that if the true superpopulation model is, for instance, a polynomial of some degree in
x, but the practitioner wrongly assumes a simple linear model, the BLUP derived from the simple model will still be model-unbiased if the sample is balanced on the necessary powers of x.8 This involves purposively selecting a sample with specific properties, a clear departure from the principle that the sampling mechanism is irrelevant. It demonstrates that in practice, even staunch model-based practitioners must consider sample design to ensure their inferences are robust.
●	Flexible Modeling and Robust Variance Estimation: Other strategies include using more flexible parametric models, such as the Box-Cox family of transformations, which allow the data to help determine the model's functional form.36 Additionally, robust variance estimators can be used that are less sensitive to misspecification of the model's variance structure.38

Empirical BLUP (EBLUP)

A significant practical complication arises because the BLUP formula often depends on model parameters that are unknown, particularly the variance components in a linear mixed model (which are crucial for small area estimation). In practice, these variance components must be estimated from the sample data. When these estimates are plugged into the BLUP formula, the resulting predictor is known as the Empirical BLUP (EBLUP).35
This substitution introduces an additional source of variability that is not accounted for in the standard BLUP variance formulas. Consequently, naively using the BLUP variance formula for an EBLUP leads to an underestimation of its true MSE, resulting in "overly optimistic" confidence intervals that are too narrow and have lower than nominal coverage rates.35 This is a major practical hurdle, especially in the field of small area estimation where EBLUPs are the standard tool. It necessitates the use of more complex MSE estimation techniques (e.g., jackknife or bootstrap methods) that can properly account for the uncertainty in the estimated variance components.

Hybrid Estimators: A Synthesis of Methodologies and Data Sources

The term "hybrid estimator" is one of the most ambiguous in the survey sampling literature, used to describe a wide array of techniques that combine different elements. Its meaning is entirely context-dependent. To bring clarity to the discussion, it is useful to establish a taxonomy based on what is being combined:
1.	Hybrid Inferential Philosophies: Combining elements of design-based and model-based inference in a single estimation procedure.
2.	Hybrid Data Sources: Combining data from two or more independent samples or data sources, a practice often referred to as composite estimation.
3.	Hybrid Sampling Designs: Combining probability sampling mechanisms with non-probability elements like quotas.
The motivation behind nearly all hybrid methods is pragmatism. They are born from the necessity of addressing real-world constraints such as declining response rates, budget cuts, the availability of new data sources (e.g., non-probability panels, administrative records), and situations where neither a pure design-based nor a pure model-based approach is feasible.17

Hybrid Inferential Philosophies

These estimators attempt a deeper and more intricate integration of the two main paradigms than is found in the model-assisted approach. They are often employed in complex situations, such as spatio-temporal modeling, where a purely design-based or model-based approach is impractical.
A prominent example is the approach developed by Brus and de Gruijter for estimating temporal trends.42 In their framework, a
design-based probability sample is used to estimate spatial means at discrete points in time. Then, a model-based time-series model (a linear mixed model) is used to analyze the sequence of these estimated spatial means to estimate the temporal trend. The final inference relies on both the randomization distribution from the spatial sampling and the stochastic assumptions of the temporal model.44 Such estimators are often the only viable option when, for instance, auxiliary variables are known only for the sampled units, rendering a full model-based prediction for non-sampled units impossible, or when the variable of interest itself is not observed on any units.44

Hybrid Data Sources: Composite Estimation

Composite estimation is arguably the most mature and practically significant area of hybrid estimation. It refers to a class of techniques that improve the precision of estimates by combining information from multiple data sources. This field is undergoing a significant evolution, shifting from the classical approach of "composite estimation" (blending final results) to the more modern and powerful approach of "composite weighting" (blending the raw data).39

Application 1: Repeated Surveys with Rotating Panels

In many longitudinal surveys, such as national labor force surveys, a rotating panel design is used where a fraction of the sample is retained from one period to the next.45 This sample overlap induces a positive correlation in the responses from common units over time. Composite estimation exploits this correlation to produce more precise estimates of both current level and period-to-period change.46
The classic composite estimator is a weighted average of two components 47:
1.	A direct estimate based solely on data from the current time period.
2.	An indirect estimate formed by taking the estimate from the previous period and updating it with an estimated change derived from the overlapping portion of the sample.
The composite estimator for the current period, t, can be written as:

Y^comp,t​=(1−K)Y^direct,t​+KY^indirect,t​

The weighting factor K, where 0≤K≤1, is chosen to minimize the variance of the composite estimator. The optimal K is a function of the correlation between periods (ρ) and the proportion of sample overlap (δ).47 Significant gains in efficiency (i.e., variance reduction) are possible when both
ρ and δ are large.45

Application 2: Combining Probability and Non-Probability Samples

Driven by the escalating costs of traditional probability surveys and the proliferation of inexpensive (but potentially biased) non-probability online panels, there is immense practical interest in combining a small, high-quality probability sample with a large non-probability sample.39
●	Traditional Composite Estimation: This involves calculating an estimate from each survey separately and then blending them with a weight that attempts to balance the low bias of the probability sample estimate with the low variance of the non-probability sample estimate.39 This piecemeal approach is computationally intensive and inferentially limited, as a separate composite estimate must be produced for every variable of interest.39
●	Composite Weighting: This more modern and superior method involves pooling the data from both samples into a single file. A single set of "composite weights" is then developed for this combined file, typically using methods like propensity score adjustment or calibration to known population totals. The goal is to adjust the non-probability sample to make it resemble the probability sample and, by extension, the target population.39 This approach is computationally far more efficient and allows for more robust and detailed analysis of the integrated dataset, as researchers work with a single file and a single set of weights.39

Application 3: Combining Administrative Data and Survey Data

Another powerful application of hybrid estimation is the integration of large-scale administrative datasets with smaller, more accurate probability surveys.40 For example, a national Health Information System (HIS) may provide complete coverage but suffer from unknown errors and biases. A carefully conducted probability survey, while smaller, provides estimates with known accuracy.
The hybrid estimator is typically a weighted average of the estimate from the administrative data and the estimate from the survey. This combined estimator can produce a single, more accurate final estimate with a narrower confidence interval than either source could provide alone.40 Furthermore, this framework can be used to validate the administrative data and produce a statistical measure of its error, a crucial piece of information for data users.40

Hybrid Sampling Designs: Model-Aided Sampling (MAS)

This class of hybrid methods combines sampling mechanisms. Model-Aided Sampling (MAS) integrates a probability sampling design with non-probability quota targets.17 The process begins with a standard probability-based selection mechanism. However, as data collection proceeds, a model is used to monitor the accumulating sample against predefined quotas for various population subgroups. Once the target number of completes for a specific quota cell is reached, data collection efforts for that cell are ceased, even if the full probability sample has not been exhausted.17
The goal of MAS is to control costs and reduce respondent burden by preventing oversampling in easy-to-reach subgroups, while still striving to maintain the representativeness afforded by the initial probability design. It is particularly useful in surveys where very low response rates are anticipated or where ensuring minimum sample sizes for rare subgroups is critical.17

The Conditional Approach: Inference for the Realized Sample

The conditional approach to inference is not a distinct class of estimators in the same vein as model-assisted or model-based. Rather, it is a fundamental principle for evaluating and improving the properties of any given estimator. It arises from a powerful critique of the unconditional nature of the classical design-based framework. Its application often leads to estimators that are familiar, such as the GREG, but provides a deeper and more rigorous justification for their use.

The Rationale for Conditioning

The core of the critique is that design-based properties, such as design-unbiasedness and design variance, are long-run averages over all possible samples that could have been selected under a given design.7 However, a survey practitioner is confronted with only one specific sample—the one that was actually realized. If this realized sample has features that make it clearly different from the "average" sample (e.g., the realized sample size in a particular domain is much larger or smaller than its expected value), then relying on properties that average over all possibilities, including many that did not occur, seems misguided.7
This point was articulated forcefully by Durbin (1969): "If the sample size is determined by a random mechanism and one happens to get a large sample one knows perfectly well that the quantities of interest are measured more accurately than they would have been if the sample size had happened to be small. To average over variations in sample size which might have occurred but did not occur, when in fact the sample size is exactly known, seems quite wrong from the standpoint of the analysis of the data actually observed".7
The goal of conditional inference, therefore, is to make statistical statements that are more relevant to the data at hand. This is achieved by narrowing the reference set for inference from the set of all possible samples to a specific subset of samples that share the observed feature of the realized sample.7

What to Condition On?

The choice of the conditioning statistic, T, is the most critical step in a conditional analysis. Ideally, T should be an ancillary statistic (or approximately ancillary), meaning it provides information about the precision of the estimate but contains little to no information about the population parameter of interest itself.7 Common choices for conditioning variables include:
●	Realized Sample Sizes: This is particularly relevant in designs where sample sizes in domains or post-strata are random variables, such as stratified simple random sampling where strata are defined after selection, or Poisson sampling.7
●	Sample Auxiliary Information: One can condition on the realized sample means or totals of auxiliary variables, X^. This is relevant when evaluating the properties of an estimator given that the sample is, for example, imbalanced with respect to known population controls.51
●	A Specific Unit's Inclusion: To measure the influence of a single, potentially outlying unit i on an overall estimate, one can condition on the event that unit i was included in the sample (i.e., condition on Ii​=1).53

Conditional Bias and Conditional Variance

Conditioning on a statistic T leads to the concepts of conditional bias and conditional variance.
●	Conditional Bias: The conditional bias of an estimator θ^ is the difference between its expected value within the conditional reference set and the true parameter value: ​(θ^)=Ep​(θ^∣T)−θ. A crucial finding in survey theory is that an estimator can be unconditionally design-unbiased, yet be substantially biased conditional on T.7 For instance, the HT estimator is unconditionally unbiased, but if the realized sample size
nd​ in a domain is much smaller than its expected value E(nd​), the HT estimator for that domain total will be conditionally biased downwards.7
●	Conditional Variance: The conditional variance, Varp​(θ^∣T), measures the variability of the estimator only among those samples that share the characteristic defined by T. This provides a more relevant measure of uncertainty for the specific sample that was observed.7
The unconditional variance can be decomposed using the Law of Total Variance:

Varp​(θ^)=Ep​+Varp​

The second term, Varp​, is the variance of the conditional expectation. If the estimator is conditionally biased, its conditional expectation varies with T, and this term will be positive, inflating the overall unconditional variance. This provides a formal justification for preferring conditionally unbiased estimators: they can lead to greater precision.

Constructing Conditional Estimators

The practical goal of conditional inference is to construct an estimator that has low or zero conditional bias. A theoretical approach is to weight the sample observations by the inverse of their conditional inclusion probabilities, πk∣T​=P(k∈s∣T).54 An estimator of the form
Y^cond​=∑i∈s​yi​/πi∣T​ would be conditionally unbiased by construction.
However, the direct calculation of these conditional probabilities is often mathematically intractable for all but the simplest sampling designs.51 This practical difficulty is a major reason why pure conditional inference has remained a largely theoretical topic.
Fortunately, it has been shown that this approach often leads back to well-known estimators. For example, conditioning on realized post-stratum counts is equivalent to the standard practice of poststratification.7 More generally, seeking a conditionally unbiased estimator often results in an estimator that is algebraically equivalent to a GREG estimator where the conditioning variables are included as auxiliaries in the assisting model.54 This provides a deep theoretical justification for regression adjustment: it is a practical method for achieving approximate conditional unbiasedness.
The rise of modern computational methods offers a path forward. Replication techniques, particularly the bootstrap, can be adapted to estimate conditional properties like conditional bias directly from the data, even when analytical formulas are unavailable.53 This development has the potential to move conditional inference from a theoretical ideal to a practical diagnostic tool for everyday survey analysis.

Comparative Analysis and Recommendations for Practice

The four approaches to estimation—model-assisted, model-based, hybrid, and conditional—represent a spectrum of tools available to the survey statistician. They are not mutually exclusive competitors so much as different solutions tailored to different problems, priorities, and philosophical stances. The choice among them depends critically on the goals of the survey, the nature of the available data, and the practitioner's tolerance for different types of error.

A Unified Comparative Framework

The fundamental distinctions between these approaches, which have been elaborated in the preceding sections, can be synthesized into a comparative framework. Table 1 provides a side-by-side comparison across key dimensions of inferential philosophy, statistical properties, and practical application.

Feature	Model-Assisted (e.g., GREG)	Model-Based (e.g., BLUP)	Hybrid (e.g., Composite)	Conditional Principle	
Inferential Basis	Primarily Design-Based 6	Purely Model-Based 12	Combined/Pragmatic 43	Design-Based (as an evaluation tool) 7	
Source of Randomness	Sampling Design p(s) 5	Superpopulation Model ξ 5	Varies: can be design, model, or both 43	Sampling Design p(s) 7	
Role of Model	Assistant (for efficiency) 6	Foundation (for all inference) 13	Component (part of a combined structure) 44	Diagnostic (to reveal conditional bias) 50	
Primary Unbiasedness Property	Asymptotic Design-Unbiasedness 9	Model-Unbiasedness 11	Varies; often aims for design-unbiasedness 57	Conditional Design-Unbiasedness 7	
Primary Variance/MSE Property	Design Variance 9	Model Variance / Model MSE 8	Design Variance or overall MSE 39	Conditional Design Variance 7	
Robustness to Model Misspecification	High (for bias); model choice affects efficiency 9	Low; high risk of severe bias 8	Varies widely by method; often a key concern 58	N/A (it is a tool to assess robustness)	
Reliance on Sampling Design	High (design weights are essential) 6	Low (design is ignorable for inference) 2	Varies; often uses design weights from one or more samples 39	High (evaluates properties within the design) 53	
Primary Use Case	Official statistics, large-scale multipurpose surveys 21	Small area estimation, prediction problems 11	Combining data sources (e.g., repeated surveys, prob/non-prob) 39	Diagnostic tool, justifying poststratification/GREG 51	
Key Strength	Robustness against bias with potential for high efficiency 9	Maximum efficiency if model is correct 8	Pragmatism; ability to solve real-world data integration problems 39	Provides inference relevant to the specific sample obtained 7	
Key Weakness	Can be inefficient if the assisting model is poor 25	High risk of bias from model failure 9	Can be ad-hoc; theory may be complex or underdeveloped 39	Often computationally intractable for complex designs 51	

Choosing the Right Tool for the Job: A Practitioner's Guide

Based on the analysis, the following recommendations can be made for survey practitioners:
●	Use Model-Assisted (GREG) as the Default: For large-scale, general-purpose surveys, particularly those conducted by national statistical offices or other institutions where producing defensible, robust estimates is paramount, the model-assisted GREG estimator is the standard and recommended choice. It provides the safety of the design-based framework while allowing for significant efficiency gains through the use of auxiliary data. It is the ideal tool when high-quality auxiliary information is available for the entire population frame.6
●	Use Model-Based (BLUP/EBLUP) for Prediction and Small Areas: The model-based approach is indispensable when the primary goal is prediction rather than the description of a fixed finite population. Its most critical application is in small area estimation, where sample sizes within specific domains of interest are too small to yield reliable direct design-based estimates.11 In these cases, one must "borrow strength" from other areas via a model. This approach should be used when there is strong prior knowledge or theoretical justification for a specific superpopulation model and when the analyst is prepared to undertake extensive model validation and diagnostics to mitigate the high risk of bias from misspecification.8
●	Use Hybrid Estimators to Solve Practical Data Integration Problems: Hybrid estimators are tools of pragmatism, designed to address specific, often messy, real-world challenges.
○	For repeated surveys with rotating panels, composite estimation is a standard and effective method to improve the efficiency of estimates of level and change.45
○	When needing to integrate probability and non-probability samples to balance cost and quality, composite weighting is a powerful and increasingly necessary technique.39
○	When needing to combine survey data with administrative records, hybrid methods offer a principled way to produce a single, improved estimate.40
●	Apply the Conditional Principle as a Diagnostic and Justification: The conditional principle should be used as a lens through which to evaluate any chosen estimator. Practitioners should consider whether their realized sample exhibits any notable characteristics (e.g., unbalanced auxiliary means, anomalous domain sample sizes). If so, they should be aware that unconditional properties may be misleading. The detection of potential conditional bias provides a strong justification for moving to a more sophisticated estimator that explicitly accounts for the conditioning variable—for example, moving from a simple HT estimator to a post‑stratified or GREG estimator.50

The Future: A Convergence of Paradigms?

The historical conflict between the design-based and model-based schools of thought, while still present in theoretical discussions, is giving way to a more pragmatic synthesis in practice. The lines between the paradigms are increasingly blurred. Model-based proponents advocate for design principles like balanced sampling to ensure robustness.8 Design-based proponents are incorporating ever more complex and flexible models from the world of machine learning to maximize efficiency.22 Hybrid and conditional approaches, by their very nature, seek to bridge the gap and find a middle ground.
The future of survey estimation does not appear to lie in the ultimate triumph of one pure paradigm over the other. Instead, it points toward a continued, thoughtful integration of principles from both. The modern survey statistician must be fluent in the language of both design and models, capable of selecting and blending techniques to create estimators that are efficient, robust, and tailored to the unique challenges and practical constraints of the task at hand.1
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