Chain Estimators in Sample Survey Statistics: An In-Depth Analysis
1. Introduction
The fundamental objective of estimation in sample surveys is to draw valid and reliable inferences about the characteristics of a large population by examining a smaller subset of that population, known as a sample.1 Achieving precision and accuracy in these estimations is paramount for informed decision-making across various domains, from public policy to market research.1 To enhance the efficiency of estimators, particularly in scenarios where resources are limited or the population is hard to reach, the incorporation of auxiliary information has become a widely adopted strategy.4 Auxiliary variables, which are correlated with the variable of interest (the study variable), can be leveraged at different stages of the survey process to yield more precise and reliable estimates.7
Building upon the foundation of using auxiliary information, advanced estimation techniques have emerged as a means to further refine the accuracy of population parameter estimates.8 Among these techniques are chain estimators, which offer a sophisticated approach to utilizing auxiliary information, especially in complex survey designs or when dealing with incomplete data.11 Chain estimators often find significant application within the framework of double sampling, a cost-effective strategy employed when obtaining complete information on all relevant variables for the entire population is impractical.11 The increasing availability of auxiliary data from diverse sources, such as census records, previous surveys, and administrative databases, has spurred the development and application of techniques like chain estimation.6 This readily accessible information provides researchers with opportunities to enhance the precision of their estimates by exploiting the relationships between the study variable and these supplementary variables. However, traditional single-stage sampling and basic estimators might not always be the most effective tools, particularly when dealing with rare or hard-to-reach populations, or in the presence of non-response.1 These situations often necessitate more sophisticated approaches that can leverage additional information or adapt to multi-stage sampling designs to produce reliable population inferences.14
2. Defining the Chain Estimator
A chain estimator in sample surveys can be formally defined as an estimator that utilizes auxiliary information sequentially, where the estimate of one parameter or variable is used to refine the estimate of another, ultimately leading to an improved estimate of the study variable's population parameter.11 The core characteristic of these estimators lies in their layered approach to incorporating auxiliary data, creating a "chain" of information flow that aims to enhance the precision of the final estimate.12 Typically, chain estimators involve the use of two or more auxiliary variables, each contributing to the refinement process.11 This sequential application distinguishes them from methods that directly incorporate auxiliary variables, such as standard ratio or regression estimators, where auxiliary information is used in a single step to adjust the estimate of the study variable.7
The "chaining" mechanism is specifically designed to exploit the inter-correlations that exist between the study variable and the auxiliary variables, and potentially between the auxiliary variables themselves, in a more nuanced manner than traditional estimation methods.12 By leveraging these relationships sequentially, chain estimators aim to extract more information from the available data, leading to more efficient and accurate population parameter estimates.11 The complexity of chain estimators can vary considerably, ranging from relatively simple chain ratio types to more intricate forms that incorporate additional statistical parameters or are specifically designed to address survey challenges such as non-response.11 This flexibility allows researchers to tailor the estimator to the specific characteristics of their survey data and the particular challenges they face in obtaining accurate population estimates.
3. Chain Ratio Estimators in Double Sampling
3.1. Understanding Double Sampling (Two-Phase Sampling)
Double sampling, also known as two-phase sampling, is a survey design technique that involves collecting sample data in two distinct phases.11 In the first phase, a larger initial sample is drawn from the population, and information is collected on one or more auxiliary variables that are known or suspected to be correlated with the study variable.11 The primary purpose of this initial phase is to obtain estimates of population parameters related to these auxiliary variables, such as the population mean.11 In the second phase, a smaller subsample is selected from the initial larger sample (Case-I) or independently from the population (Case-II), and for the units in this subsample, information is collected on both the study variable (the variable of primary interest) and the auxiliary variables.11
The primary motivation for employing double sampling arises when the population mean of a crucial auxiliary variable is unknown or when obtaining this information for the entire population is prohibitively costly or logistically impractical.11 By using a large initial sample, a reliable estimate of the auxiliary variable's population mean can be obtained, which can then be used in the second phase to improve the estimation of the study variable's population mean.19 Double sampling, therefore, provides a cost-effective approach to leveraging auxiliary information that would otherwise be inaccessible for direct ratio or regression estimation, allowing researchers to achieve greater precision in their estimates without incurring the expense of collecting detailed data on the entire population.11
3.2. Formulation of the Basic Chain Ratio Estimator
A typical chain ratio estimator in double sampling, utilizing two auxiliary variables, can be formulated as follows 11:
Yˉ^CR​=yˉ​(xˉxˉ0​​)(zˉ0​Zˉ​)
Where:
●	Yˉ^CR​ represents the chain ratio estimator of the population mean of the study variable (Ȳ).
●	yˉ​ is the sample mean of the study variable, calculated from the second-phase sample of size n.
●	xˉ is the sample mean of the first auxiliary variable, calculated from the second-phase sample of size n.
●	xˉ0​ is the sample mean of the first auxiliary variable, calculated from the first-phase sample of size n₀.
●	Zˉ is the known population mean of the second auxiliary variable.
●	zˉ0​ is the sample mean of the second auxiliary variable, calculated from the first-phase sample of size n₀.
The intuition behind this formula lies in the sequential use of the auxiliary variables to adjust the sample mean of the study variable.11 The ratio xˉxˉ0​​ acts as a correction factor based on the information about the first auxiliary variable obtained from both phases of sampling. If the sample mean of the first auxiliary variable in the subsample (xˉ) deviates from its mean in the larger initial sample (xˉ0​), this ratio adjusts the study variable's mean (yˉ​) to account for this difference.13 The second part of the estimator, (zˉ0​Zˉ​), further refines the estimate by incorporating information about the second auxiliary variable. Since the population mean of the second auxiliary variable (Zˉ) is known, the ratio of this known mean to its estimated mean from the first phase (zˉ0​) provides an additional adjustment factor.13 This adjustment leverages the correlation between the second auxiliary variable and the first auxiliary variable (which is correlated with the study variable) to indirectly improve the estimate of the study variable's population mean.13
In practice, the chain ratio estimator is applied following the two phases of sampling.13 First, a large sample of size n₀ is selected, and data on both auxiliary variables (x and z) are collected to calculate xˉ0​ and zˉ0​. Second, a subsample of size n is drawn, and data on the study variable (y) and the first auxiliary variable (x) are collected to calculate yˉ​ and xˉ. Finally, these values, along with the known population mean Zˉ, are plugged into the chain ratio estimator formula to obtain an improved estimate of the population mean of the study variable.13 The effectiveness of this estimator heavily relies on the strength of the correlation between the study variable and the first auxiliary variable, and the correlation between the two auxiliary variables.13 Stronger correlations generally lead to more precise adjustments and, consequently, a more efficient estimator.
3.3. An Illustrative Example
Consider a survey aimed at estimating the average yield of a particular fruit crop in a region.13 Obtaining the yield (study variable) for every farm in the region might be resource-intensive. However, information on the area under cultivation for this crop (first auxiliary variable) might be easier to obtain through satellite imagery or agricultural records for a large sample of farms. Furthermore, data on the total agricultural landholding of each farm (second auxiliary variable) might be available from government databases, and the average total agricultural landholding for the region might be known (Zˉ).
In this scenario, a double sampling approach could be employed.13 In the first phase, a large sample of farms (n0​) is selected using satellite imagery and agricultural records to estimate the average area under the fruit crop (xˉ0​) and the average total agricultural landholding (zˉ0​). In the second phase, a smaller subsample of these farms (n) is selected, and the actual yield of the fruit crop (yˉ​) and the area under cultivation (xˉ) are measured through field visits. Using the known average total agricultural landholding (Zˉ), the chain ratio estimator Yˉ^CR​=yˉ​(xˉxˉ0​​)(zˉ0​Zˉ​) can be used to estimate the average yield of the fruit crop in the region. This approach leverages the readily available information on land area to improve the precision of the yield estimate obtained from the smaller, more costly sample.
4. Variations and Types of Chain Estimators
The basic chain ratio estimator serves as a foundation upon which various modifications and extensions have been developed to address different survey scenarios and data characteristics.12
4.1. Chain Product Estimators
In situations where the correlation between the study variable and the auxiliary variables is negative, chain product estimators are typically employed.22 The general formula for a chain product estimator in double sampling, analogous to the chain ratio estimator, might take the form:
Yˉ^CP​=yˉ​(xˉ0​xˉ​)(Zˉzˉ0​​)
Here, the ratios are inverted compared to the chain ratio estimator, reflecting the inverse relationship between the variables. The choice between using a chain ratio or a chain product estimator depends on the direction and strength of the correlation between the study variable and the auxiliary variables.23 If an increase in the auxiliary variable is associated with a decrease in the study variable, a chain product estimator would generally be more appropriate.
4.2. Chain Regression Estimators
Chain regression estimators utilize the regression relationships between the study variable and the auxiliary variables to improve the estimation of population parameters.12 These estimators often incorporate regression coefficients that are estimated from the sample data obtained in the second phase of double sampling.25 A general form of a chain regression estimator for the population mean (Yˉ) might be:
Yˉ^LR​=yˉ​+byx​(xˉ0​−xˉ)+byz​(zˉ0​−Zˉ)
Where byx​ is the sample regression coefficient of y on x, and byz​ is the sample regression coefficient of y on z. Chain regression estimators can offer advantages over chain ratio estimators, particularly when the relationship between the variables is not strictly proportional, meaning the regression line does not necessarily pass through the origin.26 By modeling the linear relationship more directly, regression estimators can potentially provide more accurate estimates in a wider range of situations.26
4.3. Chain Exponential Estimators
Chain exponential estimators employ exponential functions of the auxiliary variables in the estimation process.29 These estimators are often used when the relationship between the study and auxiliary variables exhibits an exponential growth or decay pattern.20 A chain ratio-type exponential estimator might be formulated as:
Yˉ^ER​=yˉ​exp(xˉ0​+xˉxˉ0​−xˉ​)exp(Zˉ+zˉ0​Zˉ−zˉ0​​)
The motivation behind using exponential forms lies in their ability to capture specific types of non-linear relationships that are better modeled by exponential curves than by simple ratios or linear regressions.7 These estimators have been shown to be more efficient than traditional ratio estimators under certain realistic conditions.7
4.4. Other Variations
Beyond these main types, other variations of chain estimators exist, often tailored to specific survey needs or data characteristics.10 For instance, some estimators incorporate known population parameters of the auxiliary variables, such as the coefficient of kurtosis, to further improve efficiency.11 The coefficient of kurtosis, which measures the "tailedness" of a distribution, can provide additional information about the shape of the auxiliary variable's distribution, allowing for more refined adjustments in the estimation process.13 Furthermore, the principle of chain estimation is not limited to estimating the population mean; it can also be applied to estimate other population parameters such as the median 12 or the variance.15 These variations demonstrate the flexibility and versatility of the chain estimation framework in addressing a wide range of estimation problems in survey sampling.25
5. Statistical Properties: Bias and Variance
5.1. Bias of Chain Estimators
Many chain ratio-type estimators are inherently biased, particularly when the sample sizes are small.11 Bias in an estimator refers to the systematic difference between the expected value of the estimator and the true value of the population parameter being estimated.37 For chain estimators, the bias often arises from the non-linear nature of the ratio or product terms involved in their formulation.20 However, a crucial property of many of these biased estimators is that their bias tends to decrease as the sample size increases.39 Under large sample approximations, the bias often becomes negligible compared to the variance of the estimator.11
Several factors can influence the magnitude of the bias in chain estimators.18 These include the strength of the correlation between the study variable and the auxiliary variables, as well as the correlation between the auxiliary variables themselves.7 Additionally, the sample sizes in both the first and second phases of double sampling play a significant role; larger sample sizes generally lead to a reduction in bias.11 Researchers have derived mathematical expressions for the bias of various chain estimators under different sampling schemes and conditions, often using Taylor series expansions to approximate the expected values of the estimators.11 While some proposed chain estimators are designed to be nearly unbiased 12, it is important to acknowledge that a trade-off often exists between bias and variance when choosing an estimator in survey sampling.37
5.2. Variance and Mean Squared Error (MSE)
The variance of a chain estimator quantifies the variability of its estimates across different possible samples that could be drawn from the population.37 A lower variance indicates that the estimator provides more consistent estimates around its expected value. In the context of chain estimators, the variance depends on factors such as the population variances of the study and auxiliary variables, the correlation coefficients between them, and the sample sizes in both phases of sampling.8
To obtain a comprehensive measure of an estimator's quality, the concept of Mean Squared Error (MSE) is often used.37 The MSE combines both the bias and the variance of an estimator, providing an indication of the average squared difference between the estimator and the true population parameter.37 Mathematically, the MSE is expressed as:
MSE(θ^)=Var(θ^)+2
where θ^ is the estimator of the population parameter θ.
For specific chain ratio estimators in double sampling, such as the one formulated earlier, the MSE has been derived up to the first degree of approximation for both Case-I (subsample from the initial sample) and Case-II (independent subsample) sampling scenarios.11 These MSE formulas typically involve terms related to the population variances (often expressed through coefficients of variation), the population correlation coefficients between the study and auxiliary variables, and the sample sizes in the two phases.11 For instance, for the usual chain ratio estimator (Yˉ^CR​) discussed in Section 3.2, the MSE formulas are given by 11:
For Case-I:
MSE(Yˉ^CR​)I​=Yˉ2(f1​Cy2​+f3​Cx2​−2f3​ρyx​Cy​Cx​+f2​Cz2​−2f2​ρyz​Cy​Cz​)
For Case-II:
MSE(Yˉ^CR​)II​=Yˉ2(f1​Cy2​+f2​Cx2​+f1​Cx2​+f2​Cz2​−2f1​ρyx​Cy​Cx​−2f2​ρxz​Cx​Cz​)
Where f1​=(1/n−1/N), f2​=(1/n0​−1/N), and f3​=f1​−f2​. Cy​,Cx​,Cz​ are the coefficients of variation of the study and auxiliary variables, and ρyx​,ρyz​,ρxz​ are the population correlation coefficients.
These formulas highlight how stronger correlations with auxiliary variables (higher ρ values) and larger sample sizes (smaller f1​ and f2​ values) tend to reduce the variance and, consequently, the MSE of chain estimators.8 The MSE provides a valuable metric for comparing the overall quality of different estimators, with a lower MSE indicating a better estimator that, on average, is closer to the true population parameter.42 The efficiency gains achieved by using chain estimators are often evaluated by comparing their MSE to that of simpler estimators like the usual unbiased estimator or the standard ratio estimator.4
6. Efficiency Comparisons
A key aspect of evaluating the utility of chain estimators lies in comparing their efficiency to that of other commonly used estimators in sample surveys, such as the usual unbiased estimator (sample mean), the standard ratio estimator (using a single auxiliary variable), and the regression estimator.9 Efficiency, in this context, refers to the ability of an estimator to provide precise estimates of the population parameter with a given sample size.4 An estimator is considered more efficient if it has a smaller variance or MSE compared to another estimator for the same sample size.
Research has established theoretical conditions under which chain estimators are expected to outperform these alternatives.11 For instance, chain ratio estimators often demonstrate higher efficiency than traditional ratio estimators, particularly when multiple auxiliary variables are available and exhibit strong correlations with the study variable and with each other.5 The sequential use of auxiliary information in chain estimators allows for more refined adjustments to the estimate of the study variable, leading to a reduction in MSE compared to using just a single auxiliary variable.11
Numerous empirical studies, employing both simulations and real-world datasets, have been conducted to compare the performance of chain estimators with other estimators.13 These studies often calculate the Percent Relative Efficiency (PRE) of different estimators with respect to a base estimator, such as the usual unbiased estimator. The PRE is defined as:
PRE(θ^1​,θ^0​)=MSE(θ^1​)MSE(θ^0​)​×100%
A PRE value greater than 100% indicates that the estimator θ^1​ is more efficient than the estimator θ^0​.
For example, a study by Singh et al. (2015) proposed a chain ratio-type estimator under systematic sampling and compared its performance with several existing estimators using a real dataset.17 The results of this comparison, in terms of PRE with respect to the usual simple estimator (t₀), are summarized in the table below 17:
Estimator	MSE(t<sub>α</sub>)	PRE(t<sub>α</sub>, t₀)	
t₀	1566.48	100.00	
t₁ (Swain's Ratio)	423.33	369.99	
t₂ (Shukla's Product)	894.49	175.12	
t₃ (Usual Regression)	66.63	2349.49	
t₄ (Singh et al.'s Exponential Ratio)	934.45	167.63	
t₅ (Singh et al.'s Exponential Product)	1400.03	111.89	
t₆ (Tailor et al.'s Ratio-cum-Product)	178.24	878.86	
t<sub>m</sub> (Proposed Estimator)	33.28	4706.00	
As evident from the table, the proposed chain ratio-type estimator (t<sub>m</sub>) exhibited the highest PRE (4706.00%), indicating a substantial improvement in efficiency compared to the other estimators for this particular dataset.17
However, the relative efficiency of chain estimators is not universally superior and can be influenced by various factors such as the strength of correlations, the choice of auxiliary variables, and the specific form of the chain estimator used (ratio, regression, etc.).7 In some cases, regression estimators, including chain regression types, might offer even higher efficiency than chain ratio estimators, especially when the relationship between the study and auxiliary variables is not strictly linear through the origin.24 The choice of the most efficient estimator depends on the specific characteristics of the survey and the data at hand.
7. Chain Estimators in the Presence of Non-Response
Non-response is a pervasive issue in sample surveys, occurring when selected individuals or households do not participate or fail to provide the requested information.6 Non-response can lead to biased estimates if the characteristics of non-respondents differ systematically from those of respondents.2 To mitigate the impact of non-response, chain estimation techniques have been adapted and applied in survey sampling.10
Researchers have developed chain-type estimators specifically designed for situations with missing data or non-responding units.6 These estimators often leverage auxiliary information that is available for both responding and non-responding units to adjust for the potential bias introduced by non-response.6 For example, if the population mean of an auxiliary variable is known, and sample means are available for both respondents and an initial sample (including non-respondents), this information can be incorporated into a chain-type estimator to improve the estimation of the study variable's population mean, even with missing data.18
The methodologies used in these estimators to account for non-response vary, but they often involve modifications of the standard chain ratio or regression estimation formulas.15 These modifications aim to utilize the available auxiliary information to either impute the missing values or to adjust the weights of the responding units to better represent the entire population.6 Studies have shown that these chain estimators designed for non-response scenarios can offer improved efficiency and reduced bias compared to traditional methods that do not adequately address the issue of missing data.15 For instance, Khare et al. (2012) proposed two chain ratio type estimators for the population mean in the presence of non-response, demonstrating their higher efficiency and lower cost compared to relevant estimators.18 Thus, chain estimators provide a valuable tool for enhancing the reliability of survey estimates when faced with the challenge of non-response.
8. Advantages and Limitations of Chain Estimators
8.1. Advantages
The use of chain estimators in sample surveys offers several key advantages that contribute to more accurate and efficient population parameter estimation.11 Primarily, chain estimators can lead to increased efficiency and precision in estimation, particularly when the auxiliary variables used in the "chain" are strongly correlated with the study variable and with each other.11 This enhanced precision allows researchers to make more reliable inferences about the population based on the sample data.
A significant advantage of chain estimators is their ability to utilize multiple sources of auxiliary information in a sequential manner.11 By creating a chain of information flow, these estimators can leverage the combined predictive power of several related variables, potentially extracting more information from the available data than methods that use only a single auxiliary variable.11 This is particularly beneficial in complex surveys where a variety of relevant auxiliary data might be available.
Chain estimators are especially effective in double sampling scenarios, where the population means of crucial auxiliary variables are unknown or costly to obtain directly.11 By using a larger first-phase sample to estimate the means of these auxiliary variables, and then incorporating these estimates into the chain estimation process in the second phase, researchers can overcome the limitations of unknown population parameters and still achieve improved efficiency.11
Furthermore, chain estimators are adaptable to different types of relationships that might exist between the study variable and the auxiliary variables.12 Variations such as chain ratio, chain product, chain regression, and chain exponential estimators allow researchers to choose the form that best models the underlying relationships in their data, leading to more accurate estimates.12
Chain estimators also offer potential advantages in handling non-response and missing data in surveys.6 By leveraging auxiliary information that is more completely observed, these estimators can be adapted to impute missing values or adjust for non-response bias, thereby improving the overall quality of the survey estimates.15
Finally, the flexibility of the chain estimation framework extends to the estimation of various population parameters beyond just the mean, including the median and variance.12 This versatility makes chain estimators a valuable tool in a wide range of survey applications and research objectives. The primary strength of these methods lies in their capacity to extract more information from the available data, leading to more dependable and precise population estimates, which is vital for making well-informed decisions based on survey findings.6
8.2. Limitations
Despite their advantages, chain estimators also have certain limitations that need to be considered when deciding on their use in sample surveys.12 One notable limitation is that they can be more complex to formulate and implement compared to simpler estimators like the sample mean or the standard ratio estimator.11 The mathematical derivations of their properties, such as bias and variance, can be intricate and might require a strong understanding of statistical theory.11
Another potential drawback is that many chain ratio-type estimators are inherently biased, especially when the sample sizes are small.11 While the bias often decreases with increasing sample size, researchers need to be aware of this potential for systematic error, particularly in studies with limited sample sizes.39
The efficiency gains achieved by using chain estimators are heavily dependent on the strength and pattern of correlations between the study variable and the auxiliary variables, as well as between the auxiliary variables themselves.7 If the chosen auxiliary variables are poorly correlated with the study variable, or if the correlations are weak, the use of a chain estimator might not lead to significant improvements in efficiency and could even potentially worsen the estimates compared to simpler methods.9 Therefore, careful selection of appropriate and highly correlated auxiliary variables is crucial for the successful application of chain estimators.
Furthermore, the derivation of the statistical properties of chain estimators, such as their bias and variance, can be mathematically involved, often relying on large sample approximations.11 This complexity might make it challenging for practitioners without a strong statistical background to fully understand and apply these estimators correctly.2
Finally, the optimal form of the chain estimator and the most effective choice of auxiliary variables might not always be immediately obvious.12 Determining the best approach often requires prior knowledge about the population and the relationships between the variables, or might necessitate exploratory data analysis and model selection procedures.23 In situations with outliers or extreme observations, some chain estimators might not perform efficiently.12 The increased complexity of chain estimators demands a thorough understanding of the underlying statistical theory and careful consideration of the data characteristics to ensure their appropriate and effective application.49 The reliance on auxiliary variables also means that the quality and relevance of these variables are paramount; inaccurate or poorly correlated auxiliary data can negate the benefits of using a chain estimator.51
9. Conclusion
Chain estimators represent a sophisticated class of statistical tools for enhancing the efficiency and accuracy of population parameter estimation in sample surveys.11 By sequentially incorporating auxiliary information, particularly within the framework of double sampling, these estimators offer a powerful approach to leveraging the relationships between multiple variables to refine population estimates.11 Their adaptability to different types of relationships (ratio, product, regression, exponential) and their potential for handling challenges such as non-response underscore their significance in advanced survey methodology.12
However, the application of chain estimators requires careful consideration of their statistical properties, including bias and variance, as well as the specific design of the survey and the characteristics of the data.8 The efficiency gains promised by chain estimators are contingent upon the strength and nature of the correlations between the study and auxiliary variables, and their implementation can be more complex than that of traditional estimators.7 Ongoing research continues to explore and refine chain estimation techniques, aiming to develop more robust and widely applicable methods for drawing precise inferences from sample survey data.7 The decision to employ a chain estimator should be informed by a thorough understanding of its underlying principles, its assumptions, and its potential advantages and limitations in the context of the specific survey objectives and data at hand.
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