Iterative Proportional Fitting (Raking) Methods in Sample Survey Statistics
I. Introduction to Iterative Proportional Fitting (Raking)
A. Defining Raking and its Core Purpose in Survey Weighting
Raking, a technique formally identified as Iterative Proportional Fitting (IPF), is a statistical procedure employed extensively within the domain of sample survey statistics to adjust the weights assigned to individual survey responses.1 The fundamental objective of raking is to ensure that the weighted distributions of selected auxiliary variables within the sample align with their known distributions in the target population. These known population distributions are commonly referred to as control totals or marginals.1 This adjustment process is critical for enhancing the accuracy and representativeness of survey estimates. It achieves this by rectifying discrepancies between the sample and the population that may originate from diverse sources, including inherent sampling variability, biases introduced by nonresponse, and deficiencies in population coverage.2
As articulated in one source, "With raking, a researcher chooses a set of variables where the population distribution is known, and the procedure iteratively adjusts the weight for each case until the sample distribution aligns with the population for those variables".1 This iterative nature of adjustment is a defining characteristic of the technique. Furthermore, raking is widely utilized to revise tables of data, particularly when the available information is "incomplete, inaccurate, outdated, or a sample".6
The core function of raking extends beyond a mere mechanical recalibration of weights. It represents a systematic method to integrate robust external information—typically derived from comprehensive and reliable sources such as national censuses or large-scale administrative datasets—into the survey sample. By compelling the sample's characteristics on these auxiliary variables to mirror established population benchmarks, raking aims to mitigate distortions that can arise from the sampling process itself, such as differential nonresponse patterns or inadequacies in the sampling frame.2 Consequently, the procedure can be conceptualized as an "infusion of external knowledge" into the sample data. This infusion is pivotal because it enhances the sample's capacity to accurately reflect the broader population from which it was drawn, thereby strengthening the validity and reliability of inferences made from the survey data. This understanding also highlights a crucial dependency: the success of raking in improving data quality is inextricably linked to the accuracy and reliability of the external control totals used. Any inaccuracies or biases present in these benchmarks will inevitably be reflected in, and potentially distort, the final weighted survey estimates.
B. Historical Context and Seminal Contributions (Deming & Stephan, 1940)
The methodological foundations of iterative proportional fitting were formally introduced to the statistical community by W. Edwards Deming and Frederick F. Stephan in their landmark 1940 paper, "On a Least Squares Adjustment of a Sampled Frequency Table When the Expected Marginal Totals Are Known".2 Their pioneering work was initially motivated by the practical challenge of adjusting sample data derived from the 1940 U.S. Census. The aim was to ensure that these sample-based tabulations were consistent with complete census counts available for certain demographic characteristics.2 Deming and Stephan conceptualized and proposed IPF as a systematic procedure "to make an un-biased guess at individual cells of data, based on knowledge from samples and estimated ('known') totals".7
It is a noteworthy historical detail that while Deming and Stephan originally posited that IPFP served as an algorithm leading to a minimizer of the Pearson Chi-squared statistic, Stephan later reported that the procedure does not, in fact, achieve this specific minimization objective.8 This clarification is important for a precise understanding of its theoretical underpinnings and evolution.
The genesis of raking within the context of large-scale census applications, such as the U.S. Census, is particularly telling. It underscores the method's foundational role in addressing the complexities of official statistics, where achieving consistency between different data sources and ensuring the utmost accuracy are paramount concerns.2 This historical context suggests that IPF was engineered from the outset with considerations of robustness and practical applicability to complex, high-dimensional data structures. Such characteristics likely contributed significantly to its subsequent widespread adoption and adaptation across a diverse array of survey contexts, extending far beyond its initial applications in census work. The method's ability to reconcile sample data with more comprehensive population information in high-stakes environments cemented its value in the field of survey statistics.
C. Alternative Terminology (IPF, RAS, etc.)
Iterative Proportional Fitting is recognized and referred to by a variety of names across different scientific and analytical disciplines. This multiplicity of terms reflects not only its broad applicability to diverse data adjustment problems but also instances of its independent discovery or adaptation in various fields.6
Among the common alternative terms are:
●	IPF or IPFP: These are standard abbreviations frequently encountered in statistical literature and academic discourse.
●	Raking: This term is widely used in the specific context of survey statistics and also finds currency in computer science.1
●	RAS algorithm: This designation is common in the field of economics, particularly within input-output analysis. The name is derived from the matrix notation of the model, often expressed as finding diagonal matrices R and S such that an adjusted matrix X=RZS meets the required marginals, where Z is the initial matrix.6
●	Cross–Fratar or Furness method: These names are typically associated with applications in transport engineering and traffic modeling.6
●	Matrix scaling: This is a more general term used in computer science and numerical analysis to describe procedures that adjust matrix elements to satisfy certain row and column sum constraints.8
●	Rim-weighting or structure-preserving estimation: These terms are also encountered in statistical literature, emphasizing the adjustment to marginal (rim) totals and the method's property of preserving certain structural aspects of the original data.6
●	Biproportional fitting or biproportion: These terms are used in both statistics and economics to describe the proportional adjustment process across two dimensions.8
The existence of such a diverse lexicon for what is essentially the same underlying mathematical procedure is indicative of its fundamental utility. The core problem that IPF addresses—adjusting an initial matrix or table of data to conform to new marginal constraints while remaining, in some sense, "close" to the original data structure—is a recurrent challenge across numerous analytical domains.6 The independent emergence of similar solutions under different names in fields as varied as economics, transport engineering, and computer science points to the method's inherent robustness, intuitive appeal, and its capacity to provide a practical solution to this common data adjustment problem.
II. The Mechanics of Raking: Algorithm and Iterative Adjustments
A. Step-by-Step Explanation of the Raking Process (e.g., for a two-dimensional table)
The raking algorithm is an iterative procedure designed to adjust the weights assigned to individual cases within a survey sample. This iterative adjustment continues until the weighted marginal totals for a pre-specified set of auxiliary variables match known population totals for these variables.1 The unadjusted data, often in the form of cell counts or initial weighted values in a contingency table, are frequently referred to as 'seed' cells.7
For a clearer understanding, consider the application of raking to a two-dimensional table, for example, when adjusting survey data to known population totals for age categories and gender categories simultaneously. The process typically unfolds as follows:
1.	Initialization: The process begins with a set of initial weights for each case in the sample. These might be base weights, calculated as the inverse of the probabilities of selection if a probability sampling design was used, or they could be equal weights if no design information is incorporated at this stage.2
2.	First Dimension Adjustment: The weights are first adjusted so that the weighted sum for each category of the first auxiliary variable (e.g., age groups) matches the corresponding known population total for that age group. This adjustment is performed proportionally. For each case, its current weight is multiplied by a factor equal to the ratio of the known population total for its specific age group to the current weighted sample total for that same age group.7
3.	Second Dimension Adjustment: Using the weights that resulted from the adjustment in the previous step (i.e., after adjusting for age), a similar proportional adjustment is made for the second auxiliary variable (e.g., gender). The weights are modified so that the weighted sum for each gender category matches its known population total. Again, this is achieved by multiplying the current weights by the ratio of the population total for each gender category to the current weighted sample total for that gender category.7
4.	Iteration: The crucial aspect of raking is its iterative nature. The adjustment for the second dimension (gender) might have slightly perturbed the alignment previously achieved for the first dimension (age). Therefore, the process of adjusting to the marginals of the first dimension is repeated using the weights from step 3, followed by another adjustment to the marginals of the second dimension. This cycling through each adjustment variable (or dimension) sequentially continues.1 One complete sequence of adjustments across all specified dimensions constitutes a single cycle or iteration of the algorithm.2
A clear illustration of this iterative dynamic is provided: "If the adjustment for education pushes the sex distribution out of alignment, then the weights are adjusted again so that men and women are represented in the desired proportion. The process is repeated until the weighted distribution of all of the weighting variables matches their specified targets".1 For a two-dimensional table, this involves, in the first step of an iteration, proportionally adjusting each row of seed cells to meet the marginal row totals. In the second step, each column of these already row-adjusted cells is then proportionally adjusted to meet the marginal column totals. This sequence completes one iteration and is repeated until convergence.7
This iterative back-and-forth adjustment mechanism is fundamental to raking's ability to handle multiple constraints. Each step in an iteration aims to satisfy one set of marginal constraints, potentially causing minor deviations from others that were previously met. The algorithm proceeds by successively refining the weights, gradually reducing these deviations until a state of equilibrium is approached, where all specified marginal constraints are simultaneously (or approximately) satisfied. This iterative refinement is the key to its capacity to manage multi-dimensional constraints effectively, particularly in common situations where full cross-tabular population data (i.e., population counts for every cell formed by the intersection of all auxiliary variables) are unavailable. The ability of the algorithm to converge to a stable and consistent solution under these conditions represents one of its significant practical strengths.
B. Convergence Criteria and Iterative Nature
The iterative adjustment process inherent in raking continues until the weighted sample marginals for all auxiliary variables used in the weighting scheme are deemed "sufficiently close" to their corresponding population control totals.2 The determination of "sufficiently close" is governed by pre-defined convergence criteria.
Typically, convergence is assessed against a specified tolerance level. The algorithm is programmed to terminate when the absolute or proportional difference between the weighted sample total and the population control total for each category of each raking variable falls below a small, pre-specified threshold value.7 For example, a common tolerance might be that all adjusted sample marginals must be within 0.1% or 0.001 (in proportional terms) of their respective population targets. It is important for practitioners to recognize that, as noted in 7, IPF adjustments "converge to fit marginal totals within any selected level of closeness, but in most cases will never exactly sum to all of the pre-determined marginal totals." This implies that convergence in raking is often an approximation, albeit a very close one.
The number of iterations required to achieve convergence can vary substantially. This variability is influenced by several factors, including the total number of auxiliary variables used for raking, the number of categories within each of these variables, the magnitude of the initial discrepancies between the sample marginals and the population marginals, and the stringency of the chosen convergence criterion.11 Generally, more variables, more categories per variable, larger initial differences, and tighter tolerance levels will necessitate a greater number of iterations.
In practical implementations, it is standard procedure to set an upper limit on the total number of iterations the algorithm is allowed to perform.11 This safeguard prevents excessively long computation times in cases of very slow convergence and provides a stopping point if the algorithm fails to converge due to issues such as inconsistent control totals or highly sparse data.
The concept of convergence in raking is not merely a technical detail; it has practical implications. The choice of convergence criteria can directly influence the final set of weights and, consequently, the survey estimates derived from them. A very strict tolerance might lead to a large number of iterations or even non-convergence if the constraints are nearly contradictory or the data are very sparse. Conversely, an overly loose tolerance might result in weights that do not adequately reflect the known population marginals, thereby failing to achieve the intended bias reduction. This trade-off necessitates careful consideration by the practitioner, who must often monitor the raking process to ensure that convergence is achieved in a reasonable manner and that the resulting weights are sensible.
III. Theoretical Foundations and Statistical Properties
A. Mathematical Formulation of Raking
Let (w_i^{(0)}) represent the initial base weight for survey unit (i) (e.g., the inverse of its selection probability, (1/\pi_i)). The objective of raking is to derive a set of final weights, (w_i), such that for each auxiliary variable (k) used in the raking process, and for each category (j) of that variable, the sum of the final weights for all units belonging to category (j) of variable (k) equals the known population total for that category. This can be expressed as:
i∈category j of variable k∑​wi​=Nj(k)​
where (N_j^{(k)}) is the known population total for category (j) of variable (k).
Consider a simple two-variable scenario, where the first variable has (H) classes and the second variable has (K) classes. The raking estimator for a population total (Y), denoted as (\hat{Y}r), can be formulated as 2:
Y^r​=h=1∑H​k=1∑K​i∈cell(h,k)∑​whk,i​yi​
Here, (w{hk,i}) is the final raked weight assigned to unit (i) that falls into the cell defined by class (h) of the first variable and class (k) of the second variable, and (y_i) is the value of the survey variable of interest for unit (i).
The iterative adjustment process for a two-dimensional table can be described more concretely. Let (n_{jk}^{(0)}) be the initial sample count (or sum of initial weights (w_i^{(0)})) in the cell corresponding to category (j) of the first variable and category (k) of the second variable. Let (N_{j\cdot}) be the known population total for category (j) of the first variable (summed over all categories of the second variable), and (N_{\cdot k}) be the known population total for category (k) of the second variable (summed over all categories of the first variable).
In iteration (t), starting with cell values (n_{jk}^{(t-1)}) from the previous iteration (or initial values for (t=1)):
1.	Adjust to first dimension (row) totals: njk(t,1)​=njk(t−1)​∑k​njk(t−1)​Nj⋅​​ This step scales the values in each row (j) so that their sum matches the target row total (N_{j\cdot}).
2.	Adjust to second dimension (column) totals: njk(t,2)​=njk(t,1)​∑j​njk(t,1)​N⋅k​​ This step scales the values in each column (k) (using the results from step 1) so that their sum matches the target column total (N_{\cdot k}). The values (n_{jk}^{(t,2)}) then become the starting values (n_{jk}^{(t)}) for the next iteration. The individual weights (w_i) are adjusted proportionally according to which cell ((j,k)) they belong to at each adjustment step.
A general formulation, as presented in 8, considers an initial matrix (Z) (representing, for example, unweighted sample counts (n_{ij})) and aims to find an adjusted matrix (X) that is "closest" to (Z) while ensuring that the row and column totals of (X) match those of a target specification (e.g., known population marginals (u_i) for rows and (v_j) for columns). Starting with initial values (x_{ij}^{(0)} = n_{ij}), the iterative steps for iteration (s) (where (s) indexes half-steps of an iteration) are:
Adjust rows:
xij(2s−1)​=xij(2s−2)​∑k​xik(2s−2)​ui​​
Adjust columns:
xij(2s)​=xij(2s−1)​∑k​xkj(2s−1)​vj​​
This process is repeated until convergence.
The RAS algorithm, which is mathematically equivalent to IPF, can be conceptualized as finding diagonal matrices (R = \text{diag}(r_1,..., r_I)) and (S = \text{diag}(s_1,..., s_J)) such that the adjusted matrix (X = RZS) has the desired target marginals.8 In this formulation, (R) contains row adjustment factors and (S) contains column adjustment factors. Estimating these factors (R) and (S) directly, rather than explicitly calculating the full matrix (X) at each step, can offer computational efficiencies, especially for large tables.8
The mathematical structure of these iterative proportional adjustments reveals a key characteristic of raking: at each step, the existing proportional distribution of weights or counts within a given row (or column) is preserved. Only the overall total for that row (or column) is scaled to match the target marginal. This proportional scaling mechanism is what ensures the adjusted matrix remains structurally "close" to the original matrix, a property often described as preserving the odds ratios or interaction structure of the initial table, subject to the new marginal constraints. This is a fundamental aspect of why raking is often referred to as a "structure-preserving" estimation technique.
B. Underlying Assumptions
The effective and statistically valid application of raking (Iterative Proportional Fitting) rests upon several key assumptions and preconditions regarding the data and the weighting process itself.
●	Availability and Accuracy of Population Control Totals: This is perhaps the most critical prerequisite. Raking fundamentally relies on the existence of accurate and reliable population marginal totals for the chosen auxiliary variables. The quality of the raked survey estimates is directly dependent on the quality of these external control totals; any errors, biases, or outdatedness in the control totals will be systematically incorporated into the weighted sample estimates.1
●	Non-Zero Marginal Totals and Seed Cells:
○	The IPF algorithm, in its standard form, cannot operate if any of the target marginal control totals are zero. A common practical workaround for this issue is to substitute a very small positive value (e.g., 0.001) for the zero marginal, allowing the algorithm to proceed.7
○	Similarly, initial sample counts (seed cells) in the cross-classification of raking variables that have a value of zero will typically not be adjusted by the raking process; they will remain zero. If it is desired that such cells receive some weight (implying the possibility of their occurrence in the population), a small initial value can be assigned to these seed cells before starting the iteration.7
●	Non-Contradictory Constraints: The set of marginal constraints (i.e., the population control totals for all raking variables) must be internally consistent and logically achievable from the sample data.11 For instance, the sum of marginal totals for one variable must equal the sum of marginal totals for another if they are to represent the same overall population size. Contradictory or inconsistent constraints can lead to the algorithm failing to converge or producing nonsensical results.11
●	Sufficient Sample Sizes in Categories: While raking is generally more flexible with smaller sample sizes than methods like cell weighting 14, the presence of very small sample sizes within the categories being adjusted can lead to unstable weights (excessively large or small) or can slow down the convergence of the algorithm.11 Some sources suggest that categories containing fewer than, for example, 5% of the total sample cases might impede the convergence process.11
●	Correlation between Auxiliary Variables and Survey Variables of Interest: The primary statistical benefit of raking, particularly in terms of reducing bias and improving the precision (reducing variance) of survey estimates, is most pronounced when the auxiliary variables used for weighting are highly correlated with the key outcome variables of the survey. Additionally, effectiveness is enhanced if these auxiliary variables are also related to the underlying mechanisms of nonresponse or coverage error.2 If the chosen auxiliary variables are unrelated to the survey outcomes or the processes leading to sample imbalances, raking may offer little substantive improvement in the accuracy of the estimates and could, in some cases, unnecessarily increase variance.15
●	Missing at Random (MAR) Assumption (Implicit for Nonresponse Bias Correction): For raking to be effective in reducing nonresponse bias for a particular survey outcome, it often relies on an implicit assumption that, conditional on the set of raking variables, the nonresponse mechanism is "Missing at Random" (MAR). The MAR assumption posits that the probability of a unit being a nonrespondent depends only on the observed auxiliary variables (the raking variables) and not on the unobserved value of the survey outcome itself (or other unobserved variables).16 If this assumption holds, then within the strata or cells defined by the raking variables, respondents and nonrespondents are assumed to be similar with respect to the survey's key outcomes. If, however, nonresponse is related to the outcome variable even after conditioning on the raking variables (a situation known as Missing Not At Random - MNAR), then raking may not fully eliminate the nonresponse bias. The definition of MAR, as provided in 17, is that "conditional to a set of auxiliary variables, missingness is unrelated to survey outcomes," with the raking variables serving as these conditioning auxiliary variables.
The MAR assumption serves as a critical, though often unstated, theoretical foundation for raking's ability to adequately correct for nonresponse bias. If the selected raking variables do not effectively capture the primary dimensions along which nonresponse occurs and along which nonresponse is correlated with the survey's variables of interest, then the adjustments made by raking might provide a superficial alignment with population demographics without addressing the underlying substantive biases. For instance, if a survey aims to measure income, and high-income individuals are less likely to respond, raking on age and gender alone will not correct the income-related nonresponse bias unless age and gender are very strong proxies for income and its relationship with nonresponse. This underscores the paramount importance of judiciously selecting raking variables that are substantively relevant to both the nonresponse process and the key survey outcomes.
C. Relationship to the Horvitz-Thompson Estimator
The Horvitz-Thompson (HT) estimator is a cornerstone in the theory of survey sampling, providing a general method for estimating population totals (or means) from probability samples. In the HT framework, each sampled unit (k) is assigned a weight (d_k) equal to the inverse of its probability of inclusion in the sample ((\pi_k)), i.e., (d_k = 1/\pi_k).13 Under the condition that all inclusion probabilities are known and non-zero for all units in the population, the HT estimator is unbiased for the population total.13
Raking, as a member of the broader class of calibration estimators, typically begins with a set of initial weights. These initial weights are often the base weights derived from the sampling design, which are precisely the Horvitz-Thompson weights ((d_k)).2 The raking procedure then adjusts these initial weights iteratively to ensure that they satisfy the specified marginal constraints based on known population auxiliary information.
Calibration estimators, including raking, are designed to improve upon the properties of the basic HT estimator. By incorporating auxiliary information about the population structure, these methods aim to achieve reductions in both the variance and the potential bias of survey estimates compared to what might be obtained using the HT estimator alone.2 It is recognized that the HT estimator, while unbiased, can be inefficient if relevant auxiliary population data are available but not utilized in the estimation process.21 Raking directly addresses this by leveraging such auxiliary data.
In this context, the Horvitz-Thompson estimator can be viewed as a foundational element upon which raking builds. It provides the initial, design-based weighting that accounts for differential selection probabilities. Raking then refines this initial weighting by aligning the sample to known population characteristics, thereby seeking to correct for deviations of the sample from the population that may not have been fully addressed by the sampling design itself (e.g., due to nonresponse or sampling variability). Thus, raking is not typically seen as a replacement for design-based inference principles embodied by the HT estimator, but rather as an enhancement or an additional layer of adjustment that leverages post-sampling information to improve estimate quality. It aims to make the final weighted sample "look like" the population on key known dimensions, a property that the HT estimator alone does not guarantee beyond what is achieved by the probabilistic nature of the sampling design.
IV. Applications of Raking Across Diverse Survey Domains
A. Use Cases in Public Opinion Polls, Market Research, Health Surveys, and Demography
The versatility and practical utility of raking (Iterative Proportional Fitting) are evident in its widespread application across a multitude of survey research domains. Its ability to adjust sample data to known population characteristics makes it an invaluable tool for enhancing the reliability and representativeness of survey findings.
●	Public Opinion Polls: Raking is a standard and often essential weighting methodology employed by numerous public polling organizations, including prominent institutions like the Pew Research Center.1 In this context, raking is used to ensure that the demographic composition of the polled sample (e.g., with respect to age, gender, race, educational attainment, geographic region) accurately reflects that of the broader population. This alignment is crucial for producing poll results that are considered representative of public opinion on various social, political, and economic issues.
●	Market Research: Market research firms frequently utilize raking to adjust their survey samples to match the demographic and other relevant characteristics of their target markets.3 By doing so, they can obtain more accurate insights into consumer preferences, purchasing behaviors, brand perceptions, and market trends. These insights, in turn, inform strategic business decisions related to product development, marketing campaigns, and customer segmentation.
●	Political Campaigns: In the realm of political campaigning, accurate survey data is vital for gauging voter sentiment, identifying key voter segments, and developing effective campaign strategies.3 Raking helps ensure that survey results are representative of the electorate by adjusting for demographic imbalances in the sample, thereby providing campaign strategists with more reliable data for decision-making.
●	Health Surveys: Health researchers extensively use raking to adjust data from health-related surveys to match known population characteristics, such as age, sex, socioeconomic status, and insurance coverage.3 This adjustment is critical for ensuring that findings regarding disease prevalence, health behaviors, access to care, and the effectiveness of health interventions can be generalized to the broader population. For instance22 explicitly mentions the use of raked weights to correct for differential nonresponse in beneficiary surveys. Similarly, studies like the Brazilian National Health Survey have employed raking to produce generalizable estimates.23
●	Demography and Geography: In demographic and geographic research, IPF is a widely adopted technique for a variety of estimation tasks. These include estimating population attributes for small geographic areas, updating age and sex structures of populations between censuses, estimating migration flows between regions, and creating synthetic microdata populations for use in simulation models when only marginal population data (e.g., from censuses or administrative records) are available.6 Examples cited include estimating populations by age and sex, and modeling inter-area migration flows.6
●	Official Statistics: Given its historical origins in census adjustments 2, raking continues to be a relevant and important tool for national statistical offices and other agencies involved in the production of reliable official statistics. It helps ensure consistency and accuracy in data reported about the population.
●	Observational Studies: More recent research has begun to explore the application of IPF as a direct balancing method in non-experimental observational studies. In this context, IPF is used to create balanced comparison groups by adjusting weights so that the distribution of covariates is similar between treated and control groups. Some studies suggest that IPF can compare favorably to traditional propensity score methods, partly because it can utilize all available observations and thereby reduce certain types of bias associated with case exclusion.24
The pervasiveness of raking across such diverse fields underscores its fundamental utility as a general-purpose data adjustment tool. Its capacity to align sample data with known population structures, regardless of the specific subject matter of the survey, makes it highly valuable. A key factor contributing to its widespread adoption is the relative simplicity of its data requirements: raking primarily needs only marginal population totals for the adjustment variables.1 This is a significant practical advantage, as detailed cross-tabulated population data (i.e., population counts for every unique combination of all adjustment variables) are often unavailable or difficult to obtain, whereas marginal totals are more commonly accessible from census data or other large-scale population registers. This practicality has enabled its application in a vast range of research scenarios.
V. Advantages and Strengths of Employing Raking
A. Reduction of Nonresponse and Coverage Bias
One of the principal motivations for employing raking in survey weighting is its capacity to mitigate biases arising from unit nonresponse and deficiencies in population coverage.1 Unit nonresponse occurs when selected individuals in a sample do not participate in the survey, and if nonrespondents differ systematically from respondents on key survey variables, this can lead to biased estimates. Similarly, coverage bias arises when the sampling frame used to draw the sample does not adequately represent all segments of the target population.2
Raking addresses these issues by adjusting the initial survey weights. If certain demographic groups (or groups defined by other auxiliary variables) are underrepresented in the responding sample relative to their known proportion in the population (perhaps due to higher nonresponse rates or undercoverage for those groups), raking will increase the weights of the respondents from these groups. Conversely, if other groups are overrepresented, their weights will be decreased. This process forces the weighted marginal distributions of the auxiliary variables in the sample to align with their corresponding population benchmarks.5 By achieving this alignment, raking aims to correct for imbalances in the sample that are attributable to differential nonresponse or coverage errors, provided these errors are related to the auxiliary variables used in the raking process.
The effectiveness of raking in reducing such biases, however, is not absolute. It is contingent upon a crucial condition: the auxiliary variables chosen for raking must be related to the mechanisms driving nonresponse or coverage deficiencies and, ideally, also related to the primary survey variables of interest.2 If the chosen raking variables (e.g., standard demographics like age and gender) are not good proxies for the factors that truly differentiate respondents from nonrespondents with respect to the survey's core measures, then the bias reduction achieved by raking might be minimal or, in some cases, even misleading. For instance, if nonresponse is primarily driven by income level, and income is the main variable of interest, simply raking on geographic region may not adequately correct for the income-related nonresponse bias unless income itself is included as a raking variable or is very highly correlated with geographic region and its pattern of nonresponse. Thus, while bias reduction is a key objective, its successful attainment depends heavily on the judicious selection of appropriate auxiliary variables.
B. Enhancement of Sample Representativeness and Generalizability
A direct consequence of adjusting sample weights to match known population distributions on key auxiliary variables is the enhancement of the sample's overall representativeness.1 When the demographic composition (or the distribution of other characteristics used for raking) of the weighted sample mirrors that of the target population, the sample is considered to be more representative of that population, at least with respect to those specific characteristics.
This improved representativeness, in turn, strengthens the generalizability of the survey findings. Researchers can make inferences about the entire target population with greater confidence when they know that the sample from which these inferences are drawn has been adjusted to reflect the population's known structure.3 For example, if a survey on healthcare utilization is raked to match population distributions of age, sex, and socioeconomic status, the resulting estimates of healthcare usage are more likely to be generalizable to the overall population than estimates from an unweighted sample that might over- or underrepresent certain groups.
It is important to qualify, however, that the representativeness achieved through raking is specifically with respect to the variables that were included in the raking process. The weighted sample may align well with the population on, say, age and education if these were raking variables, but it may still not be representative on other unobserved variables or on variables that were not included in the weighting scheme, particularly if these omitted variables are also related to nonresponse or coverage issues and are not highly correlated with the raking variables. Therefore, while raking significantly improves representativeness on the chosen dimensions, it does not guarantee perfect representativeness on all conceivable characteristics.
C. Simplicity in Data Requirements (Marginal Totals)
A significant practical advantage of the raking method is its relatively modest data requirements compared to some other weighting techniques, particularly full post-stratification or cell weighting.1 Raking primarily requires knowledge of only the marginal population proportions (or totals) for each individual auxiliary variable that is being used in the weighting adjustment.1
This means that if a researcher wishes to adjust a sample based on age, gender, and educational attainment, they would need to know the population distribution for age categories, the population distribution for gender categories, and the population distribution for education categories separately. Crucially, raking does not necessitate knowledge of the population counts for every single cell in a multi-way cross-classification of all these auxiliary variables (e.g., the population count of 18-24-year-old males with a college degree). This contrasts sharply with cell weighting (a form of post-stratification), which would require such detailed joint population counts for every cell defined by the intersection of all stratification variables.2
This "marginals only" requirement makes raking a highly pragmatic and flexible tool, especially in common research situations where detailed, multi-way cross-tabulated population data are unavailable or unreliable. While census data, for example, might provide accurate marginal totals for age and education separately for a given geography, the joint age-by-education population counts might not be readily available, especially for very specific population subgroups or for intercensal periods. Raking allows researchers to leverage the available marginal information for multiple variables without the need for this more complex joint distribution data. This significantly expands the range and number of auxiliary variables that can be feasibly incorporated into the weighting process, thereby potentially improving the quality of the adjustments. This practical advantage is a key reason for raking's widespread adoption in survey research.
D. Improved Accuracy of Survey Estimates
By effectively reducing biases stemming from nonresponse and coverage issues, and by enhancing the representativeness of the sample, raking generally contributes to the improved accuracy of survey estimates.2 Accuracy, in a statistical sense, is often conceptualized in terms of the Mean Squared Error (MSE) of an estimator, which combines both bias and variance (MSE = Bias² + Variance). Raking aims to reduce the bias component of MSE.
The extent to which raking improves accuracy is closely tied to the strength of the relationship between the auxiliary variables used for weighting and the survey outcome variables of interest.2 If the auxiliary variables are highly correlated with the survey outcomes (and/or with the propensity to respond), the adjustments made by raking are more likely to lead to a substantial reduction in bias, and thus a lower MSE.
However, while raking can be effective in reducing bias, it is also important to consider its potential impact on the variance of the estimates. The process of adjusting weights can sometimes lead to an increase in variance, particularly if some weights become extremely large or small (an issue discussed in Section VI.A). If the increase in variance is substantial, it could potentially offset some or all of the gains achieved from bias reduction, meaning the overall impact on MSE (and thus on "accuracy" in its comprehensive statistical sense) might be less favorable or even negative in some instances.5 Therefore, while improved accuracy through bias reduction is a primary goal and a frequent outcome of raking, it is not an automatic guarantee. A careful balance must be struck, and the impact on both bias and variance should ideally be considered when evaluating the effectiveness of a raking procedure.
VI. Limitations, Challenges, and Considerations in Raking
A. Potential for Extreme Weights and Impact on Variance (Design Effect)
A notable challenge associated with the raking procedure is its potential to produce extreme weights for certain survey respondents. This occurs when some cases are assigned very large weights while others receive very small weights.5 Such situations often arise when the sample contains very few respondents from a particular subgroup that constitutes a significant proportion of the target population, or when there are substantial discrepancies between the initial (unweighted) sample distributions and the target population marginals that the raking algorithm is attempting to match.
Extreme weights are problematic primarily because they can significantly increase the variance of the survey estimates. This increased variance translates into wider confidence intervals around the estimates and reduced statistical precision, making it more difficult to detect statistically significant differences or to make precise statements about population parameters.5 Respondents who are assigned very large weights can exert an undue influence on the final survey estimates, meaning that the characteristics or responses of a small number of individuals can disproportionately affect the overall results. Conversely, respondents with very small weights contribute minimally to the estimates, effectively diminishing their voice in the survey findings.5
The "design effect" (DEFF) is a commonly used metric to quantify this increase in variance due to complex sampling designs and weighting adjustments like raking. It is typically calculated as the ratio of the variance of an estimate under the actual (weighted) design to the variance that would have been obtained from a simple random sample of the same size. Mathematically, the weighted variance can be expressed as the unweighted variance multiplied by the DEFF (Weighted Variance = Unweighted Variance * DEFF).14 A DEFF substantially greater than 1.0 indicates an inflation of variance and a loss of precision due to the weighting.
To mitigate the problem of extreme weights, practitioners sometimes employ techniques such as weight trimming or capping. These methods involve setting predetermined limits (minimum and/or maximum values) for the weights. Weights that fall outside these limits are then adjusted to the boundary values.5 However, while trimming can help control the variance inflation, it is not without its own drawbacks. The act of trimming weights can reintroduce some degree of bias into the estimates (as the weights no longer perfectly sum to the control totals) or may affect the convergence properties of the raking algorithm if trimming is done iteratively.5 Some advanced SAS macros, for instance, offer options for integrating weight trimming directly into the raking iterations.5
The issue of extreme weights highlights a fundamental tension in survey weighting: the effort to reduce bias by forcing the sample to align closely with known population totals can inadvertently lead to an inflation of variance. This necessitates careful diagnostic checks of the weight distribution after raking and a considered judgment about the potential trade-off between bias reduction and variance increase.
B. Convergence Issues (Slow or Non-convergence, Factors Affecting It)
The iterative nature of the raking algorithm, while powerful, can sometimes lead to convergence problems. The algorithm may take an unexpectedly large number of iterations to converge to a stable solution (slow convergence), or, in some circumstances, it may fail to converge at all within a practical timeframe or to the desired level of precision.8
Several factors can contribute to these convergence difficulties 11:
●	Complexity of the Raking Scheme: A large number of auxiliary variables used for raking, or a large number of categories within these variables, can significantly increase the computational burden and slow down the convergence process.
●	Sparsity of Data: Small sample sizes in some of the marginal categories being adjusted can be problematic. If a particular category that has a non-zero population target is represented by very few (or no) sample cases, the algorithm may struggle to find appropriate weight adjustments. Some sources suggest that categories with fewer than 5% of the total sample cases may particularly hinder convergence.11
●	Initial Discrepancies: Large initial differences between the unweighted sample proportions and the target population proportions for the control variables can lead to a greater number of iterations being required to achieve convergence.
●	Over-Constraint: Imposing too many marginal constraints relative to the effective sample size can make it difficult for the algorithm to find a solution that satisfies all constraints simultaneously.
●	Inconsistent or Contradictory Control Totals: This is a critical factor. If the provided population control totals are internally inconsistent (e.g., the sum of marginal totals for different variables implies different overall population sizes, or the combination of marginals is logically impossible given the sample structure), the algorithm may fail to converge.7 For example, if one marginal total indicates 50% of the population is male, while another set of marginals (perhaps from a different data source or subject to rounding) implies through other constraints that only 48% can be male, convergence might become impossible.
●	Empty Cells: The presence of "structural" empty cells in the underlying cross-classification of raking variables (cells that are empty in the sample and are also expected to be zero or near-zero in the population) can sometimes affect convergence, although raking is generally more robust to this than cell weighting.13
To address or mitigate convergence issues, practitioners may employ several strategies. These include setting a reasonable upper limit on the number of iterations, relaxing the convergence criteria (i.e., accepting a slightly larger tolerance for deviations from control totals), collapsing sparse categories of raking variables into larger, more stable groups, or, most importantly, carefully re-examining the consistency and accuracy of the population control totals themselves.11
Convergence problems should not be viewed merely as a technical inconvenience. They can often serve as a diagnostic indicator of more fundamental issues with the chosen raking scheme, the quality or consistency of the control data, or an over-specification of the weighting model. In such cases, a substantive review of the raking setup is required, rather than solely relying on algorithmic adjustments to force convergence.
C. Sensitivity to Choice of Control Variables
The effectiveness and validity of the raking procedure are heavily dependent on the judicious selection of the auxiliary (control) variables that are used to align the sample with the population.2 The choice of these variables is not a trivial step but a critical analytical decision that can significantly impact the quality of the final survey estimates.
Ideally, the variables chosen for raking should possess several key characteristics 2:
1.	Correlation with Key Survey Outcomes: The auxiliary variables should be substantially correlated with the primary variables of interest in the survey. If the raking variables are strong predictors of the survey outcomes, adjusting the sample to match the population on these variables is more likely to reduce bias in the estimates of those outcomes.
2.	Relationship to Nonresponse or Coverage Mechanisms: The auxiliary variables should also be related to the likelihood of unit nonresponse or to any known coverage deficiencies in the sampling frame. If, for example, younger individuals are less likely to respond to a survey, including age as a raking variable can help correct for the resulting imbalance.
3.	Accurate Measurement: The auxiliary variables must be accurately measured in both the survey sample and the external data source from which the population control totals are derived. Discrepancies in measurement or definition can lead to erroneous adjustments.
4.	Availability of Reliable Population Totals: Accurate and up-to-date population control totals must be available for the chosen auxiliary variables.
Using inappropriate control variables (e.g., variables that are weakly related to survey outcomes or nonresponse patterns) or employing control variables that are poorly measured or for which population totals are inaccurate may not only fail to improve the survey estimates but could potentially introduce new biases or unnecessarily inflate the variance of the estimates.
The selection of control variables, therefore, requires more than just statistical consideration; it demands subject-matter expertise and a thorough understanding of the survey's specific objectives, its target population, and potential sources of error. Simply raking on a standard set of available demographic variables without careful consideration of their relevance to the survey's core measures or the specific nonresponse patterns observed might be an ineffective exercise. In worse-case scenarios, if the chosen variables are not appropriate or if they interact in unexpected ways, raking could even distort underlying relationships in the data. This underscores the importance of a theoretically informed and empirically guided approach to selecting control variables.
D. Conditions Leading to Biased or Inefficient Results
While raking is a powerful tool for improving survey estimates, certain conditions can lead to it producing biased or inefficient results. Understanding these conditions is crucial for its appropriate application.
●	Ignoring Joint Distributions (When Relevant and Available): Standard raking procedures adjust sample weights to match the marginal distributions of auxiliary variables. If reliable information on the joint distribution of these variables in the population is available (e.g., population counts for age-by-gender cells) but is not used (perhaps because only marginal raking is performed), this omission can be a source of bias, especially if strong interactions exist between the auxiliary variables that are not adequately captured by sequential marginal adjustments.14 However, it is important to note that raking is often chosen precisely because such detailed joint population information is unavailable, and its strength lies in its ability to operate with only marginal data.
●	Measurement Error in Auxiliary Variables or Control Totals: The accuracy of raking is fundamentally dependent on the accuracy of its inputs. If the auxiliary variables used for raking are measured with error in the survey sample, or if the population control totals obtained from external sources are themselves inaccurate or outdated, the raking process can propagate these errors, leading to biased final estimates. The adage "garbage in, garbage out" applies directly.
●	Inefficiency Due to Weak Correlation between Auxiliary and Target Variables: The statistical efficiency gained from raking (i.e., reduction in variance for a given level of bias) is closely linked to the strength of the (linear) correlation between the auxiliary variables used for adjustment and the true target variables of interest in the survey. If this correlation is weak, the raking adjustments may do little to improve the precision of the estimates for the target variables, and the resulting estimators can be inefficient.15 This issue has been particularly noted in the context of generalized raking estimators applied to complex data, such as Electronic Health Records (EHR), where error-prone versions of target variables are used as auxiliaries. If, for example, a misclassified event indicator (from EHR data) is used to form an auxiliary variable, and this misclassified indicator has a low correlation with the true event status, the raking estimator based on it may be inefficient.15
●	Violation of the Missing at Random (MAR) Assumption: As discussed earlier, for raking to effectively correct for nonresponse bias concerning a specific survey variable, the nonresponse mechanism for that variable should ideally be Missing at Random (MAR) conditional on the set of raking variables. If nonresponse is actually Missing Not At Random (MNAR) – meaning the probability of nonresponse depends on the unobserved value of the survey variable itself, even after accounting for the raking variables – then raking will likely produce biased estimates for those variables affected by this MNAR mechanism. Raking cannot correct for biases that are unrelated to the variables it is adjusting for.
Raking is a statistical tool that operates based on the available data and a set of underlying assumptions. Its success in yielding unbiased and efficient estimates is fundamentally contingent upon the quality of its inputs (accurate sample data and population control totals) and the validity of these assumptions (such as the MAR assumption for nonresponse correction and the substantive relevance of the chosen control variables). It is not a panacea capable of rectifying all forms of survey error or creating information that is not present in the data or the control totals. It cannot, for instance, correct for biases stemming from poorly worded survey questions or systematic measurement errors unrelated to the raking dimensions. An understanding of these limitations is essential for its responsible application.
VII. Raking in the Context of Calibration Estimation: A Comparative Analysis
A. Raking as a Calibration Technique
Raking is recognized as a specific and widely used technique within the broader class of statistical methods known as calibration estimation.2 Calibration estimators, in general, function by adjusting an initial set of survey weights (often the base weights derived from the sampling design) to produce a new set of weights. These revised weights are constrained to satisfy certain conditions, most commonly that the weighted sum of selected auxiliary variables in the sample must equal known population totals for those same variables.2
The overarching goal of calibration is to make the revised weights, (w_i), as "close" as possible (according to some distance function) to the original weights, (d_i), while simultaneously satisfying a set of calibration equations. These equations are typically of the form (\sum_{i \in s} w_i \mathbf{x}_i = \mathbf{X}), where (\mathbf{x}_i) is a vector of auxiliary variable values for sample unit (i), and (\mathbf{X}) is the corresponding vector of known population totals for these auxiliary variables.2 Raking achieves this calibration by iteratively adjusting the weights until they conform to the specified marginal population totals for multiple auxiliary variables.20
Situating raking within the general framework of calibration estimation is important because it allows for a more unified theoretical understanding of weight adjustment procedures. This framework, notably developed by Deville and Särndal (1992) 2, provides common principles for evaluating and comparing different weighting methods. It allows for differentiation based on aspects such as the implicit or explicit distance function being minimized to measure the "closeness" of adjusted weights to initial weights, and the specific types of constraints (e.g., marginal vs. cell-based) that a particular method is designed to handle. Raking's iterative proportional adjustment mechanism is its distinct approach to solving this constrained optimization problem.
B. Comparison with Post-stratification
Post-stratification is another common weighting technique, closely related to raking, and also considered a form of calibration. However, there are key differences in their operational mechanics and data requirements.
●	Data Requirements:
○	Post-stratification: This method requires known population totals for each individual cell formed by the complete cross-classification of all auxiliary variables used in the stratification scheme.2 For example, if stratifying a sample by age (e.g., 5 categories) and gender (2 categories), post-stratification would necessitate having reliable population counts for all 10 unique age-by-gender cells.
○	Raking: In contrast, raking primarily requires only the known population marginal totals for each auxiliary variable independently.1 Using the same age/gender example, raking would need the 5 population totals for the age categories and the 2 population totals for gender, but not necessarily the 10 joint age-by-gender population counts.
●	Flexibility and Handling Multiple Variables:
○	Post-stratification: This method becomes increasingly difficult to implement, or even impossible, as the number of auxiliary variables (and their categories) increases. The number of cells in the full cross-tabulation grows exponentially with each additional variable, leading to a large number of cells, many of which may have very small or zero sample cases. This also demands highly detailed, multi-dimensional population data that is often unavailable.2
○	Raking: Raking offers greater flexibility when dealing with multiple adjustment variables simultaneously precisely because it operates on marginal totals. It is often described as a multivariate version of post-stratification, capable of adjusting to several dimensions without requiring full cross-tabular control data.2
●	Precision and Bias:
○	Post-stratification: When correctly implemented with accurate population cell counts and sufficiently large sample sizes within each cell, post-stratification can be a very effective method for reducing both bias and variance, particularly if the post-strata are homogeneous with respect to the survey variables. However, it can produce unreliable or unstable weights if sample sizes within some cells are too small.2
○	Raking: Raking can generally be used with smaller overall sample sizes than cell weighting because it avoids the problem of empty or sparsely populated cells in a high-dimensional cross-classification.14 However, by not explicitly using joint population distribution information (if such information were available and reliable), raking might theoretically introduce some bias compared to a perfectly executed post-stratification, especially if there are strong interaction effects between the auxiliary variables that are not fully captured by sequential adjustments to marginals.14 In a comparative study on a Brazilian health survey, raking demonstrated a slight advantage in some evaluation measures, such as the coefficient of variation for g-weights (weight adjustment factors), compared to post-stratification, with both methods successfully producing generalizable estimates.23
●	When Raking is Preferred:
○	Raking is generally the preferred method when reliable population totals for the full cross-classification of all desired auxiliary variables are not available, but accurate marginal totals for each variable can be obtained.2 This is a very common scenario in practice.
○	It is also favored when using a substantial number of auxiliary variables, which would make full post-stratification impractical due to an excessive number of cells, many of which would likely have very small or zero sample counts.2
○	Raking may also be chosen if there is a desire to preserve the overall association structure (e.g., odds ratios) observed in the original sample data, as its proportional adjustment mechanism tends to maintain these relationships, subject to meeting the marginal constraints.2
The decision between employing raking versus post-stratification is frequently a pragmatic one, largely dictated by the availability and granularity of the population control data and the number of auxiliary variables the researcher wishes to incorporate into the weighting scheme. Raking's principal strength lies in its adaptability to situations with more limited auxiliary information and its ability to handle a greater number of adjustment dimensions simultaneously.
C. Comparison with Other Calibration Methods (Linear, Logit)
Beyond post-stratification, raking can be compared with other specific forms of calibration estimators, such as those based on linear or logit models for the weight adjustments. As outlined in 21 and 21, three basic types of calibration are often distinguished:
●	Linear (or regression) calibration: This approach is mathematically identical to the generalized regression (GREG) estimator. It uses a linear model to relate the survey variables to the auxiliary variables, and the weight adjustments are derived based on this linear relationship.
●	Raking (multiplicative calibration): As discussed, raking employs a multiplicative model for adjusting the weights. It is particularly popular in the United States and has the important property of guaranteeing that all final adjusted weights ((g_i w_i^{(0)}), where (g_i) is the adjustment factor) will be positive, provided the initial weights are positive.21
●	Logit calibration: This method uses a logit link function to model the weight adjustments. It is more commonly used in Europe and offers the flexibility of allowing for the specification of explicit upper and lower bounds for the final weights, which can help in controlling extreme weight values.
A distinct advantage of raking, when compared to unconstrained linear calibration, is its inherent guarantee of producing positive weights.21 Negative weights, which can sometimes arise from unconstrained linear calibration, are generally undesirable in practice as they are difficult to interpret and can lead to nonsensical estimates for certain parameters (e.g., negative estimates of counts or proportions).
While these different calibration methods have distinct mathematical formulations and properties for the resulting weights, it has been observed that the choice between them may sometimes have less of an impact on the final survey estimates than the careful selection of the auxiliary variables themselves.21 The quality and relevance of the auxiliary information used for calibration are often the more critical determinants of the success of the weighting adjustment. Nevertheless, properties like guaranteed positivity (for raking) or explicit weight bounding (for logit calibration) can be important practical considerations in choosing a specific calibration method.
Table 1: Comparison of Raking, Post-stratification, and Other Calibration Estimators
Feature	Raking (Iterative Proportional Fitting)	Post-stratification (Cell Weighting)	General Linear Calibration (e.g., GREG)	General Logit Calibration	
Primary Data Requirement (Control Totals)	Marginal totals for each auxiliary variable	Cell totals for the full cross-classification of auxiliary variables	Totals for each auxiliary variable (can be continuous)	Totals for each auxiliary variable (can be continuous)	
Handling of Multiple Variables	Flexible; well-suited for many variables	Becomes impractical with many variables due to numerous cells	Flexible; handles multiple variables through a regression model	Flexible; handles multiple variables	
Weight Properties (e.g., positivity, bounds)	Guarantees positive weights (if initial weights are positive)	Weights are positive if cell counts are positive	Weights can be negative unless constrained	Allows explicit upper and lower bounds for weights	
Iterative Nature	Yes, iterative proportional adjustments	No, typically a one-step adjustment within cells	No, typically a direct calculation based on a model	Often involves iteration to solve for bounded weights	
Flexibility with Small Samples in Cells	More robust; does not require counts for all cells	Can be unstable or impossible if cell sample sizes are too small	Can be applied, but model estimates may be unstable	Can be applied, but model estimates may be unstable	
Use of Joint Distribution Information	Implicitly uses some through iteration, but primarily marginals	Explicitly uses full joint distribution (cell totals)	Uses relationships via a model, not direct joint totals	Uses relationships via a model, not direct joint totals	
Typical Use Case Scenario	Marginals known, many variables, detailed cell data unavailable	Full cell population counts known, fewer variables, cells well-populated	Auxiliary variables available, linear relationship assumed	Desire to control weight range explicitly	
This table provides a comparative overview, highlighting how raking's reliance on marginal totals and its iterative nature offer a practical approach when detailed cross-classified population data are scarce or when numerous adjustment variables are employed.
VIII. Software and Practical Implementation of Raking Procedures
A. Overview of Software Packages (R, SAS, Python, Stata) and Specific Libraries/Macros
The widespread adoption and practical importance of raking (Iterative Proportional Fitting) in survey research are reflected in the availability of dedicated tools and functions across all major statistical software packages.
●	R:
○	The survey package, authored by Thomas Lumley, is a comprehensive toolkit for survey analysis in R. It includes the powerful calibrate() function, which can perform various types of calibration, including raking (by specifying calfun="raking" or using the alias rake()). It also offers the postStratify() function for post-stratification.21 These functions are well-integrated with the survey design objects used by the package, facilitating complex survey data analysis.
○	Additionally, specialized R code for performing two-, three-, and even four-dimensional IPF has been developed by researchers like Nels Tomlinson and Eddie Hunsinger, often made available with supporting documentation and examples.27
●	SAS:
○	SAS users have access to various macros specifically designed for raking. For instance5 and 11 describe an enhanced SAS raking macro that incorporates features for diagnosing and remedying convergence problems, as well as options for weight trimming during the iterative process. Such diagnostic capabilities are crucial for complex raking schemes.
○	A SAS macro for two-dimensional IPF, developed by Webb Sprague, has also been noted as a useful tool for practitioners.27
●	Python:
○	For Python users, the ipfn library, developed by Damien Forthomme, stands out as a significant contribution. It implements N-dimensional IPF and offers versions that leverage the efficiencies of NumPy and pandas for data handling and computation.27
○	The development of ipfn and discussions within the scientific Python community regarding the potential inclusion of IPF functionalities into core libraries like SciPy (specifically scipy.stats) or statsmodels underscore the growing demand for robust and well-maintained IPF tools in the Python ecosystem.28 Statsmodels, for example, has an open issue related to finding the "closest matrix with given marginals," a problem directly addressed by IPF.28
●	Stata:
○	Stata users can utilize the user-written ipfweight module. This module is specifically designed to perform iterative proportional fitting (raking) to adjust survey sampling weights to match known population margins. It provides options for specifying convergence tolerance, maximum number of iterations, and thresholds for controlling the range of weighting factors.12
○	Stata's extensive suite of svy commands for survey data analysis can readily use weights created by procedures like raking. Raking is often discussed as a preliminary step to generate appropriate post-stratification or calibration weights for subsequent analysis with these svy commands.29
●	Other Software:
○	Beyond these major packages, IPF routines have also been implemented in other environments. For example, an MS Excel macro for performing IPF on two-dimensional tables, developed by Paul Norman, has been mentioned 27, catering to users who may prefer spreadsheet-based tools for simpler applications.
The availability of these diverse software implementations signifies the broad acceptance and practical necessity of raking in the field of survey research. The ongoing development efforts and community discussions, particularly in the Python space, indicate that raking remains a relevant and evolving technique, with continued interest in enhancing its accessibility and functionality for data analysts.
B. Practical Guidance for Implementing Raking
Effectively implementing raking involves more than just executing an algorithm; it requires careful planning, data preparation, diligent monitoring, and thorough validation. The following practical guidelines can help ensure a robust and successful raking procedure:
●	Variable Selection: The choice of auxiliary variables is paramount. Select variables that are:
○	Accurately measured in both the survey sample and the source of population control totals.
○	Strongly related to the key survey outcome variables and/or to the mechanisms of nonresponse or undercoverage.2
○	Available with reliable and up-to-date population control totals.
●	Data Preparation:
○	Ensure that the population control totals are internally consistent. For example, the sum of marginal totals across different dimensions should ideally reconcile to the same overall grand total.7
○	Address issues of zero seed cells (initial sample counts in the cross-classification of raking variables) or zero marginal control totals. This might involve collapsing categories to ensure sufficient sample sizes or substituting very small positive values to allow the algorithm to proceed, though these decisions should be made judiciously and documented.7
●	Convergence Monitoring:
○	Set appropriate convergence criteria (tolerance levels) that define how closely the weighted sample marginals must match the population targets.
○	Specify a maximum number of iterations to prevent excessively long runtimes or to handle cases of non-convergence.11
○	Actively monitor the convergence process. Many software tools provide diagnostics on the iteration history, which can help identify variables or categories that are converging slowly or causing problems.
●	Weight Diagnostics:
○	After the raking process has converged, thoroughly examine the distribution of the final weights.
○	Check for the presence of extreme weights (both very large and very small). Calculate summary statistics of the weight distribution (e.g., mean, median, minimum, maximum, coefficient of variation).
○	Compute the design effect (DEFF) due to weighting to understand the extent of variance inflation.5
○	If extreme weights are problematic, consider strategies like weight trimming (capping weights at certain thresholds). However, be mindful that trimming can reintroduce some bias or affect the exact matching to control totals, so it involves a trade-off.5
●	Validation:
○	Validate the results of the raking procedure. At a minimum, confirm that the weighted sample marginals for the control variables closely match the target population totals.
○	If possible, compare weighted estimates for other variables (not used in raking but for which external benchmarks might exist) to assess the broader impact of the weighting.3
●	Documentation: Maintain comprehensive documentation of the entire raking process. This should include:
○	The specific auxiliary variables used.
○	The sources and reference dates of the population control totals.
○	The convergence criteria and maximum iterations set.
○	Details of any category collapsing, handling of zero cells/marginals, or weight trimming procedures employed.
○	Summary statistics of the final weight distribution and the design effect.
The effective application of raking is itself an iterative process. It often involves not just a single run of an algorithm but also careful data scrutiny, diagnostic checks of intermediate and final results, and potentially the refinement of the weighting model (e.g., by changing variables, collapsing categories, or adjusting convergence parameters). It is a blend of statistical science and analytical judgment, requiring the analyst to be actively engaged in setting up the procedure correctly, evaluating its performance, and interpreting the results in the context of the specific survey. It should not be treated as an automated "push-button" solution.
Table 2: Overview of Software Packages and Functions for Raking

Software Package	Key Function(s)/Macro(s)/Library	Notable Features/Considerations	
R	survey::calibrate() (rake())	Handles N-dimensions, various calibration distance functions, integrated with survey design objects, allows custom functions.	
	survey::postStratify()	Specific function for post-stratification.	
	Custom scripts (e.g., Tomlinson/Hunsinger)	May offer specific implementations for 2D, 3D, 4D IPF.	
SAS	Custom Raking Macros	Often include advanced convergence diagnostics, iterative weight trimming options, detailed output logs.5	
Python	ipfn library	N-dimensional IPF, supports NumPy and pandas for efficiency, actively discussed in scientific Python community.27	
	Potential in scipy.stats or statsmodels	Under discussion; statsmodels has related matrix adjustment functionalities.28	
Stata	ipfweight (user-written module)	Stepwise IPF, allows tolerance/iteration control, min/max weight factor thresholds, simple missing value replacement.12	
	svy command suite	Uses weights created by raking for survey analysis; supports post-stratification directly.	
This table serves as a practical reference for researchers, guiding them to appropriate tools within their preferred software environments and highlighting key capabilities relevant to implementing raking procedures.
IX. Advanced Topics and Current Research in Raking
A. Generalized Raking Estimators
The fundamental principles of raking are being extended and adapted to address more complex data challenges encountered in modern research, leading to the development of "generalized raking estimators." These estimators often arise in contexts where data are imperfect, such as when dealing with measurement error, a common issue in large observational datasets like Electronic Health Records (EHR).15
Generalized raking techniques can leverage error-prone auxiliary variables (which might be imperfect measurements of the true variables of interest, available for a large sample) to improve the efficiency of analyses conducted on a smaller, validated subsample where error-free measurements are available. A key appeal of these methods is their potential to yield consistent estimates without requiring explicit modeling of the complex error structure, which is often unknown in practice.15
However, the statistical efficiency of these generalized raking estimators, much like traditional raking, depends critically on the strength of the (linear) correlation between the auxiliary variables used for raking and the true target variable that is being estimated. In specific settings, such as time-to-event studies using EHR data where the event indicator itself is subject to misclassification, this linear correlation can be quite low. When error-prone data are used to construct auxiliary variables, and these auxiliaries are only weakly correlated with the true, error-free outcome, the resulting generalized raking estimators can be inefficient, meaning they may have larger variances than alternative approaches or fail to realize substantial gains over simpler estimators.15
Current research in this area explores methods to enhance the efficiency of generalized raking estimators in such challenging scenarios. One promising avenue involves the use of multiple imputation techniques. These techniques can be employed to create improved auxiliary variables, either by imputing the target variables themselves or by directly imputing the auxiliary variables, with the aim of strengthening their linear correlation with the true outcome and thereby improving the efficiency of the subsequent raking estimation.15 The development of generalized raking signifies an important evolution of the core raking concept, adapting it to the nuances of contemporary data environments where issues like measurement error are pervasive and require robust analytical solutions.
B. Stability of the Raking Procedure
Beyond its practical applications, recent academic research has also focused on the theoretical properties of the Iterative Proportional Fitting procedure, including rigorous investigations into its stability. The stability of a statistical algorithm refers to its behavior under small perturbations or changes in the input data. A stable algorithm is one whose output does not change drastically in response to minor variations in its inputs.
For instance, a notable contribution by Deligiannidis, de Bortoli, and Doucet (2024) establishes the quantitative uniform stability of the IPF procedure (which is also known as Sinkhorn's algorithm in other contexts, particularly in optimal transport).30 Their work demonstrates this stability with respect to perturbations of the marginal control totals, providing quantitative bounds in terms of the 1-Wasserstein metric. Such findings have theoretical implications, for example, in the context of Schrödinger bridges, which are related to entropy-regularized optimal transport problems where IPF is a key computational tool.30
This line of research into the stability and other theoretical properties of raking is crucial because it enhances the fundamental understanding of the algorithm's reliability and robustness. Demonstrating that the raking procedure is stable under certain conditions provides greater mathematical confidence in its application across diverse data scenarios. If the output of raking (the adjusted weights) is relatively insensitive to small errors or uncertainties in the input control totals (which themselves might be estimates from larger surveys or subject to minor inaccuracies), then practitioners can have more assurance in the dependability of the results obtained from raking. Such theoretical investigations complement the extensive practical use of raking by providing a firmer mathematical foundation for its application.
C. Addressing Complex Data Issues with Raking
The core iterative adjustment mechanism of raking is proving to be adaptable for addressing an expanding range of complex data issues beyond its traditional role in correcting for demographic imbalances in surveys. This demonstrates the enduring utility and flexibility of the underlying IPF principles.
●	Measurement Error: As highlighted in the discussion of generalized raking, IPF concepts are being actively applied to develop estimators that can handle scenarios where both covariates and outcome variables are subject to measurement error, a common problem in fields relying on administrative or passively collected data like EHRs.15
●	Dependent Covariate and Outcome Error: Research is specifically targeting situations where measurement errors in covariates and complex outcomes (such as failure-time data in survival analysis) are not independent but are correlated. Improved generalized raking estimators are being proposed to address these intricate error structures more effectively.15
●	Balancing in Observational Studies: There is growing interest in using IPF as a direct method for achieving covariate balance between treatment and control groups in observational studies. This application seeks to create comparable groups for causal inference by adjusting weights so that the marginal distributions of important pre-treatment covariates are similar across groups. Some research suggests that IPF may offer advantages over traditional propensity score methods, such as utilizing all available observations (thereby avoiding bias from excluding unmatched units) and potentially achieving better external balance due to perfect alignment on specified covariates.24 This represents a conceptual expansion of raking from survey representativeness to causal inference contexts.
●	Combining National and State Survey Data for Small Area Estimation: Raking ratio estimators provide a methodological approach for linking data from large-scale national surveys with smaller state-level surveys. This can be particularly useful for improving estimates for rare outcomes (e.g., drug use prevalence) at smaller geographic levels (e.g., counties). By post-stratifying using auxiliary variables known to be associated with the outcome and then using raking to calibrate cell estimates to known marginal totals, it is possible to produce more stable and reliable estimates for these small areas than could be obtained from any single survey alone.9
The application of raking principles to these increasingly complex data challenges—ranging from sophisticated measurement error models to the demands of causal inference and small area estimation—showcases the remarkable adaptability of the core iterative adjustment algorithm. It signifies a methodological evolution where raking is moving beyond its conventional role of simple demographic weighting in surveys to become a versatile component in a broader array of advanced statistical modeling and data integration techniques. This ongoing research and application ensure the continued relevance and development of IPF methods in addressing contemporary analytical problems.
X. Concluding Remarks and Best Practices
A. Summary of Key Insights
Iterative Proportional Fitting, or raking, stands as a cornerstone technique in sample survey statistics, valued for its capacity to enhance the quality and reliability of survey estimates. Its primary strength lies in its ability to reduce biases stemming from nonresponse and coverage errors by aligning the weighted sample distributions of key auxiliary variables with known population marginal totals. This alignment, in turn, improves the representativeness of the sample with respect to these characteristics, thereby bolstering the generalizability of the survey findings. A significant practical advantage contributing to its widespread applicability across diverse survey domains—from public opinion polling and market research to health surveys and demographic analysis—is the simplicity of its data requirements, needing only marginal population totals rather than more complex joint distribution data.
However, the application of raking is not without its challenges. Practitioners must be vigilant about the potential for the procedure to generate extreme weights, which can inflate the variance of estimates and reduce statistical precision, an effect quantified by the design effect. Convergence issues, where the algorithm may converge slowly or not at all, can also arise, particularly with complex raking schemes, sparse data, or inconsistent control totals. The effectiveness of raking is highly sensitive to the careful selection of appropriate auxiliary variables and the accuracy of the population control totals used as benchmarks. Furthermore, for raking to effectively address nonresponse bias, it often relies on the underlying (and often untestable) assumption that nonresponse is Missing at Random (MAR) conditional on the raking variables.
Despite these considerations, raking remains a highly pragmatic and flexible calibration technique. Its iterative nature allows for adjustment across multiple dimensions simultaneously, making it an indispensable tool in the survey statistician's repertoire.
B. Recommendations for Effective Use of Raking in Survey Practice
To maximize the benefits of raking while mitigating its potential drawbacks, survey practitioners should adhere to a set of best practices:
1.	Judicious Selection of Auxiliary Variables: The choice of variables for raking is critical. These variables should be demonstrably related to the key survey outcomes and/or to the mechanisms of nonresponse or undercoverage. Furthermore, it is essential that accurate and up-to-date population control totals are available for these selected variables.
2.	Thorough Data Preparation and Consistency Checks: Before initiating the raking process, carefully verify the consistency of population control totals (e.g., ensuring that marginals for different dimensions sum to the same grand total). Address any issues related to zero counts in crucial sample cells or zero values in marginal targets, perhaps by collapsing categories or using small replacement values, with all such decisions carefully documented.
3.	Systematic Monitoring of Convergence: Implement clear convergence criteria (tolerance levels) and set a maximum number of iterations. Actively monitor the convergence process, utilizing diagnostic outputs from software where available, to identify any variables or categories that may be causing slow convergence or instability.
4.	Diligent Examination of Resulting Weights: After convergence, conduct a thorough diagnostic examination of the final weight distribution. Assess the range of weights, identify any extreme values, and calculate measures such as the coefficient of variation of the weights and the design effect (DEFF) to understand the impact on variance.
5.	Consideration of Weight Trimming: If extreme weights are found to be problematic (e.g., leading to an unacceptably high DEFF), consider the use of weight trimming or capping procedures. However, this should be done with caution, as trimming can reintroduce some bias or affect the precise alignment with control totals. The trade-off between variance reduction and potential bias increase must be carefully weighed.
6.	Transparency in Reporting: Fully document the raking methodology employed in any research report or publication. This documentation should include a clear statement of the auxiliary variables used, the sources and reference dates of the population control totals, the convergence parameters, and details of any adjustments made for issues like zero cells or extreme weights.
7.	Acknowledgement of Limitations: Recognize that raking, while a powerful tool, is not a panacea for all survey errors. Its effectiveness is contingent upon the validity of underlying assumptions (such as MAR for nonresponse adjustment) and the quality of the input data. It cannot correct for biases stemming from poor questionnaire design, interviewer effects, or measurement errors unrelated to the raking dimensions.
In essence, the effective application of raking demands a thoughtful combination of statistical expertise, subject-matter knowledge, and meticulous attention to practical implementation details. It is an analytical process that often requires iteration and refinement, rather than being a fully automated, "push-button" solution. By adhering to these best practices, researchers can harness the power of raking to significantly improve the quality and credibility of their survey-based inferences.
Table 3: Summary of Key Advantages and Disadvantages of Raking

Aspect	Description	
Advantages		
Bias Reduction (Nonresponse/Coverage)	Adjusts sample weights to correct for imbalances due to nonresponse and coverage errors, aligning sample with population benchmarks on key auxiliary variables.2	
Improved Representativeness	Enhances the degree to which the weighted sample reflects the known characteristics of the target population, particularly for the raking variables.3	
Simplicity of Data Needs (Marginals Only)	Requires only marginal population totals for auxiliary variables, not full cross-tabulations, making it highly practical when detailed population data are scarce.1	
Flexibility with Multiple Variables	Can simultaneously adjust for multiple auxiliary variables without requiring their joint population distribution, making it suitable for complex weighting schemes.2	
Improved Accuracy (Potential)	Can lead to more accurate survey estimates by reducing bias, especially when auxiliary variables are correlated with survey outcomes and nonresponse.2	
Disadvantages/Challenges		
Potential for Extreme Weights	May produce very large or very small weights for some cases, particularly with sparse sample subgroups or large initial discrepancies.5	
Variance Inflation (Design Effect)	Extreme weights can significantly increase the variance of survey estimates, reducing precision and widening confidence intervals; quantified by the Design Effect (DEFF).5	
Convergence Issues	The iterative algorithm may converge slowly or fail to converge, especially with many variables/categories, inconsistent controls, or sparse data.8	
Sensitivity to Choice and Quality of Control Variables/Totals	Effectiveness heavily depends on the relevance and accuracy of chosen auxiliary variables and their population control totals; errors in controls are propagated.2	
Potential Bias if Joint Distributions Ignored or MAR Violated	Ignoring available joint distribution information (if strong interactions exist) or if the Missing at Random (MAR) assumption for nonresponse is violated can lead to biased estimates.14	
Requires Accurate and Consistent Population Control Totals	The entire procedure relies on the availability of high-quality, consistent, and up-to-date population benchmarks for the auxiliary variables.1	
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