No full-text available
Request the article directly
from the author on ResearchGate.
Rescaled bootstrap for stratified multistage sampling
Abstract
In large scaled sample surveys it is common practice to employ stratified multistage designs where units are selected using simple random sampling without replacement at each stage. Variance estimation for these types of designs can be quite cumbersome to implement, particularly for non-linear estimators. Various bootstrap methods for variance estimation have been proposed, but most of these are restricted to single-stage designs or two-stage cluster designs. An extension of the rescaled bootstrap method (Rao and Wu 1988) to stratified multistage designs is proposed which can easily be extended to any number of stages. The proposed method is suitable for a wide range of reweighting techniques, including the general class of calibration estimators. A Monte Carlo simulation study was conducted to examine the performance of the proposed multistage rescaled bootstrap variance estimator.
... We study two bootstrap calibration methods proposed in survey sampling (see e.g. [1,31]) and their extension to centered calibration [37]. Our results show different bootstrap calibration techniques lead to different bootstrap asymptotic distributions of WLEs in a general semiparametric model. ...
... A standard method for calibration in a bootstrap sample is calibration to the phase I average (see e.g. [1,31]). We call this method bootstrap calibration. ...
Two-phase stratified sampling without replacement yields generally smaller
asymptotic variance of estimators than under the convenient assumption of
independence often made in practice. Motivated by the variance estimation
problem, we propose and study a nonparametric bootstrap procedure for two-phase
sampling. We establish conditional weak convergence of bootstrap inverse
probability weighted empirical processes with several variants of calibration.
Two main theoretical difficulties are the dependent observations due to
sampling without replacement, and the complex limiting processes of the linear
combinations of Brownian bridge processes. To address these issues, the
proposed bootstrap weights take the form of the product of two weights
corresponding to randomness from each phase and stratum. We apply our bootstrap
to weighted likelihood estimation and establish two $Z$-theorems for a general
semiparametric model where a nuisance parameter can be estimated either at a
regular or a non-regular rate. We show different bootstrap calibration methods
proposed in the survey sampling literature yield different bootstrap asymptotic
distributions.
... If the sampling design is a simple random sample, then random with replacement samples may be used as bootstrap replicates. For more complex sampling designs, a number of resampling methods have been developed [16][17][18][19]. For the application presented in this paper, which involves the analysis of complex sample survey data, we used the multistage rescaled bootstrap method [18]. ...
... For more complex sampling designs, a number of resampling methods have been developed [16][17][18][19]. For the application presented in this paper, which involves the analysis of complex sample survey data, we used the multistage rescaled bootstrap method [18]. ...
Estimating incidence from cross-sectional data sources is both important to the understanding of the HIV epidemic and challenging from a methodological standpoint. We develop a new incidence estimator that measures the size of the undiagnosed population and the amount of time spent undiagnosed in order to infer incidence and transmission rates. The estimator is calculated using commonly collected information on testing history and HIV status and, thus, can be deployed in many HIV surveys without additional cost. If ART biomarker status and/or viral load information is available, the estimator can be adjusted for biases in self-reported testing history. The performance of the estimator is explored in two large surveys in Kenya, where we find our point estimates to be consistent with assay-derived estimates, with much smaller standard errors.
... The two cohorts of data (one from the target present and the other from the target absent) also offer two ways of resampling. [25][26][27][28] One is to resample the composite data from the two cohorts constituting simple bootstrapping while retaining their labels or tags (target absent and target present) or resample the individual cohorts separately constituting stratified bootstrapping. 7,16 These two forms of bootstrapping are nonparametric because we are not estimating any specific parameters associated with the data. ...
Students pursuing baccalaureate degrees in electrical engineering and computer engineering are required to take a course in probability and statistics. While the course continues to be mostly conceptual, author started initiatives to introduce data analytics in this course with special emphasis on machine vision applications. Topics such as receiver operating characteristics curves and hypothesis testing are covered through examples and exercises with students having individual datasets. Continuing with this theme, bootstrapping and associated methodologies have now been introduced to facilitate interpretation of machine vision experiments. A demo created that illustrates simple, stratified, and parametric bootstrapping as a means to understand the statistics of a machine vision sensor is presented. It encompasses a number of conceptual topics such as random variables, densities, parameter estimation, chi square testing, etc. alongside data analytics offering a holistic picture of machine learning and machine vision to the undergraduate students. A demo created that illustrates simple, stratified and parametric bootstrapping as a means to understand the statistics of a machine vision sensor is presented. It encompasses a number of conceptual topics suchas random variables, densities, parameter estimation, chi square testing, etc.alongside data analytics offering a holistic picture of machine learning and machine vision to the undergraduate students. The figure provides the summary results of a receiver operating characteristics curve (ROC) analysis providing the values of the mean, standard deviation and 95% confidence interval of the mean.
... To remedy this situation, we estimated bootstrap sampling weights for a mixed sampling scheme and further evaluated our bootstrap variance using a simulation study. 38,39 The variances for parameter estimates in the models used 1000 bootstrap samples and used these variances to estimate the corresponding P values. All models were fitted using R version 2.15.3 using the nlm function in the lme package, and bootstrap variances were estimated with the coding program using the sampling package. ...
OBJECTIVES
To examine the effects of age and race on the association of apolipoprotein E (APOE) genotypes with cognitive decline in a population sample.
DESIGN
Longitudinal study of 18 years’ duration.
SETTING
Biracial urban US population sample.
PARTICIPANTS
There were a total of 5807 participants, 60% African American (AA) and 40% European American (EA).
MEASUREMENTS
A composite cognitive function based on individual tests of episodic memory, perceptual speed, and the Mini‐Mental State Examination.
RESULTS
The frequencies of APOE ε2/ε3 (14% vs 12%), ε2/ε4 (4% vs 2%), ε3/ε4 (29% vs 22%), and ε4/ε4 (4% vs 2%) genotypes were higher among AAs than EAs. After adjusting for demographic factors, the rate of decline in global cognition was twice as high among participants with the APOE ε4/ε4 genotype compared to participants with the APOE ε3/ε3 genotype (0.097 vs 0.048 SD units [SDUs] per year; P < .0001). This doubling was not different between AAs (0.091 vs 0.045 SDUs per year) and EAs (0.118 vs 0.059 SDUs per year) (Pinteraction = .63). The APOE ε3/ε4 genotype was associated with a higher rate of decline with age (Pinteraction = .021), while the APOE ε2/ε4 genotype (Pinteraction = .016) and the APOE ε2/ε3 genotype (Pinteraction = .043) were associated with a lower rate of decline with higher age. The APOE ε2/ε2 genotype was associated with a lower rate of decline in episodic memory, while the APOE ε2/ε4 was associated with a higher rate of decline in episodic memory and perceptual speed.
CONCLUSIONS
The association of the APOE genotypes with cognitive decline was not different between AAs and EAs. However, individuals with different APOE genotypes showed a lower or a higher rate of decline with age. J Am Geriatr Soc, 1–7, 2018.
... We accounted for the sample's complex multistage design by applying the NLSY79 survey weights and its clustered design with design-based estimates of the standard error. 21 We estimated standard errors by using a survey-specific bootstrap approach, 22 with the survey package in R, which has performed well in simulation. Table 1 presents characteristics of the observed sample, weighted to represent women in the US population with at least 1 child born to singleton pregnancies during the study period. ...
Objectives:
To model the hypothetical impact of preventing excessive gestational weight gain on midlife obesity and compare the estimated reduction with the US Healthy People 2020 goal of a 10% reduction of obesity prevalence in adults.
Methods:
We analyzed 3917 women with 1 to 3 pregnancies in the prospective US National Longitudinal Survey of Youth, from 1979 to 2012. We compared the estimated obesity prevalence between 2 scenarios: gestational weight gain as reported and under the scenario of a hypothetical intervention that all women with excessive gestational weight gain instead gained as recommended by the Institute of Medicine (2009).
Results:
A hypothetical intervention was associated with a significantly reduced estimated prevalence of obesity for first (3.3 percentage points; 95% confidence interval [CI] = 1.0, 5.6) and second (3.0 percentage points; 95% CI = 0.7, 5.2) births, and twice as high in Black as in White mothers, but not significant in Hispanics. The population attributable fraction was 10.7% (95% CI = 3.3%, 18.1%) in first and 9.3% (95% CI = 2.2%, 16.5%) in second births.
Conclusions:
Development of effective weight-management interventions for childbearing women could lead to meaningful reductions in long-term obesity. (Am J Public Health. Published online ahead of print July 20, 2017: e1-e7. doi:10.2105/AJPH.2017.303881).
... Since the microcensus is a sample without replacement drawn from a finite population, the "naïve" bootstrap procedure described above can not be applied in exactly this form. Instead, the "rescaled" bootstrap procedure introduced by Rao and Wu (1988) with the adjustment of using rescaled weights instead of rescaled survey data values (see Rao, Wu, and Yue 1992) is used with the additional modification of selecting bootstrap samples without replacement (see Chipperfield and Preston 2007;Preston 2009), also incorporating the stratification by region r (see Section 3.2). To be more specific, instead of drawing c bootstrap samples with replacement of the same size m r as the original sample, subsamples without replacement of size m j r = m r /2 are drawn. ...
The Austrian microcensus is the biggest sample survey of the Austrian population, itis a regionally stratied cluster sample with a rotational pattern. The sampling fractionsdier signicantly between the regions, therefore the sample size of the regions is quitehomogeneous. The primary sampling unit is the household, within each household allpersons are surveyed. The design weights are the input for the calibration on populationcounts and household forecasts. It is performed by iterative proportional tting. Untilthe third quarter of 2014 only demographic, regional and household information wereused in the weighting procedure. From the fourth quarter 2014 onwards the weightingprocess was improved by adding an additional dimension to the calibration, namely alabour status generated from administrative data and available for the whole population.Apart from that some further minor changes were introduced. This paper describes themethodological and practical issues of the microcensus weighting process and the varianceestimation applied from 2015 onwards. The new procedure was used for the rst timefor the forth quarter of 2014, published at the end of March 2015. At the same time, allprevious microcensus surveys back to 2004 were reweighted according to the new approach.
... Bootstrap for multistage sampling under without-replacement sampling of PSUs has been considered, for example, in [14,24,27,30,31,33], among others. In this section, we consider the so-called with-replacement bootstrap of PSUs (see [33]). ...
Multistage sampling is commonly used for household surveys when there exists
no sampling frame, or when the population is scattered over a wide area.
Multistage sampling usually introduces a complex dependence in the selection of
the final units, which makes asymptotic results quite difficult to prove. In
this work, we consider multistage sampling with simple random without
replacement sampling at the first stage, and with an arbitrary sampling design
for further stages. We consider coupling methods to link this sampling design
to sampling designs where the primary sampling units are selected
independently. We first generalize a method introduced by [Magyar Tud. Akad.
Mat. Kutat\'{o} Int. K\"{o}zl. 5 (1960) 361-374] to get a coupling with
multistage sampling and Bernoulli sampling at the first stage, which leads to a
central limit theorem for the Horvitz--Thompson estimator. We then introduce a
new coupling method with multistage sampling and simple random with replacement
sampling at the first stage. When the first-stage sampling fraction tends to
zero, this method is used to prove consistency of a with-replacement bootstrap
for simple random without replacement sampling at the first stage, and
consistency of bootstrap variance estimators for smooth functions of totals.
... However, the resampling procedures available in PLS-PM software assume that the data are collected via simple random sampling, such that observations are both independent and identically distributed (e.g., Kock, 2013;Ringle, Wende, & Will, 2005;Sanchez & Trinchera, 2013). When complex survey data is involved, the resampling strategy must mimic the original sampling scheme (Aidara, 2013;Antal & Tillé, 2011;Pal, 2009;Preston, 2009), and PLS resampling routines should be modified accordingly. Similarly, the new permutation approach to PLS-PM inference (cf. ...
The purpose of the present article is to take stock of a recent exchange in Organizational Research Methods between critics and proponents of partial least squares path modeling (PLS-PM). The two target articles were centered around six principal issues, namely whether PLS-PM: (a) can be truly characterized as a technique for structural equation modeling (SEM), (b) is able to correct for measurement error, (c) can be used to validate measurement models, (d) accommodates small sample sizes, (e) is able to provide null hypothesis tests for path coefficients, and (f) can be employed in an exploratory, model-building fashion. We summarize and elaborate further on the key arguments underlying the exchange, drawing from the broader methodological and statistical literature to offer additional thoughts concerning the utility of PLS-PM and ways in which the technique might be improved. We conclude with recommendations as to whether and how PLS-PM serves as a viable contender to SEM approaches for estimating and evaluating theoretical models.
... To remedy this situation, we estimated bootstrap sampling weights for mixture of two-stage sampling and population sampling schemes. 30,31 We estimated variances for parameter estimates in the BMI-specific and combined models using this resampling procedure. All models were fitted using R version 2.15.3 using the lmer function in the lme4 package, and bootstrap variances were estimated with coding program using the sampling package. ...
Genetic variation alone may not account for common chronic disease susceptibility. Rather, an interaction between genetic and environmental factors may clarify the underlying disease mechanism. Hence, we tested whether body mass index (BMI) modified the genetic association of the apolipoprotein E ε4 allele with cognitive decline. The data came from a longitudinal population-based sample of 4055 participants interviewed at 3-year intervals from 1993 to 2012. Cognitive function was assessed using a standardized global cognitive score and BMI was assessed at baseline and classified as normal, overweight, and obese. There were 1374 (34%) participants with the ε4 allele. In normal BMI participants, cognitive decline was 0.048 units/y without the ε4 allele, and increased by an additional 0.031 units/y with the ε4 allele. In overweight participants, cognitive decline was 0.038 units/y without the ε4 allele, and increased by an additional 0.026 units/y with the ε4 allele. Finally, in obese participants, cognitive decline was 0.038 units/y without the ε4 allele, and increased by an additional 0.014 units/y with the ε4 allele. The association of ε4 allele with cognitive decline was significantly lower in obese participants compared with normal BMI participants (P=0.003), thereby suggesting significant gene-environment interaction on cognitive decline.
Social and economic studies are often implemented as complex survey designs. For example, multistage, unequal probability sampling designs utilised by federal statistical agencies are typically constructed to maximise the efficiency of the target domain level estimator (e.g. indexed by geographic area) within cost constraints for survey administration. Such designs may induce dependence between the sampled units; for example, with employment of a sampling step that selects geographically indexed clusters of units. A sampling‐weighted pseudo‐posterior distribution may be used to estimate the population model on the observed sample. The dependence induced between coclustered units inflates the scale of the resulting pseudo‐posterior covariance matrix that has been shown to induce under coverage of the credibility sets. By bridging results across Bayesian model misspecification and survey sampling, we demonstrate that the scale and shape of the asymptotic distributions are different between each of the pseudo‐maximum likelihood estimate (MLE), the pseudo‐posterior and the MLE under simple random sampling. Through insights from survey‐sampling variance estimation and recent advances in computational methods, we devise a correction applied as a simple and fast postprocessing step to Markov chain Monte Carlo draws of the pseudo‐posterior distribution. This adjustment projects the pseudo‐posterior covariance matrix such that the nominal coverage is approximately achieved. We make an application to the National Survey on Drug Use and Health as a motivating example and we demonstrate the efficacy of our scale and shape projection procedure on synthetic data on several common archetypes of survey designs.
Resampling methods are a common measure to estimate the variance of a statistic of interest when data consist of nonresponse and imputation is used as compensation. Applying resampling methods usually means that subsamples are drawn from the original sample and that variance estimates are computed based on point estimators of several subsamples. However, newer resampling methods such as the rescaling bootstrap of Chipperfield and Preston [Efficient bootstrap for business surveys. Surv Methodol. 2007;33:167–172] include all elements of the original sample in the computation of its point estimator. Thus, procedures to consider imputation in resampling methods cannot be applied in the ordinary way. For such methods, modifications are necessary. This paper presents an approach applying newer resampling methods for imputed data. The Monte Carlo simulation study conducted in the paper shows that the proposed approach leads to reliable variance estimates in contrast to other modifications.
It is routine practice for survey organizations to provide replication weights as part of survey data files. These replication weights are meant to produce valid and efficient variance estimates for a variety of estimators in a simple and systematic manner. Most existing methods for constructing replication weights, however, are only valid for specific sampling designs and typically require a very large number of replicates. In this paper we first show how to produce replication weights based on the method outlined in Fay (1984) such that the resulting replication variance estimator is algebraically equivalent to the fully efficient linearization variance estimator for any given sampling design. We then propose a novel weight-calibration method to simultaneously achieve efficiency and sparsity in the sense that a small number of sets of replication weights can produce valid and efficient replication variance estimators for key population parameters. Our proposed method can be used in conjunction with existing resampling techniques for large-scale complex surveys. Validity of the proposed methods and extensions to some balanced sampling designs are also discussed. Simulation results showed that our proposed variance estimators perform very well in tracking coverage probabilities of confidence intervals. Our proposed strategies will likely have impact on how public-use survey data files are produced and how these data sets are analyzed.
According to the United Nations International Panel on Climate Change good practice guidance, an annual forest biomass carbon balance (AFCB) can be estimated by either the stock-difference (SD) or the gain–loss (GL) method. An AFCB should be accompanied by an analysis and estimation of uncertainty (EU). EUs are to be practicable and supported by sound statistical methods. Sampling and model errors both contribute to an EU. As sample size increases, the sampling error decreases but not the error due to errors in model parameters. Uncertainty in GL AFCB estimates is dominated by model-parameter errors. This study details the delta technique for obtaining an EU with the SD and the GL method applicable to the carbon in aboveground forest biomass. We employ a Brownian bridge process to annualize the uncertainty in SD AFCBs. A blend of actual and simulated data from three successive inventories are used to demonstrate the application of the delta technique to SD- and GL-derived AFCBs during the years covered by the three inventories (SD) and rescaled national wood volume harvest statistics (GL). Examples are limited to carbon in live trees with a stem diameter of 7 cm or greater. We confirm that a large contribution to the uncertainty in an AFCB comes from models used to estimate biomass. Application of the delta technique to summary statistics can significantly underestimate uncertainty as some sources of uncertainty cannot be quantified from the available information. We discuss limitations and problems with the Monte Carlo technique for quantifying uncertainty in an AFCB.
Objective
Depressive symptoms and the apolipoprotein E (APOE) ε4 allele are independent risk factors for cognitive decline. However, it is not clear whether the presence of both depressive symptoms and the APOE ε4 allele increases cognitive decline.MethodsA prospective study of a population-based sample of 4150 (70% African American and 63% women) participants 65 years and older who were interviewed at 3-year intervals was conducted. Depressive symptoms were measured using the 10-item version of the Center for Epidemiologic Studies Depression scale, with each item coded as presence or absence of a symptom. The APOE genotype was ascertained by DNA samples collected during follow-up. Cognitive function was assessed at the initial and follow-up interviews (average follow-up of 9.2 years), using a standardized global cognitive score.ResultsThere were 1405 (34%) participants with one or more copies of the APOE ε4 allele. In participants with no depressive symptoms, cognitive function decreased by 0.0412 unit per year among those with no copies and 0.0704 unit per year among those with one or more copies of the APOE ε4 allele. For each additional symptom of depression, cognitive decline increased by 0.0021 unit per year among those with no copies and 0.0051 unit per year among those with one or more copies of the APOE ε4 allele. The three-way interaction of depressive symptoms, APOE ε4 allele, and time was significant (p = .021).Conclusions
The association of depressive symptoms on cognitive decline was increased among participants with one or more copies of the APOE ε4 allele compared with those without the allele.
Various bootstrap methods for variance estimation and confidence intervals in complex survey data, where sampling is done without replacement, have been proposed in the literature. The oldest, and perhaps the most intuitively appealing, is the without-replacement bootstrap (BWO) method proposed by Gross (1980). Unfortunately, the BWO method is only applicable to very simple sampling situations. We first introduce extensions of the BWO method to more complex sampling designs. The performance of the BWO and two other bootstrap methods, the rescaling bootstrap (Rao and Wu 1988) and the mirror-match bootstrap (Sitter 1992), are then compared through a simulation study. Together these three methods encompass the various bootstrap proposals.Différentes variantes de la méthode du bootstrap ont été proposées afin d'estimer la variance et construire des intervalles de confiance dans le contexte de sondages complexes où l'échantillonnage se fait sans remise. La plus ancienne et probablement la plus naturelle est le bootstrap sans remise (BWO) proposé par Gross (1980). Malheureusement cette méthode n'est applicable qu'à des plans d'échantillonnage très simples. Nous proposons une généralisation de la méthode BWO à des plans d'échantillonnage plus complexes. Cette nouvelle méthode et deux autres variantes du bootstrap, proposées respectivement par Rao et Wu (1988) et Sitter (1992), sont comparées à l'aide de simulations. Ces trois méthodes englobent plusieurs des différentes variantes proposées.
Variance estimation techniques for nonlinear statistics, such as ratios and regression and correlation coefficients, and functionals, such as quantiles, are reviewed in the context of sampling from stratified populations. In particular, resampling methods such as the bootstrap, the jackknife, and balanced repeated replication are compared with the traditional linearization method for nonlinear statistics and a method based on Woodruff's confidence intervals for the quantiles. Results of empirical studies are presented on the bias and stability of these variance estimators and on confidence-interval coverage probabilities and lengths.Nous réexaminons dans le contexte d'échantillons tirés de populations stratifiées les techniques d'estimation de la variance pour statistiques non linéaires, telles que quotients, coefficients de régression et de corrélation, de měme que pour fonctionnelles, telles que quantiles. Nous comparons en particulier des méthodes de rééchantillonnages comme les méthodes d'auto-amorçage, de Quenouille-Tukey et de répliques équilibrées répétées avec la méthode traditionnelle de linéarisation pour les statistiques non linéaires et une méthode basée sur des intervalles de confiance de type Woodruff pour les quantiles. Nous présentons les résultats d'études expérimentales sur le biais et la stabilité de ces estimateurs de variance de même que sur les probabilités de recouvrement et les longueurs des intervalles de confiance construits à partir de ces derniers.
This paper is about the asymptotic distribution of linear combinations of stratum means in stratified sampling, with and without replacement. Both the number of strata and their size is arbitrary. Lindeberg conditions are shown to guarantee asymptotic normality and consistency of variance estimators. The same conditions also guarantee the validity of the bootstrap approximation for the distribution of the $t$-statistic. Via a bound on the Mallows distance, situations will be identified in which the bootstrap approximation works even though the normal approximation fails. Without proper scaling, the naive bootstrap fails.
In the last decade, calibration estimation has developed into an important field of research in survey sampling. Calibration is now an important methodological instrument in the production of statistics. Several national statistical agencies have developed software designed to compute calibrated weights based on auxiliary information available in population registers and other sources.
This paper reviews some recent progress and offers some new perspectives. Calibration estimation can be used to advantage in a range of different survey conditions. This paper examines several situations, including estimation for domains in one-phase sampling, estimation for two-phase sampling, and estimation for two-stage sampling with integrated weighting. Typical of those situations is complex auxiliary information, a term that we use for information made up of several components. An example occurs when a two-stage sample survey has information both for units and for clusters of units, or when estimation for domains relies on information from different parts of the population.
Complex auxiliary information opens up more than one way of computing the final calibrated weights to be used in estimation. They may be computed in a single step or in two or more successive steps. Depending on the approach, the resulting estimates do differ to some degree. All significant parts of the total information should be reflected in the final weights. The effectiveness of the complex information is mirrored by the variance of the resulting calibration estimator. Its exact variance is not presentable in simple form. Close approximation is possible via the corresponding linearized statistic. We define and use automated linearization as a shortcut in finding the linearized statistic. Its variance is easy to state, to interpret and to estimate. The variance components are expressed in terms of residuals, similar to those of standard regression theory. Visual inspection of the residuals reveals how the different components of the complex auxiliary information interact and work together toward reducing the variance.
Depuis une dizaine d'années, l'estimation par le calage occupe un rôole important dans la théorie et la pratique des enquêetes par sondage. Cet article survole quelques développements importants en ce domaine et en présente quelques aspects nouveaux. L'estimation par le calage est avantageuse dans différents contextes. C'en est ainsi pour les trois types de sondage abordés dans cet article: l'estimation pour des sous-populations (domaines) pour unéchantillonnage en une seule phase, l'estimation pour l'échantillonnage en deux phases et l'estimation pour l'échantillonnage à deux degrés avec une pondération intégrée.
Dans le cadre de ces exemples, l'information auxiliaire est typiquement d'une certaine complexité, en ce sens qu'elle peut comporter plusieurs composantes. Cette structure polyvalente se refl ète dans le calcul des poids de calage. Par exemple, pour un sondage prévoyant un échantillonnage à deux degrés, on peut disposer et d'information auprès des unités primaires et d'information auprès des unités secondaires. Ainsi, lors du calage, il convient de profiter, simultanément et de façon efficace, des deux types d'information.
L'information auxiliaire complexe permettra, dans nos exemples, plus d'une faç on d'effectuer le calage. On peut calculer les poids par un calage direct, sur l'ensemble de l'information, ou bien, le calage peut se faire en deuxétapes, dont la première se sert d'une partie de l'information pour arriverà des poids préliminaires qu'on utilise ensuite dans un calcul de poids finaux. Pour connaïitre l'efficacité des différents estimateurs par calage, une évaluation de leurs variances respectives s'impose.
A cause de la nature non-linéaire d'un estimateur par calage, sa variance ne possède pas une forme simple et explicite. On procède à une linéarisation de l'estimateur. Normalement, c'est une procédure fastidieuse, comportant un développement en série de Taylor avec une évaluation d'un nombre de dérivées partielles. Pour les fins de cet article, il faut trouver la forme linéarisée d'un bon nombre de différents estimateurs par calage. Pour cette raison, nous suivons une procédure simplifiée, la “linéarisation automatisée”, qui amène rapidement au résultat appropriée.
Suite à la linéarisation d'un estimateur par calage, il est facile d'obtenir une proche approximationà la variance. Dans plusieurs de nos exemples, la variance de l'estimateur par calage se présente comme une somme de deux composantes, chacune donnée en fonction de certains résidus de régression ou de régression généralisée. Nous montrons comment une inspection visuelle de ces résidus fournit des clefs importantes pour identifier et interpréter les sources de la variabilité.