A Self Documenting Programming Environment for Weighting

A Self Documenting ProgrammingEnvironment for WeightingWilfried Grossmann and Petra OfnerDepartment of Statistics and Decision Support Systems, University Vienna, A-1010 Vienna, Universitätsstrasse 5, AustriaAbstract. The paper presents a proposal for a statistical computing environment for weighting, which processes statistical data and description of data simultaneously. The developed model is of interest also for other types of computing in official statistics.Keywords. Programming environments, Weighting, Metadata1 IntroductionWeighting plays an important role in survey computing, mainly for improving quality of estimates with respect to bias and variance but also for adjusting of estimates in such way that they match controls. Various weighting algorithms have been developed, which need besides the actual data additional information about the data, in particular information about underlying populations and information about the sampling design, stored frequently separate from survey data. Hence, only data with computed weights are delivered outside the production environment. Consequently, in case of reuse of the data it is difficult to evaluate the adequacy of existing weights, in some cases even impossible.The paper presents a proposal for an environment, which supports weighting as well as structured documentation of the weighting process. Besides weighting the proposed approach seems to be useful for integration of processing and documentation in other areas of statistical computing. Section 2 discusses the requirements on the environment from the statistical point of view. In section 3 we consider a data model fulfilling the requirements outlined in section 2 and section 4 describes processing inside the environment in case of weighting.2 Statistical requirements for a weighting environmentLet be the vector of the values of a variable Y of interest in a finite population and denote by )y ,,y ,y (N 21′=K y )y ,,y ,y (ns s 2s 1′=K s y isis y w ∑= the observed values of the variable in a sample s . One of the primary goals in survey computing is calculationof a linear estimate for population totals or means. The coefficients are called weights for the observations . Various methods are known for calculation of the weights, which may be classified into the following three approaches.)Y (ˆΘis w s yThe first one is the design-based approach assuming known sampling probabilities for the observation units. Based on the sampling probabilities is πfor the observed units one can define the weights proportional to is π/1. Such weights are often called base weights valid for all variables in the survey dataset and the corresponding estimator is known as Horvitz-Thompson estimator.The second approach closer to traditional statistical modeling is the model-based approach. It assumes that the values of auxiliary variables )X ,,X ,X (X p K 21= are available for all population units prior to sampling. Based on the auxiliary information one defines a linear model for the target variable Y of the following form:[]V )Y (Var ,X Y E ==β. Using the observations one can estimate the parameter vector s y β from the sample and define an optimal estimator for the quantity of interest (cf. Valliant et al. (2000)). Usually these weights are specific for each target variable Y .A third alternative is the model-assisted approach, which tries to bridge the two alternatives by using information from the sampling design for calculation of base weights and auxiliary variables for modification of base weights. These new weights are calibrated in such sense that the weighted sums of population totals for the auxiliary variables reproduce the known population totals. Depending on the type of the auxiliary variables a number of computational procedures are available. Well known examples are generalized regression estimates (GREG), based on a universal regression model for all target variables, or calibration weights, defined by minimizing a predefined distance between base weights and the new calibrated weights (Deville and Särndal (1992)). In case of qualitative auxiliaries a rather simple calculation method known as raking can be used (cf. Kalton et al. (1998) for an introductory survey). Usually these weights are considered as universal for the whole data set.Although the model based approach seems to be more favorable from a statistical point of view, the appealing fact of the model-assisted approach is that it can be integrated into statistical databases rather easily. The property of reproduction of population totals for summary tables produced with the calibration weights is sometimes called numerical consistency (cf. also Renssen et al. (2001) for analyzing numerical consistency in connection with output databases produced from more than one survey).From the exposition it is obvious that calculation of weights presupposes the combination of data from a number of different sources: besides survey data we need also data for the sampling design and data for auxiliary variables from a population database. These data sources are often not fully compatible with respect to their structure, for example case by variates survey data may be used together with a sampling plan defined in a table of selection probabilities for strata and auxiliary variables represented in tables of marginal counts for the population. Consequently, calculation of weights needs (as usual in applied statistics) a number of tedious preprocessing steps for aligning the data according to the requirements of the algorithm. Hence, an integrated repository for all data occurring in connection with survey data, which supports preprocessing and computation, may be useful. From the data user perspective it seems important tohave access not only to the weights but also to the documentation of the weighting process.3 Data Structures for a weighting environmentIn order to fulfill the requirements stated at the end of the previous section we need a data structure which allows a unified and integrated representation of different types of data (survey data, sampling data, population data) occurring in different formats (case data, summary tables). Moreover for supporting also preprocessing we need not only representation of the data itself (first order data), but also a data representation of the description of these data (second order data or metadata). A rather simple data structure fulfilling both requirements is a so-called container structure (cf. Denk et al (2002) for a theoretical foundation). Basically a container consists of a bucket carrying first order data as a relational table and a bucket schema containing the description of the data bucket. In agreement with the main statistical data structures we discern two bucket formats: case type buckets for case level data and summary type buckets for dimensional data (tables or more general multidimensional cubes). All buckets occurring in connection with a data set are pooled together into a composite. The integrated representation necessary for a data source is schematically given in figure 3.1. Bullet nodes in the graph correspond to directories and circled nodes correspond to physical datasets with B denoting the buckets and S the corresponding bucket schemas.CompositeB B B B B BS S S S S SFigure 3.1. Composite structure for statistical dataCorresponding to the requirement analysis we discern statistical composites carrying all information about survey data and population composites for population data. Inside the statistical composite we have containers of four different types: a data container for survey data, a sampling container for sampling information, a weight container for the weights available for the datasetand a method container for the results of the calculations applied to the data (cf.section 4). The following figure shows an example of a bucket schema for a data container.Container variable Variable reference Grouping levelid (key) V1 -sex V20 education V3 1income V41Fig 3.2. Bucket schema for a data containerThe column “variable reference” in the examples refers to the container attributeslisting all variables used inside the composite together with a number important properties. Besides the usual specification for statistical variables like quantitative, qualitative or string variables it gives also a reference to the admissible ranges forthe variable. These ranges are stored in the grouping level sets and the selectedrange is indicated in the grouping level column of the bucket schema. The grouping levels support a number of predefined recoding and transformation operations for variables (cf. Papageorgiou et al. (2001) for details in a similar model). For our purpose the most important descriptive element for variables is therole specification of a variable, informing about the (statistical) meaning of the variable inside the composite. Typical roles in a data bucket may be identifying variable (key inside the bucket), auxiliary variable or observation variable. In the sampling bucket possible roles are selection probability or stratum count. Figure3.3 shows an example for a bucket in a sampling container using the qualitative variable V5 for defining the strata and for the other variables in the column headings roles are indicated.V5 V6(stratum count) V7(selection probability) Stratum 1 880 0,25Stratum 2 144 0,30Fig 3.3. Bucket for a sampling containerThe roles of variables in a weight bucket correspond to the weighting method and inside a method bucket the roles are defined according to the applied transformation. A typical role in the context of weighting may be variance of the estimate for the population total. Note that these roles are global roles inside the composite. Besides these global roles we have to consider also roles of variables in the context of compassion, for example explanatory variables in a regression model.The population composite is structured similar to the data composite: the data bucket is reserved for population data available either as a register (bucket formatcase) or marginal counts (bucket format summary). The structure bucket informs about the population structure, usually defined by stratification.The implementation of the structure could be done in different environments. In order to be fully compatible with some standard in statistical computing the prototype was implemented in the SAS environment, i.e. all containers as well as the mentioned directories are realized as SAS datasets. Such an implementation is convenient for performing statistical computing with the data but rather cumbersome with respect to data administration inside the structure.4 Transformation of compositesUsing the structure outlined in section 3 computation of weights (as any other statistical computation) can be interpreted as a sequence of transformations T producing from a number of input composites C a new output composite . Description of the transformation encompasses besides the input and output composites the definition of an operator together with specification of operator parameters, a variable list to which the operator applies and a variable list which is generated by the operator. Based on these specifications the transformation is executed by the following main processing activities:k C ,,C ,K 21)C ,,C ,C (T C k new K 21=(i) Feasibility check for the transformation(ii) Definition of a detailed computation plan(iii) Statistical computation according to the plan(iv) Generation of the output composite and documentationThe feasibility check infers whether the required computation is possible using only second order data, i.e. information contained in the bucket schemas and the description of container attributes, in particular the roles of the variables inside the composite. Checking presupposes a careful analysis of a family of transformations from a statistical point of view. In case of weighting we have to analyze the data requirements for different types of weighting. Let us consider two examples:Calculation of base weights is feasible, provided that the variables describing the sampling design in the sampling schema are available also in the data container and the grouping levels are in both containers compatible. For instance in the example given by figures 3.2 and 3.3 base weights are not feasible, because variable V5 is not part of the bucket schema in the data container.Checks for calibration weights have to take into account the type of the auxiliary variables. Consider for instance the case of two categorial auxiliaries and a desired adjustment according to the marginals with respect of these two variables (defined in the parameter specification of the transformation). Calibration is only feasible if the two auxiliaries are used in the data container in compatible form and calculation of base weights can be done in advance. From computer science point of view such checks can be treated formally using the theory of semistructured data sets. Details in the context of this model may be found in Denk et al (2002).Based on the results of the feasibility check one can define a detailed computation plan by specifying a sequence of conventional statistical algorithms, together witha data structure obtained as a view on the input composites. For example, in the above sketched case of calibration weighting the plan encompasses following steps: Calculation of base weights using data contained in the sampling container, calculation of calibration factors using data from the population composite and determination of weights in a form compatible to the structure of the survey data in the statistical composite.Execution of this plan by statistical computation and generation of the new output composite is rather straight forward, provided that the various numerical procedures are accessible.Documentation of the transformations is given in the transformation repository, which is shown schematically in figure 4.1. Trafo history informs about the applied transformation, details about composites and variables are kept in Origin Composite and Origin Variables, structural information about the operators are contained in the Operator Directory. Also these structures are implemented as SAS data sets, the outlined processing steps are realized as SAS macros.HistoryOrigin CompositesTransformationsVariablesParametersFig 4.1. Structure of the transformation repositoryReferencesDenk, M., Froeschl, K.A., Grossmann, W. (2002). Statistical Composites: A Transformation-bound Representation of Statistical Data. Submitted to Statistical and Scientific Databases 2002, Edinburgh.Deville, J.C., Särndal, C.E. (1992). Calibration estimators in survey sampling. J.American Statistical Association, 87, 376-382.Kalton, G. Flores-Cervantes I. (1998). Weighting Methods. In: New Methods in Survey Research (A. Westlake et al. eds.), 77-93. Association for Survey Computing, Chesham Bucks, UK.Papageorgiou, H. Pentaris, F. Theodorou, E. Vardaki, M. Petrakos, M. (2001). A statistical metadata model for simultaneous manipulation of both data and metadata. J. of Intelligent Information Systems 17, 169 – 192.Renssen, R.H., Kroese, A.H., Willeboordse, A.J. (2001). Aligning Estimates by Repeated Weighting. Working paper, Statistics Netherlands (Project no: RSM-70110).Valliant, R., Dorfmann, A.H., Royall, R.M. (2000). Finite Population Sampling and Inference. Wiley Series in Probability and Statistics, New York.。