Page 78 - Contributed Paper Session (CPS) - Volume 2

CPS1437 Thanyani M.

Formulating sample weight calibration as an
optimisation problem solved using linear
programming techniques
1,2
Thanyani Maremba
1 Statistics South Africa
2 University of Limpopo

Abstract
In the analysis of survey samples, the sample elements are assigned a number,
most commonly referred to as a ‘sampling weight’ which is subsequently used
to produce estimates. Calibration is a technique that adjusts these sample
weights with the aim of improving estimates. The creation of final weights is
an iterative process. In addition, the process employs weight optimisation
models using linear programming. Mathematically, calibration is a large-scale
convex optimisation problem with linear constraints. The purpose of using
optimisation methods is attempting to minimise the difference between the
weighted sample distributions of known population distributions across a set
of control variables at both the household and person-levels. Such constraints
can be embedded into a linear programming problem, with an artificially
chosen linear objective function, and solved (at least in principle) by general
linear programming method, that is, the Simplex Algorithm. While the relaxed
methods of formulating constraints are available, it is of utmost importance
that it does not distort sampling distribution. Within StatMx linear
programming was used to achieve the above objective.

Keywords
Calibration; weighting; optimisation; linear programming; generalised
regression

1. Introduction
Calibration is a technique that adjusts sample weights with the aim of
improving estimates. To adjust the sample weights, additional information is
used. Based upon Census data, population register, or other large surveys, it
is often possible to know the population total, mean or proportion for other
variables measured in the survey, as well as the values recorded for the
members of the sample. Typical examples include the proportion of males and
females, the mean number of bedrooms in a household or total income.
The variables for which this additional information is known are referred
to as auxiliary variables. The mathematical problem of calibration can be
defined informally as follows. Suppose there are some initial weights (often
referred to as design weights … , assigned to objects of a survey.

Furthermore assume that there are auxiliary variables and that for these
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