Multiway Sample Allocation

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The contents related to Multiway Sample Allocation are shown in the following sections:


R-package Multiway Sample Allocation allows to allocate the sample of multi-way stratified sampling design.

The multi-way stratification is achieved by combining the variables defining the marginal cells of a multi-way contingency table. In the context of the sampling theory the marginal cells are identified by the categories of the variables defining the interest domain partitions being independent of each other. For example, the stratification obtained by combining the categories of two partitions into domains of interest defines a two-way stratification.
A partition is denoted as independent or marginal when the domains cannot be got as aggregation of other domains defined in other partitions. The allocation of the sample in the dependent domains is done by aggregating the sample sizes of the marginal domains. In case of multi-way stratified designs, the sample size is fixed for each stratum. In case of incomplete stratified sampling design the sample size is fixed for each domain of interest.

The Multiway.Sample.Allocation package defines the allocation of the sample in accordance with the precision constraints of the estimates for:

  • different parameters of interest (total) (multivariate problem) at sub-population or reference domain level (multi-domain issue);
  • one stage stratified sampling designs simple or with varying inclusion probabilities variables within the multi-way strata, in which the sample size is fixed at the stratum level; incomplete stratified sampling designs in which the sample size is fixed at the domain level (except for rounding to the above/below integer);
  • varying inclusion probability designs where the sample size is fixed (approximately) at the domain level (the performance of the allocation process subject to computational constraints associated with the population size and the number of the estimates to be considered).

The algorithm performing the allocation (Falorsi and Righi, 2015) is an extension of the Chromy (1987) and Bethel (1989) algorithm. The algorithm solves an optimization problem formalized according to a general expression of the variance of the estimates (Falorsi and Righi, 2015) depending on:

  • the superpopulation model used to define input parameters (mean and variance of the variable defining the parameters of interest);
  • the implemented sampling design.

The main output of the package is the inclusion probability of the population units.


Status: validated
Author: Istat
Licence: EUPL-1.1
GSBPM code: 2.4 Design frame and sample
Programming language: R
Keywords: sample allocation, multi-way stratification, incomplete stratified sampling design
Contact: name: Paolo Righi

Software and documentation


R (version ≥ 3.1.1), R-package MASS.


Copyright 2016 Istat

Licensed under the European Union Public Licence (EUPL), version 1.1 or subsequent. You may not use this work except in compliance with the Licence. You may obtain a copy of the Licence at: Unless required by applicable law or agreed to in writing, software distributed under the Licence is distributed on an “AS IS” basis, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the Licence for the specific language governing permissions and limitations under the Licence.


Istat assumes no responsibility for the results arising from use of the instrument that is inconsistent with the methodological guidance contained in the documentation available.


Version 1.0 – Windows binaries

Version 1.0 – Package source


Reference manual – Multiway Sample Allocation v. 1.0


Last edit: 19 November 2020