An optimisation algorithm applied to the one-dimensional stratification problem
Section 6. Conclusions

Table of contents

As already mentioned, stratified sampling is a very important idea in survey sampling design, in helping to achieve improved precision of survey estimates for a given sample size or survey budget. This is particularly true for skewed or heterogeneous populations often found in business or establishment surveys. The potential gains due to stratification are strongly dependent on the delimitation of the strata and on the allocation of the sample to the strata, given a specified stratification variable and sample selection method.

The present paper presented a new optimization method for the stratification problem based on the Biased Random-Key Genetic Algorithm (BRKGA). Our approach (named BR) couples BRKGA for the definition of the stratum boundaries with the formulation for optimum sample allocation proposed in de Moura Brito et al. (2015), which is efficient with respect to computing time for large populations (large $N) .$

The results reported here for the comparison of this approach with the five competitors considered suggest that BR offers a good alternative for addressing the stratification and allocation problems in a practical situation.

Our approach can be easily generalised to cases where the stratification variable $x$ is not “measured”, but instead represents a summary of several covariates in the form of a predicted $y$ variable. The same is true for generalising to two or more numeric $x$ variables, which can be easily accomplished by changing the decoding function used to retrieve feasible solutions from the BRKGA algorithm with the R package stratbr (see de Moura Brito et al., 2017a).

Future work will focus on developing and evaluating alternative decoding procedures which may be used in BR, aiming to produce solutions with superior quality when compared to those produced using the decoding procedure considered here. This research may also focus on solving the dual problem of minimising the total sample size for a specified precision, such as did Lavallée and Hidiroglou (1988). Finally, additional empirical work may focus on considering varying sample sizes across the various study populations, as did Kozak (2004) and Gunning and Horgan (2004).

Appendix A

Table A.1
Description of the 27 populations considered in the numerical experiment
Table summary
This table displays the results of Description of the 27 populations considered in the numerical experiment. The information is grouped by Population (appearing as row headers), Description (appearing as column headers).
Population	Description
AgrMinas	Agricultural production of municipalities in Minas Gerais State, Brazil, from 2006 Agricultural Census.
AgrMinas	Stratification variable: planted area.
BeefFarms	Australian beef farms, stratified into seven industrial regions, as considered by (Chambers and Dunstan, 1986).
BeefFarms	Stratification variable: farm size.
Beta103	Simulated population generated from a Beta distribution with parameters $a = 10$ and $b = 3$ as considered by (Keskintürk and Er, 2007).
Chi5	Simulated population generated from a Chi-square distribution with $d f = 5$ as considered by (Keskintürk and Er, 2007).
Coffee	Coffee farms in the state of Paraná, Brazil, in the 1996 Agricultural Census, as considered by (de Moura Brito et al., 2015).
Coffee	Stratification variable: number of coffee trees.
CensoCO	Data from the 2012 school census in Brazil for the mid-west region.
CensoCO	Stratification variable: number of classrooms.
Debtors	Population of debtors of an Irish firm as considered by (Er, 2011).
Debtors	Stratification variable: Irish debtors’ stated liabilities.
HHinctot	Population of gross family income values (income before tax) from a Family Expenditure Survey 2001 carried out by Statistics Canada, as considered by (Er, 2011).
Iso2004	Data on net sales of 487 Turkish Industrial Enterprises out of the 500 largest enterprises in 2004, obtained by the Istanbul Industrial Chamber, as considered by (Keskintürk and Er, 2007).
Iso2004	Stratification variable: net sales.
Kozak1, Kozak3, Kozak4	Populations considered by (Kozak and Verma, 2006). Stratification variable: were generated based on following formula: $X = \exp (Z),$ where $Z$ is a realization of a normal random variable.
ME84	This data is from Särndal, Swensson and Wretman (1992) as considered by (Er, 2011).
ME84	Stratification variable: number of municipal employees in 1984.
MRTS	Population simulated from the Monthly Survey on Sales in Retail Trade from Statistics Canada, as considered by (Er, 2011).
MRTS	Stratification variable: the size measure used for Canadian retailers in the Monthly Retail Trade Survey (MRTS) carried out by Statistics Canada. This size measure is created using a combination of independent survey data and three administrative variables from the corporation tax return.
P75	Population in thousands of the 284 Swedish municipalities in 1975, as considered by (Er, 2011).
P75	Stratification variable: population in thousands.
P100e10	Population simulated from a Normal distribution with $μ = 100$ and $σ = 10$ as considered by (Keskintürk and Er, 2007).
pop1076	Population extracted from the Brazilian Annual Manufacturing Survey as considered by (de Moura Brito et al., 2017b).
pop1076	Stratification variable: number of employees.
pop1616	Population extracted from the Brazilian Annual Manufacturing Survey as considered by (de Moura Brito et al., 2017b).
pop1616	Stratification variable: number of employees.
pop2911	Population extracted from the Brazilian Annual Manufacturing Survey as considered by (de Moura Brito et al., 2017b).
pop2911	Stratification variable: number of employees.
Pop500	Population with $N = 500$ simulated from the Log-Normal Distribution $X = e^{z}$ where $Z$ is Normal with $μ = 4$ and $σ^{2} = 2 .7$ as considered by (Hedlin, 2000).
Pop800	Population with $N = 800$ simulated from the Log-Normal Distribution $X = e^{z}$ where $Z$ is Normal with $μ = 4$ and $σ^{2} = 2 .7$ as considered by (Hedlin, 2000).
REV84	Value of buildings in million Swedish Crown for the 284 Swedish municipalities in 1984, as considered by (Er, 2011).
REV84	Stratification variable: the revenues from the 1985 municipal taxation.
SugarCaneFarms	Australian sugar cane farms as considered by (Chambers and Dunstan, 1986).
SugarCaneFarms	Stratification variable: total cane harvested.
USbanks	Assets in millions of US Dollars for the large north American commercial banks, as considered by (Er, 2011).
USbanks	Stratification variable: the resources in millions of dollars of large commercial US banks.
UScities	Population in thousands for North American cities in 1940, as considered by (Er, 2011).
UScities	Stratification variable: population in thousands.
UScolleges	Numbers of students in four-year US faculties in 1952-1953 as considered by (Er, 2011).
UScolleges	Stratification variable: number of students.
Swiss	Data on Swiss municipalities in 2003, as available from the SamplingStrata package in R.
Swiss	Stratification variable: area under cultivation.

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An optimisation algorithm applied to the one-dimensional stratification problem
Section 6. Conclusions

Appendix A

References

An optimisation algorithm applied to the one-dimensional stratification problem Section 6. Conclusions

Appendix A

References

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

An optimisation algorithm applied to the one-dimensional stratification problem
Section 6. Conclusions