Optimum Strata Boundaries Research Articles

Constructing efficient strata boundaries in stratified sampling using survey cost

For maximum precision in population parameter estimation under the Stratified sampling design, the optimum strata boundaries (OSB) could be constructed based on a continuous study variable rather than a set of categorical variables. If constructed optimally, the OSB results in homogenous units within each stratum leading to optimal stratum sample sizes (OSS) as well. The OSB and OSS may not remain optimum if the problem is considered in terms of a fixed total sample size, especially when a survey design involves a fixed budget. This article suggests a methodology for computing the OSB and OSS when the per unit stratum measurement costs for the survey or its probability density function are known. To plan for such a stratified survey, we demonstrate a design-based stratification empirically by using Wave 18 of the HILDA Survey general release dataset where we estimate the mean level of Gamma-distributed annual total disposable income in Australia, which could potentially be an important variable for policy decision-making. We also provide numerical illustrations for hypothetical study variables that follow exponential and right-triangular distributions respectively. The findings indicate that the suggested method is satisfactory in the sense that it is either more efficient or relatively comparable with other methods aimed at improving the accuracy of population parameter estimates. The proposed technique has been implemented in the updated stratifyR package.

Methods of Stratification for a Generalised Auxiliary Variable Optimum Allocation

In stratified sampling, ever since Dalenius [1] undertook the problem of optimum stratification, the research in the area has been progressing in various perspectives and dimensions till date. Amidst the multifaceted developments in the trend of the research, consideration of the topic by taking into account various aspects such as different sample selection methods and allocations, study variable based stratification, auxiliary variable based stratification, superpopulation models, extension to two study variables for a single auxiliary variable, extension to two stratification variables for a single study variable etc., are a few noteworthy ones. However, with regard to considering optimum stratification of heteroscedastic populations, as live populations are generally heteroscedastic, it was Gupt and Ahamed [2,3] who considered the problem for a few allocations under a heteroscedastic regression superpopulation (HRS) model. As a sequel to the work of the authors, in this paper, the problem of optimum stratification for an objective variable y based on a concomitant variable x under the HRS model is considered for an allocation proposed by Gupt [4,5] and termed as Generalised Auxiliary Variable Optimum Allocation (GAVOA). Methods of stratification in the form of equations and approximate solutions to the equations which stratify populations at optimum strata boundaries (OSB) and approximately optimum strata boundaries (AOSB) respectively are obtained. Mathematical analysis is used in minimizing sampling variance of the estimator of population mean and deriving all the proposed methods of stratification. The proposed equations divide heteroscedastic populations, symmetrical or moderately skewed or highly skewed, at OSB, but, the equations are implicit in nature and not easy in solving. Therefore, a few methods of finding AOSB are deduced from the equations through analytically justified steps of approximation. The methods may provide practically feasible solutions in survey planning in stratifying heteroscedastic population of any level of heteroscedasticity and the work may contribute, to some extent, theoretically in the research area. The methods are empirically examined in a few generated heteroscedastic data of varied shapes with some assumed levels of heteroscedasticity and found to perform with high efficiency. The proposed methods of stratification are restricted to the particular allocation used.

Choice of Strata Boundaries for Allocation Proportional to Stratum Cluster Totals in Stratified Cluster Sampling

In survey planning, sometimes, there arises situation to use cluster sampling because of nature of spatial relationship between elements of population or physical feature of land over which elements are dispersed or unavailability of reliable list of elements. At the same time, there requires technique and strategy for ensuring precision of the sample in representing the parent population. Although several theoretical cum practical works have been done in cluster sampling, stratified sampling and stratified cluster sampling, so far, the problem of stratified cluster sampling for a study variable based on an auxiliary variable, which is required in practice, has never been approached. For the first time, this paper deals with the problem of optimum stratification of population of clusters in cluster sampling with clusters of equal size of a characteristic y under study based on highly correlated concomitant variable x for allocation proportional to stratum cluster totals under a super population model. Equations giving optimum strata boundaries (OSB) for dividing population, in which sampling unit of the population is a cluster, are obtained by minimising sampling variance of the estimator of population mean. As the equations are implicit in nature, a few methods of finding approximately optimum strata boundaries (AOSB) are deduced from the equations giving OSB. In deriving the equations, mathematical tools of calculus and algebra are used in addition to statistical methods of finding conditional expectation of variance. All the proposed methods of stratification are empirically examined by illustrating in live data, population of villages in Lunglei and Serchhip districts of Mizoram State, India, and found to perform efficiently in stratifying the population. The proposed methods may provide practically feasible solution in planning socio-economic survey.

StratifyR: An R Package for optimal stratification and sample allocation for univariate populations

SummaryThis R package determines optimal stratification of univariate populations under stratified sampling designs using a parametric‐based method. It determines the optimum strata boundaries (OSB), optimum sample sizes (OSS) and multiple other quantities for the study variable, y, using the best‐fit probability density function of a study variable available from survey data. The method requires the parameters and other characteristics of the distribution of the study variable to be known, either from available data or from a hypothetical distribution if the data are not available. In the implementation, the problem of determining the OSB is formulated as a mathematical programming problem and solved by using a dynamic programming technique. If the data of the population (i.e. the study variable) are available to the surveyor, the method estimates its best‐fit distribution and determines the OSB and OSS under Neyman allocation, directly. When the dataset is not available, stratification is made based on the assumption that the values of the study variable, y, are available as hypothetical realisations of proxy values of y from past/recent surveys. Thus, it requires certain distributional assumptions about the study variable. At present, the package handles stratification for the populations where the study variable follows a continuous distribution: namely, Pareto, Triangular, Right‐triangular, Weibull, Gamma, Exponential, Uniform, Normal, Lognormal and Cauchy distributions. In this paper, applications of major functionalities in the package are illustrated with a number of real/simulated as well as some hypothetical populations.

A MATHEMATICAL PROGRAMMING APPROACH FOR OBTAINING OPTIMUM STRATA BOUNDARIES USING TWO AUXILIARY VARIABLES UNDER PROPORTIONAL ALLOCATION

Abstract Optimum stratification is the method of choosing the best boundaries that make the strata internally homogenous. Many authors have attempted to determine the optimum strata boundaries (OSB) when a study variable is itself a stratification variable. However, in many practical situations fetching information regarding the study variable is either difficult or sometimes not available. In such situations we find help in the variable (s) closely related to the study variable. Using auxiliary information many authors have formulated the problem as a MPP by redefining the problem as the problem of optimum strata width, and developed a solution procedure using dynamic programming technique. By using many distributions they worked out the optimum strata boundary points for the population under different allocation. In this paper, under proportional allocation OSBs are determined for the study variable using two auxiliary variables as the basis of stratification with uniform, right-triangular, exponential and lognormal frequency distribution by formulating the problems which are executed by using dynamic programming. Empirical studies are presented to illustrate the computation details of the solution procedure and its comparison with the existing literature.

Optimum strata boundaries and sample sizes in health surveys using auxiliary variables.

Using convenient stratification criteria such as geographical regions or other natural conditions like age, gender, etc., is not beneficial in order to maximize the precision of the estimates of variables of interest. Thus, one has to look for an efficient stratification design to divide the whole population into homogeneous strata that achieves higher precision in the estimation. In this paper, a procedure for determining Optimum Stratum Boundaries (OSB) and Optimum Sample Sizes (OSS) for each stratum of a variable of interest in health surveys is developed. The determination of OSB and OSS based on the study variable is not feasible in practice since the study variable is not available prior to the survey. Since many variables in health surveys are generally skewed, the proposed technique considers the readily-available auxiliary variables to determine the OSB and OSS. This stratification problem is formulated into a Mathematical Programming Problem (MPP) that seeks minimization of the variance of the estimated population parameter under Neyman allocation. It is then solved for the OSB by using a dynamic programming (DP) technique. A numerical example with a real data set of a population, aiming to estimate the Haemoglobin content in women in a national Iron Deficiency Anaemia survey, is presented to illustrate the procedure developed in this paper. Upon comparisons with other methods available in literature, results reveal that the proposed approach yields a substantial gain in efficiency over the other methods. A simulation study also reveals similar results.

Construction of Optimum Strata Boundaries for Uniform and Exponential Auxiliary Variable

Construction of Optimum Strata Boundaries for Uniform and Exponential Auxiliary Variable

Statistical investigation through stratified random sampling for apple production in Himachal Pradesh

The present study focuses on standardization of sampling technique and comparison of different types of sample allocation methods in combination with various stratification tools (optimum strata boundaries, number of strata and optimum sample size etc.) for obtaining efficient estimators of area and production of apple in Himachal Pradesh. Forth is purpose, various aspects involved in optimum stratification with reference todata collected from the selected or chardists in district Shimla, during the year 2011-12 have been analyzed. The variable "Area under Apple" was chosen as the stratification variable as it had high correlation(r=0.96) with the estimation variable "Production of Apple". Four methods of construction of strata viz., equalization of strata total, equalization of cumulative of ?f(y) , equalization of cumulative of ½{r(y)+f(y)} and equalization of cumulative of were used and their relative efficiencies for estimating total production of apple in the study district of the state have been analyzed. The critical examination of the result revealed that with the increase in number of strata from 2 to 4 and sample size from 10 to 40, equalization of cumulative of method along with Neyman allocation resulted in least variance (0.89) and maximum percentage gain in efficiency (20418.16). Thus, equalization of cumulative of method with L>2 can profitably be applied for the estimation of apple production in the study district of the state Himachal Pradesh, India.

PENENTUAN TITIK-TITIK BATAS OPTIMUM STRATA PADA PENARIKAN CONTOH ACAK BERLAPIS DENGAN PEMROGRAMAN DINAMIK (Kasus : Pengeluaran per Kapita Propinsi Jawa Timur Tahun 2008)

Optimum stratification is the method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation. In order to make the strata internally homogenous, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. This could be achieved effectively by having the distribution of the main study variable known and create strata by cutting the range of the distribution at suitable points. The problem of finding Optimum Strata Boundaries (OSB) is considered as the problem of determining Optimum Strata Widths (OSW). The problem is formulated as a Mathematical Programming Problem (MPP), which minimizes the variance of the estimated population parameter under Neyman allocation subject to the restriction that sum of the widths of all the strata is equal to the total range of the distribution. The distributions of the study variable are considered as continuous with standard normal density functions. The formulated MPPs, which turn out to be multistage decision problems, can then be solved using dynamic programming technique proposed by Buhler and Deutler (1975). After the counting process using C++ program received the width of each stratum. From these results the optimal boundary point can be determined for each stratum. For the two strata to get the optimal point on the boundary x 1 = 0.002. For the formation of three strata obtained the optimal point on the boundary x 1 = -0.546 and x 2 = 0.552. For the formation of four strata obtained optimal boundary point is x 1 = -0.869, x 2 = 0.003 and x 3 = 0.878. In forming five strata obtained optimal boundary point x 1 = -1.096, x 2 = -0.331, x 3 = 0.339 and x 4 = 1.107. The establishment of a total of six strata obtained the optimal point on the boundary x 1 = -1.267, x 2 = -0.569, x 3 = 0.005, x 4 = 0.579 and x 5 = 1.281.

Determining Optimum Strata Boundaries and Sample Sizes for Skewed Population with Log-Normal Distribution

The method of choosing the best boundaries that make strata internally homogenous as far as possible is known as optimum stratification. To achieve this, the strata should be constructed in such a way that the strata variances for the characteristic under study be as small as possible. If the frequency distribution of the study variable x is known, the optimum strata boundaries (OSB) could be obtained by cutting the range of the distribution at suitable points. If the frequency distribution of x is unknown, it may be approximated from the past experience or some prior knowledge obtained at a recent study. Many skewed populations have log-normal frequency distribution or may be assumed to follow approximately log-normal frequency distribution. In this article, the problem of finding the OSB and the optimum sample sizes within the stratum for a skewed population with log-normal distribution is studied. The problem of determining the OSB is redefined as the problem of determining optimum strata widths (OSW) and is formulated as a Nonlinear Programming Problem (NLPP) that seeks minimization of the variance of the estimated population mean under Neyman allocation subject to the constraint that the sum of the widths of all the strata is equal to the range of the distribution. The formulated NLPP turns out to be a multistage decision problem that can be solved by dynamic programming technique. A numerical example is presented to illustrate the application and computational details of the proposed method. A comparison study is conducted to investigate the efficiency of the proposed method with other stratification methods, viz., Dalenius and Hodges’ cum method, geometric method by Gunning and Horgan, and Lavallée–Hidiroglou method using Kozak’s algorithm available in the literature. The study reveals that the proposed technique is efficient in minimizing the variance of the estimate of the population mean and is useful to obtain OSB for a skewed population with log-normal frequency distribution.

Optimum Stratification of a Skewed Population

The focus of this paper is to develop a technique of solving a combined problem of determining Optimum Strata Boundaries(OSB) and Optimum Sample Size (OSS) of each stratum,when the population understudy is skewed and the study variable has a Pareto frequency distribution. The problem of determining the OSB is formulated as a Mathematical Programming Problem (MPP) which is then solved by dynamic programming technique. A numerical example is presented to illustrate the computational details of the proposed method. The proposed technique is useful to obtain OSB and OSS for a Pareto type skewed population, which minimizes the variance of the estimate of population mean.

Determining the Optimum Stratum Boundaries Using Mathematical Programming

The method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation, is known as optimum stratification. In order to make the strata internally homogeneous, the strata are constructed in such a way that the strata variances should be as small as possible for the characteristic under study. In this paper the problem of determining optimum strata boundaries (OSB) is discussed when strata are formed based on a single auxiliary variable with a varying measurement cost per units across strata. The auxiliary variable considered in the problem is a size variable that holds a common model for a whole population. The OSB are achieved effectively by assuming a suitable distribution of the auxiliary variable and creating strata by cutting the range of the distribution at optimum points. The problem of finding the OSB, which minimizes the variance of the estimated population mean under a weighted stratified balanced sampling, is formulated as a mathematical programming problem (MPP). Treating the formulated MPP as a multistage decision problem, a solution procedure using dynamic programming technique is developed. A numerical example using a hospital population data is presented to illustrate the computational details of the solution procedure. A software program coded in JAVA is written to carry out the computation. The distribution of the auxiliary variable in this example is considered to be continuous with an exponential density function.

Optimum stratification for allocation proportional to strata totals for scrambled response

The paper considers the problem of finding optimum strata boundaries to optimize the estimation of sensitive/stigmatized characters when sample sizes from different strata are allocated proportionally to strata totals of the auxiliary variable. Assuming the form of the regression of the estimation variable y on the auxiliary variable x as y = η(x)+e and the form of the conditional variance V (y|x), the rule for finding approximate optimum strata boundaries is proposed. The limiting expression of the variance of the estimator of mean in optimum stratification has also been obtained.

Minimum variance stratification for scrambled response

In this paper we have considered the problem of finding optimum strata boundaries (OSB) when the samples from different strata are selected with simple random sampling with replacement (SRSWR) and data are collected by scrambled randomized response technique on the sensitive variable. In the present paper a procedure which is applicable in more general situation is suggested for obtaining approximate optimum strata boundaries (AOSB). A limiting expression for the variance of the estimator of population mean, optimum number of strata, approximate expression for (nh) and expression for optimum cost allocated to L strata have also been suggested. The paper concludes with a numerical illustration.

Optimum stratification for scrambled response with ratio and regression methods of estimation

The paper considers the problem of optimum stratification when the information on the auxiliary variable x is used to estimate the population mean yof the sensitive study variable y using ratio and regression methods of estimation and the data are collected by scrambled randomized response technique on the sensitive character. The paper proposes cum. 3 p4(x) rule for finding approximate optimum strata boundaries (AOSB).

Determination of optimum strata boundaries for scrambled randomized response

The paper considers the problem of finding optimum strata boundaries (OSB) when the samples from different strata are selected with simple random sampling and with replacement (SRSWR) and the data are collected by scrambled randomized response technique on the sensitive character. The paper proposes a rule for finding approximately optimum strata boundaries (AOSB).

Optimum stratification for equal allocation

The problem of optimum stratification on the auxiliary variablex for equal allocation has been considered. A cum\(f\sqrt \varphi \) rule of finding approximately optimum strata boundaries (AOSB) has been suggested. A numerical investigation into the relative efficiency of equal allocation with respect to optimum allocation has also been made.

Optimum stratification with ratio and regression methods of estimation

The paper considers the problem of optimum stratification on an auxiliary variablex when the information on the auxiliary variablex is also used to estimate the population mean\(\bar Y\) using ratio or regression methods of estimation. Assuming the form of the regression of the estimation variabley on the auxiliary variablex as also the form of the conditional variance function V(y/x), the problem of determining optimum strata boundaries (OSB) is shown to be a particular case of optimum stratification on the auxiliary variable for stratified simple random sampling estimate. A numerical investigation has also been made to study the amount of gain in efficiency that can be brough about by stratifying the population.

Optimum stratification in sampling with varying probabilities

The paper considers the problem of optimum stratification on an auxiliary variablex when the units from the different strate are selected with probability proportional to the value of the auxiliary variable. Under a super-population set-up assuming the form, of the regression of the estimation variabley on the auxiliary variablex as also the form of the variance functionV(y/x), minimal equations giving optimum strata boundaries have been obtained for the Neyman allocation method. As the minimal equations cannot be solved easily, methods to find approximate solutions have been given. A numerical illustration has also been given to study the effect of optimum stratification.