Optimum Allocation in Multivariate Stratified Random Sampling: A Modified Prékopa’s Approach

Numerous optimization problems arise in survey designs. The problem of obtaining an optimal (or near optimal) sampling design can be formulated and solved as a mathematical programming problem. In multivariate stratified sample surveys usually it is not possible to use the individual optimum allocations for sample sizes to various strata for one reason or another. In such situations some criterion is needed to work out an allocation which is optimum for all characteristics in some sense. Such an allocation may be called an optimum compromise allocation. This paper examines the problem of determining an optimum compromise allocation in multivariate stratified random sampling, when the population means of several characteristics are to be estimated. Formulating the problem of allocation as an all integer nonlinear programming problem, the paper develops a solution procedure using a dynamic programming technique. The compromise allocation discussed is optimal in the sense that it minimizes a weighted sum of the sampling variances of the estimates of the population means of various characteristics under study. A numerical example illustrates the solution procedure and shows how it compares with Cochran's average allocation and proportional allocation.

The problem of allocating the sample numbers to the strata in multivariate stratified surveys, where, apart from the cost involved in enumerating the selected individuals in the sample, there is an overhead cost associated with each stratum, has been formulated as a non-linear programming problem. The variances of the posterior distributions of the means of various characters are put to restraints and the total cost is minimized. The main problem is broken into subproblems for each of which the objective function turns out to be convex. When the number of subproblems happens to be large an approach has been indicated for obtaining an approximate solution by solving only a small number of subproblems.

This paper deals with the problem of allocation of sample size in stratified random sampling if there are several characters under study. Different generalized measures of dispersion are proposed and Neyman-like optimum solutions are calculated and compared.

In multivariate sampling, the major interest is on the problem of estimation of several population characteristics which often make conflicting demands on the sampling procedure. In this type of survey, the best allocation for one item may not in general be the best for another. There is the need to come up with compromise solution in a survey with many characteristics under study. This paper focuses on comparing some techniques of optimum sample allocation which are Yates/Chatterjee, Booth and Sedransk and Vector Maximum Criterion (VMC) on five sets of real life data stratified into six strata and two variates with desired variances using: (i.) Method of maximum variances with fixed n and (ii.) arbitrary fixing of variances. The stratum sample size nh among the classes are obtained to ascertain the criterion that will produce the smallest n. Based on the set of data collected and used for the empirical study it was discovered that Vector Maximum Criterion (VMC), Booth and Sedransk are superior to Yates/Chatterjee.

ABSTRACTIn multivariate stratified sample survey with L strata, let p-characteristics are defined on each unit of the population. To estimate the unknown p-population means of each characteristic, a random sample is taken out from the population. In multivariate stratified sample survey, the optimum allocation of any characteristic may not be optimum for others. Thus the problem arises to find out an allocation which may be optimum for all characteristics in some sense. Therefore a compromise criterion is needed to workout such allocation. In this paper, the procedure of estimation of p-population means is discussed in the presence of nonresponse when the use of linear cost function is not advisable. A solution procedure is suggested by using lexicographic goal programming problem. The numerical illustrations are given for its practical utility.

In stratified random sampling, Ahsan et al. [Mixed allocation in stratified sampling. Aligarh J Stat. 2005;25:87–97] introduced the concept of ‘Mixed allocation’ for fixed cost by minimizing the variance of the stratified sample mean which was based on the work of Clark and Steel [Optimum allocation of sample to strata and stages with simple additional constraints. Statistician. 2000;49(2):197–207] for two-stage univariate sampling design. The present manuscript studies the problem of obtaining a ‘compromise mixed allocation’ in multivariate stratified sampling using Fuzzy Programming Technique. To exhibit the application of the defined approach, a numerical illustration is considered and solved using Lingo 13.0. In order to demonstrate the superiority of proposed approach, the obtained results have been compared with four other existing approaches.

In sample surveys, stratified sampling is useful if the strata weights are known for each stratum. If they are not known double sampling may be used by selecting a large preliminary sample to estimate the strata weights. Then a stratified sample may be selected independently or from the initial sample. If the problem of non-response is also present then the strata are to be virtually divided into two disjoint and exhaustive groups of respondents and non-respondents. A subsamples from non-respondents is then selected and second more extensive attempt is to be group is to be made for obtaining the required information. The problem of obtaining a compromise allocation for first and second phase of sampling is the formulated as a multi-objective non- linear programming problem that minimizes the sum of variances of the stratified sample mean subject to the non-linear cost constraint. The formulated problem is solved using fuzzy programming and fuzzy goal programming based on piecewise linear approximation. A numerical example is presented to illustrate the computational details. Keywords: Non-response, Multivariate Stratified Sampling, Multiobjective Integer non-linear programming problem, Fuzzy Programming, Travel cost

Inefficiency in the use of available scarce resources has been the bane on increased food production hence low income among the cream of farmers across the nation. The study examined resource use efficiency (with water as the key variable) in irrigated maize production in the Northern part of Taraba State. The study revealed that water was over-utilized in irrigated maize production in the area of study since it had an MVP of less than unity. Multi-stage stratified random and purposive random sampling techniques involving six wards and eighteen villages were used to select respondents. One hundred and twenty (120) questionnaires were administered on one hundred (120) irrigated maize growers. Only one hundred and seventeen (117) of the questionnaires were retrieved for analysis. Data were analyzed using descriptive statistics, gross margin and net farm income analyses and production functions. Data were fitted to four functional forms based on the OLS techniques. The Cobb-Douglas (power function) production function gave the best fit. All the co-efficients except those of agro-chemical and education carried the expected positive signs. The specific objectives of the study were to determine resource use efficiency, describe socio-economic characteristics of irrigated maize growers, estimate cost and returns and identify constraints to irrigated maize production among respondents. The study revealed the cost and returns per hectare as N 55, 152.61 and N 105,937.50 respectively. About 73.1% variability in the dependent variable was accounted for by the independent variables. The result revealed that all the scarce resources were not used efficiently hence not to optimum economic advantage. This was attested to by the high ratios (greater than unity) of MVP/ MFC for fertilizer, seeds, labour and land and low ratio (less than unity) for the key variable, water. For optimum resource allocation to water about 233% decrease in MVP was required while seeds, labour, fertilizer and land required 92.8%, 87.1%, 71.8% and 98.7% increase in MVP respectively. The estimate of elasticity of production summed up to 0.961 meaning decreasing return to scale. Irrigational facilities such as water pumps, sprinklers, pipes and relevant accessories should be made available to farmers to encourage them expand the scale of production thereby boosting increased food production

Objective To study doctor-patient interest demands satisfaction and its influencing factors of the payment system reform of the new rural cooperative medical care scheme to provide reference for the reform. Methods Cross-sectional survey was conducted from September 2016 to February 2017. Multi-stage stratified random sampling was used in six counties of three provinces in the eastern, middle and western regions of China, and mathematical statistics was applied to analyze the data. Results The doctor-patient overall interest demands satisfaction was high, but the satisfaction was lower both with the income and ability improvement of medical staff and with the benefits of farmers. The influencing factors of the satisfaction of managers in medical institutions included the type of payment, educational level and work unit(P<0.05). The influencing factors of medical staff′s satisfaction included the type of payment, work unit, and working years among others(P<0.05). The influencing factors of farmers′ satisfaction included the type of payment and the average annual income, etc(P<0.05). Conclusions The core interest demands of both doctors and patients should be valued to enhance their satisfaction. Diseases related groups should be promoted and applied scientifically, and appropriately integrated with other methods of payment. Both doctors and patients′ understanding of the payment reform should be improved by propaganda and training, to get their support and cooperation. Key words: Health services administration; New rural cooperative medical care scheme; Payment system reform; Interest demands; Satisfaction

The present research study was conducted in Wokha district of Nagaland. The study comprises of 60 sample farmers selected by following a multi stage stratified random sampling technique. Socio economic status of the chilli growers in the study area reveals a literacy percentage of 95.74 per cent and the average family size of chilli growers in the study area was found to be 4.35. Earners constitute about 33.33 per cent of the total sample population of the working population, 58.15 per cent of the sampled respondent depends on agricultural and allied activities as their main occupation. From the study on economic analysis of chilli cultivation revealed that on an average per hectare total cost of chilli cultivation was ₹. 1,38,596.67 with a gross income of ₹. 3,00,440.00. The average yield of chilli was found to be 75.00 q per ha. Considering the prevailing price of chilli in the study area which is ₹ 4,000.00 per q the gross income was found that the average net return from chilli cultivation ₹ .1,52 888.66. Out of the total cost, the share of variable cost and fixed cost constitute 93.59 per cent and 6.41 per cent respectively. whereas the benefit cost ratio was found out to be 2.24.

Objective: To understand the status quo of smoking and change pattern of smoking related behaviors in the past 20 years in the urban adults in Tianjin. Methods: Data was from the study of Chinese Chronic Disease and Risk Factors (2015). Multi stage stratified random sampling method was used to conduct a questionnaire survey in 7 surveillance sites in Tianjin. Results: in the urban residents aged>20 years were compared with those in surveys in 1996 and 2010. Results In 2015, the smoking rate in men and women aged>20 years in Tianjin were 41.1% and 4.5% respectively. The average age of starting smoking was (19.2±4.7) years for men and (24.8±10.9) years for women, showing a younger age trend. Compared with the data from 1996 and 2010, the smoking rate in the urban residents was in decrease, but the rate of successful smoking cessation was in increase in 2015, the differences were significant. Conclusions: Since 1996, the rate of smoking in the urban residents of Tianjin has been in decline, however it is still at a high level, especially in women. The average age of starting smoking remains to be younger. It is necessary to strengthen the health education about harm of smoking in adolescents.

The study was conducted to evaluate factors that significantly influence indigenous poultry production among smallholder farmers in Tigania west Mer County using the production function approach. The study relied on primary data collected by administering structured questionnaires to 359 respondents randomly selected from 5 wards using the multi stage stratified random sampling and probability proportionate to size techniques. A cross sectional survey design and primary data on poultry yields, inputs and farmer demographics were collected. A one step Cobb Douglas production function in logarithmic form was used to estimate the frontier production. From the analyses, results showed that over 50% of the deviations in poultry output resulted from production constraints. Further, off farm income, technology adoption, flock size, extension access and gender of respondents positively influenced indigenous poultry production, while land under cultivation had a negative influence. Results revealed a possibility to increase indigenous poultry production using improved feeds and vaccinations since they were the most limiting resources. In addition, policy interventions aimed at enhancing technology adoption and improving provision of extension services would serve as a motivation to persuade farmers increase stock and apportion more credits to poultry production. This would increase output leading to economies of scale thus increased returns among smallholder farmers. Key words: Indigenous chicken, production, smallholder farmer.

To explore the prevalence of weight misperception and related influencing factors among adult residents in Guangdong province so as to provide information for prevention and control on weight misperception. A multi-stage stratified random sampling method was used to select the sample. Forty-two streets/villages were selected from 21 counties/districts through randomly sampling. Four communities were then chosen from every selected town or district, followed by 40 families chosen from every village or community. Questionnaire was used to collect data on weight perception and its related risk factors. SPSS 16.0 was used for data analysis. There were 6625 respondents participating in the study. Out of them, 50.2% participants misperceived their weight status, among which 35.9% of them underestimated while 14.3% overestimated their weights. Females aged 15 - 24 were more likely to overestimate weights than males in the same age group (38.6% vs. 18.5%), while males were more likely to underestimate weights than females (25.8% vs. 8.5%). The prevalence of underestimation on weights increased with the increase of age in both males and females but the prevalence of overestimation on weights decreased. Data from multivariate results from logistic analysis showed that rural residents, males, being elderly, residents with low education level, manual occupations (agriculture, forestry, animal husbandry and fishery), low family income and with anxiety were the major risk factors on underestimation of weight. However, factors as being urban residents, females, adolescents, minority and never having received weight measurement etc. were the major risk factors of overestimated on weight. Misperceptions of weight status in Guangdong province exhibited a high prevalence with complicated influencing factors, calling for more psychological research to be carried out to prevent and reduce the misperceptions on weight status.

A child safety seat protects children from injury during motor vehicle accidents (MVAs). However, there is a lack of enforcement of regulation regarding its use in Saudi Arabia. This study aimed to determine the use of child safety seat and its determinants and barriers in Jeddah, Saudi Arabia. This cross-sectional study was based on a structured face-to-face interview across Jeddah among families who drove in cars with children aged ≤5 years. The study used stratified multistage random sampling across the population of the governorate of Jeddah. A total of 675 parents were included, of which 311 (46.1%) reported having a child safety seat in their vehicle, and only 165 (24.4%) reported its regular use. Awareness levels among parents regarding child safety seat use and its benefits, parents' level of education, family size, family income, and the belief in the need for laws governing child safety seat were key determinants of child safety seat use. Poor awareness among parents regarding the utility of child safety seat in reducing child injuries, low education levels among parents, larger families with multiple children, poor affordability, low family income, and a clear lack of laws mandating the use of child safety seat were identified as key barriers. The regular use of child safety seat for children aged ≤5 years is low in even one of the most urban populations of Saudi Arabia. This indicates the need for large-scale awareness drives and stricter implementation of laws enforcing the use of child safety seat in Saudi Arabia.

This cross-sectional study was carried out to determine the prevalence of overweight and obesity among secondary school children in the city of Mashhad, Iran and its association with dietary patterns. A total of 1189 secondary school children (579 males and 610 females) aged 12- 14 years old were selected through a stratified multistage random sampling. All adolescents were measured for weight and height. Household socio-demographic information were self-reported by parents. Adolescents were classified as overweight or obese based on BMI-for age Z-score. Dietary patterns were assessed using a validated Iranian food frequency questionnaire (FFQ) included 121 food items. The principal component factor analysis (PCA) was applied to derive dietary patterns, and Logistic Regression (LR) was applied to examine the association between dietary pattern and adolescents BMI. The overall prevalence of overweight and obesity among the study population was 17.2% and 11.9%, respectively. The PCA analysis revealed the presence of two dietary patterns that were labeled as ‘Healthy dietary pattern’ (HP), and ‘Unhealthy dietary pattern’ (UP). LR analysis showed that HP was significantly associated with BMI (OR: 1.28, 95% CI: 1.124-1.47). Similarly, UP was significantly associated with BMI (OR: 0.861, 95% CI: .725-.968). In Iran, nutritional transition has taken place in the context of urbanization and has changed lifestyle, and dietary patterns. Policies must be formulated and circulated in the society to reach every family in the form of healthy dietary pattern.