Strata Variances Research Articles

Designs for half-diallel experiments with commutative orthogonal block structure

In some experiments, the experimental units are all pairs of individuals who have to undertake a given task together. The set of such pairs forms a triangular association scheme. Appropriate randomization then gives two non-trivial strata. The design is said to have commutative orthogonal block structure (COBS) if the best linear unbiased estimators of treatment contrasts do not depend on the stratum variances. There are precisely three ways in which such a design can have COBS. We give a complete description of designs for which all treatment contrasts are in the same stratum. Then we give a very general construction for designs with COBS which have some treatment contrasts in each stratum.

Adaptive radial importance sampling under directional stratification

We establish radial importance sampling under directional stratification and construct its easy-to-implement algorithm for estimating the probability of failure in structural reliability analysis. The proposed algorithm is expected to run in a fully adaptive manner for averaging the increasing realizations towards the unknown stratum probability of failure, along with updating parameterized importance sampling and adjusting the allocation of computing budget only occasionally yet effectively, fully considering the decreasing stratum variances, all on a single set of replications. The formulation does not require a monotonicity condition on the radial distance in the polar coordinate system to justify a deterministic numerical procedure, such as the root finding of directional simulation in its standard form. A wide variety of numerical results were provided for illustrating the applicability and effectiveness of the proposed framework and algorithm.

Adaptive stratified sampling for structural reliability analysis

An adaptive framework of stratified sampling is established for estimating the probability of failure in structural reliability analysis in combination with variance reduction employed by strata. Among a variety of variance reduction techniques, stratification of Latin hypercube type makes it possible to apply importance sampling and control variates by strata in an adaptive manner without the need to track differences in computing cost across strata. By batching the resulting paralleled tasks, the proposed algorithm adjusts the allocation of computing budget only occasionally yet effectively taking full account of decreasing stratum variances. A variety of structural reliability problems with sparse structures lie within the scope of the proposed framework in the sense that stratification with gradual adjustment of computing budget avoids wasting the resource on possibly many empty strata, while importance sampling and control variates improve Monte Carlo methods on non-empty strata. We provide numerical results to illustrate the wide applicability and effectiveness of the proposed framework.

A Bayesian approach to the selection of two-level multi-stratum factorial designs

In a multi-stratum factorial experiment, there are multiple error terms (strata) with different variances that arise from complicated structures of the experimental units. For unstructured experimental units, minimum aberration is a popular criterion for choosing regular fractional factorial designs. One difficulty in extending this criterion to multi-stratum factorial designs is that the formulation of a word length pattern based on which minimum aberration is defined requires an order of desirability among the relevant words, but a natural order is often lacking. Furthermore, a criterion based only on word length patterns does not account for the different stratum variances. Mitchell, Morris and Ylvisaker [Statist. Sinica 5 (1995) 559–573] proposed a framework for Bayesian factorial designs. A Gaussian process is used as the prior for the treatment effects, from which a prior distribution of the factorial effects is induced. This approach is applied to study optimal and efficient multi-stratum factorial designs. Good surrogates for the Bayesian criteria that can be related to word length and generalized word length patterns for regular and nonregular designs, respectively, are derived. A tool is developed for eliminating inferior designs and reducing the designs that need to be considered without requiring any knowledge of stratum variances. Numerical examples are used to illustrate the theory in several settings.

Multivariate Multi-Objective Allocation in Stratified Random Sampling: A Game Theoretic Approach.

We consider the problem of multivariate multi-objective allocation where no or limited information is available within the stratum variance. Results show that a game theoretic approach (based on weighted goal programming) can be applied to sample size allocation problems. We use simulation technique to determine payoff matrix and to solve a minimax game.

Nonparametric Variance Estimation Under Fine Stratification: An Alternative to Collapsed Strata

Fine stratification is commonly used to control the distribution of a sample from a finite population and to improve the precision of resulting estimators. One-per-stratum designs represent the finest possible stratification and occur in practice, but designs with very low numbers of elements per stratum (say, two or three) are also common. The classical variance estimator in this context is the collapsed stratum estimator, which relies on creating larger “pseudo-strata” and computing the sum of the squared differences between estimated stratum totals across the pseudo-strata. We propose here a nonparametric alternative that replaces the pseudo-strata by kernel-weighted stratum neighborhoods and uses deviations from a fitted mean function to estimate the variance. We establish the asymptotic behavior of the kernel-based estimator and show its superior practical performance relative to the collapsed stratum variance estimator in a simulation study. An application to data from the U.S. Consumer Expenditure Survey illustrates the potential of the method in practice.

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 Multivariate Stratified Sampling Designs with Travel Cost: A Multiobjective Integer Nonlinear Programming Approach

The problem of optimum allocation in stratified sampling and its solution is well known in sampling literature for univariate populations (see Cochran, 1977; Sukhatme et al., 1984). In multivariate populations where more than one characteristics are to be studied on every selected unit of the population the problem of finding an optimum allocation becomes more complex due to conflicting behaviour of characteristics. Various authors such as Dalenius (1953, 1957), Ghosh (1958), Yates (1960), Aoyama (1963), Gren (1964, 1966), Folks and Antle (1965), Hartley (1965), Kokan and Khan (1967), Chatterjee (1972), Ahsan and Khan (1977, 1982), Chromy (1987), Wywial (1988), Bethel (1989), Kreienbrock (1993), Jahan et al. (1994), Khan et al. (1997), Khan et al. (2003), Ahsan et al. (2005), Díaz-García and Ulloa (2006, 2008), Ahsan et al. (2009) etc. used different compromise criteria to work out a compromise allocation that is optimum for all characteristics in some sense. Almost all the previous authors used some function of the sampling variances of the estimators of various characteristics to be measured as an objective that is to be minimized for a fixed cost given as a linear function of sample allocations. Because the variances are not unit free it is more logical to consider the minimization of some function of squared coefficient of variations as an objective. Previously this concept was used by Kozok (2006). Furthermore, investigators have to approach the sampled units in order to get the observations. This involves some travel cost. Usually this cost is neglected while constructing a cost function. This travel cost may be significant in some surveys. For example if the strata consist of some geographically difficult-to-approach areas. The authors problem of optimum allocation in multivariate stratified sampling is discussed with an objective to minimize simultaneously the coefficients of variation of the estimators of various characteristics under a cost constraint that includes the measurement as well as travel cost. The formulated problem of obtaining an optimum compromise allocation turns out to be a multiobjective all-integer nonlinear programming problem. Three different approaches are considered: the value function approach, ∈ –constraint method, and Distance–based method, to obtain compromise allocations. The cost function considered also includes the travel cost within stratum to reach the selected units. Additional restrictions are placed on the sample sizes to avoid oversampling and ensure the availability of the estimates of the strata variances. Numerical examples are also presented to illustrate the computational details of the proposed methods.

Multistratum fractional factorial designs

Recent work on multistratum fractional factorial designs is set in a general and unified framework, and a criterion for selecting multistratum fractional factorial designs that takes stratum variances into account is proposed. Application of the general theory is illustrated on designs of experiments with multiple processing stages, including split-lot designs, blocked strip-plot designs, and post-fractionated strip-block designs. In particular, this helps elucidate the relationship between the three different design settings studied by Miller (1997), Bingham et al. (2008), and Vivacqua and Bisgaard (2009). The construction and selection of two-stage designs in these settings are shown to be equivalent. Good designs based on our criterion are found and compared with those tabulated in Vivacqua and Bisgaard (2009). Multistratum experiments refer to those with multiple sources of errors. The error structure of an experiment is determined by the structure of experimental units, called the block structure. The block structure, treatment structure, and design (assignment of the treatments to experimental units) together specify a linear model based on which estimates of the treatment contrasts of interest are computed from the data. In two important papers, Nelder (1965a,b) developed a unified theory for the analysis of randomized experiments with what he called simple block structures, which cover most of the block structures encountered in practice. Speed and Bailey (1982) and Tjur (1984) further developed the theory to cover the more general orthogonal block structures. An excellent account can be found in Bailey (2008). In a series of papers, Brien and Bailey (2006, 2009, 2010) discussed experiments involving multiple randomizations. Multistratum experiments are common in agriculture. Federer and King (2007) provided a comprehensive treatment of the design and analysis of split plot and split block experiments. In recent years, we have seen rising interest in design of industrial experiments with multiple strata. Some treatment factors may require larger experimental units than others since their levels are more dif- ficult to change, or in experiments with multiple processing stages the levels of

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.

Empirical Bayes Estimation of Undercount in the Decennial Census

Abstract On April 1, 1990, the decennial census for the United States will be conducted by the U.S. Bureau of the Census. By December 31, 1990, the Census Bureau is specified by law to submit state population counts for the purpose of reapportionment of the U.S. House of Representatives, and by March 31, 1991, to submit small-area population counts for the purpose of redistricting. Census counts are used in a variety of other ways: for revenue-sharing formulas between different levels of government, for demographic projections, as a base for morbidity and mortality statistics, and so forth. Inaccurate census counts should be cause for concern for the whole nation. It is universally acknowledged that certain groups of people (e.g., young black males, illegal aliens, etc.) are harder to count than others. If the hard-to-count groups are distributed in equal proportions throughout the United States, there would be far less controversy over what to do about the uncounted people. As it is, many large American cities such as Chicago, Detroit, New York, and Los Angeles feel they are losing federal funds because their cities contain larger numbers of the groups that are less well counted. And certain states such as New York and California feel they are underrepresented in Congress, to the benefit of Midwestern states such as Indiana and Iowa. Census undercount is defined simply as the difference between the true count and the census count, expressed as a percentage of the true count. Small-area estimation of this undercount is considered here, using empirical Bayes methods based on a new and, it is argued, more realistic model than has been used before. Grouping of like subareas from areas such as states, counties, and so on into strata is a useful way of reducing the variance of undercount estimators. By modeling the subareas within a stratum to have a common mean and variances inversely proportional to their census counts, and by taking into account sampling of the areas (e.g., by dual-system estimation), empirical Bayes estimators that compromise between the (weighted) stratum average and the sample value can be constructed. The amount of compromise is shown to depend on the relative importance of stratum variance to sampling variance. These estimators are evaluated at the state level (51 states, including Washington, D.C.) and stratified on race/ethnicity (3 strata) using data from the 1980 postenumeration survey (PEP 3–8, for the noninstitutional population).

Empirical Bayes Estimation of Undercount in the Decennial Census

Empirical Bayes methods are used to estimate the extent of the undercount at the local level in the 1980 U.S. census. "Grouping of like subareas from areas such as states, counties, and so on into strata is a useful way of reducing the variance of undercount estimators. By modeling the subareas within a stratum to have a common mean and variances inversely proportional to their census counts, and by taking into account sampling of the areas (e.g., by dual-system estimation), empirical Bayes estimators that compromise between the (weighted) stratum average and the sample value can be constructed. The amount of compromise is shown to depend on the relative importance of stratum variance to sampling variance. These estimators are evaluated at the state level (51 states, including Washington, D.C.) and stratified on race/ethnicity (3 strata) using data from the 1980 postenumeration survey (PEP 3-8, for the noninstitutional population)."

Estimation of stratum variances in planning of crop acreage surveys

A modelling approach is used to obtain initial estimates of stratum crop acreage variances for designing crop surveys, particularly those using the remote sensing technology. The present methodology is developed based on the concept of stratum variance as a function of the sampling unit size and it uses primarily the historical crop statistics which are commonly available in most countries. Methods are proposed for the determination of stratum variances corresponding to unit sizes different from the sampling unit size and for which the historical crop statistics can be used. The methodology is applied to estimate stratum variances for wheat in the U.S. Great Plains. An evaluation of these estimates made by comparing them to those obtained from certain satellite sample data shows that the proposed method leads to reliable stratum variance estimates for a fairly large size (5×6 nautical miles area segment) sampling unit.

Recovery of Row and Column Information in Row-Column Designs with Adjusted Orthogonality

SUMMARY The analysis of row-column designs which satisfy the property of adjusted orthogonality is considered. For such designs a series of results are derived which relate the information matrix, its inverse, the estimator of the treatment effects and the adjusted treatment sum of squares to those of its component designs. It is also shown that estimators of the stratum variances required for the recovery of information from between rows and between columns can be obtained by a relatively simple method.

An analysis of correlation matrices: Equal correlations

SUMMARY We study the large-sample joint distribution of Z, the -fp(p - 1) Fisher z-transforms of the elements in a p variable correlation matrix. Under the null hypothesis of equal population correlations the variance matrix of Z has just three projector matrices in its spectral decomposition. These define three mutually orthogonal invariant subspaces of sample space, or 'error strata' as they would be called in the analysis of variance. The squared lengths of the projections of the sample vector onto each of these subspaces, when divided by the stratum variance, provide a natural partition for the large-sample chi-squared test for equality of correlations. A linear model is given which provides a statistical interpretation for the error strata, and hence the components in the partition. As well as providing a simple test for equality of correlations, the procedure indicates how familiar techniques in the spirit of analysis of variance can be used to investigate correlation matrices.

Optimum Allocation of Sample Size and Prior Distributions: A Review

Summary In this review paper, the problem of optimum allocation of sample size to strata is examined in the light of a priori distributions. We first consider Neyman's optimum allocation in the case of the Simple Random Sampling (SRS) scheme and review the results concerning the justification for the assumption that the unknown proportionate values of within stratum variances ca's can be replaced by the known proportionate values of aVc's which are the corresponding within stratum variances based on the auxiliary information. We next review the problem of optimum stratification with proportional and optimum allocations in the case of SRS in the light of a priori distributions and discuss the role of arranging the auxiliary variate in increasing order for stratification. We then turn our attention to varying probability sampling schemes and present a review of the results based on a study discussing whether a stratified ,rPS (PPS) strategy with various non-optimal allocations is likely to be worth while in practice.

The Combination of Information in Generally Balanced Designs

Summary A general method is proposed for combining information in generally balanced designs when estimates of treatment effects are available from more than one stratum. Estimates of stratum variances are obtained by equating residual sum of squares in each stratum to their expectations, using the ML estimates of treatment effects. Weights may be given, the solution then being explicit, or fitted, in which case iteration is required. The results of a Monte Carlo study are given for a BIB design to test the distribution of the variance estimates.

Multivariate Stratified Surveys

In this paper we consider the allocation problem for multivariate stratified surveys. If the stratum variances for the different variates are not distributed in the same way Neyman allocation optimizing the measurement of one variate is of limited value. In our formulation we determine the allocation such that sample estimates meet stated levels of precision or tolerance at minimum cost. Solution of the allocation problem is shown to be a programming problem and an example is given to illustrate it. By obtaining the solution to one plan a sampler essentially obtains the solution to a whole series of plans. The problem of tolerance setting is then discussed. An emprical solution, based on practical rather than some over-riding theoretical consideration, to the problem is given. A set of coefficients which elucidate the cost implications of each of the tolerances are derived.