Optimum Allocation in Multivariate Double Sampling for Biomass Estimation

Abstract

Double sampling or two-phase sampling involves sampling of any population in 2 phases. The first phase yields data on any desired factor by direct measurements as well as by some indirect method. In the second phase, data are collected by the indirect method only. The estimated variance in double sampling with regression and ratio estimators are described in detail by Cochran (1963). The first sample is a simple random sample of size n'. The second sample of size n is a random subsample from the first sample, but may be drawn independently if this is more convenient. The first step is to set up the estimate and to determine its variance. The auxiliary variate (xi) is used to make a regression estimate of y. It is assumed that the population is finite but very large and that the relation between yi and xi is linear. In the first (large) sample (size n'), only xi is measured. In the second (small) sample (size n), both xi and yi are measured. One of the major problems in double sampling is determining the number of samples required in each phase to give the desired accuracy for the maximum economy. The efficiency of double sampling depends on two things: (1) the precision of the mathematical relationship, and (2) the cost of direct measurements compared to indirect estimates. If too many direct samples are taken, the cost of sampling becomes unnecessarily high, while the use of too few direct samples results in an unreliable mathematical relationship. Thus, it is desirable to estimate the size of the two samples; the large sample (n') and the small sample (n). In its application to estimating crop biomass, the first phase of double sampling involves estimation of the plant biomass ocularly or by capacitance meter. In the second phase, the plant biomass is estimated as in the first phase followed by clipping and weighing of the plants. For a detailed discussion on the statistical aspects of the double sampling, the reader is referred to Schumacher and Chapman (1948), Hansen et al. (1953), National Research Council (1962), and Cochran (1963). With a single factor under study and for a given sampling procedure, optimum allocation of resources to direct and indirect methods of estimation is well defined. However, a simple procedure for optimum allocation in multivariate double sampling is not available. A technique is described which enables the investigator to find optimum allocation in multivariate double sampling which minimizes cost or variance and also gives achievable variances of the estimated means. For sampling involving a single independent variable, the procedure for optimum allocation in double sampling is reviewed. According to Cochran (1963), the cost of double sampling is