The problem with which this book is concerned originated in the need for improved sampling methods in the social sciences, particularly in large-scale investigations. Research on sampling problems includes both extensive study of probability designs in the form of models and experimentation with these designs under operating conditions. A sampling plan shows not only the way in which the sample is drawn but also how it is to be analyzed. Sampling plans focus not only on accurately representing the population but also on fairly appraising stability.

The design for the study involves a comparison and contrast of five multistage sampling...

Before entering into the mathematical structure of the multi-stage sampling models, we shall have to fix upon an appropriate notation. It is shown in Table 1.

Partial expectations and variances at a given stage, keeping the stages above fixed, are suffixed by that stage number

32.\[\begin{array}{ll} \text{i.e.}, & E_{2}=E_{2}/_{1}\\ & E_{5}= E_{5}/_{1,2,3,4}\\ \text{and} & V_{5} = V_{5}/_{1,2,3,4} \end{array}\]

The following identity is used repeatedly in the mathematical derivations of the various models.

33.\[V_{s,s+1,\cdots, k} = E_{s}E_{s+1}E_{s+2}E_{s+3}E_{s+4}\cdots E_{k-1}V_{k}+E_{s}\cdots E_{k-2}V_{k-1}E_{k}+ \cdots + E_{s}V_{s+1}E_{s+2}E_{s+3}\cdots E_{k}+V_{s}E_{s+1}E_{s+2}\cdots E_{k}\ (1 \leq s \leq k)\]

The following probabilities of selection are repeatedly used in the different models because they were most suitable for our practical applications. However, in other practical situations the probabilities of selection could be different from these.

34.\[\begin{align*} P_{i_1 } i_2 \cdots i_{k - 2} i_{k - 1} &= \frac{{N_{i_1 } \cdots i_{k - 1} k}}{{\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }} \\ P_{i_1 \cdots i_{k - 2} } &= \frac{{\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }}{{\sum\limits_{i_{k - 2} = 1}^{N_{i_1 \cdots i_{k - {\rm{s}}} (k - 2)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }} \\ P_{i_1 \cdots i_{k - s} } &= \frac{{\sum\limits_{i_{k - s + 1} = 1}^{N_{i_1 \cdots i_{k - s} (k - s + 1)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }}{{\sum\limits_{i_{k - s} = 1}^{N_{i_1 \cdots i_{k - s - 1} (k - s)} } {\sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } } }}(s = 2, \cdots ,k - 1) \\ P_{i_1 } &= \frac{{\sum\limits_{i_2 = 1}^{N_{i_1 2} } \cdots \sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 2} (k - 1)} } {N_{i_1 \cdots i_{k - 1} k} } }}{N} \end{align*} \]${P_{{i_1} \cdots {i_k}}}$is any...

The following cost function could be considered to arrive at optimum sample size allocations at different stages\[\begin{align*} C = n_1 c_1 + c_2 \sum\limits_{i_1 = 1}^{n_1 } {n_{i_1 2} + c_s } \sum\limits_{i_1 = 1}^{n_1 } {\sum\limits_{i_2 = 1}^{n_{i_i 2} } {n_{i_1 i_2 {\rm{s}}} + \cdots + c_s } } \sum\limits_{i_1 = 1}^{n_1 } {\sum\limits_{i_2 = 1}^{n_{i_i 2} } \cdots } \\ \cdots \sum\limits_{i_{s - 1} = 1}^{n_{i_1 \cdots i_{s - 2} (s - 1)} } {n_{i_1 i_2 \cdots i_{s - 1} s} + \cdots + c_k } \sum\limits_{i_1 = 1}^{n_1 } \cdots \sum\limits_{i_{k - 1} = 1}^{n_{i_1 i_2 \cdots i_{k - 2} (k - 1)} } {n_{i_1 i_2 \cdots i_{k - 1} k} } \end{align*} \caption{(3.1)}whereCis the over-all cost,c₁ is the average cost per 1st-stage sampling unit, andc_sis the average cost persth-stage sampling unit. Minimization with respect to sample sizes of linear functions ofCin (3.1) could be cumbersome due to the presence of sample sizes atop the various summation signs.

However, a second cost function, more wieldy than the function in (3.1), is:\[\begin{align*}C^* = n_1 c_1 + n_2 c_2 \sum\limits_{i_1 = 1}^{n_1 } {\tau _{i_1 } n_{i_1 2} + n_1 c_s } \sum\limits_{i_1 = 1}^{N_1 } {\tau _{i_1 } n_{i_1 2} } \sum\limits_{i_2 = 1}^{N_{i_i 2} } {\tau _{i_1 i_2 } n_{i_1 i_2 s} + \cdots } \\ \cdots + c_s n_1 \sum\limits_{i_1 = 1}^{N_1 } {\tau _{i_1 } n_{i_1 2} } \sum\limits_{i_2 = 1}^{N_{i_1 2} } {\tau _{i_1 i_2 } n_{i_1 2} } \cdots \sum\limits_{i_{s - 1} = 1}^{N_{i_1 \cdots i_{s - 2} (s - 1)} } {\tau _{i_1 \cdots i_{s - 1} } n_{i_1 \cdots i_{s - 1} s} + \cdots } \\ \cdots + c_k n_1 \sum\limits_{i_1 = 1}^{N_1 } {\tau _{i_1 } n_{i_1 2} } \sum\limits_{i_2 = 1}^{N_{i_1 2} } {\tau _{i_1 i_2 } n_{i_1 2} } \cdots \sum\limits_{i_{k - 1} = 1}^{N_{i_1 \cdots i_{k - 1} k} } {\tau _{i_1 \cdots i_{k - 1} } n_{i_1 \cdots i_{k - 1} k} } \end{align*} \]where\[\begin{array}{rl} {\tau _{i_1 \cdots i_{s - 1} } = P_{i_1 \cdots i_{s - 1} } } & {{\text{if}}\ P\ {\text{is used at the}}\ (s - 1){\rm{th}}\ {\text{stage}}} \\ { = \frac{1}{{N_{i_1 \cdots i_{s - 2} (s - 1)} }}} & {{\text{if}}\ R\ {\text{is used at the}}\ (s - 1){\rm{th}}\ {\text{stage}}\;(s = 2, \cdots ,k)} \\ \end{array}\]and\[\begin{array}{ll} {{E_1}({n_1}) = {n_1}} & \text{is the number of} \ 1\text{st} - \text{stage sampling units} \\ {{E_{12}}_{\vdots} \left[ {\sum\limits_{{i_1} = 1}^{{n_1}} {{n_{{i_1}2}}} } \right] = {n_1}\sum\limits_{{i_1} = 1}^{{N_1}} {{\tau _{{i_1}}}{n_{{i_1}2}}} } & \text{is the average number of} \ 2\text{nd} - \text{stage sampling units} \\ {{E_{12 \cdots s}}\left[ {\sum\limits_{{i_1} = 1}^{{n_1}} {\sum\limits_{{i_2} = 1}^{{n_{{i_1}2}}} { \cdots \sum\limits_{{i_{s - 1}} = 1}^{{n_{{i_1} \cdots {i_{s - 2}}(s - 1)}}} {{n_{{i_1} \cdots {i_{s - 1}}s}}} } } } \right]} & \text{is the average number of the} \ s\text{th}-\text{stage sampling units} \\ = {n_1}\sum\limits_{{i_1} = 1}^{{N_1}} {{\tau _{{i_1}}}} {n_{{i_1}2}} \cdots \sum\limits_{{i_{s - 1}} = 1}^{{N_{{i_1} \cdots {i_{s - 2}}(s - 1)}}} {{\tau _{{i_1} \cdots {i_{s - 1}}}}{n_{{i_1} \cdots {i_{s - 1}}s}}} & \end{array} \]and\[\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {E\left[ {{C_s}\sum\limits_{{i_1} = 1}^{{n_1}} {\sum\limits_{{i_2} = 1}^{{n_{{i_1}2}}} { \cdots \sum\limits_{{i_{s - 1}} = 1}^{{n_{{i_1} \cdots {i_{s - 2}}(s - 1)}}} {{n_{{i_1} \cdots {i_{s - 1}}s}}} } } } \right]} \\ { = {C_s}{n_1}\sum\limits_{{i_1} = 1}^{{N_1}} {{\tau _{{i_1}}}{n_{{i_1}2}} \cdots \sum\limits_{{i_{s - 1}} = 1}^{{N_{{i_1} \cdots {i_{s - 2}}(s - 1)}}} {{\tau _{{i_1} \cdots {i_{s - 1}}}}{n_{{i_1} \cdots {i_{s - 1}}s}}} } } \\ \end{array} } & \text{is the total cost for the average number of the} \ s\text{th}-\text{stage sampling units} \\ \end{array} \]One of the two following procedures should be adopted as dictated by the demands...

This chapter aims to enable the research worker to put into practice the mathematical models which were discussed in Chapter 2 and also to evaluate their relative efficiencies. The criteria for evaluating relative efficiencies of the different sampling models have been presented earlier on page 7 of Chapter 1.

We begin by presenting in Table 2 the parameters for the population consisting of 432 public schools. The parameter values enabled us (1) to examine how close they were to the sample estimates ӯ obtained under the different models and (2) to deriveV(ӯ), which under normal circumstances could not have...

First, we defined the various mathematical sampling models and a design for comparing and contrasting the various models with respect to estimation and efficiency. We presented the three basic formulas under each model: the unbiased estimate of the population, the multi-component variance of the estimate, and the unbiased estimate of the variance of the estimate. We then dealt with the appropriate cost function which plays a leading role in current sampling models.

Finally, we described the practical application of all the models studied. The basic unit for this study was a population composed of 22,209 high school juniors. We gave...