Optimum allocation for a dual-frame telephone survey 2. Take-all protocolOptimum allocation for a dual-frame telephone survey 2. Take-all protocol

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In the take-all protocol, one conducts survey interviews for all units in both samples $s_A$ and $s_B.$ Therefore, variable data collection costs can be approximated by the model

$$C_{TA}=c_A{n}_A+c_B{n}_B,(2.1)$$

where $c_A$ is the cost per completed interview in sample $s_A$ and $c_B$ is the cost per completed interview in sample $s_B.$ The expected numbers of survey interviews in the cell-phone sample are $\left(1-\beta \right)n_B$ CPO units and $\beta n_B$ dual-user units.

The unbiased estimator of the population total (Hartley 1962) is given by

$$\ddot{Y}={\widehat{Y}}_a+p{\widehat{Y}}_{ab}+q{\widehat{Y}}_{ba}+{\widehat{Y}}_b ,(2.2)$$

where $p$ is a mixing parameter, $q=1-p,$ ${\widehat{Y}}_a=\left(N_A/n_A\right)y_a$ is an estimator of the LLO total, ${\widehat{Y}}_{ab}=\left(N_A/n_A\right)y_{ab}$ is an estimator of the dual-user total derived from the landline sample, ${\widehat{Y}}_{ba}=\left(N_B/n_B\right)y_{ba}$ is an estimator of the dual-user total derived from the cell-phone sample, ${\widehat{Y}}_b=\left(N_B/n_B\right)y_b$ is an estimator of the CPO total, $y_a$ is the sum of the variable of interest for the observations in $s_A$ and in domain $U^a,y_{ab}$ is the sum of the variable of interest for the observations in $s_A$ and in domain $U^{ab},y_{ba}$ is the sum of the variable of interest for the observations in $s_B$ and in domain $U^{ab},$ and $y_b$ is the sum of the variable of interest for the observations in $s_B$ and in domain $U^b.$ We examine the choice of $p$ in Section 4.

Given fixed $p,$ we find that the variance of $\ddot{Y}$ is

$$\text{Var}\left\{\ddot{Y}\right\}=N^2\left(\frac{Q_A^2}{n_A}+\frac{Q_B^2}{n_B}\right) ,(2.3)$$

where $W_A=N_A/N,W_B=N_B/N,$

$$Q_A^2=W_A^2\left\{\left(1-\alpha \right)S_a^2+\alpha p^2{S}_{ab}^2+\alpha \left(1-\alpha \right){\left({\overline{Y}}_a-p{\overline{Y}}_{ab}\right)}^2\right\},$$

and

$$Q_B^2=W_B^2\left\{\left(1-\beta \right)S_b^2+\beta q^2{S}_{ab}^2+\beta \left(1-\beta \right){\left({\overline{Y}}_b-q{\overline{Y}}_{ab}\right)}^2\right\}.$$

The classical optimum allocation of the total sample to the two sampling frames (Cochran 1977) is defined by

$$\begin{array}{lll}{n}_{A,opt}\hfill & =\hfill & \frac{K{Q}_A}{\sqrt{c_A}}\hfill \\ n_{B,opt}\hfill & =\hfill & \frac{K{Q}_B}{\sqrt{c_B}},\hfill \end{array}(2.4)$$

where $K$ is a constant that depends upon whether the objective of the allocation is to minimize cost subject to a constraint on variance, or to minimize variance subject to a constraint on cost. The minimum variance subject to fixed cost $C_{TA}$ is given by

$$\min \left[\text{Var}\left\{\ddot{Y}\right\}\right]=\frac{{\left(\sqrt{c_A}{Q}_A+\sqrt{c_B}{Q}_B\right)}^2}{C_{TA}} ,(2.5)$$

while the minimum cost subject to fixed variance $V_0$ is

$$\min \left[C_{TA}\right]=\frac{{\left(\sqrt{c_A}{Q}_A+\sqrt{c_B}{Q}_B\right)}^2}{V_0}.(2.6)$$

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