Optimum allocation for a dual-frame telephone survey 3. Screening protocolOptimum allocation for a dual-frame telephone survey 3. Screening protocol

In the screening protocol, one conducts survey interviews for all units in the landline sample $s_{A} .$ One conducts screening interviews (for telephone status) for all units in the cell-phone sample $s_{B}$ and then conducts the survey interviews only for the units that screen-in as CPO. Therefore, expected data collection costs arise according to the model

$\begin{array}{l} C_{S C} & = c_{A} n_{A} + {c^{'}}_{B} β n_{B} + {c^{″}}_{B} (1 - β) n_{B} \\ = c_{A} n_{A} + {c^{‴}}_{B} n_{B}, \end{array} (3.1)$

where ${c^{'}}_{B}$ is the cost per completed screener (to ascertain telephone status) in sample $s_{B},$ ${c^{″}}_{B}$ is the cost per completed screener and interview in sample $s_{B},$ and ${c^{‴}}_{B} = {c^{'}}_{B} β + {c^{″}}_{B} (1 - β) .$ In this notation, $n_{A}$ is the number of survey interviews completed amongst landline respondents and $n_{B}$ is the number of completed interviews (telephone screener only for non-CPO respondents, and screener plus survey interview for CPO respondents) amongst cell-phone respondents. That is, the expected total number of completed survey interviews is $n_{A} + (1 - β) n_{B} .$

The unbiased estimator of the overall population total is

$\hat{Y} = {\hat{Y}}_{A} + {\hat{Y}}_{b}, (3.2)$

where ${\hat{Y}}_{A} = (N_{A} / n_{A}) y_{A},$ ${\hat{Y}}_{b} = (N_{B} / n_{B}) y_{b},$ and $y_{A} = y_{a} + y_{a b} .$ The variance of the estimator is

$Var {\hat{Y}} = N^{2} (\frac{R_{A}^{2}}{n_{A}} + \frac{R_{B}^{2}}{n_{B}}), (3.3)$

where

$R_{A}^{2} = W_{A}^{2} S_{A}^{2}$

and

$R_{B}^{2} = W_{B}^{2} S_{b}^{2} {1 - β + β (1 - β) \frac{{\bar{Y}}_{b}^{2}}{S_{b}^{2}}} .$

The optimal allocation of the total sample is

$\begin{array}{l} n_{A, o p t} & = & L R_{A} / \sqrt{c_{A}} \\ n_{B, o p t} & = & L R_{B} / \sqrt{{c^{‴}}_{B}}, \end{array}$

where $L$ is a constant that depends on the fixed constraint: cost or variance. The minimum variance subject to fixed cost is given by

$\min [Var {\hat{Y}}] = {\frac{{(\sqrt{c_{A}} R_{A} + \sqrt{{c^{‴}}_{B}} R_{B})}^{2}}{C_{S C}}}_{}, (3.4)$

and the minimum cost subject to fixed variance is

$\min [C_{S C}] = \frac{{(\sqrt{c_{A}} R_{A} + \sqrt{{c^{‴}}_{B}} R_{B})}^{2}}{V_{0}} . (3.5)$

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Catalogue no. 12-001-X

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Date modified:: 2017-09-20

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Optimum allocation for a dual-frame telephone survey 3. Screening protocolOptimum allocation for a dual-frame telephone survey 3. Screening protocol