Non-response follow-up for business surveys
Section 3. Some theoretical properties of the proposed follow‑up strategy

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Let $C$ be the total budget allocated for non-response follow-up, which could be defined in terms of monetary or time units. A cost is incurred for each call attempt and depends on the call outcome. We denote by $c^{\left(1\right)},$ $c^{\left(2\right)}$ and $c^{\left(3\right)},$ the cost per call attempt for a “response”, “final non-response” and “still-in-progress” outcome, respectively. To simplify our derivations, we assume that these costs are the same for each sample unit and do not vary during data collection. Let $c_{hi}={\displaystyle {\sum }_{k=1}^K{c}_{hik}}$ be the cost of either resolving unit $i\in s_{2h}$ or reaching the maximum number of call attempts for that unit, where $c_{hik}$ is the cost of the $k^{\text{th}}$ call attempt for unit $i\in s_{2h}.$ If a unit $i\in s_{2h}$ is resolved at the $l^{\text{th}}$ attempt, $c_{hik}$ is defined to be zero for all $k>l.$ Therefore, the cost $c_{hik}$ is either zero, if unit $i\in s_{2h}$ has been resolved before the $k^{\text{th}}$ attempt, or $c^{\left(1\right)},$ $c^{\left(2\right)}$ or $c^{\left(3\right)},$ depending on the call outcome. For a given sample size $n_2$ and a fixed value of $K,$ the total follow-up cost, ${\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{2h}}{c}_{hi}}},$ is a random variable when each sample unit is followed up until it is resolved or the maximum number of call attempts has been reached. Taking the expectation of the total cost with respect to the follow-up sampling design and non-response mechanism, conditionally on $s_{1,\text{nr}},$ we obtain the expected follow-up cost:

$$\tilde{C}\left(n_2,K\right)={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}{\tilde{c}}_{hi}\left(K\text{}\right)}},(3.1)$$

where ${\tilde{c}}_{hi}\left(K\text{}\right)={\displaystyle {\sum }_{k=1}^K{\tilde{c}}_{hik}}$ is the expected cost of either resolving unit $i\in s_{1h,\text{nr}}$ or reaching the maximum number of call attempts, when that unit is selected in $s_2,$ and ${\tilde{c}}_{hik}$ is the expected cost of the $k^{\text{th}}$ attempt, $k\le K,$ for that unit. Given $c_{hik}\ne 0$ only if unit $i$ has not been resolved before the $k^{\text{th}}$ attempt, it is easy to see that the expected cost ${\tilde{c}}_{hik}$ is

$${\tilde{c}}_{hik}={\left(P_{2hi}^{\left(3\right)}\right)}^{k-1}\left(c^{\left(1\right)}{P}_{2hi}^{\left(1\right)}+c^{\left(2\right)}{P}_{2hi}^{\left(2\right)}+c^{\left(3\right)}{P}_{2hi}^{\left(3\right)}\right).$$

The expected cost ${\tilde{c}}_{hi}\left(K\text{}\right)$ reduces to

$$\begin{array}{l}{\tilde{c}}_{hi}\left(K\text{}\right)={\displaystyle {\sum }_{k=1}^K{\tilde{c}}_{hik}}\\ =\left(c^{\left(1\right)}{P}_{2hi}^{\left(1\right)}+c^{\left(2\right)}{P}_{2hi}^{\left(2\right)}+c^{\left(3\right)}{P}_{2hi}^{\left(3\right)}\right){\displaystyle {\sum }_{k=0}^{K-1}{\left(P_{2hi}^{\left(3\right)}\right)}^k}\\ =\left(c^{\left(1\right)}{P}_{2hi}^{\left(1\right)}+c^{\left(2\right)}{P}_{2hi}^{\left(2\right)}+c^{\left(3\right)}{P}_{2hi}^{\left(3\right)}\right)\frac{1-{\left(P_{2hi}^{\left(3\right)}\right)}^K}{1-P_{2hi}^{\left(3\right)}}.(3.2)\end{array}$$

Using ${\pi }_{2hi}=n_2{\pi }_{2hi}^{*}$ along with condition (2.2), we can determine the follow-up sample size necessary to expend the budget $C,$ on average, while ensuring each unit is resolved or has reached the maximum number of attempts, $K.$ That is, we can determine the follow-up sample size such that the expected follow-up cost (3.1) is exactly equal to the budget $C.$ This sample size is

$$n_2\left(C,K\right)=\frac{C}{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}{\tilde{c}}_{hi}\left(K\text{}\right)}}},(3.3)$$

where ${\tilde{c}}_{hi}\left(K\text{}\right)$ is given in (3.2). For a fixed budget $C,$ the sample size $n_2\left(C,K\right)$ is inversely related to $K$ and is a minimum when $K=\infty ;$ i.e., when there is no upper limit on the number of calls. This means that, for a fixed cost $C,$ choosing a sample size larger than $n_2\left(C,\infty \right)$ has an effect similar to reducing the value of $K,$ thereby increasing the expected number of unresolved units. Also, if a sample size smaller than $n_2\left(C,\infty \right)$ is chosen, the expected cost (3.1) is smaller than the budget $C;$ i.e., on average, the budget is not entirely expended. The sample size $n_2\left(C,\infty \right)$ is thus the minimum sample size that expends the budget $C,$ on average.

From the sample size $n_2\left(C,K\right)$ in (3.3), the expected number of respondents to the follow-up survey is

$$\begin{array}{l}{\tilde{n}}_{2r}\left(C,K\right)={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}{p}_{2hi}\left(K\text{}\right)}}\\ =C\frac{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}{p}_{2hi}\left(K\text{}\right)}}}{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}{\tilde{c}}_{hi}\left(K\text{}\right)}}},(3.4)\end{array}$$

where $p_{2hi}\left(K\text{}\right)$ is given in (2.7), and the expected response rate is

$$\frac{{\tilde{n}}_{2r}\left(C,K\right)}{n_2\left(C,K\right)}={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}{p}_{2hi}\left(K\text{}\right)}}.(3.5)$$

From (2.7) and (3.5), we observe that the expected response rate does not depend on the budget $C$ and decreases as $K$ decreases. It was noted above that choosing a sample size larger than the minimum sample size $n_2\left(C,\infty \right),$ for a fixed cost $C,$ has an effect similar to reducing the value of $K.$ Consequently, choosing a sample size larger than $n_2\left(C,\infty \right)$ would also have the effect of reducing the expected response rate.

We can also obtain the expected number of resolved units in a way similar to (3.4) as

$${\tilde{n}}_{2,\text{res}}\left(C,K\right)=C\frac{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}\left(1-{\left(P_{2hi}^{\left(3\right)}\right)}^K\right)}}}{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}{\tilde{c}}_{hi}\left(K\text{}\right)}}}.(3.6)$$

It can be easily seen that ${\tilde{n}}_{2,\text{res}}\left(C,K\right)\le n_2\left(C,K\right)$ and that ${\tilde{n}}_{2,\text{res}}\left(C,\infty \right)=n_2\left(C,\infty \right).$ If the follow-up sample size is chosen to be smaller than $n_2\left(C,\infty \right)$ then the expected cost ${\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}{\tilde{c}}_{hi}}\left(\infty \text{}\right)}=C^{*},$ with $C^{*}<C,$ and, from (3.4) and (3.6), both the expected number of respondents and resolved units decrease.

If the probability $P_{2hi}^{\left(3\right)}$ is very close to 1 for a few units $i\in s_{1h,\text{nr}},$ $h=1,\dots ,L,$ the minimum sample size $n_2\left(C,\infty \right)$ could become very small. In this situation, it may be appropriate to choose a finite value of $K$ to avoid spending too large a portion of the budget on a few units. This would reduce the expected response rate, as noted above, and possibly increase the bias of estimates. However, using a finite value of $K$ might also significantly increase the expected number of respondents and reduce the variance of estimates. Plots of the expected response rate and the expected number of respondents as a function of $K$ may be useful to determine a suitable trade-off between the maximization of the expected response rate $\left(K=\infty \right)$ and the maximization of the expected number of respondents, which could be reached at a finite value of $K.$ A small reduction of the expected response rate might be tolerated if it yields a significant increase in the expected number of respondents.

Under uniform follow-up response, we have: $P_{2hi}^{\left(1\right)}=P_2^{\left(1\right)},$ $P_{2hi}^{\left(2\right)}=P_2^{\left(2\right)}$ and $P_{2hi}^{\left(3\right)}=P_2^{\left(3\right)},$ for each unit $i\in s_{1h,\text{nr}},$ $h=1,\dots ,L.$ The follow-up sample size (3.3), the expected number of respondents (3.4), the expected response rate (3.5) and the expected number of resolved units (3.6) reduce to

$$n_2\left(C,K\right)=\frac{C}{\left(c^{\left(1\right)}{P}_2^{\left(1\right)}+c^{\left(2\right)}{P}_2^{\left(2\right)}+c^{\left(3\right)}{P}_2^{\left(3\right)}\right)}\frac{1-P_2^{\left(3\right)}}{1-{\left(P_2^{\left(3\right)}\right)}^K},(3.7)$$

$${\tilde{n}}_{2r}\left(C,K\right)=\frac{C}{\left(c^{\left(1\right)}{P}_2^{\left(1\right)}+c^{\left(2\right)}{P}_2^{\left(2\right)}+c^{\left(3\right)}{P}_2^{\left(3\right)}\right)}{P}_2^{\left(1\right)},(3.8)$$

$$\frac{{\tilde{n}}_{2r}\left(C,K\right)}{n_2\left(C,K\right)}=P_2^{\left(1\right)}\frac{1-{\left(P_2^{\left(3\right)}\right)}^K}{1-P_2^{\left(3\right)}},(3.9)$$

and

$${\tilde{n}}_{2,\text{res}}\left(C,K\right)=\frac{C}{\left(c^{\left(1\right)}{P}_2^{\left(1\right)}+c^{\left(2\right)}{P}_2^{\left(2\right)}+c^{\left(3\right)}{P}_2^{\left(3\right)}\right)}\left(1-P_2^{\left(3\right)}\right),(3.10)$$

respectively. It is worth pointing out that the expected number of respondents (3.8) and the expected number of resolved units (3.10) no longer depend on $K.$ The expected number of resolved units, ${\tilde{n}}_{2,\text{res}}\left(C,K\right),$ is therefore equal to the minimum sample size to expend the budget $C,$ $n_2\left(C,\infty \right),$ for every value of $K.$ As noted for the general expected response rate (3.5), the expected response rate (3.9) does not depend on the budget $C$ and decreases as $K$ decreases. Given the above observations, the value of $K$ that maximizes both the expected response rate and the expected number of respondents is $K=\infty$ under uniform response, which leads to choosing the sample size $n_2\left(C,\infty \right).$

The probabilities $P_{2hi}^{\left(1\right)},$ $P_{2hi}^{\left(2\right)}$ and $P_{2hi}^{\left(3\right)}$ are unknown. In practice, these probabilities must be replaced with estimates in the above expressions. Because they are needed before selecting the follow-up sample and collecting data, estimates of $P_{2hi}^{\left(1\right)},$ $P_{2hi}^{\left(2\right)}$ and $P_{2hi}^{\left(3\right)}$ could be obtained from previous survey data.

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