Non-response follow-up for business surveys
Section 2. Proposed follow-up strategy

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$${\widehat{Y}}_{\text{FULL}}={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h}}{w}_{1hi}{y}_{hi}}},(2.1)$$

where $w_{1hi}=1/{\pi }_{1hi}$ denotes the design weight associated with $i\in s_{1h}.$

In practice, not all sampled units respond to the mail-out. Suppose that, after a certain period of time, $n_{1hr}$ of the $n_{1h}$ sampled units respond in stratum $h.$ We denote the set of respondents in stratum $h$ by $s_{1hr},$ and the response probability for unit $i\in s_{1h}$ by $p_{1hi}.$ A sample of $n_2$ units, $s_2,$ is then selected from the set of all non-respondents to the mail-out, $s_{1,\text{nr}}.$ We denote by $s_{2h},$ the set of $n_{2h}$ units selected for a follow-up in stratum $h$ among the set of non-respondents to the mail-out in stratum $h,$ $s_{1h,\text{nr}}.$ We denote the probability that the mail-out non-respondent $i\in s_{1h,nr}$ is selected in the follow-up sample $s_2$ by ${\pi }_{2hi}.$ We assume that this probability can be written as ${\pi }_{2hi}=n_2{\pi }_{2hi}^{*},$ where ${\pi }_{2hi}^{*}$ does not depend on the follow-up sample size $n_2$ and satisfy the condition

$${\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,\text{nr}}}{\pi }_{2hi}^{*}}=1}.(2.2)$$

This condition is satisfied for simple random sampling, stratified simple random sampling, with proportional or Neyman allocation, and probability proportional to size sampling.

Units of the sample $s_2$ are followed up via telephone. If all $n_{2h}$ units respond to the follow-up, $h=1,\dots ,L,$ the unbiased Hansen and Hurwitz (1946) estimator of the population total $Y$ can be used:

$${\widehat{Y}}_{\text{HH}}={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1hr}}{w}_{1hi}{y}_{hi}}}+{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{2h}}{w}_{1hi}{w}_{2hi}{y}_{hi}}},(2.3)$$

where $w_{2hi}=1/{\pi }_{2hi}$ is the follow-up design weight of unit $i\in s_{2h}.$ The objective of the sample $s_2$ is to estimate the unknown total ${\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1h,nr}}{w}_{1hi}{y}_{hi}}}.$ If a variable $x$ strongly related to the variable of interest $y$ is available before sample selection for all the mail-out non-respondents, it seems natural to use $w_{1hi}{x}_{hi}$ as an auxiliary variable for stratification or as a size measure for probability proportional to size sampling.

As pointed out by a reviewer, it is important to wait until mail-out data collection is closed before selecting the follow-up sample. If units respond to the mail-out after the follow-up sample has been selected, some decisions on how to handle these late respondents are required. If they are not discarded, it may be difficult to obtain an unbiased estimator like (2.3) without introducing model assumptions (see Beaumont, Bocci and Hidiroglou, 2014). This issue may also have implications on the length of the collection period.

As pointed out in the introduction, it is unlikely that all the follow-up sample units will respond. Suppose that after the end of the data collection period, $n_{2hr}$ units have responded to the follow-up in stratum $h.$ We denote by $s_{2hr}$ the set of the $n_{2hr}$ respondents in stratum $h.$ We consider the non-response-adjusted version of the Hansen and Hurwitz (1946) estimator:

$${\widehat{Y}}_{\text{HH}-\text{NA}}={\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{1hr}}{w}_{1hi}{y}_{hi}}}+{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{i\in s_{2hr}}{w}_{1hi}{w}_{2hi}{a}_{2hi}{y}_{hi}}},(2.4)$$

where $a_{2hi}$ is a non-response weight adjustment. Under uniform non-response, a suitable weight adjustment is the inverse of the overall weighted response rate:

$$a_{2hi}=a_2=\frac{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{j\in s_{2h}}{w}_{1hj}{w}_{2hj}}}}{{\displaystyle {\sum }_{h=1}^L{\displaystyle {\sum }_{j\in s_{2hr}}{w}_{1hj}{w}_{2hj}}}},i\in s_{2hr},h=1,\dots ,L.(2.5)$$

A less restrictive assumption is uniform non-response within strata. Under this assumption, a suitable weight adjustment would be the inverse of the stratum weighted response rate:

$$a_{2hi}=a_{2h}=\frac{{\displaystyle {\sum }_{j\in s_{2h}}{w}_{2hj}}}{{\displaystyle {\sum }_{j\in s_{2hr}}{w}_{2hj}}},i\in s_{2hr},h=1,\dots ,L.(2.6)$$

Note that the non-response weight adjustment (2.6) is computable only if $n_{2hr}>0$ for all strata. Alternatively, unweighted versions of (2.5) and (2.6) could also be considered.

As mentioned earlier, follow-up of non-respondents who have been selected in $s_2$ is performed via telephone. In our proposed data collection procedure, a calling queue is first created by randomly ordering units in $s_2.$ These units are then called sequentially until the queue is empty or the entire follow-up budget has been expended, whichever comes first. Each call attempt made to units in $s_2$ results in one of these three outcomes:

1. Response: A response is obtained from the unit. The unit is removed from the calling queue so that it does not get called again.
2. Final non-response: The unit is finalized as a non-respondent; it should not be called back again and is removed from the calling queue. The most common example of this outcome is a refusal to respond to the survey.
3. Still in progress: The unit is not finalized and needs to be called again; it is therefore returned to the end of the calling queue. An example of this outcome is an attempt where no contact is made or an attempt where an appointment is made for a callback.

The “response” and “final non-response” outcomes are both final outcomes, in the sense that the unit is removed from the calling queue and the collection process. This is in contrast to the “still-in-progress” outcome where the unit is returned to the calling queue so that it can be called again. A unit that completes the data collection process with an outcome of “response” or “final non-response” is said to be finalized or resolved, otherwise, it is said to be unresolved. There are two types of non-respondents after data collection: i) Finalized units with a “final non-response” outcome; and ii) Unresolved units. Both types of non-respondents are accounted for in estimation using the non-response-adjusted estimator (2.4).

We assume that, for a given sample unit, the outcomes of the call attempts are independent, and the probability associated with each of the three possible outcomes remains constant throughout the entire data collection period. For a given sampled unit $i\in s_{2h},$ $h=1,\dots ,L,$ the probability of a “response” is denoted as $P_{2hi}^{\left(1\right)},$ the probability of a “final non-response” is denoted as $P_{2hi}^{\left(2\right)},$ and the probability of a “still-in-progress” outcome is denoted as $P_{2hi}^{\left(3\right)}.$ In practice, the independence and constant probability assumptions may not hold exactly. The independence assumption is expected to be more plausible if the probabilities are conditional on strong predictors and if the time gap between two successive call attempts on the same unit is not too short. The constant probability assumption is not satisfied when the probabilities depend on predictors that can vary during data collection, such as the time of day or day of the week of the call attempt. Although it might be possible to extend our model to time-varying predictors, it would complicate our theoretical developments and simulation study. These assumptions are made throughout the paper to simplify our analyses. This is a limitation of our investigations that should be kept in mind when interpreting our results.

Multiple phone call attempts may be necessary to reach and resolve a unit. Data collection managers may wish to impose an upper limit on the number of call attempts that can be made to any follow-up sample unit. If a unit is still in progress after reaching the limit, it is removed from the calling queue and remains unresolved at the end of data collection. Let $K$ be that upper limit on the number of call attempts. Assuming each unresolved unit at the end of data collection always reaches the maximum number of attempts $K,$ the probability that unit $i\in s_{1h,\text{nr}}$ responds when selected in the sample $s_2$ can be written as $p_{2hi}\left(K\right)={\displaystyle {\sum }_{k=1}^K{p}_{2hik}},$ where $p_{2hik}$ is the probability that unit $i\in s_{1h,\text{nr}}$ responds exactly at the $k^{\text{th}}$ attempt when selected in $s_2.$ Under our assumptions, it is easy to see that $p_{2hik}={\left(P_{2hi}^{\left(3\right)}\right)}^{k-1}{P}_{2hi}^{\left(1\right)}.$ As a result, we have

$$\begin{array}{c}{p}_{2hi}\left(K\right)={\displaystyle {\sum }_{k=1}^K{p}_{2hik}}\\ =P_{2hi}^{\left(1\right)}{\displaystyle {\sum }_{k=0}^{K-1}{\left(P_{2hi}^{\left(3\right)}\right)}^k}\\ =P_{2hi}^{\left(1\right)}\frac{1-{\left(P_{2hi}^{\left(3\right)}\right)}^K}{1-P_{2hi}^{\left(3\right)}}.(2.7)\end{array}$$

In the next section, equation (2.7) will be used to determine an appropriate follow-up sample size.

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