Reducing the response imbalance: Is the accuracy of the survey estimates improved? Section 3. ImbalanceReducing the response imbalance: Is the accuracy of the survey estimates improved? Section 3. Imbalance

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The concept of balance has been often used in statistical literature with reference to an equality of means of specified variables for two sets of units, one a subset of the other. One method to realize a probability sample $s$ from $U$ that is balanced with respect to a vector $x$ is the Cube Method, see Deville and Tillé (2004). In the context with nonresponse, we want to know how well balanced a response $r$ is, compared with the probability sample $s$ that would have given unbiased estimates. A given auxiliary vector $x$ has computable means ${\overline{x}}_r={\displaystyle {\sum }_r{d}_k{x}_k}/{\displaystyle {\sum }_r{d}_k}$ for the response and ${\overline{x}}_s={\displaystyle {\sum }_s{d}_k{x}_k}/{\displaystyle {\sum }_s{d}_k}$ for the sample. If they are equal, an unlikely outcome, the response is perfectly balanced with respect to $x.$ The contrast between response $r$ and sample $s$ can be measured by the scalar quantities

$$Q_s={\left({\overline{x}}_r-{\overline{x}}_s\right)}^{\prime }{\Sigma }_s^{-1}\left({\overline{x}}_r-{\overline{x}}_s\right);Q_r={\left({\overline{x}}_r-{\overline{x}}_s\right)}^{\prime }{\Sigma }_r^{-1}\left({\overline{x}}_r-{\overline{x}}_s\right).(3.1)$$

They differ only in the $J\times J$ weighting matrix, ${\Sigma }_s={\displaystyle {\sum }_s{d}_k{x}_k{x}_k^{\prime }}/{\displaystyle {\sum }_s{d}_k}$ as opposed to ${\Sigma }_r={\displaystyle {\sum }_r{d}_k{x}_k{x}_k^{\prime }}/{\displaystyle {\sum }_r{d}_k},$ both assumed non-singular. In particular, $Q_s$ is important for the statistic called imbalance of $r$ with respect to the specified $x‑$vector:

$$IMB\left(r,x|s\right)=P^2{\left({\overline{x}}_r-{\overline{x}}_s\right)}^{\prime }{\Sigma }_s^{-1}\left({\overline{x}}_r-{\overline{x}}_s\right)=P^2{Q}_s,(3.2)$$

where $P$ is the response rate (2.1); see for example Särndal and Lundquist (2014a). The full notation $IMB\left(r,x|s\right)$ emphasizes that imbalance depends on the realized response $r$ and on the choice of $x‑$vector. Unless required for emphasis, we use the simpler notation $IMB.$ We have $0\le IMB\le P\left(1-P\right)$ for any $r$ and vector formulation $x,$ given $s.$ $IMB$ is a descriptive measure of the response $r.$ It is related to a special case of the $\text{R}‑$indicator, whose motivation lies instead in the estimation of (the unknown) response probabilities for the population units, see for example Bethlehem et al. (2011).

The $IMB$ statistic (3.2) can be continuously computed and monitored in a data collection extending over a period of time, say several days or weeks, during which contact attempts continue with a sample unit until desired data are obtained, or, if this fails, until the unit is declared a non-respondent. As the response rate $P$ grows, $IMB$ serves as a tool for monitoring and managing the data collection to achieve in the end a response set $r$ which, if not perfectly balanced to satisfy ${\overline{x}}_r={\overline{x}}_s,$ will at least have considerably lower $IMB$ than if balancing had not been attempted in data collection. There are methods for balancing based on response propensity, such as the Threshold method and the Equal proportions method in Särndal and Lundquist (2014a, 2014b).

We consider later the particular case where $s$ is a self-weighting sample (as when $s$ is a simple random sample), the response $r$ has fixed size $m,$ and $x$ is a group vector of dimension $J$ as defined in Section 2. Then both $s$ and $r$ are split into $J$ non-overlapping groups. For the sample group $s_j,$ denote by $n_j$ the size and by $W_{js}=n_j/n$ the relative size; ${\displaystyle {\sum }_{j=1}^J{n}_j=n}.$ For the response group $r_j,$ let $m_j\le n_j$ be the size; ${\displaystyle {\sum }_{j=1}^J{m}_j=m}.$ The imbalance (3.2) is then

$$IMB={\displaystyle {\sum }_{j=1}^J{W}_{js}{\left(p_j-p\right)}^2},(3.3)$$

where the response rates are $p_j=m_j/n_j$ in group $j$ and $p=m/n$ overall. (The response rate $P$ is defined in (2.1) with general design weights $d_k;$ for a self-weighting sample, where $d_k$ is constant, we use small $p$ for the response rate.) If $IMB=0,$ we have perfect balance; all group response rates $p_j$ are then equal.

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