Reducing the response imbalance: Is the accuracy of the survey estimates improved? Section 7. The first resultReducing the response imbalance: Is the accuracy of the survey estimates improved? Section 7. The first result

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Result 1 refers to the following survey context: A self-weighting sample $s$ of size $n$ is drawn from $U=\left\{1,\dots ,k,\dots ,N\right\};$ $d_k$ is the same for all $k.$ The auxiliary vector $x$ is a group vector of dimension $J,$ so the sample $s$ and the response set $r,$ assumed to be of fixed size $m<n,$ are split into $J$ non-overlapping groups. The notation for these is given at the end of Section 3. The values $y_k$ are treated as fixed, non-random, as is usual in the design-based tradition. If $y_k$ were observed for all $k\in s,$ then ${\widehat{Y}}_{FUL}=N{\overline{y}}_s$ with ${\overline{y}}_s={\displaystyle {\sum }_s{y}_k}/n$ would be design unbiased for the population $y‑$total $Y={\displaystyle {\sum }_U{y}_k}.$ But $y_k$ is available for $k\in r$ only; the CAL estimator (5.1) becomes ${\widehat{Y}}_{CAL}=N{\displaystyle {\sum }_{j=1}^J{W}_{js}{\overline{y}}_{r_j}},$ where ${\overline{y}}_{r_j}$ is the mean of respondent values $y_k$ in group $j.$ Statistical properties $–$ expected value and variance $–$ of $\left({\widehat{Y}}_{CAL}-{\widehat{Y}}_{FUL}\right)/N={\Delta }_r$ with ${\Delta }_r={\left(b_r-b_s\right)}^{\prime }{\overline{x}}_s={\displaystyle {\sum }_{j=1}^J{W}_{js}{\overline{y}}_{r_j}-{\overline{y}}_s}$ are given in Result 1 for the following probabilistic setting: All $\left(\begin{array}{c}{n}_{ }\\ m_{ }\end{array}\right)$ response sets $r$ of fixed size $m$ are assumed a priori equally probable. Given $s,$ the imbalance $IMB$ is determined by ${\overline{x}}_r=\left(1/m\right){\left(m_1,\dots ,m_j,\dots ,m_J\right)}^{\prime }.$ Given ${\overline{x}}_r,$ we are left with $R={\prod }_{j=1}^J\left(\begin{array}{c}{n}_j\\ m_j\end{array}\right)$ sets $r,$ all with the same non-zero probability $1/R$ and the same $IMB,$ given by (3.3). The other sets $r$ of size $m$ are no longer in scope. Conditioning on ${\overline{x}}_r$ allows us to study the properties of CAL as a function of $IMB.$ Result 1 involves the variance of the study variable $y,$ within-group and combined over groups:

$$S_{yj}^2={\displaystyle {\sum }_{s_j}{\left(y_k-{\overline{y}}_{s_j}\right)}^2}/\left(n_j-1\right);S_y^2={\displaystyle \sum _{j=1}^J{W}_{js}{S}_{yj}^2}.(7.1)$$

Result 1. Let $s$ be a self-weighting sample of size $n$ and let $x_k$ be a group vector of dimension $J.$ Assume that all $\left(\begin{array}{c}{n}_{ }\\ m_{ }\end{array}\right)$ response sets $r$ of fixed size $m$ are a priori equally probable. Then

$$\overline{\Delta }=\text{E}\left({\Delta }_r|{\overline{x}}_r,m,s\right)=0(7.2)$$

$$S_{\Delta }^2=\text{E}\left({\left({\Delta }_r-\overline{\Delta }\right)}^2|{\overline{x}}_r,m,s\right)=\left(\frac{1}{m}-\frac{1}{n}\right)S_y^2+\frac{1}{m}{\displaystyle \sum _{j=1}^J{W}_{js}\left(\frac{p}{p_j}-1\right)S_{yj}^2}(7.3)$$

where $W_{js}=n_j/n$ and $p_j=m_j/n_j$ are relative size and response rate, respectively, for group $j,$ $p=m/n$ is the overall response rate, and $S_y^2$ and $S_{yj}^2$ are given in (7.1). If response rates $p_j$ and variances $S_{yj}^2$ vary by little only over the groups, then

$$S_{\Delta }^2\approx \left(1-p+\frac{IMB}{p^2}\right)\frac{S_y^2}{m}(7.4)$$

where $IMB$ is given by (3.3).

For full response, when $r=s,$ the right hand sides of (7.3) and (7.4) are zero; the approximation in (7.4) is exact: $S_{\Delta }^2=0.$ To interpret Result 1, note that the first term on the right hand side of (7.3) is a constant, given $m.$ It states the conditional variance for a perfectly balanced response, where $p_j$ is the same for all groups. The second is the penalty term, namely the penalty for failing to get perfect balance in data collection. Its size depends on how well an adaptive design succeeds in generating group response rates $p_j$ that vary little only. It is zero if all $p_j$ can be made equal.

Formula (7.4) states that the variance $S_{\Delta }^2$ is decreasing with $IMB$ in a roughly linear fashion. Thus low imbalance brings improved chances for a small deviation ${\widehat{Y}}_{CAL}-{\widehat{Y}}_{FUL}=\widehat{N}{\Delta }_r.$ This is important for practice. To illustrate, for a nonresponse of $1-p=40$ per cent, $S_{\Delta }^2\approx 0.57S_y^2/m$ if $IMB=0.06,$ but if $IMB=0,$ as in perfect balance, that variance is reduced to $S_{\Delta }^2\approx 0.40S_y^2/m.$ The improvement is clear but cannot be claimed to be very large. This is because with most data, $IMB/p^2$ is small compared with a nonresponse $1-p$ of the order of 30 to 60 per cent, cases that we are mainly concerned with here. Thus taking action to reduce imbalance has a desirable effect, although modest rather than strong.

In (7.2) and (7.3), the expectation $\text{E}(\cdot )$ is taken by averaging over the $R={\displaystyle {\prod }_{j=1}^J\left(\begin{array}{c}{n}_j\\ m_j\end{array}\right)}$ equi-probable sets $r$ that remain out of $\left(\begin{array}{c}{n}_{ }\\ m_{ }\end{array}\right)$ after fixing ${\overline{x}}_r.$ It should also be noted that more than one ${\overline{x}}_r$ can give the same value $IMB.$ Hence there may be more than one value $S_{\Delta }^2$ for the same $IMB.$ The linearly increasing function of $IMB$ in (7.4) is nevertheless their common approximation.

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