Reducing the response imbalance: Is the accuracy of the survey estimates improved? Section 4. The regression aspectReducing the response imbalance: Is the accuracy of the survey estimates improved? Section 4. The regression aspect

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The imbalance (IMB) is determined by the auxiliary vector $x$ with no attention paid to the study variable $y.$ But the relation of $x$ to $y$ is also important for the bias of estimated $y‑$totals. Strong regression of $y$ on $x$ is likely to give small bias, intuitively because regression predicted $y‑$values can then give close substitutes for those missing. For some survey data, the strength of the regression may be modest but nevertheless important in its effect on bias. The ordinary linear regression coefficient vectors for the whole sample $s$ and for the response $r$ are, respectively,

$$b_s={\left({\displaystyle {\sum }_s{d}_k{x}_k{x}_k^{\prime }}\right)}^{-1}{\displaystyle {\sum }_s{d}_k{x}_k{y}_k};b_r={\left({\displaystyle {\sum }_r{d}_k{x}_k{x}_k^{\prime }}\right)}^{-1}{\displaystyle {\sum }_r{d}_k{x}_k{y}_k}.(4.1)$$

Under nonresponse, $b_r$ is computable but not $b_s.$ The $J\times J$ matrices to invert are assumed non-singular. Normally $b_r\ne b_s,$ perhaps with considerable (but unknown) difference. The regression based on the response is inconsistent.

The imbalance in the $y‑$variable is ${\overline{y}}_r-{\overline{y}}_s,$ where the means are ${\overline{y}}_s={\displaystyle {\sum }_s{d}_k{y}_k}/{\displaystyle {\sum }_s{d}_k}$ for the sample (unknown) and ${\overline{y}}_r={\displaystyle {\sum }_r{d}_k{y}_k}/{\displaystyle {\sum }_r{d}_k}$ for the response (computable). The decomposition

$${\overline{y}}_r-{\overline{y}}_s={\left({\overline{x}}_r-{\overline{x}}_s\right)}^{\prime }{b}_r+{\left(b_r-b_s\right)}^{\prime }{\overline{x}}_s,(4.2)$$

highlights two undesirable differences, ${\overline{x}}_r-{\overline{x}}_s$ (due to imbalance in the $x‑$vector), and $b_r-b_s$ (due to inconsistent regression); to obtain (4.2) note that ${\overline{x}}_r^{\prime }{b}_r={\overline{y}}_r$ and ${\overline{x}}_s^{\prime }{b}_s={\overline{y}}_s,$ which are consequences of the $x‑$vector condition (2.2).

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