2. Modified regression estimation

John Preston

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Consider a finite population $U^{\left(t\right)}$ at time $t$ partitioned into $H$ non-overlapping strata $U_1^{\left(t\right)},\dots ,U_h^{\left(t\right)},\dots ,U_H^{\left(t\right)},$ where $U_h^{\left(t\right)}$ is comprised of $N_h^{\left(t\right)}$ units. A simple random sample without replacement $s_h^{\left(t\right)}$ of $n_h^{\left(t\right)}$ units is selected with inclusion probabilities ${\pi }_i^{\left(t\right)}=n_h^{\left(t\right)}/N_h^{\left(t\right)}\left(i\in U_h^{\left(t\right)}\right)$ within each stratum $h$ at time $t,$ leading to a total sample $s^{\left(t\right)}={\displaystyle {\cup }_{h=1}^H{s}_h^{\left(t\right)}}$ of size $n^{\left(t\right)}={\displaystyle {\sum }_{h=1}^H{n}_h^{\left(t\right)}}.$ An unbiased estimate of the population total $Y^{\left(t\right)}={\displaystyle {\sum }_{h=1}^H{\displaystyle {\sum }_{i\in U_h^{\left(t\right)}}{y}_i^{\left(t\right)}}}$ is given by the Horvitz-Thompson (HT) estimator ${\widehat{Y}}_{\text{HT}}^{\left(t\right)}={\displaystyle {\sum }_{h=1}^H{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}}{w}_i^{\left(t\right)}{y}_i^{\left(t\right)}}},$ where $w_i^{\left(t\right)}=1/{\pi }_i^{\left(t\right)}$ is the design weight for unit $i$ at time $t$ and $y_i^{\left(t\right)}$ is the value for the variable of interest $y$ for unit $i$ at time $t.$ Assume that there exists a set of auxiliary variables $x^{\left(t\right)}$ at time $t$ for which the population totals $X^{\left(t\right)}={\displaystyle {\sum }_{i\in U^{\left(t\right)}}{x}_i^{\left(t\right)}}$ are known and $x_i^{\left(t\right)}$ are known for every $i\in s^{\left(t\right)}.$

The generalised regression (GR) estimator (Särndal, Swensson and Wretman 1992) is a model assisted estimator, designed to improve the accuracy of the estimates by using auxiliary variables that are correlated with the variable of interest. The GR estimator is given by:

$${\widehat{Y}}_{\text{GR}}^{\left(t\right)}={\widehat{Y}}_{\text{HT}}^{\left(t\right)}+{\left(X^{\left(t\right)}-{\widehat{X}}_{\text{HT}}^{\left(t\right)}\right)}^T{\widehat{\beta }}_{\text{GR}}^{\left(t\right)}(2.1)$$

where ${\widehat{\beta }}_{\text{GR}}^{\left(t\right)}$  is the vector of linear regression model parameters given by:

$${\widehat{\beta }}_{\text{GR}}^{\left(t\right)}={\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}{x}_i^{\left(t\right)}{x}_i^{\left(t\right)}{}^T}{c_i^{\left(t\right)}}}\right)}^{-1}\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}{x}_i^{\left(t\right)}{y}_i^{\left(t\right)}}{c_i^{\left(t\right)}}}\right)(2.2)$$

and $c_i^{\left(t\right)}$ are specified factors that relate to the variance structure of the linear regression model associated with the GR estimator $y_i^{\left(t\right)}=x_i^{\left(t\right)}{}^T{\widehat{\beta }}_{\text{GR}}^{\left(t\right)}+{\epsilon }_i^{\left(t\right)},$ with $E\left({\epsilon }_i^{\left(t\right)}\right)=0,$ $\text{Var}\left({\epsilon }_i^{\left(t\right)}\right)=c_i^{\left(t\right)}{\sigma }^2$ and $\text{Cov}\left({\epsilon }_i^{\left(t\right)},{\epsilon }_j^{\left(t\right)}\right)=0$ for all $i\ne j.$ The GR estimator can also be written as:

$${\widehat{Y}}_{\text{GR}}^{\left(t\right)}={\displaystyle \sum _{i\in s^{\left(t\right)}}{\tilde{w}}_i^{\left(t\right)}{y}_i^{\left(t\right)}}(2.3)$$

where ${\tilde{w}}_i^{\left(t\right)}=w_i^{\left(t\right)}{\tilde{g}}_i^{\left(t\right)}$ and ${\tilde{g}}_i^{\left(t\right)}$ is the $g-$ weight for unit $i$ at time $t$ given by:

$${\tilde{g}}_i^{\left(t\right)}=1+{\left(X^{\left(t\right)}-{\widehat{X}}_{\text{HT}}^{\left(t\right)}\right)}^T{\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}{x}_i^{\left(t\right)}{x}_i^{\left(t\right)}{}^T}{c_i^{\left(t\right)}}}\right)}^{-1}\frac{x_i^{\left(t\right)}}{c_i^{\left(t\right)}}.(2.4)$$

At time $t>1$ define a set of composite auxiliary variables $z^{\left(t\right)}$ for which "pseudo-benchmark�, totals ${\tilde{Z}}^{\left(t\right)}$ (based on the key survey estimates at time $t-1$) are known and $z_i^{\left(t\right)}$ can be derived for every $i\in s^{\left(t\right)}.$ The modified regression (MR) estimator is the GR estimator where the variables in the regression model are the auxiliary variables $x^{\left(t\right)}$ and the composite auxiliary variables $z^{\left(t\right)}.$ The MR estimator given by:

$${\widehat{Y}}_{\text{MR}}^{\left(t\right)}={\widehat{Y}}_{\text{HT}}^{\left(t\right)}+{\left(\left(X^{\left(t\right)},{\tilde{Z}}^{\left(t\right)}\right)-\left({\widehat{X}}_{\text{HT}}^{\left(t\right)},{\widehat{Z}}_{\text{HT}}^{\left(t\right)}\right)\right)}^T{\widehat{\beta }}_{\text{MR}}^{\left(t\right)}(2.5)$$

where ${\widehat{\beta }}_{\text{MR}}^{\left(t\right)}$ is the vector of linear regression model parameters given by:

$${\widehat{\beta }}_{\text{MR}}^{\left(t\right)}={\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right){\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right)}^T}{c_i^{\left(t\right)}}}\right)}^{-1}\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right)y_i^{\left(t\right)}}{c_i^{\left(t\right)}}}\right).(2.6)$$

The MR estimator can also be written as:

$${\widehat{Y}}_{\text{MR}}^{\left(t\right)}={\displaystyle \sum _{i\in s^{\left(t\right)}}{\stackrel{⌣}{w}}_i^{\left(t\right)}{y}_i^{\left(t\right)}}(2.7)$$

where ${\stackrel{⌣}{w}}_i^{\left(t\right)}=w_i^{\left(t\right)}{\stackrel{⌣}{g}}_i^{\left(t\right)}$ and ${\stackrel{⌣}{g}}_i^{\left(t\right)}$ is the $g-$ weight for unit $i$ at time $t$ given by:

$$\begin{array}{lll}{\stackrel{⌣}{g}}_i^{\left(t\right)}\hfill & =\hfill & 1+{\left(\left(X^{\left(t\right)},{\tilde{Z}}^{\left(t\right)}\right)-\left({\widehat{X}}_{\text{HT}}^{\left(t\right)},{\widehat{Z}}_{\text{HT}}^{\left(t\right)}\right)\right)}^T\hfill \\ \hfill & \times \hfill & {\left({\displaystyle \sum _{i\in s^{\left(t\right)}}\frac{w_i^{\left(t\right)}\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right){\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right)}^T}{c_i^{\left(t\right)}}}\right)}^{-1}\frac{\left(x_i^{\left(t\right)},z_i^{\left(t\right)}\right)}{c_i^{\left(t\right)}}.(2.8)\hfill \end{array}$$

The key to the effectiveness of MR estimator is the definition of the composite auxiliary variables. Ideally, the values for the composite auxiliary variables at time $t,$ would be equal to the values for the key survey variables at time $t-1.$ However, due to the rotation of units into and out of sample from one time period to the next, values for the key survey variables at time $t-1$ will be missing by design for those units in the sample at time $t$ which were not in the sample at time $t-1.$

There are several possible techniques available to define the composite auxiliary variables. The earliest modified regression estimators were the MR1 estimator (Singh and Merkouris 1995; Singh 1996), and the MR2 estimator (Singh, Kennedy, Wu and Brisebois 1997) which used values for the composite auxiliary variables given respectively by:

$$z_{\left(\text{MR}1\right)i}^{\left(t\right)}=\{\begin{array}{ll}{y}_i^{\left(t-1\right)},\hfill & \text{if }i\in s_h^{\left(t\right)}\cap s_h^{\left(t-1\right)}\hfill \\ {\overline{Y}}_{\left(\text{MR}\right)h}^{\left(t-1\right)},\hfill & \text{if }i\in s_h^{\left(t\right)}\backslash s_h^{\left(t-1\right)}\hfill \end{array}(2.9)$$

$$z_{\left(\text{MR}2\right)i}^{\left(t\right)}=\{\begin{array}{ll}{y}_i^{\left(t\right)}+\left({\displaystyle {\sum }_{i\in s_h^{\left(t\right)}}{w}_i^{\left(t\right)}}/{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}\cap s_h^{\left(t\right)-1}}{w}_i^{\left(t\right)}}\right)\left(y_i^{\left(t-1\right)}-y_i^{\left(t\right)}\right),\hfill & \text{if }i\in s_h^{\left(t\right)}\cap s_h^{\left(t-1\right)}\hfill \\ y_i^{\left(t\right)},\hfill & \text{if }i\in s_h^{\left(t\right)}\backslash s_h^{\left(t\right)-1}\hfill \end{array}(2.10)$$

and ${\overline{Y}}_{\left(\text{MR}\right)h}^{\left(t-1\right)}$ are the composite regression estimators of the population mean in stratum $h$ for key survey variables at time $t-1.$

The MR1 values for the composite auxiliary variables use a mean imputation method to impute for the missing values, while the MR2 values use a reverse historical imputation method to impute for the missing values and then modify the non-imputed values so that the HT estimator of the composite auxiliary variables ${\widehat{Z}}_{\text{HT}}^{\left(t\right)}={\displaystyle {\sum }_{h=1}^H{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}}{w}_i^{\left(t\right)}{z}_{\left(\text{MR}2\right)i}^{\left(t\right)}}}$ at time $t$ is unbiased for the corresponding key survey variables $Y^{\left(t-1\right)}$ at time $t-1.$

The MR1 estimator has been found to perform better for point-in-time estimates, while the MR2 estimator has been found to perform better for movement estimates. Fuller and Rao (2001) proposed an alternative estimator that provides a compromise between improving point-in-time estimates and improving movement estimates by using values for the composite auxiliary variables given by:

$$z_{\left(\text{MR}\right)i}^{\left(t\right)}=\left(1-\alpha \right)z_{\left(\text{MR}1\right)i}^{\left(t\right)}+\alpha z_{\left(\text{MR}2\right)i}^{\left(t\right)}.(2.11)$$

The composite auxiliary variable (2.11) requires a decision on the choice of $\alpha ,$ which will depend on the correlations over time for the key survey variables, and the relative importance of the point-in-time and movement estimates.

Beaumont and Bocci (2005) proposed a refinement to the composite auxiliary variable, which they proffered did not require an arbitrary choice of $\alpha :$

$$z_{\left(\text{MRR}\right)i}^{\left(t\right)}=\{\begin{array}{ll}{y}_i^{\left(t-1\right)},\hfill & \text{if }i\in s_h^{\left(t\right)}\cap s_h^{\left(t-1\right)}\hfill \\ y_i^{\left(t\right)}+\left({\displaystyle {\sum }_{i\in s_h^{\left(t\right)}\cap s_h^{\left(t-1\right)}}{w}_i^{\left(t\right)}\left(y_i^{\left(t-1\right)}-y_i^{\left(t\right)}\right)}/{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}\cap s_h^{\left(t-1\right)}}{w}_i^{\left(t\right)}}\right),\hfill & \text{if }i\in s_h^{\left(t\right)}\backslash s_h^{\left(t-1\right)}.\hfill \end{array}(2.12)$$

The MRR values for the composite auxiliary variables use a reverse historical imputation method to impute for the missing values and then modify the imputed values so that the HT estimator of the composite auxiliary variables ${\widehat{Z}}_{\text{HT}}^{\left(t\right)}={\displaystyle {\sum }_{h=1}^H{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}}{w}_i^{\left(t\right)}{z}_{\left(\text{MRR}\right)i}^{\left(t\right)}}}$ at time $t$ is unbiased for the corresponding key survey variables $Y^{\left(t-1\right)}$ at time $t-1.$

The MR estimators can deviate from the GR estimator over time (Fuller and Rao 2001). In a repeated survey this "drift� problem will be characterized by a substantial deviation which extends over time between the MR estimator and the GR estimators, while in a simulation study it will be characterized by a reduction over time in the relative efficiency of the MR estimator compared to the GR estimators. A potential solution to the "drift� problem would be to use a weighted average of the MR estimator and the GR estimator (Bell 1999) given by:

$${\widehat{Y}}_{\text{MRC}}^{\left(t\right)}=\alpha {\widehat{Y}}_{\text{GR}}^{\left(t\right)}+\left(1-\alpha \right){\widehat{Y}}_{\text{MR}}^{\left(t\right)}.(2.13)$$

The compromise modified regression (MRC) estimator should also provide a compromise between the efficiency gains in the point-in-time and movement estimates, as the MR estimators will generally perform better than the GR estimator for movement estimates, but will not always perform better for point-in-time estimates; in particular the MR2 and MRR estimators.

The MRC estimator requires a decision on the choice of $\alpha .$ Using linearization (or Taylor series) methods to approximate the variance of (2.13), a relatively straight forward expression for $\alpha$ can be found which minimises the variance on the movement estimates while maintaining the variance on the point-in-time estimates produced using GR estimator.

The current MR estimators perform best when units in the population are unchanged between the previous and current time periods. If there are significant changes in the population over time, then these modified regression estimators will be unsuitable in their present form, as these estimators can accrue serious biases over time. While a simple factor $\left({\displaystyle {\sum }_{i\in s_h^{\left(t-1\right)}}{w}_i^{\left(t-1\right)}}/{\displaystyle {\sum }_{i\in s_h^{\left(t\right)}}{w}_i^{\left(t\right)}}\right)$ could be applied to the MR1, MR2 and MRR values to account for the changes in the population size in stratum $h$ between time $t-1$ and time $t,$ these modified regression estimators still can accrue considerable biases over time.

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