4. Two-step calibration weighting

Phillip S. Kott and Dan Liao

Previous | Next

4.1 Calibration weighting in two steps

In practice, the components of $x_k$  are often 0/1 group-membership identifiers, and the groups are mutually exclusive and exhaustive. In that situation, $g^T{x}_k$  can only take on $P$  values. Almost any weight-adjustment function, $\alpha \left(g^T{x}_k\right),$  will yield equivalent results. An example is the linear function, $\alpha \left(g^T{x}_k\right)=1+g^T{x}_k,$  of Lundström and Särndal (1999).

One popular weight-adjustment function that sometimes cannot be used (note the italicized “almost” in the previous paragraph) is $\alpha \left(g^T{x}_k\right)=1+\exp \left(g^T{x}_k\right),$  which assumes response is a logistic function of $x_k.$  The problem is that this weight-adjustment function cannot return values less than unity. We noted in the previous section, that sometimes one may need ${\alpha }_k$  to be less than 1. A routine that tries to use $\alpha \left(g^T{x}_k\right)=1+\exp \left(g^T{x}_k\right)$  and fit the calibration equations will fail.

This can be a particular problem when assuming a logistic response model and trying to calibrate to the population in a single step. There may be a component of $z_k,$  say $z_{ka},$  that is always nonnegative, but the original sample and response set are such that ${\sum }_R{d}_k{z}_{ka}>{\sum }_U{z}_{ka}$  even though ${\sum }_R{d}_k{z}_{ka}$  cannot exceed ${\sum }_S{d}_k{z}_{ka}.$  Thus, calibrating to the population will always fail because no ${\alpha }_k$  can be less than 1.

Calibrating to the original sample, by contrast, need not fail, since ${\sum }_R{d}_k{z}_{ka}\le {\sum }_S{d}_k{z}_{ka}.$  This suggest that one calibrates first to the original sample, which removes the response bias if the assumed response model holds, and then to the population, which removes the response bias if the prediction model holds. Estevao and Särndal (2002) discuss a variety of ways to calibrate in steps, but we focus on a single method here.

A second advantage of calibration weighting in two steps can be realized even when the calibration variables used in both steps are the same or a subset of those used in the single step. This happens when the response model holds, and the linear prediction model is only roughly true. Some version or “optimal” estimation can then be used in the second calibration-weighting step to increase efficiency. Rao (1994) introduced the notion of the optimal regression estimator. It was put into calibration-weighting form and discussed further in Bankier (2002) and Kott (2009, Section 4.2). Detail and how this can be done are provided in Sections 4.2 and 5.

4.2 Estimation and variance estimation when calibrating in two steps

In this subsection, we start with a fairly general two-step calibration estimator for a total and then address estimating its variance. The first calibration-weighting step, which is to the original sample, employs $x_{1k}$  as the vector of response-model variables and $z_{1k}$  as the calibration vector. Each has $P_1$  components. The weight-adjustment function has the form described in equation (2.4) with $g_1$  now replacing $g.$  The calibration equation is ${\sum }_R{d}_k\alpha \left(g_1^T{x}_{1k}\right)z_{1k}={\sum }_S{d}_k{z}_{1k}.$

The second calibration-weighting step, which is to the population, employs $x_{2k}$  and $z_{2k},$  each with $P_2$  components. The nonresponse bias under the response model is removed in the first step. For the weight-adjustment function for the second step, we propose using

$$h_k\left(g_2^T{x}_{2k}\right)=\frac{{\ell }_k+\exp \left(g_2^T{x}_{2k}\right)}{1+\exp (g_2^T{x}_{2k})/u_k},(4.1)$$

where $u_k>{\ell }_k>0$  may be set almost at whim (but see below). The right-hand side of equation (4.1) can vary across the $k$  (and so can depend on $d_k$  and ${\alpha }_k),$  yet $h_k\left(0\right)={h^{\prime }}_k\left(0\right)=1,$  making it asymptotically indistinguishable from the linear function: $1+g_2^T{x}_{2k}.$  For simplicity, we will call $h_k\left(g_2^T{x}_{2k}\right)$  and ${h^{\prime }}_k\left(g_2^T{x}_{2k}\right),h_k$  and ${h^{\prime }}_k$  respectively. From a quasi-sampling-design viewpoint, both are asymptotically identical to unity. The second calibration equation is ${\sum }_S{d}_k{h}_k\left(g_2^T{x}_{2k}\right)z_{2k}={\sum }_U{z}_{2k}.$  Because this equation must hold, there are limits on the available choices for $u_k$  and ${\ell }_k$  in equation (4.1).

A good simultaneous variance estimator for $t_y={\sum }_R{w}_k{y}_k={\sum }_R{d}_k\alpha \left(g_1^T{x}_{1k}\right)h_k\left(g_2^T{x}_{2k}\right)y_k$  is (as we shall see)

$$\begin{array}{ll}v\left(t_y\right)=\hfill & {\displaystyle \sum _{k,j\in S}\left(1-\frac{{\pi }_k{\pi }_j}{{\pi }_{kj}}\right)\left[d_k\left(z_{1k}^T{b}_1+{\alpha }_k{h}_k{e}_{1k}\right)\right]}\left[d_j\left(z_{1j}^T{b}_1+{\alpha }_j{h}_j{e}_{1j}\right)\right]\hfill \\ \hfill & +{\displaystyle \sum _{k\in R}{d}_k\left(h_k^2{\alpha }_k^2-h_k{\alpha }_k\right)e_{1k}^2,}\hfill \end{array}(4.2)$$

where

$$e_{2k}=y_k-z_{2k}^T{\left({\displaystyle {\sum }_S{d}_j{\alpha }_j{h^{\prime }}_j{x}_{2j}{z}_{2j}^T}\right)}^{-1}{\displaystyle {\sum }_S{d}_j{\alpha }_j{h^{\prime }}_j{x}_{2j}{y}_j},(4.3)$$

$$b_1={\left({\displaystyle {\sum }_S{d}_f{{\alpha }^{\prime }}_f{x}_{1f}{z}_{1f}^T}\right)}^{-1}{\displaystyle {\sum }_S{d}_f{{\alpha }^{\prime }}_f{h}_f{x}_{1f}{e}_{2f},(4.4)}$$

and

$$e_{1k}=e_{2k}-x_{1k}^T{b}_1.(4.5)$$

Let $x_k$  now be the vector composed of the non-duplicated components of $x_{1k}$  and $x_{2k}$  and define $z_k$  analogously. Sufficient conditions for (4.2) to be a simultaneous variance estimator include the corresponding components of equation (4.1) depending on whether either the response model in equation (2.4) holds with $x_{1k}$  replacing $x_k$  or the prediction model is $\text{E}\left(y_k|x_k,z_k\right)=z_{2k}^T{\beta }_2,$  whether or not $k$  is sampled or responds if sampled, and the ${\epsilon }_{2k}=y_k-z_{2k}^T{\beta }_2$  are uncorrelated random variables with variances equal to ${\sigma }_{2k}^2=z_{2k}^T{\eta }_2,$  where ${\eta }_2$  need not be specified other than having finite components. Now, both $N^{-1}{\sum }_R{d}_k{\alpha }^{\prime }\left(g_1^T{x}_{1k}\right)z_{1k}{x}_{1k}^T$  and $N^{-1}{\sum }_R{d}_k{h^{\prime }}_k\left(g_2^T{x}_{2k}\right)z_{2k}{x}_{2k}^T$  are assumed to be of full rank and bounded as the sample size grows arbitrarily large.

The variance estimator in equation (4.2) is almost the same as the estimator in (3.1): $x_k$  has been replaced with $x_{1k}$  and $z_k$  with $z_{1k},$  while $h_k{e}_{2k}$  substitutes for $y_k$  (we will get to a small difference shortly). Observe that $e_{2k}$  is effectively an expression of the “residual” from the second calibration-weighting step. This residual is multiplied by the weight-adjustment factor $h_k,$  which is asymptotically unity from the quasi-sampling-design-based perspective and a constant from the prediction-model viewpoint. The product is then used to create the first-step “regression-coefficient” $b_1$  in equation (4.4) and its accompanying “residual” $e_{1k}$  in equation (4.5). We do the second step regression first because $t_y-T_y={\sum }_R{w}_k{y}_k-{\sum }_U{y}_k={\sum }_R{w}_k{e}_{2k}-{\sum }_U{e}_{2k}.$

It is for estimating the prediction model of $t_y$  as an estimator of $T_y,{\sum }_S\left(w_k^2-w_k\right){\sigma }_{2k}^2,$  that the last appearance of $h_k$  on the right-hand side of equation (4.2) is not squared, as it would be if $h_k{e}_{2k}$  substituted for $y_k$  everywhere. From a quasi-design viewpoint, $h_k$  is asymptotically identical to unity, so whether or not it is squared makes no asymptotic difference.

Observe that the ${h^{\prime }}_j$  have been inserted in equation (4.3) for the same reason as ${\alpha }^{\prime }$  was inserted into $b$  in equation (3.1). Since the ${h^{\prime }}_j$  are asymptotically unity, however, they are not really needed (and serve no function whatever from a prediction-model viewpoint). A similar argument applies to the $h_f$  in equation (4.4): they are asymptotically unity from the quasi-sampling-design viewpoint (and part of an estimate of 0 from a prediction-model viewpoint).

Previous | Next