2. One-step calibration weighting

Phillip S. Kott and Dan Liao

Previous | Next

2.1 Calibration weighting and unit nonresponse

In the absence of nonresponse (or frame errors), calibration weighting is a sampling-weight-adjustment method that creates a set of weights $\left\{w_k;k\in S\right\},$  asymptotically close to the original design weights, $d_k=1/{\pi }_k,$  that satisfy a set of calibration equations (one for each component of $z_k):$

$${\displaystyle {\sum }_S{w}_k{z}_k}={\displaystyle {\sum }_U{z}_k,}$$

where $S$  denotes the sample, ${\pi }_k$  the sample-selection probability of unit $k,U$  the population of size $N,z_k$  a vector with $P$  components each having a known population total, and ${\sum }_A$  means ${\sum }_{k\in A}.$

Kott (2009) describes a conservative set of mild conditions under which $t_y={\sum }_S{w}_k{y}_k$  is a nearly unbiased estimator for the population total $T_y={\sum }_U{y}_k$  (i.e., the relative bias of $t_y$  is asymptotically zero). Most importantly, each ${\pi }_{k}N/n$  is assumed to be bounded from below by a positive value as $N$  and the (expected) sample size, $n,$  grow arbitrarily large (we add the parenthetical “expected” in case the sample size is random).

In addition, the first four central population moments of each component of $z_k$  is assumed to be bounded from above, while $N^{-1}{\sum }_U{z}_k{z}_k^T$  converges to a positive definite matrix.

Using calibration-weighting will tend to reduce mean squared error relative to the expansion estimator, $t_y^E={\sum }_S{d}_k{y}_k,$  when $y_k$  is correlated with some components of $z_k.$  One should keep in mind, however, that most surveys have many $y_k’\text{s}\text{.}$

A simple way to compute calibration weights is linearly with the following formula:

$$\begin{array}{lll}{w}_k\hfill & =\hfill & d_k\left[1+{\left({\displaystyle {\sum }_U{z}_j}-{\displaystyle {\sum }_S{d}_j{z}_j}\right)}^T{\left({\displaystyle {\sum }_S{d}_j{z}_j{z}_j^T}\right)}^{-1}{z}_k\right]\hfill \\ \hfill & =\hfill & d_k\left[1+g^T{z}_k\right].\hfill \end{array}$$

Fuller et al. (1994) and later Lundström and Särndal (1999) argued that this linear calibration can also be used to handle unit nonresponse. The sample $S$  is replaced by the respondent sample $R,$  while

$$g=\left[\left(1-\theta \right){\left({\displaystyle {\sum }_U{z}_j}-{\displaystyle {\sum }_R{d}_j{z}_j}\right)}^T+\theta {\left({\displaystyle {\sum }_S{d}_j{z}_j}-{\displaystyle {\sum }_R{d}_j{z}_j}\right)}^T\right]{\left({\displaystyle {\sum }_R{d}_j{z}_j{z}_j^T}\right)}^{-1},$$

depending on whether the respondent sample is calibrated to the population $\left(\theta =0\right)$  or calibrated to the original sample $\left(\theta =1\right).$  Either way, the estimate is nearly unbiased under the quasi-sample-design that treats response as a second phase of random sampling so long as each unit’s probability of response has the form:

$$p_k=1/\left(1+{\gamma }^T{z}_k\right),(2.1)$$

and $g$  is a consistent estimator for the unknown parameter vector $\gamma$  in equation (2.1).

The problem with the response function in equation (2.1) is that the implicit estimator for $p_k,{\widehat{p}}_k=1/\left(1+g^T{z}_k\right)$  can be negative. A nonlinear form of calibration weighting avoiding this possibility was suggested by Kott and Liao (2012) based on the generalized exponential form of Folsom and Singh (2000). It uses Newton’s method (iterative Taylor-series approximations) to find a $g$  such that the calibration equation (from here on, we refer to the vector of component calibration equations as the calibration equation):

$${\displaystyle {\sum }_R{w}_k{z}_k}={\displaystyle {\sum }_R{d}_k\alpha \left(g^T{z}_k\right)z_k}=\left(1-\theta \right){\displaystyle {\sum }_U{z}_k+\theta {\displaystyle {\sum }_S{d}_k{z}_k}(2.2)}$$

holds, where $\theta =0$  or $1,$

$$\alpha \left(g^T{z}_k\right)=\frac{\ell +\exp \left(g^T{z}_k\right)}{1+\exp \left(g^T{z}_k\right)/u},(2.3)$$

$\ell ,$  the lower bound of $\alpha (\cdot ),$  is nonnegative (so that calibration weights are likewise nonnegative), and the upper bound of $\alpha (\cdot ),u>\ell ,$  can be either finite or infinite.

Although there are other reasonable forms the weight-adjustment function $\alpha \left(g^T{z}_k\right)$  can take, we will restrict our attention to functions in the form in equation (2.3). This is a generalization of both raking where $\ell =0,u=\infty ,$  and the implicit estimation of a logistic response model, where $\ell =1,u=\infty .$  In Deming and Stephan’s original (1940) iterative-proportional-fitting algorithm for raking, the components of $z_k$  were restricted to indicator functions. We use “raking” more broadly here to mean calibration weighting with a weight-adjustment function of the form $\alpha \left(g^T{z}_k\right)=\exp \left(g^T{z}_k\right).$

When $\ell <1,$  equation (2.3) becomes the generalized-raking adjustment introduced in Deville and Särndal (1992) and discussed further in Deville, Särndal and Sautory (1993). Generalized raking not only lets the components of $z_k$  be continuous but also allows the range of the $\alpha \left(g^T{z}_k\right)$  to be constrained between a positive $\ell$  and a (possibly) finite $u.$

Deville and Särndal (1992) required $\alpha \left(0\right)={\alpha }^{\prime }\left(0\right)=1.$  Since the authors were not treating samples with nonresponse (or incorrect frames), $g^T{z}_k$  needed to converge to 0 and $\alpha \left(g^T{z}_k\right)$  to 1 as the (expected) sample size grew arbitrarily large. When adjusting design weights for nonresponse, however, setting $\ell \ge 1$  is a more sensible strategy, so that the implicit estimated probability of response does not exceed 1.

Although the original definition of calibration weighting in Deville and Särndal (1992) involved minimizing the differences between the $w_k$  and $d_k$  in $R$  as measured by some loss function, later formulations (e.g., Estevao and Särndal 2000) removed the loss function from the definition. Forcing $w_k$  and $d_k$  to be close makes little sense when calibration weighting is used to adjust for unit nonresponse since if a sampled $k$  has a relatively small probability of response, then the difference between $w_k$  and $d_k$  should be relatively large.

Rather than assuming a response model with a particular functional form, an alternative justification for using calibration weighting as a mean of removing unit-nonresponse bias assumes a prediction model in which the survey variable $y_k$  is itself a random variable such that $\text{E}\left(y_k|z_k\right)=z_k^T\beta$  for some unknown $\beta$  whether or not $k$  is sampled or whether it responds when sampled. Kott (2006) and others have observed the calibration-weighted estimator for $T_y={\sum }_U{y}_k$  will be nearly unbiased under the prediction model when calibration is done to the population (when $\theta =0$  in equation (2.2)) and under the combination of the prediction model and the original sample-selection mechanism when calibration is done to the original sample (when $\theta =1).$

The property that a calibration-weighted estimator is nearly unbiased in some sense when either an assumed response model or an assumed prediction model holds has been called “double protection against nonresponse bias” by Kim and Park (2006). It is known as “double robustness” in the biostatics literature (Bang and Robins 2005) and attributed to Robins, Rotnitzky and Zhao (1994), which dealt with item rather than unit nonresponse.

The distribution of $y_k|z_k$  under the prediction model is often assumed to be the same for sampled and nonsampled population members. That is to say, the sampling mechanism is assumed to be ignorable. In addition, the distribution of $y_k|z_k$  is often assumed to be the same whether or not a population member responds when sampled, that is, that the response mechanism is also assumed to be ignorable (Little and Rubin 2002). Here, we make weaker analogous assumptions under the prediction model, namely, that $\text{E}\left(y_k|z_k\right)$  does not depend on whether $k$  is sampled or when sampled responds. Let us say that the sampling and response mechanisms are assumed to be “first-moment ignorable”.

2.2 Instrumental variables

Deville (2000) observed that instrumental-variable calibration can be used to adjust for potential nonresponse bias by assuming a response model that depended on $x_k,$

$$p_k={\left[\alpha \left({\gamma }^T{x}_k\right)\right]}^{-1}=\frac{1+\exp \left({\gamma }^T{x}_k\right)/u}{\ell +\exp \left({\gamma }^T{x}_k\right)},(2.4)$$

but fitting calibration equations with $z_k:$

$${\displaystyle {\sum }_R{w}_k{z}_k}={\displaystyle {\sum }_R{d}_k\alpha \left(g^T{x}_k\right)z_k}=\left(1-\theta \right){\displaystyle {\sum }_U{z}_k+\theta {\displaystyle {\sum }_S{d}_k{z}_k}},(2.5)$$

where the $g$  satisfying equation (2.5) with $\theta =0$  or $1$ a consistent estimator of unknown parameter vector $\gamma$  in equation (2.4). Some mild conditions are needed for this. Sufficient are the following: $N^{-1}{\sum }_R{d}_k\alpha \left({\gamma }^T{x}_k\right)z_k$  is a consistent and bounded estimator for $N^{-1}\left[\left(1-\theta \right){\sum }_U{z}_k+\theta {\sum }_S{d}_k{z}_k\right],$   $\alpha \left(\varphi \right)$  is everywhere twice differentiable, and $N^{-1}{\sum }_R{d}_k{\alpha }^{\prime }\left(\varphi \right)z_k{x}_k^T$  is always invertible and bounded as the sample grows arbitrarily large.

Let $R_k=1$  when $k\in R,0$  otherwise. It is not hard to show that

$$\begin{array}{lll}g-\gamma \hfill & =\hfill & -{\left({\displaystyle {\sum }_S{d}_k{R}_k}{\alpha }^{\prime }\left(c_k\right)z_k{x}_k^T\right)}^{-1}\left\{{\displaystyle {\sum }_S{d}_k{R}_k}\alpha \left({\gamma }^T{x}_k\right)z_k-\left[\left(1-\theta \right){\displaystyle {\sum }_U{z}_k+\theta {\displaystyle {\sum }_S{d}_k{z}_k}}\right]\right\}\hfill \\ \hfill & \hfill & -{\left(N^{-1}{\displaystyle {\sum }_S{d}_k{R}_k}{\alpha }^{\prime }\left(c_k\right)z_k{x}_k^T\right)}^{-1}\left\{N^{-1}{\displaystyle {\sum }_S{d}_k{R}_k}\alpha \left({\gamma }^T{x}_k\right)z_k-N^{-1}\left[\left(1-\theta \right){\displaystyle {\sum }_U{z}_k+\theta {\displaystyle {\sum }_S{d}_k{z}_k}}\right]\right\}\hfill \end{array}$$

for some $c_k$  between $g^T{x}_k$  and ${\gamma }^T{x}_k,$  as Kott and Liao (2012) demonstrated when $x_k=z_k.$

Deville also noted that it is possible for components of the $x_k$  to be survey variables with values known only for respondents. Chang and Kott (2008) extended the notion of calibration weighting to allow the dimension of the $z_k-$ vector to be greater than that of the $x_k-$ vector. We will not treat either possibility in the following sections.

Kim and Shao (2013) in treating nonignorable nonresponse call the components of $z_k$  not wholly functions of the components of $x_k$  “instrumental variables”. To limit future confusion, we will henceforth use to term “model variables” to refer to the components of $x_k.$

Previous | Next