Calibration estimation in dual-frame surveys

Abstract

Survey statisticians make use of auxiliary information to improve estimates. One important example is calibration estimation, which constructs new weights that match benchmark constraints on auxiliary variables while remaining “close” to the design weights. Multiple-frame surveys are increasingly used by statistical agencies and private organizations to reduce sampling costs and/or avoid frame undercoverage errors. Several ways of combining estimates derived from such frames have been proposed elsewhere; in this paper, we extend the calibration paradigm, previously used for single-frame surveys, to calculate the total value of a variable of interest in a dual-frame survey. Calibration is a general tool that allows to include auxiliary information from two frames. It also incorporates, as a special case, certain dual-frame estimators that have been proposed previously. The theoretical properties of our class of estimators are derived and discussed, and simulation studies conducted to compare the efficiency of the procedure, using different sets of auxiliary variables. Finally, the proposed methodology is applied to real data obtained from the Barometer of Culture of Andalusia survey.

Appendices

Appendix 1: Other examples of the definition of auxiliary variable vector according to available auxiliary information

1.1 Population totals for group membership indicators are known

Let the population $\mathcal {U}$ be divided into H mutually exclusive groups $\mathcal {U}_h$, for $h=1,\ldots ,H$ such that $\bigcup _{h=1}^H \mathcal {U}_h=\mathcal {U}$ and let $\delta _k(h)$ be the indicator variable that takes value 1 if unit $k\in \mathcal {U}_h$ and 0 otherwise, for $k=1,\ldots ,N$ and $h=1,\ldots ,H$. Then, $\sum _{k=1}^N\delta _k(h)=N_h$ and $\sum _{h=1}^{H}N_h=N$. Now, consider the situation in which we know the population total of such indicator variables for each of the four domains, i.e. $N_{a,h}=\sum _{k\in a}\delta _k(h)$, $N_{ab,h}=\sum _{k\in ab}\delta _k(h)$, $N_{ba,h}=\sum _{k\in ba}\delta _k(h)=N_{ab,h}$, $N_{b,h}=\sum _{k\in b}\delta _k(h)$, for $h=1,\ldots ,H$. Note that $N_{a,h}=\sum _{k\in a}\delta _k(h)=\sum _{k=1}^N\delta _k(a)\delta _k(h)$ and similarly for the other cases. Of course, this type of auxiliary information implies that we also know the dimensions of the three sets $N_A$, $N_B$ and $N_{ab}$ considered in Sect. 3.1. Indeed, it is a special case of the present one.

In this case the vector of auxiliary variables is defined for $k=1,\ldots ,N$ by

$$\begin{aligned} {\varvec{x}}_k=\{(\delta _k(a)\delta _k(h),\delta _k(ab)\delta _k(h),\delta _k(ba)\delta _k(h),\delta _k(b)\delta _k(h)\}_{h=1,\ldots ,H} \end{aligned}$$

and the vector of known totals is ${\varvec{t}}_x=\{(N_{a,h},\eta N_{ab,h},(1-\eta )N_{ba,h},N_{b,h})\}_{h=1,\ldots ,H}$. As in Sect. 3.1 the minimization problem has an analytic solution irrespective of the distance function employed. This solution is given by

$$\begin{aligned} w_k^{\circ } = \left\{ \begin{array}{l@{\quad }l} d_{Ak}{N_{a,h}}/{\hat{N}_{a,h}} &{} \text {if } k \in \{s_a \cap \mathcal {U}_h\}\\ \eta \, d_{Ak} {N_{ab,h}}/{\hat{N}_{ab,h}} &{} \text {if } k \in \{s_{ab} \cap \mathcal {U}_h\}\\ (1-\eta )\, d_{Bk} {N_{ba,h}}/{\hat{N}_{ba,h}} &{} \text {if } k \in \{s_{ba} \cap \mathcal {U}_h\}\\ d_{Bk}{N_{b,h}}/{\hat{N}_{b,h} } &{} \text {if } k \in \{s_b \cap \mathcal {U}_h\}\\ \end{array} \right. \quad \text{ for }\, h=1,\ldots ,H, \end{aligned}$$

(22)

where $\hat{N}_{a,h}=\sum _{k\in s_a}d_{Ak}\delta _k(h)$ and similarly for the other size estimators. This is another case of complete post-stratification. The final estimator is more efficient than the Hartley estimator to the extent that groups collect units with a similar value of the variable of interest.

On the other hand, when we only know the population total in frame A and in frame B, i.e. we do not know the distribution for the intersection domain ab, then we are again in a situation of incomplete post-stratification, like that of Sect. 3.2. Here,

$$\begin{aligned} {\varvec{x}}_k=\{[\delta _k(a)+\delta _k(ab)+\delta _k(ba)]\delta _k(h),[\delta _k(b)+\delta _k(ab)+\delta _k(ba)]\delta _k(h)\}_{h=1\ldots ,H} \end{aligned}$$

and ${\varvec{t}}_x=\{(N_{A,h},N_{B,h})\}_{h=1\ldots ,H}$. We have an analytic solution for the form of the weights only for the Euclidean distance case, but it does not take a simple tractable form such as that considered in Sect. 3.2. A similar situation also arises when, as in the case considered later in the application (Sect. 7), we do not know the distribution for, say, age-sex groups, but only the total for age and the total for sex, in each of the two frames A and B. This is another example of incomplete post-stratification, which employs a form of raking (depending on the distance function employed) to obtain the final set of weights (see also examples in Sect. 4).

1.2 $N_A$, $N_B$, $N_{ab}$ known and X known

Suppose that we know the frame sizes $N_A$, $N_B$ and $N_{ab}$, and let the population total of an auxiliary numerical variable be available for the whole population $X= \sum _{k=1}^N x_{k}$ and not only for frame A as in the previous section. The auxiliary vector is thus ${\varvec{x}}_k=(\delta _k(a),\delta _k(ab),\delta _k(ba),\delta _k(b), x_{k})$ and the calibration constraints are those in (13) plus $\sum _{k\in s}w_k^{\circ }x_{k}=X.$

1.3 $N_A$, $N_B$, known and $X_A$ and $Z_B$ known

Suppose that we know $N_A$, $N_B$ and the population total $X_A$ defined in Sect. 3.3. In addition, we also know the population total of another auxiliary numerical variable $z_B$ relative to frame B, whose total is $Z_B=\sum _{k \in \mathcal {B}} z_{B}$. The auxiliary vector is

$$\begin{aligned} {\varvec{x}}_k= & {} (\delta _k(a)+\delta _k(ab)+\delta _k(ba),\delta _k(b)+\delta _k(ab)+\delta _k(ba),\\&[\delta _k(a)+\delta _k(ab)+\delta _k(ba)] x_{Ak},[\delta _k(b)+\delta _k(ab)+\delta _k(ba)] z_{Bk}) \end{aligned}$$

and the vector of known totals in this case is ${\varvec{t}}_x=(N_A,N_B,X_A,Z_B)$, which allows us to write the following calibration constraints

$$\begin{aligned}&\sum _{k\in s_a}w_k^{\circ }+\sum _{k\in s_{ab}}w_k^{\circ }+\sum _{k\in s_{ba}}w_k^{\circ }=N_A \nonumber \\&\sum _{k\in s_b}w_k^{\circ }+\sum _{k\in s_{ab}}w_k^{\circ }+\sum _{k\in s_{ba}}w_k^{\circ }=N_B,\nonumber \\&\sum _{k\in s_a}w_k^{\circ }x_{Ak}+\sum _{k\in s_{ab}}w_k^{\circ }x_{Ak}+\sum _{k\in s_{ba}}w_k^{\circ }x_{Ak}=X_A \nonumber \\&\sum _{k\in s_b}w_k^{\circ }z_{Bk}+\sum _{k\in s_{ab}}w_k^{\circ }z_{Bk}+\sum _{k\in s_{ba}}w_k^{\circ }z_{Bk}=Z_B. \end{aligned}$$

(23)

1.4 $N_A$, $N_B$, $N_{ab}$ known and $X_A$, $X_B$ known

When we know the frame sizes $N_A$, $N_B$ and $N_{ab}$ and the population totals of the same auxiliary variable x in the two frames $X_A$ and $X_B$, the auxiliary vector is

$$\begin{aligned} {\varvec{x}}_k= & {} (\delta _k(a),\delta _k(ab),\delta _k(ba),\delta _k(b), [\delta _k(a)+\delta _k(ab)+\delta _k(ba)] x_{k},\\&[\delta _k(b)+\delta _k(ab)+\delta _k(ba)] x_{k}) \end{aligned}$$

and the vector of known totals in this case is ${\varvec{t}}_x=(N_a,\eta N_{ab},(1-\eta )N_{ba},N_b,X_A,X_B)$.

Appendix 2: Technical assumptions and Proofs

1.1 Assumptions

A 1

Let ${\varvec{B}}_U=(\sum _{k=1}^N {\varvec{x}}_k^{T}{\varvec{x}}_k)^{-1}\sum _{k=1}^N{\varvec{x}}_k ^Ty_k$. Assume that ${\varvec{B}}=\lim _{N\rightarrow \infty } {\varvec{B}}_U$ exists; the distribution of ${\varvec{x}}_k$ and of $y_k$, and the sampling designs are such that $\sum _{k=1}^N {\varvec{x}}_k^T {\varvec{x}}_k$ is consistently estimated by $\sum _{k \in s} d_k^{\circ }{\varvec{x}}_k^T {\varvec{x}}_k$ and $\sum _{k=1}^N {\varvec{x}}_k^T y_k$ is consistently estimated by $\sum _{k \in s}d_k^{\circ }{\varvec{x}}_k^T y_k$.

A 2

The limiting design covariance matrix of the normalized Hartley estimators,

$$\begin{aligned} {\varvec{{\varSigma }}}= \begin{bmatrix} {\varSigma }_{yy}&\quad {\varvec{{\varSigma }}}_{xy} \\ {\varvec{{\varSigma }}}_{xy}^{T}&\quad {\varvec{{\varSigma }}}_{xx}\\ \end{bmatrix} =\lim _{N\rightarrow \infty } \frac{n_N}{N^2} \begin{bmatrix} V(\hat{Y}_H)&\quad {\varvec{C}}({\hat{{\varvec{t}}}}_{xH},\hat{Y}_H) \\ {\varvec{C}}({\hat{{\varvec{t}}}}_{xH},\hat{Y}_H)^{T}&\quad {\varvec{V}}({\hat{{\varvec{t}}}}_{xH})\\ \end{bmatrix} \end{aligned}$$

is positive defined.

A 3

The normalized Hartley estimators of ${\varvec{t}}_x$ and Y are such that a central limit theorem holds:

$$\begin{aligned} \frac{\sqrt{n_N}}{N} \begin{bmatrix} \sum _{k \in s} d_k^{\circ } y_k - Y \\ \sum _{k \in s} d_k^{\circ } {\varvec{x}}_k^{T} - {\varvec{t}}_x^T\\ \end{bmatrix} \rightarrow ^{\mathcal {L}} N({\varvec{0}},{\varvec{{\varSigma }}}). \end{aligned}$$

A 4

The estimated covariance matrix for the Hartley estimator is design consistent in the sense that

$$\begin{aligned} \frac{n_N}{N^2} \begin{bmatrix} v(\hat{Y}_H)&{\varvec{c}}({\hat{{\varvec{t}}}}_{xH},\hat{Y}_H) \\ {\varvec{c}}({\hat{{\varvec{t}}}}_{xH},\hat{Y}_H)^{T}&{\varvec{v}}({\hat{{\varvec{t}}}}_{xH})\\ \end{bmatrix} -{\varvec{{\varSigma }}}=o_p(1), \end{aligned}$$

where $v(\hat{Y}_H)=v(\hat{Y}_a+\eta \hat{Y}_{ab})+v((1-\eta )\hat{Y}_{ba}+\hat{Y}_b)$ and similarly for the others.

A 5

$\phi _s({\varvec{{\uplambda }}})$ is defined on $C=\bigcap _{k=1}^N\{{\varvec{{\uplambda }}}:{\varvec{x}}_k{\varvec{{\uplambda }}} \in \text{ Im }_k(d_k^{\circ })\}$. C is an open neighbourhood of ${\varvec{0}}$.

A 6

As $N\rightarrow \infty$, $\max ||{\varvec{x}}_k||=M<\infty$, $k=1,\ldots ,N$, and $\max F_k^{\prime \prime }(0)=M^\prime <\infty$, where $F_k^{\prime \prime }(\cdot )$ is the second derivative of $F_k(\cdot )$.

Proof of Theorem 1

By assumptions A1 and A2 we have

$$\begin{aligned} \hat{Y}_{\mathrm{GREG}}-Y&=\hat{Y}_H+({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH}){\varvec{B}}_U-Y+({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH})({\hat{{\varvec{\beta }}}}^{\circ }-{\varvec{B}}_U)\\&=\hat{Y}_H+({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH}){\varvec{B}}_U-Y+O_p(Nn_N^{-1/2})o_p(1). \end{aligned}$$

Now, $\hat{Y}_H+({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH}){\varvec{B}}_U$ is such that a central limit theorem holds for A2 and A3, i.e.

$$\begin{aligned} \frac{\sqrt{n_N}}{N}(\hat{Y}_H+({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH}){\varvec{B}}_U-Y)\rightarrow ^{\mathcal {L}}N(0,\nu ^2) \end{aligned}$$

where $\nu ^2={\varSigma }_{yy}-2{\varvec{{\varSigma }}}_{xy}{\varvec{B}}+{\varvec{B}}^{T}{\varvec{{\varSigma }}}_{xx}{\varvec{B}}$. Now, $N^2n_NV(\hat{t}_{ eH })\rightarrow \nu ^2$ as $N\rightarrow \infty$, so that $\hat{Y}_H+ ({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH}){\varvec{B}}_U-Y=O_p(Nn_N^{-1/2})$ and the result follows. $\square$

Proof of Theorem 2

Let $\tilde{y}_k={\varvec{x}}_k{\varvec{B}}_U$ and $\hat{y}_k={\varvec{x}}_k{\hat{{\varvec{\beta }}}}^{\circ }$. Then

$$\begin{aligned} v(\hat{t}_{\hat{e}H})&=v(\hat{t}_{\hat{e}H}+\hat{t}_{ eH }-\hat{t}_{ eH })\nonumber \\&=v\left( \sum _{k \in s}d_k^{\circ }\hat{e}_k+\sum _{k \in s}d_k^{\circ }e_k-\sum _{k \in s}d_k^{\circ }e_k\right) \nonumber \\&=v\left( \sum _{k \in s}d_k^{\circ }e_k+\sum _{k \in s}d_k^{\circ }(y_k-\hat{y}_k-y_k+\tilde{y}_k)\right) \nonumber \\&=v(\hat{t}_{ eH })+v(\hat{t}_{\tilde{y}-\hat{y},H})+2c(\hat{t}_{ eH },\hat{t}_{\tilde{y}-\hat{y},H}). \end{aligned}$$

(24)

Now, for A1, A2 and A4, we have

• 1. $v(\hat{t}_{ eH })=V(\hat{t}_{ eH })+o_p(N^2n_N^{-1})$,

• 2. $v(\hat{t}_{\tilde{y}-\hat{y},H})=v(\sum _{k \in s}d_k^{\circ }{\varvec{x}}_k({\varvec{B}}_U-{\hat{{\varvec{\beta }}}}^{\circ }))=({\varvec{B}}_U-{\hat{{\varvec{\beta }}}}^{\circ })^{T}v({\hat{{\varvec{t}}}}_{xH}) ({\varvec{B}}_U-{\hat{{\varvec{\beta }}}}^{\circ }) =o_p(1)O_p(N^2n_N^{-1})o_p(1)$,

• 3. $c(\hat{t}_{ eH },\hat{t}_{\tilde{y}-\hat{y},H})=c\left( \sum _{k \in s}d_k^{\circ }e_k,\sum _{k \in s}d_k^{\circ }{\varvec{x}}_k({\varvec{B}}_U-{\hat{{\varvec{\beta }}}}^{\circ })\right) ={\varvec{c}}(\hat{t}_{ eH },{\hat{{\varvec{t}}}}_{xH})({\varvec{B}}_U-{\hat{{\varvec{\beta }}}}^{\circ }) =O_p(N^2n_N^{-1})o_p(1)$.

$\square$

Proof of Theorem 3

Using Result 3 in Deville and Särndal (1992)

$$\begin{aligned} {\varvec{{\uplambda }}}=\left( \sum _{k \in s}d_k^{\circ }q_k\,{\varvec{x}}_k^T{\varvec{x}}_k\right) ^{-1}({\varvec{t}}_x-{\hat{{\varvec{t}}}}_{xH})^{T}+O_p(n_N^{-1}), \end{aligned}$$

$w_k=d_k^{\circ }F(q_k{\varvec{x}}_k{\varvec{{\uplambda }}})=:d_k^{\circ }(1+q_k{\varvec{x}}_k{\varvec{{\uplambda }}})+\epsilon _k(q_k{\varvec{x}}_k{\varvec{{\uplambda }}})$. Assumption A6 ensures that $\epsilon _k(u)=O_p(u^2)$, therefore

$$\begin{aligned} \hat{Y}_{\mathrm{CAL}}=\hat{Y}_{\mathrm{GREG}}+O_p(Nn_N^{-1})+O_p(Nn_N^{-2}). \end{aligned}$$

$\square$