A few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 5. Estimation using calibration and generalized calibrationA few remarks on a small example by Jean-Claude Deville regarding non-ignorable non-response Section 5. Estimation using calibration and generalized calibration

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5.1 Notation

To define calibration, we will establish the following notation. Let $U\mathrm{<mo>=</mo>}\left\{\mathrm{1,}\dots \mathrm{,}k\mathrm{,}\dots \mathrm{,}N\right\}$ be the set of people interviewed (here, $N\mathrm{<mo>=</mo>\; 600})$ and $R\subset U$ be the set of respondents to the question regarding drug use. As well, we define the following:

$$x_k\mathrm{<mo>=</mo>}\{\begin{array}{ll}{\left(10\right)}^{{\rm T}}\hfill & \text{if individual}k\text{is male}\hfill \\ {\left(01\right)}^{{\rm T}}\hfill & \text{if individual}k\text{is}\text{female}\mathrm{.}\hfill \end{array}$$

and

$$z_k\mathrm{<mo>=</mo>}\{\begin{array}{ll}{\left(10\right)}^{{\rm T}}\hfill & \text{if}\text{individual}k\text{reported using drugs}\hfill \\ {\left(01\right)}^{{\rm T}}\hfill & \text{if}\text{individual}k\text{reported not using drugs}\mathrm{.}\hfill \end{array}$$

Using the notation defined above,

$$\begin{array}{l}{\displaystyle \sum _{k\in U}}{x}_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{n}_{H\mathrm{.}}\\ n_{F\mathrm{.}}\end{array}\right)\mathrm{,}{\displaystyle \sum _{k\in R}}{x}_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{n}_{H\mathrm{.}}-m_H\\ n_{F\mathrm{.}}-m_F\end{array}\right)\mathrm{,}{\displaystyle \sum _{k\in R}}{z}_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{r}_{\mathrm{.}D}\\ r_{\mathrm{.}S}\end{array}\right)\mathrm{,}\hfill \\ {\displaystyle \sum _{k\in R}}{x}_k{x}_k^{{\rm T}}\mathrm{<mo>=</mo>}\left(\begin{array}{cc}{n}_{H\mathrm{.}}-m_H& 0\\ 0& n_{F\mathrm{.}}-m_F\end{array}\right)\mathrm{,}{\displaystyle \sum _{k\in R}}{x}_k{z}_k^{{\rm T}}\mathrm{<mo>=</mo>}\left(\begin{array}{cc}{r}_{HD}& r_{HS}\\ r_{FD}& r_{FS}\end{array}\right)\mathrm{,}\hfill \end{array}$$

and

$${\displaystyle \sum _{k\in R}}{z}_k{z}_k^{{\rm T}}\mathrm{<mo>=</mo>}\left(\begin{array}{cc}{r}_{\mathrm{.}D}& 0\\ 0& r_{\mathrm{.}S}\end{array}\right)\mathrm{.}$$

5.2 Estimation using simple calibration

Using simple calibration as described in Deville and Särndal (1992), we seek a weight that is expressed as

$$w_k\mathrm{<mo>=</mo>}F\left(x_k^{{\rm T}}\lambda \right)\mathrm{,}$$

where $\lambda \mathrm{<mo>=</mo>}\left({\lambda }_1\mathrm{,}{\lambda }_2\right)$ is a parameter vector and $F\left(\mathrm{.}\right)$ is a calibration function, that is, a strictly increasing function such that $F\left(0\right)\mathrm{<mo>=</mo>\; 1}$ and whose derivative $F^{\prime }\left(\mathrm{.}\right)$ is such that $F^{\prime }\left(0\right)\mathrm{<mo>=</mo>\; 1.}$

Vector $\lambda$ is determined by using the Newton method to solve the system of equations

$${\displaystyle \sum _{k\in R}}F\left(x_k^{{\rm T}}\lambda \right)x_k\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}}{x}_k\mathrm{.}(5.1)$$

Finally, the calibration estimator is given by

$$\left(\begin{array}{c}{\widehat{n}}_{\mathrm{.}D}\\ {\widehat{n}}_{\mathrm{.}S}\end{array}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in R}}{w}_k{z}_k\mathrm{.}$$

In our application, equation (5.1) becomes

$${\displaystyle \sum _{k\in R}}F\left(x_k^{{\rm T}}\lambda \right)x_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}\left(n_{H\mathrm{.}}-m_H\right)F\left({\lambda }_1\right)\\ \left(n_{F\mathrm{.}}-m_F\right)F\left({\lambda }_2\right)\end{array}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}}{x}_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{n}_{H\mathrm{.}}\\ n_{F\mathrm{.}}\end{array}\right)\mathrm{.}$$

We directly obtain the following:

$$w_k\mathrm{<mo>=</mo>}F\left(x_k^{{\rm T}}\lambda \right)\mathrm{<mo>=</mo>}\{\begin{array}{ll}{n}_{H\mathrm{.}}/\left(n_{H\mathrm{.}}-m_H\right)\hfill & \text{if individual}k\text{is male}\hfill \\ n_{F\mathrm{.}}/\left(n_{F\mathrm{.}}-m_F\right)\hfill & \text{if}\text{individual}k\text{is female}\mathrm{.}\hfill \end{array}$$

Therefore, the calibrated estimators are

$$\begin{array}{ll}{\widehat{n}}_{\mathrm{.}D}\hfill & \mathrm{<mo>=</mo>}{r}_{HD}\frac{n_{H\mathrm{.}}}{n_{H\mathrm{.}}-m_H}+r_{FD}\frac{n_{F\mathrm{.}}}{n_{F\mathrm{.}}-m_F}\hfill \\ {\widehat{n}}_{\mathrm{.}S}\hfill & \mathrm{<mo>=</mo>}{r}_{HS}\frac{n_{H\mathrm{.}}}{n_{H\mathrm{.}}-m_H}+r_{FS}\frac{n_{F\mathrm{.}}}{n_{F\mathrm{.}}-m_F}\mathrm{,}\hfill \end{array}$$

which is exactly the same result as that yielded by the method of moments and the maximum likelihood method. In this case, the solution does not depend on the calibration function used. Obviously, the example is especially simple. In more complex cases where the category definitions do not overlap, the result depends on the calibration function used.

5.3 Generalized calibration

For generalized calibration as defined in (Deville 2000, 2002, 2004; Kott 2006), the weights are expressed as

$$w_k\mathrm{<mo>=</mo>}F\left(z_k^{{\rm T}}\lambda \right)\mathrm{.}$$

Vector $\lambda$ is determined by solving the system of equations

$${\displaystyle \sum _{k\in R}}F\left(z_k^{{\rm T}}\lambda \right)x_k\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}}{x}_k\mathrm{.}(5.2)$$

Finally, the generalized calibration estimator is given by

$$\left(\begin{array}{c}{\widehat{n}}_{\mathrm{.}D}\\ {\widehat{n}}_{\mathrm{.}S}\end{array}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in R}}{w}_k{z}_k\mathrm{.}$$

In our application, equation (5.2) becomes

$${\displaystyle \sum _{k\in R}}F\left(z_k^{{\rm T}}\lambda \right)x_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{r}_{HD}F\left({\lambda }_1\right)+r_{HS}F\left({\lambda }_2\right)\\ r_{FD}F\left({\lambda }_1\right)+r_{FS}F\left({\lambda }_2\right)\end{array}\right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U}}{x}_k\mathrm{<mo>=</mo>}\left(\begin{array}{c}{n}_{H\mathrm{.}}\\ n_{F\mathrm{.}}\end{array}\right)\mathrm{,}$$

Which can be written as a matrix

$$\left(\begin{array}{cc}{r}_{HD}& r_{HS}\\ r_{FD}& r_{FS}\end{array}\right)\left(\begin{array}{c}F\left({\lambda }_1\right)\\ F\left({\lambda }_2\right)\end{array}\right)\mathrm{<mo>=</mo>}\left(\begin{array}{c}{n}_{H\mathrm{.}}\\ n_{F\mathrm{.}}\end{array}\right)\mathrm{.}$$

We simply solve the linear system

$$\left(\begin{array}{c}F\left({\lambda }_1\right)\\ F\left({\lambda }_2\right)\end{array}\right)\mathrm{<mo>=</mo>}{\left(\begin{array}{cc}{r}_{HD}& r_{HS}\\ r_{FD}& r_{FS}\end{array}\right)}^{-1}\left(\begin{array}{c}{n}_{H\mathrm{.}}\\ n_{F\mathrm{.}}\end{array}\right)\mathrm{<mo>=</mo>}\left(\begin{array}{c}\frac{n_{H\mathrm{.}}{r}_{FS}-n_{F\mathrm{.}}{r}_{HS}}{r_{FS}{r}_{HD}-r_{FD}{r}_{HS}}\\ \frac{n_{H\mathrm{.}}{r}_{FD}-n_{F\mathrm{.}}{r}_{HD}}{r_{FD}{r}_{HS}-r_{FS}{r}_{HD}}\end{array}\right)\mathrm{.}$$

The estimators are therefore

$$\begin{array}{l}{\widehat{n}}_{\mathrm{.}D}\mathrm{<mo>=</mo>}{r}_{\mathrm{.}D}\frac{n_{H\mathrm{.}}{r}_{FS}-n_{F\mathrm{.}}{r}_{HS}}{r_{FS}{r}_{HD}-r_{FD}{r}_{HS}}\\ {\widehat{n}}_{\mathrm{.}S}\mathrm{<mo>=</mo>}{r}_{\mathrm{.}S}\frac{n_{H\mathrm{.}}{r}_{FD}-n_{F\mathrm{.}}{r}_{HD}}{r_{FD}{r}_{HS}-r_{FS}{r}_{HD}}\mathrm{.}\end{array}$$

Again, the solution does not depend on the calibration function used. The solution is identical to the solution obtained using the method of moments and the maximum likelihood method. Here, too, this property results from the simplicity of the example. In more complex cases, the result depends on the calibration function used.

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