With-replacement bootstrap variance estimation for household surveys Principles, examples and implementation
Section 2. Notation and estimation

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In this section, we define our main notations, and we describe the sampling and estimation process. We first consider in Section 2.1 the case when a sample of households only is selected, and we describe the estimation process which includes treatment of unit non-response and calibration. We indicate in each case what is the benchmark variance estimator considered, i.e. the variance estimator that we aim at reproducing for the estimation of a total with the bootstrap method proposed in Section 3. The case when individuals are sub-sampled inside households is covered in Section 2.2. The benchmark variance estimators for this second case are given in Appendix A.

2.1   Case of a sample of households only

We consider estimation for a population $U_{hh}$ of households. We let $y_k$ denote the value taken by some variable of interest for the household $k.$ We are interested in the estimation of the total

$$Y_{hh}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in U_{hh}}{y}_k}\mathrm{.}(2.1)$$

2.1.1     Sampling design

We suppose that a sample $S_{hh}$ is selected in $U_{hh}$ by means of a stratified one-stage sampling design. The population $U_{hh}$ is partitioned into $H$ strata $U_{hh}^1\mathrm{,}\dots \mathrm{,}{U}_{hh}^H,$ the samples $S_{hh}^1\mathrm{,}\dots \mathrm{,}{S}_{hh}^H$ are selected inside independently, and the sample $S_{hh}$ is the union of these samples. We let ${\pi }_k$ denote the inclusion probability of a given household $k.$ The design weight is

$$d_k\mathrm{<mo>=</mo>}\frac{1}{{\pi }_k}\mathrm{.}(2.2)$$

In case of full response, the estimator of $Y_{hh}$ is

$${\widehat{Y}}_{hh}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_{hh}}{d}_k{y}_k}\mathrm{.}(2.3)$$

We consider as a benchmark variance estimator

$$v_{\text{mult}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_{hh}\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H\left[\frac{n_h}{n_h-1}{\displaystyle \sum _{k\in S_{hh}^h}{\left(d_k{y}_k-\frac{1}{n_h}{\displaystyle \sum _{k^{\prime }\in S_{hh}^h}{d}_{k^{\prime }}{y}_{k^{\prime }}}\right)}^2}\right]\mathrm{,}}(2.4)$$

with $n_h$ the size of the sample $S_{hh}^h.$ This variance estimator is unbiased if the samples are selected inside strata by multinomial sampling (Tillé, 2011, Section 5.4), a.k.a. sampling with replacement. It is conservative if the sampling designs used inside strata are more efficient than multinomial sampling (Särndal, Swensson and Wretman, 1992, Section 4.6), which we assume to hold true in the rest of the paper. The positive bias of this variance estimator is expected to be negligible when the sampling rates inside strata are themselves negligible, which is often the case in phone surveys. This is illustrated by the results of our simulation study, see Section 4.

2.1.2     Treatment of non-response

In practice, the sample $S_{hh}$ is prone to unit non-response, which leads to the observation of a sub-sample of respondents $S_{r\mathrm{,}hh}$ only. We let $r_k$ denote the response indicator of a household $k,$ and $p_k$ denote the response probability of the household $k.$ We suppose that the households respond independently of one another. Also, we suppose that unit non-response is handled through the method of Response Homogeneity Groups (RHGs), which is popular in practice (e.g. Brick, 2013; Juillard and Chauvet, 2018). Under this framework, it is assumed that the sample $S_{hh}$ may be partitioned into $C$ RHGs denoted as $S_{\mathrm{1,}hh}\mathrm{,}\dots \mathrm{,}{S}_{C\mathrm{,}hh}$ such that the response probability $p_k$ is constant inside a RHG.

For $c\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}C,$ we let $p_c$ denote the common response probability inside the RHG $S_{c\mathrm{,}hh}.$ It is estimated by

$${\widehat{p}}_c\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{c,hh}}{\theta }_k{r}_k}}{{\displaystyle {\sum }_{k\in S_{c,hh}}{\theta }_k}}\mathrm{,}(2.5)$$

with ${\theta }_k$ some weight attached to the household $k.$ The choice ${\theta }_k\mathrm{<mo>=</mo>}1$ leads to estimating $p_c$ by the unweighted response rate inside the RHG. The choice ${\theta }_k\mathrm{<mo>=</mo>}{d}_k$ leads to estimating $p_c$ by the response rate inside the RHG, weighted by the sampling weights (e.g. Kott, 2012).

Accounting for the estimated response probabilities leads to the weights corrected for non-response

$$d_{rk}\mathrm{<mo>=</mo>}\frac{d_k}{{\widehat{p}}_{c(k\mathrm{<mo\; stretchy="false">)</mo>}}}\mathrm{,}(2.6)$$

with $c\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}$ the RHG of the household $k.$ The estimator of $Y_{hh}$ adjusted for non-response is

$${\widehat{Y}}_{r\mathrm{,}hh}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_{r,hh}}{d}_{rk}{y}_k}\mathrm{.}(2.7)$$

Building on the multinomial variance estimator in (2.4) and on linearization for estimators reweighted for unit-non-response (Kim and Kim, 2007, Section 2), our benchmark variance estimator is

$$v_{\text{mult}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_{r\mathrm{,}hh}\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H\left[\frac{n_h}{n_h-1}{\displaystyle \sum _{k\in S_{hh}^h}{\left(d_k{u}_{1k}-\frac{1}{n_h}{\displaystyle \sum _{k^{\prime }\in S_{hh}^h}{d}_{k^{\prime }}{u}_{1k^{\prime }}}\right)}^2}\right]}\mathrm{,}(2.8)$$

with

$$u_{1k}\mathrm{<mo>=</mo>}{\theta }_k{\pi }_k{\overline{y}}_{rc\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}+\frac{r_k}{{\widehat{p}}_{c\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}}\left\{y_k-{\theta }_k{\pi }_k{\overline{y}}_{rc\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}\right\}\mathrm{,}$$

and

$${\overline{y}}_{rc}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{c,hh}}{d}_k{r}_k{y}_k}}{{\displaystyle {\sum }_{k\in S_{c,hh}}{\theta }_k{r}_k}}\mathrm{.}$$

This is a conservative estimator for the asymptotic variance of ${\widehat{Y}}_{r\mathrm{,}hh}.$ A key assumption for this to hold is that the response indicators $r_k$ are mutually independent.

2.1.3     Calibration

Lastly, the weights adjusted for non-response are calibrated on auxiliary totals known on the population. For simplicity, we describe only the Generalized REGression estimator (GREG, Särndal et al., 1992, Chapter 6). Let $x_k$ denote the vector of calibration variables at the household level, and $X_{hh}$ the total on the population $U_{hh}.$ For the sample $S_{r\mathrm{,}hh},$ this leads to the linear calibrated weights

$$w_k\mathrm{<mo>=</mo>}{d}_{rk}\left(1+x_k^{{\rm T}}{\lambda }_{hh}\right)\mathrm{,}$$

with

$${\lambda }_{hh}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{k\in S_{r,hh}}{d}_{rk}{x}_k{x}_k^{{\rm T}}}\right)}^{-1}\left(X_{hh}-{\widehat{X}}_{r\mathrm{,}hh}\right)\mathrm{,}(2.9)$$

and where ${\widehat{X}}_{r\mathrm{,}hh}$ is the estimator of $X_{hh},$ obtained by plugging $x_k$ into (2.7). The calibrated estimator is

$${\widehat{Y}}_{\text{cal}\mathrm{,}hh}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_{r,hh}}{w}_k{y}_k}\mathrm{.}(2.10)$$

The sampling and estimation steps are summarized in Figure 2.1.

Using linearization for estimators reweighted for unit-non-response and calibrated (Kim and Kim, 2007, Section 5), our benchmark variance estimator is

$$v_{\mathrm{mult}}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{Y}}_{\text{cal}\mathrm{,}hh}\mathrm{<mo\; stretchy="false">)</mo>}\mathrm{<mo>=</mo>}{\displaystyle \sum _{h\mathrm{<mo>=</mo>\; 1}}^H\left[\frac{n_h}{n_h-1}{\displaystyle \sum _{k\in S_{hh}^h}{\left(d_k{u}_{2k}-\frac{1}{n_h}{\displaystyle \sum _{k^{\prime }\in S_{hh}^h}{d}_{k^{\prime }}{u}_{2k^{\prime }}}\right)}^2}\right]}\mathrm{,}(2.11)$$

with

$$u_{2k}\mathrm{<mo>=</mo>}{\theta }_k{\pi }_k{\overline{e}}_{rc\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}+\frac{r_k}{{\widehat{p}}_{c\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}}\left\{e_k-{\theta }_k{\pi }_k{\overline{e}}_{rc\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{<mo\; stretchy="false">)</mo>}}\right\}\mathrm{,}$$

and

$${\overline{e}}_{rc}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{c,hh}}{d}_k{r}_k{e}_k}}{{\displaystyle {\sum }_{k\in S_{c,hh}}{\theta }_k{r}_k}}\mathrm{,}$$

where we let

$e_k\mathrm{<mo>=</mo>}{y}_k-{\widehat{B}}_{r\mathrm{,}hh}^{{\rm T}}{x}_k$ with ${\widehat{B}}_{r\mathrm{,}hh}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{k\in S_{r,hh}}{d}_{rk}{x}_k{x}_k^{{\rm T}}}\right)}^{-1}{\displaystyle \sum _{k\in S_{r,hh}}{d}_{rk}{x}_k{y}_k}(2.12)$

denote the estimated regression residuals of the variable of interest on the calibration variables. This is a conservative estimator for the asymptotic variance of ${\widehat{Y}}_{\text{cal}\mathrm{,}hh}.$

2.1.4     Computation of household weights on an example

To fix ideas, we describe a small example. We consider a population $U_{hh}$ of $N_{hh}\mathrm{<mo>=</mo>}100$ households. We suppose without loss of generality that a single stratum is used, and that a sample of $n_{hh}\mathrm{<mo>=</mo>}10$ households is selected.

The sample is $S\mathrm{<mo>=</mo>}\left\{A\mathrm{,}B\mathrm{,}\dots \mathrm{,}J\right\}.$ The inclusion probabilities of the selected units are (say)

${\pi }_A\mathrm{<mo>=</mo>}{\pi }_B\mathrm{<mo>=</mo>}{\pi }_C\mathrm{<mo>=</mo>}{\pi }_D\mathrm{<mo>=</mo>}\frac{1}{4}$ and    ${\pi }_E\mathrm{<mo>=</mo>}{\pi }_F\mathrm{<mo>=</mo>}{\pi }_G\mathrm{<mo>=</mo>}{\pi }_H\mathrm{<mo>=</mo>}{\pi }_I\mathrm{<mo>=</mo>}{\pi }_J\mathrm{<mo>=</mo>}\frac{1}{16}\mathrm{,}(2.13)$

resulting in the design weights

$d_A\mathrm{<mo>=</mo>}{d}_B\mathrm{<mo>=</mo>}{d}_C\mathrm{<mo>=</mo>}{d}_D\mathrm{<mo>=</mo>}4$ and    $d_E\mathrm{<mo>=</mo>}{d}_F\mathrm{<mo>=</mo>}{d}_G\mathrm{<mo>=</mo>}{d}_H\mathrm{<mo>=</mo>}{d}_I\mathrm{<mo>=</mo>}{d}_J\mathrm{<mo>=</mo>}16.(2.14)$

Among the 10 selected households, 7 only are surveyed due to non-response. It is accounted for by using the method of RHGs, with two groups: the units $A,$ $B,$ $F$ and $J$ in the first one, and the units $C,$ $D,$ $E,$ $G,$ $H,$ and $I$ in the second one. The units $B,$ $C$ and $G$ are non-respondents. Inside each RHG, we compute estimated response probabilities, weighted by the design weights $\left({\theta }_k\mathrm{<mo>=</mo>}{d}_k\right).$ This leads to

$$\begin{array}{ll}{\widehat{p}}_1\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{1,hh}}{d}_k{r}_k}}{{\displaystyle {\sum }_{k\in S_{1,hh}}{d}_k}}\mathrm{<mo>=</mo>}\frac{d_A+d_F+d_J}{d_A+d_B+d_F+d_J}\mathrm{<mo>=</mo>}\frac{9}{10}\mathrm{,}\hfill \\ {\widehat{p}}_2\hfill & \mathrm{<mo>=</mo>}\frac{d_D+d_E+d_H+d_I}{d_C+d_D+d_E+d_G+d_H+d_I}\mathrm{<mo>=</mo>}\frac{13}{18}\mathrm{.}(2.15)\hfill \end{array}$$

The weights accounting for non-response are obtained for the respondents by dividing the sampling weights by the estimated response probabilities. This leads to the weights

$$d_{rA}\mathrm{<mo>=</mo>}\frac{40}{9}{d}_{rD}\mathrm{<mo>=</mo>}\frac{72}{13}{d}_{rE}\mathrm{<mo>=</mo>}{d}_{rH}\mathrm{<mo>=</mo>}{d}_{rI}\mathrm{<mo>=</mo>}\frac{288}{13}{d}_{rF}\mathrm{<mo>=</mo>}{d}_{rJ}\mathrm{<mo>=</mo>}\frac{160}{9}\mathrm{.}(2.16)$$

Finally, the weights are calibrated to match exactly the population size $N_{hh}\mathrm{<mo>=</mo>}100$ and an auxiliary total $X_{\mathrm{1,}hh}\mathrm{<mo>=</mo>}60.$ Note that, using the sample of respondents, we obtain ${\widehat{N}}_{r\mathrm{,}hh}\mathrm{<mo>=</mo>}112$ and ${\widehat{X}}_{1r\mathrm{,}hh}\mathrm{<mo>=</mo>}$ 66.53. The calibrated weights are

$$\begin{array}{lll}{w}_A\mathrm{<mo>=</mo>}\mathrm{4.01,}\hfill & w_D\mathrm{<mo>=</mo>}\mathrm{4.87,}\hfill & w_E\mathrm{<mo>=</mo>}{w}_H\mathrm{<mo>=</mo>}\mathrm{19.98,}\hfill \\ w_F\mathrm{<mo>=</mo>}\mathrm{15.63,}\hfill & w_I\mathrm{<mo>=</mo>}\mathrm{19.49,}\hfill & w_J\mathrm{<mo>=</mo>}\mathrm{16.03.}(2.17)\hfill \end{array}$$

The sampling and estimation steps are summarized in Figure 2.2.

2.2   Case of a sample of households and individuals

We are interested in the population $U_{\text{ind}}$ of individuals associated to the population $U_{hh}$ of households considered in Section 2.1. If we let $y_l$ denote the value taken by some variable of interest for the individual $l,$ the parameter of interest is

$$Y_{\text{ind}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in U_{\text{ind}}}{y}_l}\mathrm{.}(2.18)$$

2.2.1     Sampling design

Within any sampled household $k\in S_{hh},$ a subsample $S_{\text{ind}\mathrm{,}k}$ of individuals is selected, and the sample $S_{\text{ind}}$ is the union of these samples. We let ${\pi }_{l\mathrm{<mo>|</mo>}k}$ denote the conditional inclusion probability of the individual $l$ inside the household $k.$ The conditional design weight of $l$ is

$d_{l\mathrm{<mo>|</mo>}k}\mathrm{<mo>=</mo>}\frac{1}{{\pi }_{l\mathrm{<mo>|</mo>}k}}$ for any   $l\in k\mathrm{,}(2.19)$

and the non-conditional design weight is

$d_l\mathrm{<mo>=</mo>}{d}_{l\mathrm{<mo>|</mo>}k}\times d_k$ for any   $l\in k\mathrm{.}(2.20)$

In case of full response, the estimator of $Y_{\text{ind}}$ is

$${\widehat{Y}}_{\text{ind}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_{hh}}{d}_k}{\displaystyle \sum _{l\in S_{\text{ind},k}}{d}_{l\mathrm{<mo>|</mo>}k}{y}_l}\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\in S_{\text{ind}}}{d}_l{y}_l}\mathrm{.}(2.21)$$

The benchmark variance estimator for ${\widehat{Y}}_{\text{ind}}$ is obtained from (2.4), by replacing $y_k$ with

$${\widehat{y}}_k\mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in S_{\text{ind},k}}{d}_{l\mathrm{<mo>|</mo>}k}{y}_l}\mathrm{.}(2.22)$$

2.2.2     Treatment of non-response

The weights of individuals accounting for the non-response of households are

$d_{rl}\mathrm{<mo>=</mo>}{d}_{rk\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}{d}_{l\mathrm{<mo>|</mo>}k\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}$ with   $k\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}$ the household containing   $l\mathrm{,}(2.23)$

with $d_{rk}$ the weight of household $k$ corrected for unit non-response (see equation (2.6)), and $d_{l\mathrm{<mo>|</mo>}k}$ the conditional sampling weight of individual $l$ inside the household $k$ (see equation (2.19)). We let

$$S_{r\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}{\displaystyle \underset{k\in S_{r,hh}}{\cup }{S}_{\mathrm{ind}\mathrm{,}k}}(2.24)$$

denote the set of all sampled individuals inside the responding households.

The individuals in $S_{r\mathrm{,}\text{ind}}$ are themselves prone to non-response, though it is usually expected to be to a smaller extent. This leads to the observation of a sub-sample of respondents $S_{rr\mathrm{,}\text{ind}}$ only. We let $r_l$ denote the response indicator and $p_l$ denote the response probability of the individual $l$ . We suppose that the individuals respond independently of one another. Also, we suppose that this non-response is handled through the method of RHGs: the sample $S_{r\mathrm{,}\text{ind}}$ may be partitioned into $D$ RHGs denoted as $S_{r\mathrm{1,}\text{ind}}\mathrm{,}\dots \mathrm{,}{S}_{rD\mathrm{,}\text{ind}}$ such that the response probability $p_l$ is constant inside a RHG.

For $d\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}D,$ we let $p_d$ denote the common response probability inside the RHG $S_{rd\mathrm{,}\text{ind}}.$ It is estimated by

$${\widehat{p}}_d\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in S_{rd,\text{ind}}}{\theta }_l{r}_l}}{{\displaystyle {\sum }_{l\in S_{rd,\text{ind}}}{\theta }_l}}\mathrm{,}(2.25)$$

with ${\theta }_l$ some weight attached to the individual $l.$ The choice ${\theta }_l\mathrm{<mo>=</mo>}1$ leads to estimating $p_d$ by the unweighted response rate inside the RHG. The choice ${\theta }_l\mathrm{<mo>=</mo>}{d}_l$ leads to estimating $p_d$ by the response rate inside the RHG, weighted by the individual sampling weights. The choice ${\theta }_l\mathrm{<mo>=</mo>}{d}_{rl}$ leads to estimating $p_d$ by the response rate inside the RHG, weighted by the individual sampling weights corrected of household unit non-response. We compare these different choices in the simulation study performed in Section 4.

Accounting for the estimated response probabilities leads to the individual weights corrected for household/individual non-response

$d_{rrl}\mathrm{<mo>=</mo>}\frac{d_{rl}}{{\widehat{p}}_{d\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}}}$ with   $d\mathrm{<mo\; stretchy="false">(</mo>}l\mathrm{<mo\; stretchy="false">)</mo>}$ the household containing   $l\mathrm{.}(2.26)$

The estimator of $Y_{\text{ind}}$ adjusted for household/individual non-response is

$${\widehat{Y}}_{rr\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in S_{rr,\text{ind}}}{d}_{rrl}{y}_l}\mathrm{.}(2.27)$$

2.2.3     Calibration

We let $z_l$ denote the vector of calibration variables at the individual level, and $Z_{\text{ind}}$ denote the total on the population $U_{\mathrm{ind}}.$ For the sample $S_{rr\mathrm{,}\text{ind}},$ this leads to the linear calibrated weights

$$w_l\mathrm{<mo>=</mo>}{d}_{rrl}\left(1+z_l^{{\rm T}}{\lambda }_{\text{ind}}\right)\mathrm{,}$$

with

$${\lambda }_{\text{ind}}\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{l\in S_{rr,\text{ind}}}{d}_{rrl}{z}_l{z}_l^{{\rm T}}}\right)}^{-1}\left(Z_{\text{ind}}-{\widehat{Z}}_{rr\mathrm{,}\text{ind}}\right)\mathrm{,}(2.28)$$

and where ${\widehat{Z}}_{rr\mathrm{,}\text{ind}}$ is the estimator of $Z_{\text{ind}},$ obtained by plugging $z_l$ into (2.27). The calibrated estimator is

$${\widehat{Y}}_{\text{cal,}\text{ind}}\mathrm{<mo>=</mo>}{\displaystyle \sum _{l\in S_{rr,\text{ind}}}{w}_l{y}_l}\mathrm{.}(2.29)$$

The sampling and estimation steps are summarized in Figure 2.3.

2.2.4     Computation of individual weights on an example

We continue the example initiated in Section 2.1.4. Recall that the sample of responding households is $S_{r\mathrm{,}hh}\mathrm{<mo>=</mo>}\left\{A\mathrm{,}D\mathrm{,}E\mathrm{,}F\mathrm{,}H\mathrm{,}I\mathrm{,}J\right\}.$ The set of all individuals inside the responding households is as follows (say):

We suppose that the sampling design consists in selecting one individual exactly inside each household. The set $S_{r\mathrm{,}\text{ind}}$ of all sampled individuals inside the responding households is

$$S_{r\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}\left\{i_1\mathrm{,}{i}_4\mathrm{,}{i}_6\mathrm{,}{i}_8\mathrm{,}{i}_{11}\mathrm{,}{i}_{12}\mathrm{,}{i}_{13}\right\}\mathrm{.}(2.31)$$

From equations (2.23) and (2.16), the individual weights corrected for household non-response are therefore

$$d_{r1}\mathrm{<mo>=</mo>}\frac{40}{3}\mathrm{,}{d}_{r4}\mathrm{<mo>=</mo>}\frac{72}{13}\mathrm{,}{d}_{r6}\mathrm{<mo>=</mo>}\frac{576}{13}\mathrm{,}{d}_{r8}\mathrm{<mo>=</mo>}\frac{160}{3}\mathrm{,}{d}_{r11}\mathrm{<mo>=</mo>}\frac{576}{13}\mathrm{,}{d}_{r12}\mathrm{<mo>=</mo>}\frac{288}{13}\mathrm{,}{d}_{r13}\mathrm{<mo>=</mo>}\frac{160}{9}\mathrm{.}(2.32)$$

Among these 7 selected individuals, 4 only are surveyed due to non-response, accounted for by using the method of Response Homogeneity Groups (RHGs). We suppose that there are two RHGs: the units $i_1,$ $i_6,$ $i_8$ and $i_{11}$ in the first one, and the units $i_4,$ $i_{12}$ and $i_{13}$ in the second one. The units $i_4,$ $i_{11}$ and $i_{13}$ are non-respondents. Inside each RHG, we compute unweighted estimated response probabilities $\left({\theta }_l\mathrm{<mo>=</mo>}1\right).$ This leads to

$$\begin{array}{ll}{\widehat{p}}_1\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in S_{r1,\text{ind}}}{r}_l}}{{\displaystyle {\sum }_{l\in S_{r1,\text{ind}}}1}}\mathrm{<mo>=</mo>}\frac{3}{4}\mathrm{,}\hfill \\ {\widehat{p}}_2\hfill & \mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in S_{r2,\text{ind}}}{r}_l}}{{\displaystyle {\sum }_{l\in S_{r2,\text{ind}}}1}}\mathrm{<mo>=</mo>}\frac{1}{3}\mathrm{.}(2.33)\hfill \end{array}$$

The weights accounting for household/individual non-response are obtained for the respondents by dividing the weights in (2.32) by the estimated response probabilities. This leads to the weights

$$d_{rr1}\mathrm{<mo>=</mo>}\frac{160}{9}\mathrm{,}{d}_{rr6}\mathrm{<mo>=</mo>}\frac{\text{2,304}}{39}\mathrm{,}{d}_{rr8}\mathrm{<mo>=</mo>}\frac{640}{9}\mathrm{,}{d}_{rr12}\mathrm{<mo>=</mo>}\frac{864}{13}\mathrm{.}(2.34)$$

Finally, the weights are calibrated to match the population size $N_{\text{ind}}\mathrm{<mo>=</mo>}200$ and an auxiliary total $Z_{\mathrm{1,}\text{ind}}\mathrm{<mo>=</mo>}450.$ Note that, using the sample of respondents, we obtain ${\widehat{N}}_{r\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}$ 214.4 and ${\widehat{Z}}_{1r\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}$ 451.3. The calibrated weights are

$$w_1\mathrm{<mo>=</mo>}\mathrm{19.61,}{w}_6\mathrm{<mo>=</mo>}\mathrm{53.93,}{w}_8\mathrm{<mo>=</mo>}\mathrm{78.43,}{w}_{13}\mathrm{<mo>=</mo>}\mathrm{48.04.}(2.35)$$

The sampling and estimation steps are summarized in Figure 2.4.

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