Cost optimal sampling for the integrated observation of different populations
Section 2. Background

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Let $U^A$ and $U^B$ denote two related target populations, where $U^A$ is the population with the available sampling frame, and $U^B$ the survey population for which a sampling frame may or may not be available. For the agricultural example, $U^A$ is the population of farms and $U^B$ the population of rural households. Let $s^A$ be a sample selected from $U^A$ without replacement and with fixed sample size $m^A,$ where $U^A$ contains $M^A$ units. Let ${\pi }_j^A$ represent the inclusion probability of the $j^{\text{th}}$ unit in $U^A$ with ${\pi }_j^A>0$ and ${\displaystyle {\sum }_{j\in U^A}{\pi }_j^A}=m^A$ with ${\pi }^A={\left({\pi }_1^A,\dots ,{\pi }_j^A,\dots ,{\pi }_{M^A}^A\right)}^{\prime }.$ We denote by $y_{j,v}$ the value of the $v^{\text{th}}$ $\left(v=1,\dots ,V\right)$ characteristic on unit $j$ and their total by $Y_v^A.$

We estimate the total $Y_v^A$ according to the Horvitz-Thompson (HT) estimator,

$${\widehat{Y}}_v^A={\displaystyle \sum _{j\in s^A}{w}_j^A{y}_{j,v}},\left(2.1\right)$$

where $w_j^A=1/{\pi }_j^A.$

Many practical sampling designs define planned domains that are sub-populations in which the sample sizes are fixed before selecting the sample. Denote by $U_h^A\left(h=1,\dots ,H\right)$ the planned domain of size $M_h^A={\displaystyle {\sum }_{j\in U^A}{d}_{j\left(h\right)}}$ where $d_{j\left(h\right)}=1$ if $j\in U_h^A$ and $d_{j\left(h\right)}=0$ otherwise. Let us suppose that the $d_{j\left(h\right)}$ values are known and available in the sampling frame for all population units. Fixed size sampling designs are those satisfying

$${\displaystyle \sum _{j\in s^A}{d}_j=m^A},$$

where $d_j={\left(d_{j\left(1\right)},\dots ,d_{j\left(h\right)},\dots ,d_{j\left(H\right)}\right)}^{\prime }$ and $m^A={\left(m_1^A,\dots ,m_h^A,\dots ,m_H^A\right)}^{\prime }$ is the vector of integer numbers defining the sample sizes fixed at the design stage, with ${\displaystyle {\sum }_{j\in U^A}{d}_{j\left(h\right)}{\pi }_j^A}=m_h^A.$ In our setting, the planned domains can overlap; therefore, the unit $j$ may have more than one value $d_{j\left(h\right)}=1$ $(\mathrm{for}h=1,\dots ,H).$ Several customary fixed size sampling designs may be considered as particular cases. A well-known example is the Stratified Simple Random Sampling WithOut Replacement (SSRSWOR) design where strata are the planned domains and each $d_j$ vector has $H-1$ elements equal to zero, and one element equal to 1, which implies that each unit $j$ can belong to one and only one planned domain. Furthermore, in this design all the units in the stratum $U_h^A$ have a uniform inclusion probability given by ${\pi }_j^A=\frac{m_h^A}{M_h^A}$ for $j\in U_h^A.$ If each $d_j$ vector has $H-1$ elements equal to zero and one element equal to 1, and the ${\pi }_j^A$ values can be different in the stratum, we have a stratified sampling design, without replacement with fixed sample sizes and varying probabilities in each stratum. On the basis of the Winkler’s definition (2001), if ${\displaystyle {\sum }_{h=1}^H{d}_{j\left(h\right)}}>1,$ we have an Incomplete Multi-Way Stratified Sampling design.

We suppose that the $M^A\times H$ matrix $D={\left(d_1,\dots ,d_j,\dots ,d_{M^A}\right)}^{\prime }$ is non-singular. According to this general sampling design framework, Deville and Tillé (2005) proposed an approximated expression of the variance for ${\widehat{Y}}_v^A$ based on the Poisson sampling theory given by

$$V\left({\widehat{Y}}_v^A|m^A\right)\cong \left[M^A/\left(M^A-H\right)\right]{\displaystyle \sum }_{j\in U^A}\left(\frac{1}{{\pi }_j^A}-1\right){\eta }_{j,v}^2,\left(2.2\right)$$

where ${\widehat{Y}}_v^A|m^A$ is the HT estimator based on a general fixed sample size design with $m^A$ units related to the vector ${\pi }^A,$

$${\eta }_{j,v}=y_{j,v}-{\pi }_j^A{d}_j^{\prime }{\beta }_v,(2.3)$$

and

$${\beta }_v={\Delta }^{-1}{\displaystyle \sum _{j\in U^A}{\pi }_j^A\left(\frac{1}{{\pi }_j^A}-1\right)d_j{y}_{j,v}}\left(2.4\right)$$

with

$$\Delta ={\displaystyle \sum _{j\in U^A}{d}_j{d}_j^{\prime }{\pi }_j^A\left(1-{\pi }_j^A\right)}.\left(2.5\right)$$

The variance (2.2) resembles the variance expression of the HT estimator under a Poisson sampling design, but it uses the residuals ${\eta }_{j,v},$ instead of the original value $y_{j,v}.$ In practice, when $H=1$ this is the variance approximation of the Conditional Poisson Sampling design (CPS, as introduced in Deville and Tillé, 2005). CPS selects samples by means of a Poisson sampling design without replacement until a given sample size is obtained.

To clarify the degree of approximation of (2.2), consider the SSRSWOR design. According to expression (2.2), we have

$$V\left({\widehat{Y}}_v^A|m^A\right)=\left[M^A/\left(M^A-H\right)\right]{\displaystyle \sum _{h=1}^H{\sigma }_{v,h}^2{M}_h^A}\left(\frac{M_h^A}{m_h^A}-1\right),$$

where ${\sigma }_{v,h}^2$ is the design variance of the $y_{j,v}$ values in stratum $U_h^A$ (see Appendix 4 of Falorsi and Righi, 2015). The above approximation works well when the number of domains $H$ remains small compared to the overall population size $M^A.$

Let $M^B,$ $N^B,$ $U_i^B$ and $M_i^B$ be the number of units in $U^B,$ the number of clusters in $U^B,$ the $i^{\text{th}}$ cluster of $U^B$ with ${\displaystyle {\cup }_{i=1}^{N^B}{U}_i^B}=U^B$ and the number of units in the $i^{\text{th}}$ cluster $U_i^B,$ respectively. We denote by $y_{ik,r}$ the value of the $r^{\text{th}}$ $\left(r=1,\dots ,R\right)$ characteristic for the $k^{\text{th}}$ unit of the $i^{\text{th}}$ cluster of $U^B$ and the population total of all $y_{ik,r}’\text{s}$ by

$$Y_r^B={\displaystyle \sum _{i=1}^{N^B}{\displaystyle \sum _{k=1}^{M_i^B}{y}_{ik,r}}}.$$

Let $l_{j,ik}$ be an indicator variable of link existence: $l_{j,ik}=1$ indicates that there is a link between $j^{\text{th}}$ unit in $U^A$ and $k^{\text{th}}$ unit in $U_i^B,$ while $l_{j,ik}=0$ indicates otherwise.

Suppose that we carry out an indirect sampling process: if the unit $j\in U^A$ is included in $s^A,$ then all the clusters $U_i^B,$ for which $L_{j,i}^B={\displaystyle {\sum }_{k=1}^{M_i^B}{l}_{j,ik}}>0,$ are observed $(\text{i}\text{.e}\text{.},y_{ik,r})$ in the indirect sample of population $U^B.$ Let $n^B$ be the size of the sample of clusters in population $U^B$ obtained after the indirect sampling process. We estimate $Y_r^B$ according to the estimator based on the theory of the Generalized Weight Share Method (GWSM) of Lavallée (2002, 2007):

$${\widehat{Y}}_r^B={\displaystyle \sum _{i=1}^{n^B}{y}_{i,r}{w}_i^B},\left(2.6\right)$$

where

$$y_{i,r}={\displaystyle \sum _{k=1}^{M_i^B}{y}_{ik,r}}$$

and

$$w_i^B={\displaystyle \sum _{j\in s^A}{w}_j^A{\tilde{L}}_{j,i}^B}$$

with

$${\tilde{L}}_{j,i}^B=\frac{L_{j,i}^B}{L_i^B}$$

and

$$L_i^B={\displaystyle \sum _{j=1}^{M^A}{L}_{j,i}^B}.$$

Theorem in Section 3 of Lavallée (2002, 2007) states that (2.6) provides an unbiased estimator for $Y_r^B$ provided all links $l_{j,ik}$ can be correctly identified and $L_i^B>0$ for all $i\in U^B.$ By defining

$$z_{j,r}={\displaystyle \sum _{i=1}^{N^B}{\tilde{L}}_{j,i}^B{y}_{i,r}},\left(2.7\right)$$

the estimator (2.6) can be expressed as a usual Horvitz-Thompson (HT) estimator on the $z$ values referring to the $U^A$ population,

$${\widehat{Y}}_r^B={\displaystyle \sum _{s^A}{z}_{j,r}{w}_j^A}.\left(2.8\right)$$

Therefore, the variance $V\left({\widehat{Y}}_r^B\right)$ of ${\widehat{Y}}_r^B$ may be expressed as the variance of the HT estimator on the $U^A$ population. The approximate variance of ${\widehat{Y}}_r^B$ for fixed size sampling designs is given by

$$V\left({\widehat{Y}}_r^B|m^A\right)\cong \left[M^A/\left(M^A-H\right)\right]{\displaystyle \sum }_{j\in U^A}\left(\frac{1}{{\pi }_j^A}-1\right){\eta }_{j,r}^2,\left(2.9\right)$$

where ${\widehat{Y}}_r^B|m^A$ is the HT estimator based on a general fixed sample size design with $m^A$ units and the related vector ${\pi }^A,$

$${\eta }_{j,r}=z_{j,r}-{\pi }_j^A{d}_j^{\prime }{\beta }_r$$

with

$${\beta }_r={\Delta }^{-1}{\displaystyle \sum _{j\in U^A}{\pi }_j^A}\left(\frac{1}{{\pi }_j^A}-1\right)d_j{z}_{j,r}.$$

Remark 2.1. An interesting extension of the above framework, useful in case of integrated studies, is the case of a total derived from a cross tabulation of a variable of the population $U^A$ with a variable of the population $U^B.$ In order to illustrate this extension, let $y_v$ be a variable of $U^A$ with $C$ modalities and let $y_{j,v\left(c\right)}$ denote a dichotomous variable where $y_{j,v\left(c\right)}=1$ if the unit $j$ is characterized by the modality $c\left(c=1,\dots ,C \right)$ of $y_v,$ and $y_{j,v\left(c\right)}=0$ otherwise. Furthermore, let $y_r$ be a variable of $U^B$ with $G$ modalities and let $y_{ik,r\left(g\right)}$ denote a dichotomous variable where $y_{ik,r\left(g\right)}=1$ if the unit $k$ of the cluster $i$ is characterized by the modality $g\left(g=1,\dots ,G \right)$ of $y_r,$ and $y_{ik,r\left(g\right)}=0$ otherwise. The total number of units of $U^B$ characterized by the modality $g\left(g=1,\dots ,G \right)$ of $y_r$ and linked with units of population $U^A$ characterized by the modality $c$ of the variable $y_v,$ can be defined as

$$Y_{\left(c,g\right)}^B={\displaystyle \sum }_{j=1}^{N^A}{\displaystyle \sum }_{i=1}^{N^B}{\displaystyle \sum _{k=1}^{M_i^B}{l}_{j,ik}{y}_{j,v\left(c\right)} y_{ik,r\left(g\right)}}={\displaystyle \sum _{i=1}^{N^B}{y}_{i,\left(c,g\right)}},$$

where $y_{i,\left(c,g\right)}={\displaystyle {\sum }_{j=1}^{N^A}{\displaystyle {\sum }_{k=1}^{M_i^B}{l}_{j,ik} y_{j,v\left(c\right)}{y}_{ik,r\left(g\right)}}}.$

As an example, let us consider the case, illustrated in the introduction, of an integrated analysis examining the productivity of farms and the malnutrition of households, and suppose that $Y_{\left(c,g\right)}^B$ represents the total of persons with a malnutrition problem in the households of workers of farms characterized by an high productivity. In this case $y_{j,v\left(c\right)}$ has value 1 if the productivity of the farm $j$ is high and $y_{ik,r\left(g\right)}$ has value 1 if the person $k$ of the household $i$ has a problem of malnutrition.

The GWSM estimator of $Y_{\left(c,g\right)}^B$ can be obtained directly from expression (2.8) using the transformed variable $z_{j,r}={\displaystyle {\sum }_{i=1}^{N^B}{\tilde{L}}_{j,i}^B{y}_{i,\left(c,g\right)}}.$

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