Cost optimal sampling for the integrated observation of different populations
Section 6. Conclusions

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In this paper, we studied the problem of the definition of optimal sampling designs for survey strategies aiming at observing in an integrated way different statistical populations related to each other. This is particularly relevant in the agricultural sector where the integrated observation allows measurement of global phenomena that affect different statistical populations such as farms and households. The integrated observation is realized by directly sampling the first population and indirectly observing the second population, exploiting the links existing among the units of the two populations. We studied the problem considering three different contexts concerning information about the links. These range from two contexts in which the information is very rich, to the third context considering a case in which the information is very poor. The uncertainty on variables of the two populations, on links and on the $z$ variables (built by the indirect sampling mechanism) is treated by introducing suitable superpopulation models for which expected values (of first and second order) are considered as known when launching the algorithm for the optimal sampling. Empirical studies were performed on real data of a developing country: Mozambique.

The main conclusions are summarized as follows.

Integrated vs independent observation. The integrated observation is essential to measure thoroughly global phenomena which impact on different populations. The main advantage is that it allows the cross tabulation of population $U^A$ variables with population $U^B$ variables. Furthermore, the integrated observation is necessary when the frame for the population $U^B$ does not exist and an indirect sampling mechanism is needed. This is the case examined in Context 3. However, for Contexts 1 and 2, if only aggregates are examined independently from each other in the two populations, the independent allocation will be more efficient.

Cost issues. The loss in efficiency of the integrated observation can be reduced if, as assumed with cost function (5.3), the average cost of observing the elementary unit of $U^B$ decreases when the size of the indirectly observed clusters increase. In this case, the performance of the integrated sample allocation and of the two independent allocations could be closer or similar as in the evaluation study. Nevertheless, it is complex to establish which relationship between $C^A$ and $C^B$ leads to two strategies with similar costs, since the allocations depend on not only on the cost of interview but also on the variability of the target parameters in the two populations and on the set of variance constraints.

Controlling the errors in the design phase. The integrated approach to allocation enables the CVs of the estimates for integrated populations to be controlled. If this is not done, the CVs of the indirectly observed population might be very high.

The impact on the uncertainty on the sample sizes. An increase in the model variances (on the variables or on the links) causes a significant increase in the sample sizes. This stresses the need of having good models for predicting the unknown variables or the links.

Appendix A

To obtain the model expectation $E_{M_v}\left({\eta }_{j,v}^2\right),$ let ${\eta }_v=\left\{{\eta }_{j,v}\right\}$ be the $M^A$ vector of residuals, where

$${\eta }_v=Y_v-\Pi D{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)Y_v,(\text{A}.1)$$

where $Y_v=\left\{y_{j,v}\right\}$ denotes the $M^A$ vector with the values of $v^{\text{th}}$ variable of interest and $\Pi =\text{diag}\left\{{\pi }_j^A\right\}$ indicates the diagonal matrix with the $M^A$ inclusion probabilities. According to model (4.1), the vector $Y_v$ may be expressed as $Y_v={\tilde{Y}}_v+u_v,$ where ${\tilde{Y}}_v=\left\{{\tilde{y}}_{i,v}\right\}$ and $u_v=\left\{u_{i,v}\right\}$ denotes the $M^A$ vectors of predictions and model residuals. Adopting the above matrix notation, the specific residuals ${\eta }_{j,v}$ can be expressed as ${\eta }_{j,v}=\left({\tilde{y}}_{j,v}+u_{j,v}\right)-{\pi }_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)\left({\tilde{Y}}_v+u_v\right).$ Therefore, the model expected values of the squared terms are given by:

$$\begin{array}{ll}{E}_{M_v}\left({\eta }_{j,v}^2\right)={\tilde{y}}_{j,v}^2+{\sigma }_{j,v}^2\hfill & -2{\pi }_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right){\tilde{Y}}_v-2{\pi }_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)0_{j{\sigma }_v}\hfill \\ \hfill & +{\pi }_j^2{\tilde{Y}}_v^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right){\tilde{Y}}_v\hfill \\ \hfill & +{\pi }_j^2{\sigma }_v^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right){\sigma }_v(\text{A}.2)\hfill \end{array}$$

where ${\sigma }_v=\left\{{\sigma }_{j,v}\right\}$ is the $M^A$ column vector of model standard errors of the $V$ variables and $0_{j{\sigma }_v}={\left(0,\dots ,{\sigma }_{j,v}^2,\dots ,0\right)}^{\prime }$ is a vector in which the $j^{\text{th}}$ element is equal to ${\sigma }_{j,v}^2$ and all other elements are zeroes. Using the above matrix notation and according to Falorsi and Righi (2015), the anticipated variance can be approximated by the following expression:

$$E_{M_v}\left[V\left({\widehat{Y}}_v^A|m^A\right)\right]=\left[M^A/\left(M^A-H\right)\right]\left[{\tilde{Y}}_v^{\prime }{\Pi }^{-1}{\tilde{Y}}_v+{\sigma }_v^{\prime }{\Pi }^{-1}{\sigma }_v-{\tilde{Y}}_v^{\prime }{\tilde{Y}}_v+{\sigma }_v^{\prime }{\sigma }_v+{\text{AVA}}_{3,v}\right].$$

Letting $a_v=D{\Delta }^{-1}{D}^{\prime }(I-\Pi ){\tilde{Y}}_v\text{},b_v={\sigma }_v^{\prime }D{\Delta }^{-1}{D}^{\prime }(I-\Pi ){\sigma }_v$ and $c_v={\sigma }_v^{\prime }\text{diag}[\text{}D{\Delta }^{-1}{D}^{\prime }{(I-\Pi )}^{\prime }\text{}(I-\Pi )D{\Delta }^{-1}{D}^{\prime }\text{}]{\sigma }_v\text{}.$ we then have ${\text{AVA}}_{3,v}=a_v^{\prime }\left(I-\Pi \right)\left(2{\tilde{Y}}_v-\Pi a_v\right)+1^{\prime }\left(I-\Pi \right)\left(2b_v-\Pi c_v\right),$ where the scalars defined as (A.1.4), (A.1.7) and (A.1.8) in Falorsi and Righi (2015) are respectively the elements of the vectors $a_v,$ $b_v$ and  $c_v.$

Appendix B

Adopting the matrix notation, the residuals ${\eta }_{j,r}$ can be expressed as

$${\eta }_{j,r}=l_j^{\prime }\left({\tilde{Y}}_r+u_r\right)-{\pi }_j{\delta }_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L\left({\tilde{Y}}_r+u_r\right),(\text{B}.1)$$

where $L=\left\{{\tilde{L}}_{j,i}^B\right\}$ is the $M^A\times N^B$ matrix of standardized links, and ${\tilde{Y}}_r=\left\{y_{i,r}\right\}$ and $u_r=\left\{u_{i,r}\right\}$ denote respectively the $N^B$ vectors with the values of the predictions and of the residuals of the $r^{\text{th}}$ variable of interest being $l_j^{\prime }$ is the $j^{\text{th}}$ row of the matrix $L.$ Therefore, the model expected values of the squared terms is given by:

$$\begin{array}{ll}{E}_{M_r}\left({\eta }_{j,r}^2\right)\hfill & ={\tilde{Y}}_r^{\prime }{l}_j{l}_j^{\prime }{\tilde{Y}}_r+{\sigma }_r^{\prime }{l}_j{l}_j^{\prime }{\sigma }_r\hfill \\ \hfill & -2{\pi }_j{\tilde{Y}}_r^{\prime }{d}_j{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L{\tilde{Y}}_r-2{\pi }_j{\sigma }_r^{\prime }{d}_j{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L{\sigma }_r\hfill \\ \hfill & +{\pi }_j^2{\tilde{Y}}_r^{\prime }{L}^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L{\tilde{Y}}_r\hfill \\ \hfill & +{\pi }_j^2{\sigma }_r^{\prime }{L}^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L{\sigma }_r(\text{B}.2)\hfill \end{array}$$

where ${\sigma }_r=\left\{{\sigma }_{j,r}\right\}$ is the $N^B$ column vector of model standard errors of the $y_r$ variables. Following the above notation, we have:

$$E_{M_v}\left[V\left({\widehat{Y}}_r^A|m^A\right)\right]=\left[M^A/\left(M^A-H\right)\right]\left[{\tilde{Y}}_r^{\prime }{L}^{\prime }{\Pi }^{-1}L{\tilde{Y}}_r+{\sigma }_r^{\prime }{L}^{\prime }{\Pi }^{-1}L{\sigma }_r-{\tilde{Y}}_r^{\prime }{L}^{\prime }L{\tilde{Y}}_r+{\sigma }_r^{\prime }{L}^{\prime }L{\sigma }_r+{\text{AVA}}_{3,r}\right].$$

Letting $a_r=D{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L{\tilde{Y}}_r,$ $b_r={\sigma }_r^{\prime }D{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right){\sigma }_r,$ $c_r={\sigma }_r^{\prime }D{\Delta }^{-1}{D}^{\prime }{\left(I-\Pi \right)}^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{D}^{\prime }{\sigma }_r,$ we have ${\text{AAV}}_{3,r}=a_r^{\prime }\left(I-\Pi \right)\left(2L{\tilde{Y}}_r-\Pi a_r\right)+1^{\prime }\left(I-\Pi \right)\left(2b_r-\Pi c_r\right).$

Finally, we note the following

$$\begin{array}{ll}{E}_{M_r}\left(z_{j,r}^2\right)\hfill & ={\displaystyle \sum _{i=1}^{N^B}{\left({\tilde{L}}_{j,i}^B\right)}^2\left({\tilde{y}}_{i,r}^2+{\sigma }_{i,r}^2\right)}+{\displaystyle \sum _{i=1}^{N^B}{\tilde{L}}_{j,i}^B{\tilde{y}}_{i,r}{\displaystyle \sum _{i^{{}^{\prime }}\ne i}{\tilde{L}}_{j,i^{{}^{\prime }}}^B{\tilde{y}}_{i^{{}^{\prime }}\text{},r}}}\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^{N^B}{\left({\tilde{L}}_{j,i}^B\right)}^2\left({\tilde{y}}_{i,r}^2+{\sigma }_{i,r}^2\right)}+{\displaystyle \sum _{i=1}^{N^B}{\tilde{L}}_{j,i}^B{\tilde{y}}_{i,r}}\left({\tilde{z}}_{j,r}-{\tilde{L}}_{j,i}^B{\tilde{y}}_{i,r}\right)\hfill \\ \hfill & ={\displaystyle \sum _{i=1}^{N^B}{\left({\tilde{L}}_{j,i}^B\right)}^2\left({\tilde{y}}_{i,r}^2+{\sigma }_{i,r}^2\right)}+{\tilde{z}}_{j,r}^2-{\displaystyle \sum _{i=1}^{N^B}{\left({\tilde{L}}_{j,i}^B\right)}^2{\tilde{y}}_{i,r}^2}\hfill \\ \hfill & ={\tilde{z}}_{j,r}^2+{\sigma }_{j,zr}^2.\hfill \end{array}$$

Appendix C

Starting from (B.2), we have:

$$\begin{array}{ll}{E}_{M_l}{E}_{M_r}\left({\eta }_{j,r}^2\right)\hfill & ={\tilde{Y}}_r^{\prime }{E}_{M_l}\left(l_j{l}_j^{\prime }\right){\tilde{Y}}_r+{\sigma }_r^{\prime }{E}_{M_l}\left(l_j{l}_j^{\prime }\right){\sigma }_r\hfill \\ \hfill & -2{\pi }_j{\tilde{Y}}_r^{\prime }{E}_{M_l}\left(l_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L\right){\tilde{Y}}_r\hfill \\ \hfill & -2{\pi }_j{\sigma }_r^{\prime }{E}_{M_l}\left(l_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L\right){\sigma }_r\hfill \\ \hfill & +{\pi }_j^2{\tilde{Y}}_r^{\prime }{E}_{M_l}\left(L^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L\right){\tilde{Y}}_r\hfill \\ \hfill & +{\pi }_j^2{\sigma }_r^{\prime }{E}_{M_l}\left(L^{\prime }\left(I-\Pi \right)D{\Delta }^{-1}{d}_j{d}_j^{\prime }{\Delta }^{-1}{D}^{\prime }\left(I-\Pi \right)L\right){\sigma }_r.(\text{C}.1)\hfill \end{array}$$

The above expected value can be easily derived based on the following general result. Let $A=\left\{a_{j,j^{{}^{\prime }}}\right\}$ be a generic $M^A\times M^A$ matrix. The generic element $g_{i,i^{{}^{\prime }}}$ in the position $i,i^{\prime }$ of the squared $N^B\times N^B$ matrix $L^{\prime }AL=\left\{g_{i,i^{{}^{\prime }}}\right\}$ is given by $g_{i,i^{{}^{\prime }}}={\displaystyle {\sum }_{j=1}^{M^A}{\displaystyle {\sum }_{j^{{}^{\prime }}=1}^{M^A}{\tilde{L}}_{j,i}^B{\tilde{L}}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B}}{a}_{j,j^{{}^{\prime }}}.$

We have $E_{M_l}\left(g_{i,i^{{}^{\prime }}}\right)={\displaystyle {\sum }_{j=1}^{M^A}{\displaystyle {\sum }_{j^{{}^{\prime }}=1}^{M^A}{\tilde{\Lambda }}_{j,i}^B{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B}}{a}_{j,j^{{}^{\prime }}}+{\displaystyle {\sum }_{j=1}^{M^A}{\displaystyle {\sum }_{j^{\prime }=1}^{M^A}{\text{Cov}}_{M_l}({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B)}}{a}_{j,j^{{}^{\prime }}}.$

Taylor’s series first order approximation of ${\tilde{\Lambda }}_{j,i}$ is given by ${\tilde{L}}_{j,i}^B\cong \frac{1}{{\Lambda }_i}\left(L_{j,i}^B-{\tilde{\Lambda }}_{j,i}^B{L}_i^B\right).$ Therefore, we have

$$\begin{array}{ll}{\text{Cov}}_{M_l}\left({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B\right)\hfill & \cong \frac{1}{{\Lambda }_i^B}\frac{1}{{\Lambda }_{i^{\prime }}^B}{\text{Cov}}_{M_l}\left[\left(L_{j\text{},i}^B-{\tilde{\Lambda }}_{j\text{},i}^B{L}_i^B\right),\left(L_{j^{{}^{\prime }}\text{}\text{},i^{{}^{\prime }}}^B-{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B{L}_{i^{{}^{\prime }}}^B\right)\right]\hfill \\ \hfill & =\{\begin{array}{ll}\left[V_{M_l}\text{}\left(L_{j\text{},i}^B\right)\left(1-2{\tilde{\Lambda }}_{j\text{},i}^B\right)+{\left({\tilde{\Lambda }}_{j,i}^B\right)}^2{V}_{M_l}\text{}\left(L_i^B\right)\right]/{\left({\Lambda }_i^B\right)}^2\hfill & \text{for }j=j^{\prime }\text{and }i=i^{\prime }\hfill \\ \left[{\tilde{\Lambda }}_{j\text{},i}^B{\tilde{\Lambda }}_{j\text{},i^{{}^{\prime }}}^B{V}_{M_l}\text{}\left(L_i^B\right)-{\tilde{\Lambda }}_{j\text{},i}^B{V}_{M_l}\text{}\left(L_{j\text{},i}^B\right)-{\tilde{\Lambda }}_{j\text{},i^{{}^{\prime }}}^B{V}_{M_l}\text{}\left(L_{j^{{}^{\prime }}\text{}\text{},i}^B\right)\right]/{\left({\Lambda }_i^B\right)}^2\hfill & \text{for }j\ne j^{\prime }\text{and }i=i^{\prime }\hfill \\ 0\hfill & \text{for }j=j^{\prime }\text{and }i\ne i^{\prime }\hfill \\ 0\hfill & \text{for }j\ne j^{\prime }\text{and }i\ne i^{\prime }\hfill \end{array}(\text{C}.2)\hfill \end{array}$$

where

$$V_{M_l}\left(L_{j,i}^B\right)={\displaystyle \sum _{k=1}^{M_i^b}{\lambda }_{j,ik}}\left(1-{\lambda }_{j,ik}\right),V_{M_l}\text{}\left(L_i^B\right)={\displaystyle \sum _{j=1}^{M^A}{V}_{M_l}\text{}\left(L_{j,i}^B\right)}.(\text{C}.3)$$

Equation (C.2) is derived from the following result. For $i=i^{\prime }$ and $j=j^{\prime },$ we obtain

$$\begin{array}{ll}{\text{Cov}}_{M_l}\left({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j,i}^B\right)\hfill & =V_{M_l}\left({\tilde{L}}_{j,i}^B\right)\hfill \\ \hfill & \cong \frac{1}{{\Lambda }_i^2}\left[V_{M_l}\text{}\left(L_{j,i}^B\right)+{\left({\tilde{\Lambda }}_{j,i}^B\right)}^2{V}_{M_l}\text{}\left(L_i^B\right)-2{\tilde{\Lambda }}_{j,i}^B{\text{Cov}}_{M_l}\text{}\left(L_{j,i}^B,L_i^B\right)\right]\hfill \\ \hfill & =\frac{1}{{\Lambda }_i^2}\left[V_{M_l}\text{}\left(L_{j,i}^B\right)\left(1-2{\tilde{\Lambda }}_{j,i}^B\right)+{\left({\tilde{\Lambda }}_{j,i}^B\right)}^2{V}_{M_l}\text{}\left(L_i^B\right)\right](\text{C}.4)\hfill \end{array}$$

where ${\text{Cov}}_{M_l}\left(L_{j,i}^B,L_i^B\right)=V_{M_l}\left(L_{j,i}^B\right).$ For $i=i^{\prime }$ and $j\ne j^{\prime },$ we obtain

$$\begin{array}{ll}{\text{Cov}}_{M_l}\left({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j^{{}^{\prime }}\text{},i}^B\right)\hfill & \cong \frac{1}{{\Lambda }_i^2}{\text{Cov}}_{M_l}\left[\left(L_{j,i}^B-{\tilde{\Lambda }}_{j,i}^B{L}_i\right),\left(L_{j^{{}^{\prime }}\text{},i}^B-{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i}^B{L}_i\right)\right]\hfill \\ \hfill & =\frac{1}{{\Lambda }_i^2}{V}_{M_l}\left[L_{j,i}^B{L}_{j^{{}^{\prime }}\text{},i}^B-{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i}^B{L}_{j,i}^B{L}_i-{\tilde{\Lambda }}_{j,i}^B{L}_{j^{{}^{\prime }}\text{},i}^B{L}_i+{\tilde{\Lambda }}_{j,i}^B{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i}^B{L}_i^2\right]\hfill \\ \hfill & =\frac{1}{{\Lambda }_i^2}\left[{\tilde{\Lambda }}_{j,i}^B{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i}^B{V}_{M_l}\left(L_i^B\right)-{\tilde{\Lambda }}_{j,i}^B{V}_{M_l}\left(L_{j,i}^B\right)-{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i}^B{V}_{M_l}\left(L_{j^{{}^{\prime }}\text{},i}^B\right)\right].(\text{C}.5)\hfill \end{array}$$

Let $a=\left\{a_j\right\}$ be a generic $M^A$ vector. The generic element ${}^j{g}_{i,i^{{}^{\prime }}}$ in the position $i,i^{\prime }$ of the squared $N^B\times N^B$ matrix $l_j{a}^{\prime }L=\left\{{}^{j}g{}_{i,i^{{}^{\prime }}}\right\}$ is given by ${}^j{g}_{i,i^{{}^{\prime }}}={\displaystyle {\sum }_{j^{{}^{\prime }}=1}^{M^A}{\tilde{L}}_{j,i}^B{\tilde{L}}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B}{a}_{j^{{}^{\prime }}},$ where $E_{M_l}\left({}^{j}g{}_{i,i^{{}^{\prime }}}\right)={\displaystyle {\sum }_{j^{{}^{\prime }}=1}^{M^A}{\tilde{\Lambda }}_{j,i}^B{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B}{a}_{j^{{}^{\prime }}}+{\displaystyle {\sum }_{j^{{}^{\prime }}=1}^{M^A}{\text{Cov}}_{M_l}\left({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B\right)}{a}_{j^{{}^{\prime }}}.$

Finally, denote by $\left\{{}^{jj}g{}_{i,i^{{}^{\prime }}}\right\}$ the generic element in the $i,i^{\prime \text{th}}$ position of the matrix $l_j{l^{\prime }}_j.$ Its generic expected value is given by $E_{M_l}\text{}\left({}^{jj}g{}_{i,i^{{}^{\prime }}}\right)={\tilde{\Lambda }}_{j,i}^B{\tilde{\Lambda }}_{j^{{}^{\prime }}\text{},i^{{}^{\prime }}}^B+{\text{Cov}}_{M_l}\text{}\left({\tilde{L}}_{j,i}^B,{\tilde{L}}_{j^{{}^{\prime }},i^{{}^{\prime }}}^B\right).$

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