2 Sample design and survey procedure

Anne Massiani

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The survey is conducted in Switzerland using a four-panel rotation scheme. Each year $t,$ a new sample of households $s_m^{A_1,t}$ is selected to replace an outgoing panel. The subscript $m$ is used here for samples of households; the subscript $p$ will be reserved for samples of persons. In accordance with the notation of Lavallée (2002), the superscripts $A_i$ refer to samples selected directly, while the superscript $B$ will be used in the rest of the document for samples selected indirectly. The new panel selected will be followed for four years according to the following scenario:

• In the first year $t,$ we first approach the households of $s_m^{A_1\mathrm{,}t}$ to complete a preliminary questionnaire called the "grid�. The variables identified in the grid are common to the entire household and relate primarily to its composition. The subsample of $s_m^{A_1,t}$ that responds to the grid is denoted $s_m^{A_2\mathrm{,}t}.$ The sample of individuals belonging to the households of $s_m^{A_2\mathrm{,}t}$ is denoted $s_p^{A_2\mathrm{,}t}$ and these individuals are called longitudinals. The households of $s_m^{A_2\mathrm{,}t}$ reached in the first survey year $t$ are then asked to complete a joint questionnaire for the entire household, called the household questionnaire.
• In the following three years $t+i,$ for $i=1$ to 3, an attempt is made to recontact the longitudinals of $s_p^{A_2\mathrm{,}t}$ aged 16 and over to survey their households based on the household composition in year $t+i,$ since households may change over time. All individuals in these households are integrated into the survey for year $t+i$ and are called cross-sectional individuals. Households for year $t+i$ that are reached by means of longitudinal $–$ that is, households reached indirectly $–$ are asked to again complete the grid and then the household questionnaire. However, it is not possible to recontact all the longitudinals, especially because some of them have moved, and this is a major cause of non-response and sample attrition.

We now set a given survey year $t$ and adopt a cross-sectional approach, meaning that we are interested in the estimates that can be produced for the population present in year $t.$ The sample surveyed during survey year $t$ consists of four panels contacted for the first, second, third or fourth time respectively. Let $s_m^{B\mathrm{,}\tau }$ denote the sample of households responding to the household questionnaire in the ${\tau }^{\text{th}}$ wave, for $\tau =\mathrm{1,}\dots \mathrm{,4.}$ As seen in Figure 2.1, the sample $s_m^{B\mathrm{,}\tau }$ was contacted via the longitudinals selected in year $t_{\tau }=t-\tau +1.$ The shaded parts on the right represent the samples participating in the survey in year $t$ (indirect sampling), while the samples on the left contain the initial households of the longitudinals through whom they were contacted (direct sampling).

Figure 2.1  Panels comprising the sample surveyed in year $t$

Description for Figure 2.1

Below is a more detailed description of the sampling design for $s_m^{A_2\mathrm{,}{t}_{\tau }},$ for $\tau =\mathrm{1,}\dots \mathrm{,4.}$ Households' composition may change over time, and therefore in the rest of the document, we will use the notation $k_1$ to designate households in the first survey year $t_{\tau }$ and the notation $k$ to designate those in survey year $t$ so as to distinguish them. Each sample $s_m^{A_2\mathrm{,}{t}_{\tau }}$ is obtained from two selection stages.

• Stage 1: a sample of households $s_m^{A_1\mathrm{,}{t}_{\tau }}$ is selected according to a design stratified by major region, of which there are seven in Switzerland. Within each stratum, draws are conducted according to a simple design. Here, ${\pi }_{k_1}^{A_1}$ denotes the first order of probability of inclusion of household $k_1$ and ${\pi }_{k_1{k^{\prime }}_1}^{A_1}$ denotes the second order of probability of inclusion of households $k_1$ and ${k^{\prime }}_1.$
• Stage 2: the second selection stage is based on non-response to the grid in the first survey year $t_{\tau }.$ This non-response is modeled using a Poisson design on households, and we note for any household $k_1:$

$\text{P(}{k}_1\in s_m^{A_2\mathrm{,}{t}_{\tau }}|k_1\in s_m^{A_1\mathrm{,}{t}_{\tau }}\text{)}=q_{k_1}^a\mathrm{.}$(2.1)

For all households $k_1$ and ${k^{\prime }}_1,$ we define as follows:

${\pi }_{k_1}^{A_2}=\text{P}\left(k_1\in s_m^{A_2\mathrm{,}{t}_{\tau }}\right)={\pi }_{k_1}^{A_1}{q}_{k_1}^a$(2.2)

${\pi }_{k_1{k^{\prime }}_1}^{A_2}=\text{P}\left(k_1\mathrm{,}{k^{\prime }}_1\in s_m^{A_2\mathrm{,}{t}_{\tau }}\right)=\{\begin{array}{ll}{\pi }_{k_1}^{A_1}{q}_{k_1}^a\hfill & \text{if }{k}_1={k^{\prime }}_1\hfill \\ {\pi }_{k_1{k^{\prime }}_1}^{A_1}{q}_{k_1}^a{q}_{{k^{\prime }}_1}^a\hfill & \text{if }{k}_1\ne {k^{\prime }}_1\mathrm{.}\hfill \end{array}$(2.3)

Finally, for all longitudinals $j$ and $j^{\prime }$ belonging respectively to households $k_1$ and ${k^{\prime }}_1,$ let

${\pi }_j^{A_2}=\text{P}\left(j\in s_p^{A_2\mathrm{,}{t}_{\tau }}\right)={\pi }_{k_1}^{A_2}{\pi }_{j{j}^{\prime }}^{A_2}=\text{P}\left(j\mathrm{,}{j}^{\prime }\in s_p^{A_2\mathrm{,}{t}_{\tau }}\right)={\pi }_{k_1{k^{\prime }}_1}^{A_2}\mathrm{.}$(2.4)

For practical reasons, samples $s_m^{A_1\mathrm{,}{t}_{\tau }},$ for $\tau =\mathrm{1,}\dots ,4$ are drawn in such a way as to be disjoint, which means that this is also the case with samples $s_m^{A_2\mathrm{,}{t}_{\tau }}.$ On the other hand, the four $s_m^{B\mathrm{,}\tau }$ samples are theoretically not necessarily disjoint, since there is a possibility that two longitudinals selected in two different waves will belong to the same household in the survey year. Considering the small sample sizes, this is extremely unlikely and has actually never been observed thus far. The fact is that when complete, a panel consists of approximately 3,600 households drawn with probabilities of selection ${\pi }_{k_1}^{A_1}$ for which the highest value is 0.0011. However, it should be noted that the methodology used would lend itself to processing non-disjoint $s_m^{B\mathrm{,}\tau }$ samples. Let

$s_m^B={\displaystyle \underset{\tau =1}{\overset{4}{\cup }}{s}_m^{B\mathrm{,}\tau }}$

and a weight $w_k,$ called the household cross-sectional weight, is calculated for each household $k$ of $s_m^B.$ The individuals in the households of $s_m^B$ are denoted by $s_p^B$ and each individual $j$ of a household $k$ of $s_m^B$ receives the weight $w_j=w_k.$

The information used to calculate the poverty and social exclusion indexes is based on the income of each household $k$ of $s_m^B.$ Some components of the household's income are taken from the household questionnaire completed by all households of $s_m^B.$ Other income components, such as wages, must again be obtained from questionnaires, this time from the individual questionnaires administered to members of the households of $s_m^B.$ In the case of complete or partial non-response to individual questionnaires or of partial non-response to the household questionnaire, some components of the income of $s_m^B$ households are missing. In some cases, especially where there is a lack of information on wages, missing values can be obtained from other sources based on administrative data. In the remaining cases, they are imputed. In this article, we do not take the effect of imputations into account. Under these conditions, the income of each household $k$ of $s_m^B$ is known and the non-responses observed in the different stages of the survey are household total non-responses: non-response to the successive grids and total non-response to the household questionnaire. Since the poverty and social exclusion indexes are calculated for individuals and not households, the income of each household is divided by a factor that depends on the number of persons in the household and their age. The result of this calculation, called equivalent income, is assigned to each individual $j$ of $s_p^B.$ The poverty and social exclusion indexes are calculated on the basis of equivalent income and the weights $w_j.$

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