3 Cross-sectional household weighting

Anne Massiani

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The weighting was carried out by Graf (2008). It utilizes non-response and calibration techniques (cf. Deville and Särndal 1992 for more information on calibration). It also uses the weight-share method (Lavallée 2002) to assign a weight to individuals reached indirectly. This weighting is carried out in five major steps. The objective of the four first step is to calculate a weight for the households in each panel $s_m^{B\mathrm{,}\tau },$ for $\tau =\mathrm{1,}\dots ,4.$ These weights are calculated separately for each panel. The last step then serves to combine these panels. Below, we review the basic elements of each step for the situation in which the response probabilities for the different stages of the survey are known. Explanations on how these parameters are estimated are provided in the numerical application in Section 7.

1. Calculate initial weight

   Let $\tau \in \left\{\mathrm{1,}\dots ,4\right\}$ be fixed. For any longitudinal $j\in s_p^{A_2\mathrm{,}{t}_{\tau }},$ we calculate the weight $w_j^{A_2}$ which takes account of non-response to the grid in wave 1:

   $$w_j^{A_2}=\frac{1}{{\pi }_j^{A_2}}\mathrm{.}\text{ (3.1)}$$

2. Adjust for non-response to the grid in the survey year

   Sample attrition is observed between the first wave and the survey year. First, it is not possible to recontact all the longitudinals of $s_p^{A_2\mathrm{,}{t}_{\tau }}.$ Second, not all the households that we manage to recontact via the longitudinals agree to fill out the grid again. We designate as $q_k^b$ the probability of household $k$ responding to the grid in the survey year, conditional on $s_p^{A_2\mathrm{,}{t}_{\tau }}.$ For the panel responding in the first wave, the $q_k^b$ values are all equal to 1, since there has not yet been any attrition. For any longitudinal $j$ belonging to household $k,$ let $q_j^b=q_k^b.$ We then calculate the weight:

   $$w_j^g=w_j^{A_2}\frac{1}{q_j^b}\mathrm{.}\text{ (3.2)}$$

   For reasons that will be explained in Section 7, the estimation of response probabilities $q_j^b$ is a delicate problem. Because of variations in the information available, these estimates are not all equal for the longitudinals in a given household $k,$ and this does not accord with the response mechanism. This difficulty, and how it is dealt with in estimating the variances of the indicators, will be discussed in Section 7.

3. Weight sharing

   Following the methodology of Lavallée (2002, Chapter 6), we introduce the concept of the initially present cohabitant: this is an individual who was not selected in wave 1, but who was included in the target population during the selection of wave 1. Newborns and immigrants are called initially absent cohabitants. Let $L_k$ be the number of longitudinals and $P_k$ be the number of cohabitants initially present in a household $k$ in the survey year. The weight of a household $k$ responding to the grid in the survey year is calculated using weight sharing as follows:

   $$w_k^p=\frac{1}{L_k+P_k}{\displaystyle \sum _{j=1}^{L_k}{w}_j^g}\mathrm{.}\text{ (3.3)}$$

4. Adjust for non-response to the household questionnaire

   The non-response observed between the grid for the survey year and the household questionnaire is modeled using a Poisson design on households. Let $q_k^c$ be the probability that household $k$ will respond to the household questionnaire, where we know that it responded to the grid for the survey year. For any household $k\in s_m^{B\mathrm{,}\tau },$ the weight is then computed as follows:

   $$w_k^{nr}=w_k^p\frac{1}{q_k^c}\mathrm{.}\text{ (3.4)}$$

5. Combine panels, then calibrate

   • Combine panels

     We want to obtain a weighting suitable for the amalgamation $s_m^B$ of the four samples $s_m^{B\mathrm{,}\tau },$ for $\tau =\mathrm{1,}\dots ,4.$ Accordingly, the weight $w_k^{nr}$ computed in the previous step must be divided, by order of magnitude, by a factor of 4. We have, for any household $k\in s_m^B:$

           $w_k^u={\varphi }_k^{\tau }\cdot w_k^{nr}\mathrm{,}$(3.5)

     where ${\varphi }_k^{\tau }$ values are allocation factors close to $1/4$ that must be optimized. Merkouris (2001) used variance minimization criteria to calculate the optimal values of ${\varphi }_k^{\tau }.$ These values depend on the variances, and hence on the design effects, associated with each panel. Since these design effects are unknown, we assume that within each major region, they are identical for each of the panels. This leads us to compute, within each major region, factors ${\varphi }_k^{\tau }$ that are proportional to the size of the sample of households responding in the ${\tau }^{\text{th}}$ wave.

   • Calibrate

     Weights $w_k^u$ are then calibrated on known margins on the population of households for the survey year. Let $w_k$ be the calibrated weight thus obtained for household $k.$

   • Assign weights to individuals

     For any individual $j$ belonging to a household $k$ of $s_m^B,$ let

     $w_j=w_k\mathrm{.}$

Note 1: The different weighting steps have been presented in the order in which they are actually performed. Below we describe another way to obtain exactly the same final weights when the response probabilities $q_j^b$ are known. In this case, since $q_j^b=q_k^b$ for all the longitudinals $j$ in a given household $k$ for the survey year, it is easy to see that the weight $w_k^{nr}$ given by formula (3.4) can be rewritten as follows:

$$w_k^{nr}=\frac{1}{L_k+P_k}\left({\displaystyle \sum _{j=1}^{L_k}{w}_j^{A_2}}\frac{1}{q_j^b}\right)\frac{1}{q_k^c}=w_k^{p^{*}}\frac{1}{q_k^b{q}_k^c}\mathrm{,}\text{ (3.6)}$$

where

$$w_k^{p*}=\frac{1}{L_k+P_k}{\displaystyle \sum _{j=1}^{L_k}{w}_j^{A_2}}\mathrm{.}\text{ (3.7)}$$

Formula (3.6) shows that one can obtain the same weight $w_k^{nr}$ as Graf (2008), and therefore the same final weights, by proceeding as follows:

This finding will be useful for the variance calculations in Section 5.

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