With-replacement bootstrap variance estimation for household surveys Principles, examples and implementation
Section 5. Application to the French panel for urban policy

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In this section, we present an illustration of the proposed methodology on a French panel for urban policy. The sampling design and the estimation steps for the sample of households are briefly described in Section 5.1, and three possible bootstrap confidence intervals are computed. The SAS macro developed to implement the proposed methodology for one-stage sampling is given in Appendix B, along with a small example. The additional sampling and estimation steps for the sample of individuals are described in Section 5.2, and three possible bootstrap confidence intervals are computed. The SAS macro developed to implement the proposed methodology for two-stage sampling is given in Appendix C, along with a small example.

5.1   Sample of households

The Panel for Urban Policy (PUP) is a survey in four waves, conducted between 2011 and 2014 by the French General Secretariat of the Inter-ministerial Committee for Cities (SGCIV). The survey aims at collecting information about security, employment, precariousness, schooling and health, for people living in the Sensitive Urban Zones (ZUS). We are only interested in the 2011 wave of the survey. A sample of households is selected, and all the individuals living in the selected households are theoretically surveyed.

The sample of households is obtained by two-stage sampling, see for example Chauvet (2015); Chauvet and Vallée (2018). Firstly, the population of districts is partitioned into 4 strata, and a global sample of $n_I\mathrm{<mo>=</mo>}40$ districts is selected by means of probability proportional to size sampling inside strata. A sample of households is then selected at the second-stage inside each selected district by means of simple random sampling, in such a way that the final inclusion probabilities of households are approximately equal inside strata (self-weighted sampling design). For the purpose of illustration, the two-stage selection of the households is not considered here, and the sample of households is viewed as directly selected by means of stratified simple random sampling.

The sample contains 2,971 households, but due to unit non-response only 1,256 households are observed. Non-response is accounted for by using Response Homogeneity Groups, defined with respect to five auxiliary variables: housing construction period, type of dwelling (apartment/house), number of rooms, low-income housing (yes/no), region. By using a logistic regression and the score method (e.g. Haziza and Beaumont, 2007), we obtain 8 response homogeneity groups. The five auxiliary variables used in the definition of the RHGs are also used for calibration.

We are interested in four categorical variables related to security, town planning and residential mobility. The variable $y_1$ gives the perceived reputation of the district (good, fair, poor, no opinion). The variable $y_2$ indicates if a member of the household has witnessed trafficking (never, rarely, sometimes, no opinion). The variable $y_3$ indicates if some significant roadworks have been done in the neighborhood in the twelve last months (yes, no, no opinion). The variable $y_4$ indicates if the household intends to leave the district during the next twelve months (certainly/probably, certainly not, probably not, no opinion). For each category $g$ of each variable $y,$ we are interested in the proportion

$${\beta }_{g\mathrm{,}hh}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in U_{hh}}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{N_{hh}}\mathrm{,}(5.1)$$

with $N_{hh}$ the total number of households. The estimator of ${\beta }_g$ adjusted for non-response is

$${\widehat{\beta }}_{\text{gr}\mathrm{,}hh}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{r,hh}}{d}_{rk}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{k\in S_{r,hh}}{d}_{rk}}}\mathrm{,}(5.2)$$

see equation (2.7). The calibrated estimator of ${\beta }_g$ is

$${\widehat{\beta }}_{\text{gcal,}hh}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{k\in S_{r,hh}}{w}_{k}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{k\in S_{r,hh}}{w}_k}}\mathrm{,}(5.3)$$

see equation (2.10).

For each proportion, we give the normality-based confidence interval making use of the bootstrap variance estimator, the percentile bootstrap and the basic bootstrap confidence intervals, see Section 3.5. We use the with-replacement Bootstrap presented in Algorithm 1 with $B\mathrm{<mo>=</mo>}$ 1,000 resamples. The results with a nominal one-tailed error rate of 2.5% are presented in Table 5.1. The three confidence intervals are very similar in all cases.

5.2   Sample of individuals

The sample of responding households contains 3,098 individuals who are theoretically surveyed, but due to unit non-response we observe a subset of 2,804 individual respondents only. Non-response is accounted for by using Response Homogeneity Groups, defined with respect to eight auxiliary variables: three at the individual level (sex, age, nationality), and five at the dwelling level (housing construction period, type of dwelling, number of rooms, low-income housing or not, region). By using a logistic regression and the score method, we obtain 8 response homogeneity groups. The three individual auxiliary variables used in the definition of the RHGs are also used for calibration.

We are interested in three variables of interest. The variable $y_5$ is quantitative, and gives the number of children. The variable $y_6$ indicates whether the individual has one or several jobs (one, several, none, no answer). The variable $y_7$ indicates whether the individual benefits from a complementary full medical cover (yes, no, no answer). For the variable $y_5,$ we compute the estimator of the total adjusted for non-reponse and the calibrated estimator given in equations (2.27) and (2.29), respectively. For the two other variables of interest and for each category $g,$ we are interested in the proportion

$${\beta }_{g\mathrm{,}\text{ind}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in U_{\text{ind}}}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_k\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{N_{\text{ind}}}\mathrm{,}(5.4)$$

with $N_{\text{ind}}$ the total number of individuals. The estimator of ${\beta }_{g\mathrm{,}\text{ind}}$ adjusted for non-response is

$${\widehat{\beta }}_{\text{grr,}\text{ind}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in S_{rr,\text{ind}}}{d}_{rrl}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_l\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{l\in S_{rr,\text{ind}}}{d}_{rrl}}}\mathrm{,}(5.5)$$

see equation (2.27). The calibrated estimator of ${\beta }_{g\mathrm{,}\text{ind}}$ is

$${\widehat{\beta }}_{\text{gcal,}\text{ind}}\mathrm{<mo>=</mo>}\frac{{\displaystyle {\sum }_{l\in S_{rr,\text{ind}}}{w}_{l}1\mathrm{<mo\; stretchy="false">(</mo>}{y}_l\mathrm{<mo>=</mo>}g\mathrm{<mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{l\in S_{rr,\text{ind}}}{w}_l}}\mathrm{,}(5.6)$$

see equation (2.29).

For each parameter, we give the normality-based confidence interval making use of the bootstrap variance estimator, the percentile bootstrap and the basic bootstrap confidence intervals. We use the with-replacement Bootstrap presented in Algorithm 2 with $B\mathrm{<mo>=</mo>}$ 1,000 resamples. The results with a nominal one-tailed error rate of 2.5% are presented in Table 5.2. The three confidence intervals are very similar in all cases.

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