Rescaled Bootstrap

Since EU-SILC is a stratified sample without replacement drawn from afinite population the naive bootstrap procedure, as described above,does not take into account the heterogeneous inclusion probabilities ofeach sample unit. Thus it will not yield satisfactory results. Thereforewe will use the so called rescaled bootstrap procedure introduced andinvestigated by(Rao and Wu 1988). Thebootstrap samples are selected without replacement and do incorporatethe stratification as well as clustering on multiple stages (see(Chipperfield and Preston 2007),(Preston 2009)).

For simplistic reasons we will only describe the rescaled bootstrapprocedure for a two stage stratified sampling design. For more detailson a general formulation please see(Preston2009).

Sampling design

Consider the finite population$U$which is divided into$H$non-overlapping strata$\bigcup _{h=1,\dots ,H}{U}_h=U$,of which each strata$h$contains of$N_h$clusters. For each strata$h$,$C_{hc}$,$c=1,\dots ,n_h$clusters are drawn, containing$N_{hc}$households. Furthermore in each cluster$C_{hc}$of each strata$h$simple random sampling is performed to select a set of households$Y_{hcj}$,$j=1,\dots ,n_{hc}$.

Bootstrap procedure

In contrast to the naive bootstrap procedure where for a stage,containing$n$sampling units, the bootstrap replicate is obtained by drawing$n$sampling units with replacement, for the rescaled bootstrap procedure$n^{*}=\lfloor \frac{n}{2}\rfloor$sampling units are drawn without replacement. Given a value$x$,$\lfloor x\rfloor$denotes the largest integer smaller than$x$,whereas$\lceil x\rceil$denotes the smallest integer lager then$x$.(Chipperfield and Preston 2007) have shownthat the choice of either$\lfloor \frac{n}{2}\rfloor$or$\lceil \frac{n}{2}\rceil$is optimal for bootstrap samples without replacement, although$\lfloor \frac{n}{2}\rfloor$has the desirable property that the resulting uncalibrated bootstrapweights will never be negative.

At the first stage the$i$-thbootstrap replicate,$f_{hc}^{i,1}$,for each cluster$C_{hc}$,$c=1,\dots ,n_h$,belonging to strata$h$,is defined by

$$f_{hc}^{i,1}=1-{\lambda }_h+{\lambda }_h\frac{n_h}{n_h^{*}}{\delta }_{hc}\forall c\in \{1,\dots ,n_h\}$$ with$$n_h^{*}=\lfloor \frac{n_h}{2}\rfloor$$$${\lambda }_h=\sqrt{\frac{n_h^{*}(1-\frac{n_h}{N_h})}{n_h-n_h^{*}}},$$

where${\delta }_{hc}=1$if cluster$c$is selected in the sub-sample of size$n_h^{*}$and 0 otherwise.

The$i$-thbootstrap replicate at the second stage,$f_{hcj}^{i,2}$,for each household$Y_{hcj}$,$j=1,\dots ,n_{hc}$,belonging to cluster$c$in strata$h$is defined by

$$f_{hcj}^{i,2}=f_{hc}^{i,1}-{\lambda }_{hc}\sqrt{\frac{n_h}{n_h^{*}}}{\delta }_{hc}[\frac{n_{hc}}{n_{hc}^{*}}{\delta }_{hcj}-1]\forall c\in \{1,\dots ,n_h\}$$ with$$n_{hc}^{*}=\lfloor \frac{n_{hc}}{2}\rfloor$$$${\lambda }_{hc}=\sqrt{\frac{n_{hc}^{*}{N}_h(1-\frac{n_{hc}}{N_{hc}})}{n_{hc}-n_{hc}^{*}}},$$

where${\delta }_{hcj}=1$if household$j$is selected in the sub sample of size$n_{hc}^{*}$and 0 otherwise.

Single PSUs

When dealing with multistage sampling designs the issue of singlePSUs, e.g. a single response unit is present at a stage or in a strata,can occur. When applying bootstrapping procedures these single PSUs canlead to a variety of issues. For the methodology proposed in this workwe combined single PSUs at each stage with the next smallest strata orcluster, before applying the bootstrap procedure.

Taking bootstrap replicates forward

The bootstrap procedure above is applied on the EU-SILC data for eachyear$y_t$,$t=1,\dots ,n_y$separately. Since EU-SILC is a yearly survey with rotating penal designthe$i$-thbootstrap replicate at the second stage,$f_{hcj}^{i,2}$,for a household$Y_{hcj}$is taken forward until the household$Y_{hcj}$drops out of the sample. That is, for the household$Y_{hcj}$,which enters EU-SILC at year$y_1$and drops out at year$y_{\tilde{t}}$,the bootstrap replicates for the years$y_2,\dots ,y_{\tilde{t}}$are set to the bootstrap replicate of the year$y_1$.

Split households

Due to the rotating penal design so called split households canoccur. For a household participating in the EU-SILC survey it ispossible that one or more residents move to a new so called splithousehold, which is followed up on in the next wave. To take thisdynamic into account we extended the procedure of taking forward thebootstrap replicate of a household for consecutive waves of EU-SILC bytaking forward the bootstrap replicate to the split household. Thatmeans, that also any new individuals in the split household will inheritthis bootstrap replicate.

Taking bootstrap replicates forward as well as considering splithouseholds ensures that bootstrap replicates are more comparable instructure with the actual design of EU-SILC.

Uncalibrated bootstrap weights

Using the$i$-thbootstrap replicates at the second stage one can calculate the$i$-thuncalibrated bootstrap weights$b_{hcj}^i$for each household$Y_{hcj}$in cluster$c$contained in strata$h$by

$${\tilde{b}}_{hcj}^i=f_{hcj}^{i,2}{w}_{hcj},$$ where$w_{hcj}$corresponds to the original household weight contained in thesample.

For ease of readability we will drop the subindices regarding strata$h$and cluster$c$for the following sections, meaning that the$j$-thhousehold in cluster$c$contained in strata$h$,$Y_{hcj}$,will now be denoted as the$j$-thhousehold,$Y_j$,where$j$is the position of the household in the data. In accordance to this the$i$-thuncalibrated bootstrap replicates for household$j$are thus denoted as${\tilde{b}}_j^i$and the original household weight as$w_j$.

Adjustment and trimming for$𝒫$

The uncalibrated bootstrap weight${\tilde{b}}_j^{[(n+1)k+c-1]}$for the$j$-thobservation is iteratively multiplied by a factor so that the projecteddistribution of the population matches the respective calibrationspecification$N_{p_c}$,$c=1,\dots ,n_P$.For each$c\in \{1,\dots ,n_P\}$the calibrated weights against$N_v^{p_c}$are computed as$${\tilde{b}}_j^{[(n+1)k+c]}={{\tilde{b}}_j}^{[(n+1)k+c-1]}\frac{N_v^{p_c}}{\sum _l{\tilde{b}}_l^{[(n+1)k+c-1]}},$$ where the summation in the denominatorexpands over all observations which have the same value as observation$j$for the sociodemographic variable$p_c$.If any weights${\tilde{b}}_j^{[nk+c]}$fall outside the range$[\frac{w_j}{4};4w_j]$they will be recoded to the nearest of the two boundaries. The choice ofthe boundaries results from expert-based opinions and restricts thevariance of which has a positive effect on the sampling error. Thisprocedure represents a common form of weight trimming where very largeor small weights are trimmed in order to reduce variance in exchange fora possible increase in bias ((Potter1990),(Potter 1993)).

Averaging weights within households

Since the sociodemographic variables$p_1,\dots ,p_{n_c}$include person-specific variables, the weights${\tilde{b}}_j^{[nk+n_p]}$resulting from the iterative multiplication can be unequal for membersof the same household. This can lead to inconsistencies between resultsprojected with household and person weights. To avoid suchinconsistencies each household member is assigned the mean of thehousehold weights. That is for each person$j$in household$a$with$h_a$household members, the weights are defined by$${\tilde{b}}_j^{[(n+1)k+n_p+1]}=\frac{\sum _{l\in a}{\tilde{b}}_l^{[(n+1)k+n_p]}}{h_a}$$ This can result in losing thepopulation structure performed in the previous subsection.

Adjustment and trimming for$\mathscr{H}$

After adjustment for individual variables the weights$b_j^{[nk+n_p+1]}$are updated for the set of household variables$\mathscr{H}$according to a household convergence constraint parameter${ϵ}_h$.The parameters${ϵ}_h$represent the allowed deviation from the population margins using theweights$b_j^{[nk+n_p+1]}$compared to$N_v^{h_c}$,$c=1,\dots ,n_H$,$v=1,\dots ,H_c$.The updated weights are computed as$$b_j^{[(n+1)k+n_p+c+1]}=\{\begin{array}{c}{b}_j^{[(n+1)k+n_p+1]}\frac{N_v^{h_c}}{\sum _l{b}_l^{[(n+1)k+n_p+1]}}\mathrm{\text{if}}\sum _l{b}_j^{[(n+1)k+n_p+1]}\notin ((1-0.9{ϵ}_h)N_v^{h_c},(1+0.9{ϵ}_h)N_v^{h_c})\hfill \\ b_j^{[(n+1)k+n_p+1]}\mathrm{\text{otherwise}}\hfill \end{array}$$ with the summation in the denominatorranging over all households$l$which take on the same values for$h_c$as observation$j$.As described in the previous subsection the new weight are recoded ifthey exceed the interval$[\frac{w_j}{4};4w_j]$and set to the upper or lower bound, depending of$b_j^{[(n+1)k+n_p+c+1]}$falls below or above the interval respectively.

Convergence

For each adjustment and trimming step the factor$\frac{N_v^{\left(.\right)}}{\sum _l{b}_l^{[(n+1)k+j]}}$,$j\in \{1,\dots ,n+1\}\setminus \{n_p+1\}$,is checked against convergence constraints for households,${ϵ}_h$,or personal variables${ϵ}_p$,where$\left(.\right)$corresponds to either a household or personal variable. To be moreprecise for variables in$𝒫$the constraints

$$\frac{N_v^{p_c}}{\sum _l{\tilde{b}}_l^{[(n+1)k+j]}}\in ((1-{ϵ}_p)N_v^{p_c},(1+{ϵ}_p)N_v^{p_c})$$ and for variables in$\mathscr{H}$the constraints

$$\frac{N_v^{h_c}}{\sum _l{\tilde{b}}_l^{[(n+1)k+j]}}\in ((1-{ϵ}_h)N_v^{h_c},(1+{ϵ}_h)N_v^{h_c})$$ are verified, where the sum in thedenominator expands over all observations which have the same value forvariables$h_c$or$p_c$.If these constraints hold true the algorithm reaches convergence,otherwise$k$is raised by 1 and the procedure repeats itself.

The above described calibration procedure is applied on each year$y_t$of EU-SILC separately,$t=1,\dots n_y$,thus resulting in so called calibrated bootstrap sample weights$b_j^{(i,y_t)}$,$i=1,\dots ,B$for each year$y$and each household$j$.