This vignette describes a generalized procedure making use of the methods implemented in the R package developed in the Italian National Institute, namely R2BEAT (“Multistage Sampling Allocation and PSU selection”).

This package allows to determine the optimal allocation of both Primary Stage Units (PSUs) and Secondary Stage Units (SSU), and also to perform a selection of the PSUs such that the final sample of SSU is of the self-weighting type, i.e. the total inclusion probabilities (as resulting from the product between the inclusion probabilities of the PSUs and those of the SSUs) are near equal for all SSUs, or at least those of minimum variability.

This general flow assumes that at least a previous round of the survey, whose sampling design has to be optimized, is available, and is characterized by the following steps:

1 Use of ReGenesees

Perform externally the definition of the sample design, and possibly of the calibration step, using the R package ReGenesees, and make the design object and the calibrated object available.

The workspace to be loaded (R2BEAT_ReGenesees.RData) is available at the link:

https://github.com/barcaroli/R2BEAT/tree/master/data

load("R2BEAT_ReGenesees.RData")   # ReGenesees design object

This is the ‘design’ object:

des
## Stratified 2 - Stage Cluster Sampling Design (with replacement)
## - [49] strata (collapsed)
## - [789, 2236] clusters
## 
## Call:
## e.svydesign(sample_2st, ids = ~municipality + id_hh, strata = ~stratum_sub, 
##     weights = ~d, self.rep.str = ~SR, check.data = TRUE)

and this is the calibrated object:

cal
## Calibrated, Stratified 2 - Stage Cluster Sampling Design (with replacement)
## - [49] strata (collapsed)
## - [789, 2236] clusters
## 
## Call:
## e.calibrate(design = des, df.population = pop, calmodel = ~clage:sex - 
##     1, partition = ~region, calfun = "logit", bounds = c(0.7, 
##     1.7), aggregate.stage = 2, force = FALSE)

It is advisable to check the presence of lonely strata:

# Control the presence of strata with less than two units
ls <- find.lon.strata(des)
## # No lonely PSUs found!

In case, provide to collapse and re-do the calibration.

In this example, in the ReGenesees objects there are the following variables:

str(des$variables)
## 'data.frame':    2244 obs. of  17 variables:
##  $ region               : Factor w/ 3 levels "north","center",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ municipality         : num  8 8 8 8 8 8 8 8 8 8 ...
##  $ stratum              : Factor w/ 24 levels "1000","2000",..: 9 9 9 9 9 9 9 9 9 9 ...
##  $ stratum_sub          : Factor w/ 81 levels "100001","100002",..: 81 81 81 81 81 81 81 81 81 81 ...
##  $ SR                   : Factor w/ 2 levels "0","1": 2 2 2 2 2 2 2 2 2 2 ...
##  $ id_hh                : Factor w/ 2236 levels "H100070","H100410",..: 69 43 64 49 367 27 372 373 374 368 ...
##  $ sex                  : Factor w/ 2 levels "1","2": 1 1 2 2 1 2 1 2 1 1 ...
##  $ clage                : Factor w/ 5 levels "cl0_17","cl18_34",..: 3 1 2 1 5 2 2 2 3 1 ...
##  $ income_hh            : num  43741 23284 23450 22171 19904 ...
##  $ work                 : num  1 1 1 2 0 1 1 1 1 2 ...
##  $ unemployed           : num  0 0 0 0 1 0 0 0 0 0 ...
##  $ d                    : num  1238 1238 1238 1238 1238 ...
##  $ progr_str            : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ var.PSU              : chr  "8.H12425" "8.H10738" "8.H12157" "8.H11208" ...
##  $ stratum_sub.collapsed: Factor w/ 49 levels "0.center.clps.1",..: 49 49 49 49 49 49 49 49 49 49 ...
##  $ active               : Factor w/ 2 levels "0","1": 2 2 2 1 1 2 2 2 2 1 ...
##  $ inactive             : Factor w/ 2 levels "0","1": 1 1 1 2 1 1 1 1 1 2 ...

where there are three potential target variables:

  1. income_hh (numeric);
  2. work (factor with values 0, 1, 2);
  3. unemployed (factor with values 0, 1).
summary(des$variables$income_hh)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0   11463   18516   21661   26763  532331
table(des$variables$work)
## 
##    0    1    2 
##  306 1487  451
table(des$variables$unemployed)
## 
##    0    1 
## 1938  306

Great attention must be paid to the nature of the target variables, especially of the ‘factor’ type. In fact, the procedure here illustrated is suitable only when categorical variables are binary with values 0 and 1, supposing we are willing to estimate proportions of ‘1’ in the population. If factor variables are of other nature, then an error message is printed.

2 Build ‘strata’, ‘deff’, ‘effst’ and ‘rho’ dataframes

Using ReGenesees objects as input, produce the following dataframes (function ‘input_to_beat.2st_1’):

  1. the ‘stratif’ dataframe containing:
  • STRATUM: identifier of the single stratum
  • N: total population in terms of final sampling units
  • Mi,Si: mean and standard deviation of target variables (i=1,2,..,P)
  • DOMk: domain(s) to which the stratum belongs
  1. the ‘deff’ (design effect) dataframe, containing the following information:
  • STRATUM: the stratum identifier
  • DEFFi: the design effect for each target variable i (i=1,2,…,P)
  1. the ‘effst’ (estimator effect) dataframe, containing the following information:
  • STRATUM: the stratum identifier
  • EFFSTi: the estimator effect for each target variable i (i=1,2,…,P)
  1. the ‘rho’ (intraclass coefficient of correlation) dataframe, containing the following information:
  • STRATUM: the stratum identifier
  • RHO_ARi: the intraclass coefficient of correlation in self-representative PSUs for each target variable i (i=1,2,…,P)
  • RHO_NARi: the intraclass coefficient of correlation in non self-representative PSUs for each target variable i (i=1,2,…,P)

Actually, the ‘deff’ dataframe is not used in the following steps, it just remains for documentation purposes.

Here is the way we can produce the above items:

load("pop.RData")
samp_frame <- pop
RGdes <- des
RGcal <- cal
strata_var <- c("stratum")      
target_vars <- c("income_hh",
                 "active",
                 "inactive",
                 "unemployed")   
weight_var <- "weight"
deff_var <- "stratum"            
id_PSU <- c("municipality")      
id_SSU <- c("id_hh")             
domain_var <- c("region") 
delta <- 1                   
minimum <- 25                

inp <- prepareInputToAllocation2(
        samp_frame,  # sampling frame
        RGdes,       # ReGenesees design object
        RGcal,       # ReGenesees calibrated object
        id_PSU,      # identification variable of PSUs
        id_SSU,      # identification variable of SSUs
        strata_var,  # strata variables
        target_vars, # target variables
        deff_var,    # deff variables
        domain_var,  # domain variables
        delta,       # Average number of SSUs for each selection unit
        minimum      # Minimum number of SSUs to be selected in each PSU
      )

and these are the results:

head(inp$strata)
stratum N STRATUM M1 M2 M3 M4 S1 S2 S3 S4 COST CENS DOM1 DOM2
1000 197451 1000 22266.58 0.6404431 0.2323140 0.1272429 14554.88 0.4798705 0.4223082 0.3332449 1 0 1 center
10000 106106 10000 27985.40 0.7679285 0.2114187 0.0206528 24367.97 0.4221544 0.4083146 0.1422189 1 0 1 north
11000 202700 11000 29173.85 0.8029080 0.1730880 0.0240040 39232.92 0.3978024 0.3783234 0.1530613 1 0 1 north
12000 57420 12000 26937.42 0.7764955 0.2075926 0.0159119 15743.78 0.4165936 0.4055834 0.1251347 1 0 1 north
13000 103089 13000 26357.25 0.7185271 0.2814729 0.0000000 14592.50 0.4497176 0.4497176 0.0000000 1 0 1 north
14000 84653 14000 20538.42 0.7518236 0.2131042 0.0350721 14285.81 0.4319547 0.4095007 0.1839621 1 0 1 north
head(inp$deff)
stratum STRATUM DEFF1 DEFF2 DEFF3 DEFF4 b_nar
1000 1000 0.960198 0.999984 1.015715 0.962537 56.50000
10000 10000 0.864671 1.703511 1.417543 0.827580 26.75000
11000 11000 1.820304 1.267734 1.352970 1.345746 23.77778
12000 12000 1.103866 0.510554 0.491751 0.711980 21.00000
13000 13000 1.000924 1.000924 1.000924 1.000000 95.00000
14000 14000 0.639871 0.865378 0.854025 0.684041 33.66667
head(inp$effst)
stratum STRATUM EFFST1 EFFST2 EFFST3 EFFST4
1000 1000 0.9689481 1 1 0.9420957
10000 10000 0.9500006 1 1 1.1915489
11000 11000 0.9544521 1 1 1.0546195
12000 12000 1.0429454 1 1 0.9732492
13000 13000 1.0019592 1 1 1.0000000
14000 14000 0.9829169 1 1 1.0974518
head(inp$rho)
STRATUM RHO_AR1 RHO_NAR1 RHO_AR2 RHO_NAR2 RHO_AR3 RHO_NAR3 RHO_AR4 RHO_NAR4
1000 1 -0.0007172 1 -0.0000003 1 0.0002832 1 -0.0006750
10000 1 -0.0052555 1 0.0273208 1 0.0162153 1 -0.0066959
11000 1 0.0360133 1 0.0117542 1 0.0154962 1 0.0151791
12000 1 0.0051933 1 -0.0244723 1 -0.0254124 1 -0.0144010
13000 1 0.0000098 1 0.0000098 1 0.0000098 1 0.0000000
14000 1 -0.0110244 1 -0.0041211 1 -0.0044686 1 -0.0096722
head(inp$psu_file)
PSU_ID STRATUM PSU_MOS
309 1000 50845
330 1000 146162
292 2000 24794
293 2000 19609
300 2000 13897
304 2000 36195
head(inp$des_file)
STRATUM STRAT_MOS DELTA MINIMUM
1000 197007 1 25
2000 261456 1 25
3000 115813 1 25
4000 17241 1 25
5000 101067 1 25
6000 47218 1 25

3 Check the coherence of populations in strata and PSUs

It may happen that the population in strata (variable ‘N’ in ‘inp1$strata’ dataset) and the one derived by the PSU dataset (variable ‘STRAT_MOS’ in ‘inp2$des_file’ dataset) are not the same.

We can check it by applying the function ‘check_input’ in this way:

newstrata <- check_input(strata=inp$strata,
                         des=inp$des_file,
                         strata_var_strata="STRATUM",
                         strata_var_des="STRATUM")
## 
## --------------------------------------------------
##  Differences between population in strata and PSUs  
## --------------------------------------------------
##    STRATUM N_in_strata N_in_PSUs relative_difference
## 1     1000      197451    197007              -0.002
## 12    2000      258193    261456               0.012
## 18    3000      116213    115813              -0.003
## 19    4000       17879     17241              -0.037
## 20    5000      102706    101067              -0.016
## 21    6000       47477     47218              -0.005
## 22    7000       30193     30370               0.006
## 23    8000       26580     26518              -0.002
## 24    9000       94610     92833              -0.019
## 2    10000      106106    106030              -0.001
## 3    11000      202700    205900               0.016
## 4    12000       57420     57657               0.004
## 5    13000      103089    102933              -0.002
## 6    14000       84653     83983              -0.008
## 7    15000      187343    186390              -0.005
## 8    16000      108621    108816               0.002
## 9    17000       59483     61117               0.027
## 10   18000       71642     74255               0.035
## 11   19000      145891    140383              -0.039
## 13   20000       62130     60853              -0.021
## 14   21000       51552     55144               0.065
## 15   22000       41688     41791               0.002
## 16   23000       72809     72165              -0.009
## 17   24000       12081     11567              -0.044
## 
## --------------------------------------------------
## Population of PSUs has been attributed to strata

Together with the print of the differences between the two populations, the function produces a new version of the strata dataset, where the population has been changed to the one derived by the PSUs dataset.

It is preferable to use this new version:

inp$strata <- newstrata

4 Optimal allocation of units in each stratum

Using the function ‘beat.2st’ in ‘R2BEAT’ package execute the optimization of PSU and SSU allocation in strata:

cv <- as.data.frame(list(DOM=c("DOM1","DOM2"),
                         CV1=c(0.02,0.03),
                         CV2=c(0.03,0.05),
                         CV3=c(0.03,0.05),
                         CV4=c(0.05,0.08)))
cv
DOM CV1 CV2 CV3 CV4
DOM1 0.02 0.03 0.03 0.05
DOM2 0.03 0.05 0.05 0.08
set.seed(1234)
minPSUstrat <- 2
inp$des_file$MINIMUM <- 25
alloc <- beat.2st(stratif = inp$strata, 
                  errors = cv, 
                  des_file = inp$des_file, 
                  psu_file = inp$psu_file, 
                  rho = inp$rho, 
                  deft_start = NULL, 
                  effst = inp$effst, 
                  minnumstrat = 2, 
                  minPSUstrat)
##    iterations PSU_SR PSU NSR PSU Total    SSU
## 1           0      0       0         0   7721
## 2           1    111      72       183 109498
## 3           2    155     114       269   7925
## 4           3    116     132       248   8660
## 5           4    146     122       268   8343
## 6           5    128     130       258   8580
## 7           6    143     124       267   8391
## 8           7    131     128       259   8550
## 9           8    143     124       267   8392
## 10          9    131     128       259   8550
## 11         10    143     124       267   8392
## 12         11    131     128       259   8550
## 13         12    143     124       267   8392
## 14         13    131     128       259   8550
## 15         14    143     124       267   8392
## 16         15    131     128       259   8550
## 17         16    143     124       267   8392
## 18         17    131     128       259   8550
## 19         18    143     124       267   8392
## 20         19    131     128       259   8550
## 21         20    143     124       267   8392

This is the sensitivity of the solution:

alloc$sensitivity
Type Dom V1 V2 V3 V4
2 DOM1 1 1 0 1 1
6 DOM2 1 1 0 8 1150
10 DOM2 2 1 1 245 1
14 DOM2 3 1 1 275 1

i.e., for each domain value and for each variable it is reported the gain in terms of reduction in the sample size if the corresponding precision constraint is reduced of 10%.

These are the expected values of the coefficients of variation:

alloc$expected
Type Dom V1 V2 V3 V4
2 DOM1 1 0.0121 0.0104 0.0296 0.0307
6 DOM2 1 0.0171 0.0132 0.0496 0.0800
10 DOM2 2 0.0214 0.0209 0.0499 0.0675
14 DOM2 3 0.0283 0.0242 0.0499 0.0339

5 Selection of PSUs

Using the function ‘select_PSU’ execute the selection of PSU in strata:

set.seed(1234)
sample_1st <- select_PSU(alloc, type="ALLOC", pps=TRUE, plot=TRUE)

This is the overall sample design:

sample_1st$PSU_stats
STRATUM PSU PSU_SR PSU_NSR SSU SSU_SR SSU_NSR
1000 2 2 0 294 294 0
2000 13 13 0 405 405 0
3000 10 0 10 250 0 250
4000 2 0 2 50 0 50
5000 2 2 0 185 185 0
6000 3 3 0 82 82 0
7000 2 0 2 50 0 50
8000 2 0 2 50 0 50
9000 1 1 0 810 810 0
10000 6 6 0 644 644 0
11000 53 41 12 1432 1132 300
12000 10 0 10 250 0 250
13000 1 1 0 30 30 0
14000 4 4 0 607 607 0
15000 38 22 16 1021 621 400
16000 69 31 38 1725 775 950
17000 1 1 0 151 151 0
18000 6 6 0 199 199 0
19000 22 6 16 550 150 400
20000 8 0 8 200 0 200
21000 1 1 0 142 142 0
22000 3 3 0 109 109 0
23000 6 0 6 150 0 150
24000 2 0 2 50 0 50
Total 267 143 124 9436 6336 3100

6 Selection of SSUs

Finally, we are able to select the Secondary Sample Units (the individuals) from the already selected PSUs (the municipalities). We proceed to select the sample in this way:

samp <- select_SSU(df=pop,
                   PSU_code="municipality",
                   SSU_code="id_ind",
                   PSU_sampled=sample_1st$sample_PSU)
## 
## --------------------------------
## Total PSUs =  267
## Total SSUs =  9436
## --------------------------------

To check that the total amount of selected units with respect to the initial allocation:

nrow(samp)
## [1] 9436
sum(alloc$alloc$ALLOC[-nrow(alloc$alloc)])
## [1] 8392

The difference is due to the fact that the constraint on the minimum number of SSUs to be selected for PSU has been enforced, thus resulting in an increase of the SSUs with respect to the optimal allocation.

We check also that the sum of weights equalizes the population size:

nrow(pop)
## [1] 2258507
sum(samp$weight)
## [1] 2258507

This is the distribution of weights:

par(mfrow=c(1, 2))
boxplot(samp$weight,col="orange")
title("Weights distribution (total sample)",cex.main=0.7)
boxplot(weight ~ region, data=samp,col="orange")
title("Weights distribution by region",cex.main=0.7)

boxplot(weight ~ province, data=samp,col="orange")
title("Weights distribution by province",cex.main=0.7)
boxplot(weight ~ stratum, data=samp,col="orange")
title("Weights distribution by stratum",cex.main=0.7)

It can be seen that the sample is fully self-weighted inside strata, and approximately self-weighted in aggregations of strata, that is the result we wanted to obtain.