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:
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 objectThis 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:
summary(des$variables$income_hh)##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0   11463   18516   21661   26763  532331table(des$variables$work)## 
##    0    1    2 
##  306 1487  451table(des$variables$unemployed)## 
##    0    1 
## 1938  306Great 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.
Using ReGenesees objects as input, produce the following dataframes (function ‘input_to_beat.2st_1’):
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 | 
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 strataTogether 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$strata <- newstrataUsing the function ‘beat.2st’ in ‘R2BEAT’ package execute the optimization of PSU and SSU allocation in strata:
cv <- as.data.frame(list(DOM=c("DOM1","DOM2"),
                         CV1=c(0.02,0.03),
                         CV2=c(0.03,0.05),
                         CV3=c(0.03,0.05),
                         CV4=c(0.05,0.08)))
cv| DOM | CV1 | CV2 | CV3 | CV4 | 
|---|---|---|---|---|
| DOM1 | 0.02 | 0.03 | 0.03 | 0.05 | 
| DOM2 | 0.03 | 0.05 | 0.05 | 0.08 | 
set.seed(1234)
minPSUstrat <- 2
inp$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   8392This 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 | 
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$PSU_stats| STRATUM | PSU | PSU_SR | PSU_NSR | SSU | SSU_SR | SSU_NSR | 
|---|---|---|---|---|---|---|
| 1000 | 2 | 2 | 0 | 294 | 294 | 0 | 
| 2000 | 13 | 13 | 0 | 405 | 405 | 0 | 
| 3000 | 10 | 0 | 10 | 250 | 0 | 250 | 
| 4000 | 2 | 0 | 2 | 50 | 0 | 50 | 
| 5000 | 2 | 2 | 0 | 185 | 185 | 0 | 
| 6000 | 3 | 3 | 0 | 82 | 82 | 0 | 
| 7000 | 2 | 0 | 2 | 50 | 0 | 50 | 
| 8000 | 2 | 0 | 2 | 50 | 0 | 50 | 
| 9000 | 1 | 1 | 0 | 810 | 810 | 0 | 
| 10000 | 6 | 6 | 0 | 644 | 644 | 0 | 
| 11000 | 53 | 41 | 12 | 1432 | 1132 | 300 | 
| 12000 | 10 | 0 | 10 | 250 | 0 | 250 | 
| 13000 | 1 | 1 | 0 | 30 | 30 | 0 | 
| 14000 | 4 | 4 | 0 | 607 | 607 | 0 | 
| 15000 | 38 | 22 | 16 | 1021 | 621 | 400 | 
| 16000 | 69 | 31 | 38 | 1725 | 775 | 950 | 
| 17000 | 1 | 1 | 0 | 151 | 151 | 0 | 
| 18000 | 6 | 6 | 0 | 199 | 199 | 0 | 
| 19000 | 22 | 6 | 16 | 550 | 150 | 400 | 
| 20000 | 8 | 0 | 8 | 200 | 0 | 200 | 
| 21000 | 1 | 1 | 0 | 142 | 142 | 0 | 
| 22000 | 3 | 3 | 0 | 109 | 109 | 0 | 
| 23000 | 6 | 0 | 6 | 150 | 0 | 150 | 
| 24000 | 2 | 0 | 2 | 50 | 0 | 50 | 
| Total | 267 | 143 | 124 | 9436 | 6336 | 3100 | 
Finally, we are able to select the Secondary Sample Units (the individuals) from the already selected PSUs (the municipalities). We proceed to select the sample in this way:
samp <- select_SSU(df=pop,
                   PSU_code="municipality",
                   SSU_code="id_ind",
                   PSU_sampled=sample_1st$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] 9436sum(alloc$alloc$ALLOC[-nrow(alloc$alloc)])## [1] 8392The difference is due to the fact that the constraint on the minimum number of SSUs to be selected for PSU has been enforced, thus resulting in an increase of the SSUs with respect to the optimal allocation.
We check also that the sum of weights equalizes the population size:
nrow(pop)## [1] 2258507sum(samp$weight)## [1] 2258507This 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.