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 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
<- find.lon.strata(des) ls
## # 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 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.
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")
<- pop
samp_frame <- des
RGdes <- cal
RGcal <- c("stratum")
strata_var <- c("income_hh",
target_vars "active",
"inactive",
"unemployed")
<- "weight"
weight_var <- "stratum"
deff_var <- c("municipality")
id_PSU <- c("id_hh")
id_SSU <- c("region")
domain_var <- 1
delta <- 25
minimum
<- prepareInputToAllocation2(
inp # sampling frame
samp_frame, # ReGenesees design object
RGdes, # ReGenesees calibrated object
RGcal, # identification variable of PSUs
id_PSU, # identification variable of SSUs
id_SSU, # strata variables
strata_var, # target variables
target_vars, # deff variables
deff_var, # domain variables
domain_var, # Average number of SSUs for each selection unit
delta, # Minimum number of SSUs to be selected in each PSU
minimum )
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:
<- check_input(strata=inp$strata,
newstrata des=inp$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:
$strata <- newstrata inp
Using the function ‘beat.2st’ in ‘R2BEAT’ package execute the optimization of PSU and SSU allocation in strata:
<- as.data.frame(list(DOM=c("DOM1","DOM2"),
cv 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)
<- 2
minPSUstrat $des_file$MINIMUM <- 25
inp<- beat.2st(stratif = inp$strata,
alloc errors = cv,
des_file = inp$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:
$sensitivity alloc
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:
$expected alloc
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)
<- select_PSU(alloc, type="ALLOC", pps=TRUE, plot=TRUE) sample_1st
This is the overall sample design:
$PSU_stats sample_1st
STRATUM | PSU | PSU_SR | PSU_NSR | SSU | SSU_SR | SSU_NSR |
---|---|---|---|---|---|---|
1000 | 2 | 2 | 0 | 294 | 294 | 0 |
2000 | 13 | 13 | 0 | 405 | 405 | 0 |
3000 | 10 | 0 | 10 | 250 | 0 | 250 |
4000 | 2 | 0 | 2 | 50 | 0 | 50 |
5000 | 2 | 2 | 0 | 185 | 185 | 0 |
6000 | 3 | 3 | 0 | 82 | 82 | 0 |
7000 | 2 | 0 | 2 | 50 | 0 | 50 |
8000 | 2 | 0 | 2 | 50 | 0 | 50 |
9000 | 1 | 1 | 0 | 810 | 810 | 0 |
10000 | 6 | 6 | 0 | 644 | 644 | 0 |
11000 | 53 | 41 | 12 | 1432 | 1132 | 300 |
12000 | 10 | 0 | 10 | 250 | 0 | 250 |
13000 | 1 | 1 | 0 | 30 | 30 | 0 |
14000 | 4 | 4 | 0 | 607 | 607 | 0 |
15000 | 38 | 22 | 16 | 1021 | 621 | 400 |
16000 | 69 | 31 | 38 | 1725 | 775 | 950 |
17000 | 1 | 1 | 0 | 151 | 151 | 0 |
18000 | 6 | 6 | 0 | 199 | 199 | 0 |
19000 | 22 | 6 | 16 | 550 | 150 | 400 |
20000 | 8 | 0 | 8 | 200 | 0 | 200 |
21000 | 1 | 1 | 0 | 142 | 142 | 0 |
22000 | 3 | 3 | 0 | 109 | 109 | 0 |
23000 | 6 | 0 | 6 | 150 | 0 | 150 |
24000 | 2 | 0 | 2 | 50 | 0 | 50 |
Total | 267 | 143 | 124 | 9436 | 6336 | 3100 |
Finally, we are able to select the Secondary Sample Units (the individuals) from the already selected PSUs (the municipalities). We proceed to select the sample in this way:
<- select_SSU(df=pop,
samp PSU_code="municipality",
SSU_code="id_ind",
PSU_sampled=sample_1st$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.