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 a sampling frame is available, containing, among the others, the following variables:

  • identifier of the Primary Sampling Units;
  • identifier of the Secondary Sampling Units;
  • variables identifying the sampling strata;
  • target variables, i.e. the variables from which sampling estimates will be produced.

As for the last type of variables, of course their direct availability is not possible: instead, proxy variables will be present in the sampling frame, or the same variables with predicted values.

Having this sampling frame, the workflow is based on the following steps:

  1. Loading data and pre-processing
  2. Producing the inputs for next steps (with a fine tuning of parameters)
  3. Optimal allocation of SSUs in PSUs
  4. Selection of PSUs
  5. Selection of SSUs

Loading data and pre-processing

We make use of a synthetic population data frame (pop), that is available at the link:

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

load("pop.RData")   
str(pop)
## 'data.frame':    2258507 obs. of  13 variables:
##  $ region       : Factor w/ 3 levels "north","center",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ province     : Factor w/ 6 levels "north_1","north_2",..: 1 1 1 1 1 1 1 1 1 1 ...
##  $ municipality : num  1 1 1 1 1 1 1 1 1 1 ...
##  $ id_hh        : Factor w/ 963018 levels "H1","H10","H100",..: 1 1 1 2 3 3 3 3 1114 1114 ...
##  $ id_ind       : int  1 2 3 4 5 6 7 8 9 10 ...
##  $ stratum      : Factor w/ 24 levels "1000","2000",..: 12 12 12 12 12 12 12 12 12 12 ...
##  $ stratum_label: chr  "north_1_6" "north_1_6" "north_1_6" "north_1_6" ...
##  $ sex          : int  1 2 1 2 1 1 2 2 1 1 ...
##  $ cl_age       : Factor w/ 8 levels "(0,14]","(14,24]",..: 3 7 8 5 4 6 6 4 4 1 ...
##  $ active       : num  1 1 0 1 1 1 1 1 1 0 ...
##  $ income_hh    : num  30488 30488 30488 21756 29871 ...
##  $ unemployed   : num  0 0 0 0 0 0 0 0 0 0 ...
##  $ inactive     : num  0 0 1 0 0 0 0 0 0 1 ...

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. Therefore, we have to handle the ‘work’ variable in this way: as values 0, 1 and 2 indicate respectively non labour force, active and inactive people, this is why we derived from ‘work’ the two binary variables, ‘active’ and ‘inactive’.

Producing the inputs for next steps

To prepare the inputs for the optimal allocation and selection of primary and secondary sampling units, the function ‘prepareInputToAllocation1’ is available.

For the execution of this function, it is necessary to assign values to the different parameters. Some of them can be directly derived by available data, but for others, namely ‘minimum’ (minimum number of SSUs to be interviewed in each selected PSU) the indication of the values is more difficult, without having any reference.

In order to orientate in the choice of the value to give to ‘minimum’, the function ‘sensitivity_min_SSU’ allows to perform a sensitivity analysis for this parameter.

To execute this function, the name of the parameter has to be given, together with the minimum and maximum value. On the basis of these minimum and maximum values, 10 different values will be used for carrying out the allocation. The output will be a graphical one.

This function requires also the definition of the precision constraints on the target values:

cv <- as.data.frame(list(DOM=c("DOM1","DOM2"),
                         CV1=c(0.02,0.03),
                         CV2=c(0.03,0.06),
                         CV3=c(0.03,0.06),
                         CV4=c(0.05,0.08)))
cv
##    DOM  CV1  CV2  CV3  CV4
## 1 DOM1 0.02 0.03 0.03 0.05
## 2 DOM2 0.03 0.06 0.06 0.08

The meaning of these constraints is that, once we select a sample and produce extimates, we expect a maximum coefficient of variation for the first variable (‘income_hh’) equal to 3% at national level (‘DOM1’) and to 4% at regional level (‘DOM2’); respectively 6% and 8% for the other three variables.

We can analyze the results in terms of first stage and second stage sizes by executing the following code:

sens_min_SSU <- sensitivity_min_SSU(
             samp_frame=pop,
             errors=cv,
             id_PSU="municipality",
             id_SSU="id_ind",
             strata_var="stratum",
             target_vars=c("income_hh","active","inactive","unemployed"),
             deff_var="stratum",
             domain_var="region",
             delta=1,
             deff_sugg=1,
             min=30,
             max=80,
             plot=TRUE)

By analysing the above graph we can decide which values are the most suitable for the sample design.

We therefore assign the following values to the required parameters:

samp_frame <- pop
id_PSU <- "municipality"  # only one
id_SSU <- "id_ind"        # only one
strata_var <- "stratum"   # only one
target_vars <- c("income_hh","active","inactive","unemployed")   # more than one
deff_var <- "stratum"     # only one
domain_var <- "region"    # only one
minimum <- 50     # minimum number of SSUs to be interviewed in each selected PSU
delta =  1        # average dimension of the SSU in terms of elementary survey units
deff_sugg <- 1.5  # suggestion for the deff value

With already assigned parameters, we can execute the ‘prepareInputToAllocation1’ function:

inp <- prepareInputToAllocation1(samp_frame,
                                id_PSU,
                                id_SSU,
                                strata_var,
                                target_vars,
                                deff_var,
                                domain_var,
                                minimum,
                                delta,
                                deff_sugg)

The function ‘prepareInputToAllocation1’ produces a list composed by six elements, stored in the ‘inp’ object:

  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)
  1. the ‘des_file’ dataframe, containing the following information:
  • STRATUM: stratum identifier
  • MOS: measure of size of the stratum (in terms of number of contained selection units)
  • DELTA: factor that report the average number of SSUs for each selection unit
  • MINIMUM: minimum number of units to be selected in each PSU
  1. the ‘PSU_file’ dataframe, containing the following information:
  • stratum identifier
  • PSU id
  • PSU_MOS: number of final selection units contained in a given PSU

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

Let us see the content of these objects:

head(inp$strata)
##           N       M1        M2        M3         M4       S1        S2        S3        S4 COST CENS DOM1 DOM2 STRATUM
## 1000 197007 23959.87 0.6650322 0.2285807 0.10638708 22179.08 0.4719792 0.4199185 0.3083324    1    0    1    2    1000
## 2000 261456 20966.65 0.6709886 0.2297519 0.09925953 19624.65 0.4698541 0.4206732 0.2990102    1    0    1    2    2000
## 3000 115813 19814.73 0.6644591 0.2315975 0.10394343 14754.88 0.4721792 0.4218532 0.3051871    1    0    1    2    3000
## 4000  17241 18732.72 0.6273418 0.2499275 0.12273070 13462.74 0.4835122 0.4329708 0.3281278    1    0    1    2    4000
## 5000 101067 22070.31 0.6134445 0.2338845 0.15267100 17187.98 0.4869603 0.4232996 0.3596701    1    0    1    2    5000
## 6000  47218 21069.07 0.6135796 0.2348469 0.15157355 17342.74 0.4869288 0.4239031 0.3586070    1    0    1    2    6000
inp$deff
##    STRATUM DEFF1 DEFF2 DEFF3 DEFF4      b_nar
## 1     1000   1.5   1.5   1.5   1.5 4925.17500
## 12    2000   1.5   1.5   1.5   1.5 1005.60000
## 18    3000   1.5   1.5   1.5   1.5  222.71731
## 19    4000   1.5   1.5   1.5   1.5   47.89167
## 20    5000   1.5   1.5   1.5   1.5 2526.67500
## 21    6000   1.5   1.5   1.5   1.5  786.96667
## 22    7000   1.5   1.5   1.5   1.5  168.72222
## 23    8000   1.5   1.5   1.5   1.5   69.78421
## 24    9000   1.5   1.5   1.5   1.5 4641.65000
## 2    10000   1.5   1.5   1.5   1.5  883.58333
## 3    11000   1.5   1.5   1.5   1.5  174.49153
## 4    12000   1.5   1.5   1.5   1.5   57.65700
## 5    13000   1.5   1.5   1.5   1.5 5146.65000
## 6    14000   1.5   1.5   1.5   1.5 1049.78750
## 7    15000   1.5   1.5   1.5   1.5  194.15625
## 8    16000   1.5   1.5   1.5   1.5   44.59672
## 9    17000   1.5   1.5   1.5   1.5 3055.85000
## 10   18000   1.5   1.5   1.5   1.5  618.79167
## 11   19000   1.5   1.5   1.5   1.5  189.70676
## 13   20000   1.5   1.5   1.5   1.5   55.32091
## 14   21000   1.5   1.5   1.5   1.5 2757.20000
## 15   22000   1.5   1.5   1.5   1.5  696.51667
## 16   23000   1.5   1.5   1.5   1.5  240.55000
## 17   24000   1.5   1.5   1.5   1.5   48.19583
inp$effst
##    STRATUM EFFST1 EFFST2 EFFST3 EFFST4
## 1     1000      1      1      1      1
## 2     2000      1      1      1      1
## 3     3000      1      1      1      1
## 4     4000      1      1      1      1
## 5     5000      1      1      1      1
## 6     6000      1      1      1      1
## 7     7000      1      1      1      1
## 8     8000      1      1      1      1
## 9     9000      1      1      1      1
## 10   10000      1      1      1      1
## 11   11000      1      1      1      1
## 12   12000      1      1      1      1
## 13   13000      1      1      1      1
## 14   14000      1      1      1      1
## 15   15000      1      1      1      1
## 16   16000      1      1      1      1
## 17   17000      1      1      1      1
## 18   18000      1      1      1      1
## 19   19000      1      1      1      1
## 20   20000      1      1      1      1
## 21   21000      1      1      1      1
## 22   22000      1      1      1      1
## 23   23000      1      1      1      1
## 24   24000      1      1      1      1
inp$rho
##    STRATUM RHO_AR1     RHO_NAR1 RHO_AR2         RHO_NAR2 RHO_AR3        RHO_NAR3 RHO_AR4       RHO_NAR4
## 1     1000       1 0.0032494875       1 0.00001260175649       1 0.0000003631192       1 0.000039120880
## 2     2000       1 0.0028554017       1 0.00150936389450       1 0.0007420929883       1 0.000937018761
## 3     3000       1 0.0069678726       1 0.00162968276279       1 0.0006469515878       1 0.002837431259
## 4     4000       1 0.0114552934       1 0.00578473329221       1 0.0019797687826       1 0.008962657055
## 5     5000       1 0.0002677333       1 0.00000001682475       1 0.0000029484212       1 0.000003404961
## 6     6000       1 0.0057050500       1 0.00004270905958       1 0.0000397945795       1 0.000194411580
## 7     7000       1 0.0041928513       1 0.00335650970751       1 0.0013094132833       1 0.001811885741
## 8     8000       1 0.0047238668       1 0.00232954042415       1 0.0018778264665       1 0.002170139711
## 9     9000       1 0.0000000000       1 0.00000000000000       1 0.0000000000000       1 0.000000000000
## 10   10000       1 0.0036846052       1 0.00159042535852       1 0.0008341965831       1 0.001542112124
## 11   11000       1 0.0106855950       1 0.00108932726442       1 0.0010329547671       1 0.001443153086
## 12   12000       1 0.0248038343       1 0.00234284567982       1 0.0020173346363       1 0.003133182814
## 13   13000       1 0.0000000000       1 0.00000000000000       1 0.0000000000000       1 0.000000000000
## 14   14000       1 0.0006512162       1 0.00089486265511       1 0.0005406860974       1 0.000374792839
## 15   15000       1 0.0068491200       1 0.00099219098192       1 0.0007727553419       1 0.000804727789
## 16   16000       1 0.0136596351       1 0.00303718119769       1 0.0021714455040       1 0.003432476610
## 17   17000       1 0.0000000000       1 0.00000000000000       1 0.0000000000000       1 0.000000000000
## 18   18000       1 0.0049434365       1 0.00437892168554       1 0.0002362549576       1 0.003892389733
## 19   19000       1 0.0157809669       1 0.01068408388378       1 0.0027579762296       1 0.008247327069
## 20   20000       1 0.0103417224       1 0.01400092460177       1 0.0044279712391       1 0.010470589088
## 21   21000       1 0.0000000000       1 0.00000000000000       1 0.0000000000000       1 0.000000000000
## 22   22000       1 0.0004350194       1 0.00362953805252       1 0.0011558770239       1 0.001149223429
## 23   23000       1 0.0069095241       1 0.00300292420693       1 0.0023502401768       1 0.003027075348
## 24   24000       1 0.0127711611       1 0.01070575411782       1 0.0017807607239       1 0.007809884541
inp$des_file
##    STRATUM STRAT_MOS DELTA MINIMUM
## 1     1000    197007     1      50
## 2     2000    261456     1      50
## 3     3000    115813     1      50
## 4     4000     17241     1      50
## 5     5000    101067     1      50
## 6     6000     47218     1      50
## 7     7000     30370     1      50
## 8     8000     26518     1      50
## 9     9000     92833     1      50
## 10   10000    106030     1      50
## 11   11000    205900     1      50
## 12   12000     57657     1      50
## 13   13000    102933     1      50
## 14   14000     83983     1      50
## 15   15000    186390     1      50
## 16   16000    108816     1      50
## 17   17000     61117     1      50
## 18   18000     74255     1      50
## 19   19000    140383     1      50
## 20   20000     60853     1      50
## 21   21000     55144     1      50
## 22   22000     41791     1      50
## 23   23000     72165     1      50
## 24   24000     11567     1      50
head(inp$psu_file)
##   PSU_ID STRATUM PSU_MOS
## 1      1   12000    1546
## 2      2   12000     936
## 3      3   12000     367
## 4      4   10000   13032
## 5      5   12000     678
## 6      6   11000    3193

It may happen that the population in strata (variable ‘N’ in ‘inp$strata’ dataset) and the one derived by the PSU dataset (variable ‘STRAT_MOS’ in ‘inp$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      197007    197007                   0
## 2    10000      106030    106030                   0
## 3    11000      205900    205900                   0
## 4    12000       57657     57657                   0
## 5    13000      102933    102933                   0
## 6    14000       83983     83983                   0
## 7    15000      186390    186390                   0
## 8    16000      108816    108816                   0
## 9    17000       61117     61117                   0
## 10   18000       74255     74255                   0
## 11   19000      140383    140383                   0
## 12    2000      261456    261456                   0
## 13   20000       60853     60853                   0
## 14   21000       55144     55144                   0
## 15   22000       41791     41791                   0
## 16   23000       72165     72165                   0
## 17   24000       11567     11567                   0
## 18    3000      115813    115813                   0
## 19    4000       17241     17241                   0
## 20    5000      101067    101067                   0
## 21    6000       47218     47218                   0
## 22    7000       30370     30370                   0
## 23    8000       26518     26518                   0
## 24    9000       92833     92833                   0
## 
## --------------------------------------------------
## Population of PSUs has been attributed to strata

Optimal allocation

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

inp$des_file$MINIMUM <- 50
minPSUstrat = 2
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, 
                  minPSUstrat,
                  minnumstrat = 2)
##   iterations PSU_SR PSU NSR PSU Total  SSU
## 1          0      0       0         0 7836
## 2          1     31     104       135 8281
## 3          2     39     104       143 8272
## 4          3     38     104       142 8274

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  1 1181
## 10 DOM2   2   1  0  1  245
## 14 DOM2   3 184  1 45    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 0.0101 0.0260 0.0393
## 6  DOM2   1 0.0129 0.0074 0.0258 0.0800
## 10 DOM2   2 0.0263 0.0213 0.0526 0.0798
## 14 DOM2   3 0.0300 0.0357 0.0599 0.0486

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)

This is the overall sample design:

sample_1st$PSU_stats
##    STRATUM PSU PSU_SR PSU_NSR  SSU SSU_SR SSU_NSR
## 1     1000   2      2       0  286    286       0
## 2     2000   9      3       6  452    152     300
## 3     3000   4      0       4  200      0     200
## 4     4000   2      0       2  100      0     100
## 5     5000   2      2       0  219    219       0
## 6     6000   2      0       2  100      0     100
## 7     7000   2      0       2  100      0     100
## 8     8000   2      0       2  100      0     100
## 9     9000   1      1       0  557    557       0
## 10   10000   6      6       0  587    587       0
## 11   11000  26      2      24 1300    100    1200
## 12   12000   8      0       8  400      0     400
## 13   13000   1      1       0  703    703       0
## 14   14000   4      4       0  577    577       0
## 15   15000  27      9      18 1361    461     900
## 16   16000  18      0      18  900      0     900
## 17   17000   1      1       0  154    154       0
## 18   18000   4      2       2  200    100     100
## 19   19000   7      1       6  350     50     300
## 20   20000   4      0       4  200      0     200
## 21   21000   1      1       0  125    125       0
## 22   22000   3      3       0  150    150       0
## 23   23000   4      0       4  200      0     200
## 24   24000   2      0       2  100      0     100
## 25   Total 142     38     104 9421   4221    5200

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,
                   verbose=FALSE)

To check that the total amount is practically equal to what determined in the allocation step:

nrow(samp)
## [1] 9421
sum(alloc$alloc$ALLOC[-nrow(alloc$alloc)])
## [1] 8274

and that the sum of weights equalize population size:

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

This is the distribution of weights:

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.