pps.sampling: Sampling with Probabilities Proportional to Size
In samplingbook: Survey Sampling Procedures
Description
Usage
Arguments
Details
Value
Author(s)
References
See Also
Examples
Description
The function provides sample techniques with sampling probabilities which are proportional to the size of a quantity z.
Usage
1
pps.sampling(z, n, id = 1:N, method = 'sampford', return.PI = FALSE)
Arguments
z
vector of quantities which determine the sampling probabilities in the population
n
positive integer for sample size
id
an optional vector with identification values for population elements. Default is 'id = 1:N', where 'N' is length of 'z'.
method
the sampling method to be used. Options are 'sampford', 'tille', 'midzuno' or 'madow'.
return.PI
logical. If TRUE the pairwise inclusion probabilities for all individuals in the population are returned.
Details
The different methods vary in their run time. Therefore, method='sampford' is stopped if N > 200 or if n/N < 0.3. method='tille' is stopped if N > 500.
In case of large populations use method='midzuno' or method='madow'.
Value
The function pps.sampling returns a value, which is a list consisting of the components
call
is a list of call components: z vector of quantity data, n sample size, id identification values, and method sampling method
sample
resulted sample
pik
inclusion probabilities
PI
sample second order inclusion probabilities
PI.full
full second order inclusion probabilities
Author(s)
Juliane Manitz
References
Kauermann, Goeran/Kuechenhoff, Helmut (2010): Stichproben. Methoden und praktische Umsetzung mit R. Springer.
See Also
Examples
1
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8
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## 1) simple suppositious example
data <- data.frame(id = 1:7, z = c(1.8, 2 ,3.2 ,2.9 ,1.5 ,2.0 ,2.2))
# Usage of pps.sampling for Sampford method
set.seed(178209)
pps.sample_sampford <- pps.sampling(z=data$z, n=2, method='sampford', return.PI=FALSE)
pps.sample_sampford
# sampling elements
id.sample <- pps.sample_sampford$sample
id.sample
# other methods
set.seed(178209)
pps.sample_tille <- pps.sampling(z=data$z, n=2, method='tille')
pps.sample_tille
set.seed(178209)
pps.sample_midzuno <- pps.sampling(z=data$z, n=2, method='midzuno')
pps.sample_midzuno
set.seed(178209)
pps.sample_madow <- pps.sampling(z=data$z, n=2, method='madow')
pps.sample_madow
## 2) influenza
data(influenza)
summary(influenza)
set.seed(108506)
pps <- pps.sampling(z=influenza$population,n=20,method='midzuno')
pps
sample <- influenza[pps$sample,]
sample
Example output
Loading required package: pps
Loading required package: sampling
Loading required package: survey
Loading required package: grid
Loading required package: Matrix
Loading required package: survival
Attaching package: 'survival'
The following objects are masked from 'package:sampling':
cluster, strata
Attaching package: 'survey'
The following object is masked from 'package:graphics':
dotchart
pps.sampling object: Sample with probabilities proportional to size
Method of Sampford:
PPS sample:
[1] 3 7
Sample probabilities:
[,1] [,2]
[1,] 0.41025641 0.07281474
[2,] 0.07281474 0.28205128
[1] 3 7
pps.sampling object: Sample with probabilities proportional to size
Method of Tille:
PPS sample:
[1] 1 3
Sample probabilities:
[,1] [,2]
[1,] 0.23076923 0.05955335
[2,] 0.05955335 0.41025641
pps.sampling object: Sample with probabilities proportional to size
Method of Midzuno:
PPS sample:
[1] 3 4
Sample probabilities:
[,1] [,2]
[1,] 0.41025641 0.08974359
[2,] 0.08974359 0.37179487
Warning message:
In pps.sampling(z = data$z, n = 2, method = "madow") :
Systematic Sample with zeros in 'PI': For calculating estimates use approximate methods.
pps.sampling object: Sample with probabilities proportional to size
Method of Madow:
PPS sample:
[1] 3 6
Sample probabilities:
[,1] [,2]
[1,] 0.4102564 0.2307692
[2,] 0.2307692 0.2564103
id district population cases
Min. : 1001 LK Aachen : 1 Min. : 34719 Min. : 0.00
1st Qu.: 5877 LK Ahrweiler : 1 1st Qu.: 104553 1st Qu.: 9.00
Median : 8331 LK Aichach-Friedberg: 1 Median : 145130 Median : 27.00
Mean : 8468 LK Alb-Donau-Kreis : 1 Mean : 193910 Mean : 44.58
3rd Qu.: 9778 LK Altenburger Land : 1 3rd Qu.: 244154 3rd Qu.: 59.00
Max. :16077 LK Altenkirchen : 1 Max. :1770629 Max. :410.00
(Other) :418
pps.sampling object: Sample with probabilities proportional to size
Method of Midzuno:
PPS sample:
[1] 35 83 107 109 130 140 157 210 219 223 257 273 290 294 324 342 361 371 418
[20] 423
Sample probabilities:
[,1] [,2] [,3] [,4] [,5]
[1,] 0.090052479 0.0053250174 0.0059535012 0.0047392541 0.0034975812
[2,] 0.005325017 0.0622266431 0.0040841690 0.0032027173 0.0023764993
[3,] 0.005953501 0.0040841690 0.0702093391 0.0036435201 0.0026981161
[4,] 0.004739254 0.0032027173 0.0036435201 0.0549863651 0.0020847939
[5,] 0.003497581 0.0023764993 0.0026981161 0.0020847939 0.0401586824
[6,] 0.003732237 0.0025338499 0.0028776442 0.0022220297 0.0016004173
[7,] 0.006483352 0.0045523488 0.0050892276 0.0040569473 0.0029997593
[8,] 0.008401858 0.0060389695 0.0067597276 0.0053697111 0.0039575729
[9,] 0.012398528 0.0088061102 0.0098845055 0.0078132411 0.0057404116
[10,] 0.005174948 0.0035019448 0.0039719917 0.0030984806 0.0023004465
[11,] 0.002864379 0.0019502481 0.0022124945 0.0017123914 0.0012252743
[12,] 0.008450507 0.0060727695 0.0067978960 0.0053995583 0.0039793498
[13,] 0.009647954 0.0069047173 0.0077373683 0.0061342117 0.0045153648
[14,] 0.003036693 0.0020665218 0.0023448451 0.0018140834 0.0012971039
[15,] 0.006431964 0.0045069425 0.0050384741 0.0040168511 0.0029705044
[16,] 0.004274207 0.0028953870 0.0032909440 0.0025366183 0.0018582626
[17,] 0.001019824 0.0006958571 0.0007887969 0.0006115611 0.0004389276
[18,] 0.021162722 0.0148352675 0.0166928972 0.0131373023 0.0096249324
[19,] 0.001754391 0.0011966398 0.0013566479 0.0010515130 0.0007543015
[20,] 0.007120642 0.0051154636 0.0057186561 0.0045542072 0.0033625680
[,6] [,7] [,8] [,9] [,10]
[1,] 0.0037322373 0.0064833515 0.008401858 0.012398528 0.0051749484
[2,] 0.0025338499 0.0045523488 0.006038970 0.008806110 0.0035019448
[3,] 0.0028776442 0.0050892276 0.006759728 0.009884505 0.0039719917
[4,] 0.0022220297 0.0040569473 0.005369711 0.007813241 0.0030984806
[5,] 0.0016004173 0.0029997593 0.003957573 0.005740412 0.0023004465
[6,] 0.0429213432 0.0032000875 0.004223948 0.006129724 0.0024525529
[7,] 0.0032000875 0.0776962790 0.007376869 0.010839972 0.0044258747
[8,] 0.0042239481 0.0073768695 0.101469709 0.013610984 0.0058670986
[9,] 0.0061297244 0.0108399724 0.013610984 0.145720691 0.0085497394
[10,] 0.0024525529 0.0044258747 0.005867099 0.008549739 0.0603389749
[11,] 0.0013160329 0.0024584545 0.003239456 0.004693184 0.0018882346
[12,] 0.0042472268 0.0074191704 0.009478120 0.013668384 0.0058998664
[13,] 0.0048202032 0.0084603611 0.010675567 0.015081223 0.0067064091
[14,] 0.0013934264 0.0026058836 0.003434764 0.004977611 0.0020007067
[15,] 0.0031688154 0.0055617878 0.007317016 0.010747307 0.0043818549
[16,] 0.0019798777 0.0036619353 0.004839951 0.007032667 0.0028018497
[17,] 0.0004710923 0.0008759647 0.001152750 0.001667949 0.0006738797
[18,] 0.0102821064 0.0183855203 0.023526910 0.031543156 0.0143947850
[19,] 0.0008096773 0.0015067189 0.001983242 0.002870225 0.0011588027
[20,] 0.0035879141 0.0062419697 0.008119152 0.011989184 0.0049717937
[,11] [,12] [,13] [,14] [,15]
[1,] 0.0028643788 0.008450507 0.009647954 0.0030366928 0.0064319641
[2,] 0.0019502481 0.006072769 0.006904717 0.0020665218 0.0045069425
[3,] 0.0022124945 0.006797896 0.007737368 0.0023448451 0.0050384741
[4,] 0.0017123914 0.005399558 0.006134212 0.0018140834 0.0040168511
[5,] 0.0012252743 0.003979350 0.004515365 0.0012971039 0.0029705044
[6,] 0.0013160329 0.004247227 0.004820203 0.0013934264 0.0031688154
[7,] 0.0024584545 0.007419170 0.008460361 0.0026058836 0.0055617878
[8,] 0.0032394561 0.009478120 0.010675567 0.0034347640 0.0073170160
[9,] 0.0046931835 0.013668384 0.015081223 0.0049776115 0.0107473066
[10,] 0.0018882346 0.005899866 0.006706409 0.0020007067 0.0043818549
[11,] 0.0327578552 0.003257213 0.003694280 0.0010480086 0.0024346001
[12,] 0.0032572130 0.102010224 0.010724216 0.0034536096 0.0073589162
[13,] 0.0036942799 0.010724216 0.115314393 0.0039174705 0.0083902417
[14,] 0.0010480086 0.003453610 0.003917471 0.0347627729 0.0025805668
[15,] 0.0024346001 0.007358916 0.008390242 0.0025805668 0.0769701592
[16,] 0.0015276777 0.004866735 0.005525980 0.0016180460 0.0036259547
[17,] 0.0003527623 0.001159043 0.001313939 0.0003761049 0.0008675107
[18,] 0.0078606233 0.023638836 0.026393757 0.0083392295 0.0182213615
[19,] 0.0006059566 0.001994077 0.002260750 0.0006461439 0.0014921643
[20,] 0.0027542889 0.008166424 0.009329958 0.0029198540 0.0061912162
[,16] [,17] [,18] [,19] [,20]
[1,] 0.0042742071 0.0010198236 0.021162722 0.0017543911 0.007120642
[2,] 0.0028953870 0.0006958571 0.014835268 0.0011966398 0.005115464
[3,] 0.0032909440 0.0007887969 0.016692897 0.0013566479 0.005718656
[4,] 0.0025366183 0.0006115611 0.013137302 0.0010515130 0.004554207
[5,] 0.0018582626 0.0004389276 0.009624932 0.0007543015 0.003362568
[6,] 0.0019798777 0.0004710923 0.010282106 0.0008096773 0.003587914
[7,] 0.0036619353 0.0008759647 0.018385520 0.0015067189 0.006241970
[8,] 0.0048399514 0.0011527504 0.023526910 0.0019832423 0.008119152
[9,] 0.0070326670 0.0016679492 0.031543156 0.0028702253 0.011989184
[10,] 0.0028018497 0.0006738797 0.014394785 0.0011588027 0.004971794
[11,] 0.0015276777 0.0003527623 0.007860623 0.0006059566 0.002754289
[12,] 0.0048667349 0.0011590435 0.023638836 0.0019940766 0.008166424
[13,] 0.0055259804 0.0013139393 0.026393757 0.0022607503 0.009329958
[14,] 0.0016180460 0.0003761049 0.008339229 0.0006461439 0.002919854
[15,] 0.0036259547 0.0008675107 0.018221362 0.0014921643 0.006191216
[16,] 0.0493637409 0.0005460989 0.011810244 0.0009388111 0.004108154
[17,] 0.0005460989 0.0116140248 0.002790485 0.0002041539 0.000980808
[18,] 0.0118102439 0.0027904849 0.242136509 0.0048028196 0.020421365
[19,] 0.0009388111 0.0002041539 0.004802820 0.0199937150 0.001687221
[20,] 0.0041081542 0.0009808080 0.020421365 0.0016872205 0.086701381
id district population cases
35 5554 LK Borken 370196 86
83 8117 LK Goeppingen 255807 67
107 3254 LK Hildesheim 288623 85
109 6434 LK Hochtaunuskreis 226043 8
130 3457 LK Leer 165088 5
140 3355 LK Lueneburg 176445 57
157 5770 LK Minden-Luebbecke 319401 86
210 8119 LK Rems-Murr-Kreis 417131 110
219 5382 LK Rhein-Sieg-Kreis 599042 72
223 9187 LK Rosenheim 248047 67
257 1061 LK Steinburg 134664 22
273 5978 LK Unna 419353 42
290 5170 LK Wesel 474045 8
294 15091 LK Wittenberg 142906 22
324 5314 SK Bonn 316416 11
342 16051 SK Erfurt 202929 188
361 9464 SK Hof 47744 12
371 5315 SK Koeln 995397 35
418 3405 SK Wilhelmshaven 82192 17
423 5124 SK Wuppertal 356420 62
samplingbook documentation built on April 3, 2021, 1:06 a.m.
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Description Usage Arguments Details Value Author(s) References See Also Examples
Description
The function provides sample techniques with sampling probabilities which are proportional to the size of a quantity z.
Usage
1 | pps.sampling(z, n, id = 1:N, method = 'sampford', return.PI = FALSE)
|
Arguments
z |
vector of quantities which determine the sampling probabilities in the population |
n |
positive integer for sample size |
id |
an optional vector with identification values for population elements. Default is |
method |
the sampling method to be used. Options are |
return.PI |
logical. If |
Details
The different methods vary in their run time. Therefore, method='sampford' is stopped if N > 200 or if n/N < 0.3. method='tille' is stopped if N > 500.
In case of large populations use method='midzuno' or method='madow'.
Value
The function pps.sampling returns a value, which is a list consisting of the components
call |
is a list of call components: |
sample |
resulted sample |
pik |
inclusion probabilities |
PI |
sample second order inclusion probabilities |
PI.full |
full second order inclusion probabilities |
Author(s)
Juliane Manitz
References
Kauermann, Goeran/Kuechenhoff, Helmut (2010): Stichproben. Methoden und praktische Umsetzung mit R. Springer.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 | ## 1) simple suppositious example
data <- data.frame(id = 1:7, z = c(1.8, 2 ,3.2 ,2.9 ,1.5 ,2.0 ,2.2))
# Usage of pps.sampling for Sampford method
set.seed(178209)
pps.sample_sampford <- pps.sampling(z=data$z, n=2, method='sampford', return.PI=FALSE)
pps.sample_sampford
# sampling elements
id.sample <- pps.sample_sampford$sample
id.sample
# other methods
set.seed(178209)
pps.sample_tille <- pps.sampling(z=data$z, n=2, method='tille')
pps.sample_tille
set.seed(178209)
pps.sample_midzuno <- pps.sampling(z=data$z, n=2, method='midzuno')
pps.sample_midzuno
set.seed(178209)
pps.sample_madow <- pps.sampling(z=data$z, n=2, method='madow')
pps.sample_madow
## 2) influenza
data(influenza)
summary(influenza)
set.seed(108506)
pps <- pps.sampling(z=influenza$population,n=20,method='midzuno')
pps
sample <- influenza[pps$sample,]
sample
|
Example output
Loading required package: pps
Loading required package: sampling
Loading required package: survey
Loading required package: grid
Loading required package: Matrix
Loading required package: survival
Attaching package: 'survival'
The following objects are masked from 'package:sampling':
cluster, strata
Attaching package: 'survey'
The following object is masked from 'package:graphics':
dotchart
pps.sampling object: Sample with probabilities proportional to size
Method of Sampford:
PPS sample:
[1] 3 7
Sample probabilities:
[,1] [,2]
[1,] 0.41025641 0.07281474
[2,] 0.07281474 0.28205128
[1] 3 7
pps.sampling object: Sample with probabilities proportional to size
Method of Tille:
PPS sample:
[1] 1 3
Sample probabilities:
[,1] [,2]
[1,] 0.23076923 0.05955335
[2,] 0.05955335 0.41025641
pps.sampling object: Sample with probabilities proportional to size
Method of Midzuno:
PPS sample:
[1] 3 4
Sample probabilities:
[,1] [,2]
[1,] 0.41025641 0.08974359
[2,] 0.08974359 0.37179487
Warning message:
In pps.sampling(z = data$z, n = 2, method = "madow") :
Systematic Sample with zeros in 'PI': For calculating estimates use approximate methods.
pps.sampling object: Sample with probabilities proportional to size
Method of Madow:
PPS sample:
[1] 3 6
Sample probabilities:
[,1] [,2]
[1,] 0.4102564 0.2307692
[2,] 0.2307692 0.2564103
id district population cases
Min. : 1001 LK Aachen : 1 Min. : 34719 Min. : 0.00
1st Qu.: 5877 LK Ahrweiler : 1 1st Qu.: 104553 1st Qu.: 9.00
Median : 8331 LK Aichach-Friedberg: 1 Median : 145130 Median : 27.00
Mean : 8468 LK Alb-Donau-Kreis : 1 Mean : 193910 Mean : 44.58
3rd Qu.: 9778 LK Altenburger Land : 1 3rd Qu.: 244154 3rd Qu.: 59.00
Max. :16077 LK Altenkirchen : 1 Max. :1770629 Max. :410.00
(Other) :418
pps.sampling object: Sample with probabilities proportional to size
Method of Midzuno:
PPS sample:
[1] 35 83 107 109 130 140 157 210 219 223 257 273 290 294 324 342 361 371 418
[20] 423
Sample probabilities:
[,1] [,2] [,3] [,4] [,5]
[1,] 0.090052479 0.0053250174 0.0059535012 0.0047392541 0.0034975812
[2,] 0.005325017 0.0622266431 0.0040841690 0.0032027173 0.0023764993
[3,] 0.005953501 0.0040841690 0.0702093391 0.0036435201 0.0026981161
[4,] 0.004739254 0.0032027173 0.0036435201 0.0549863651 0.0020847939
[5,] 0.003497581 0.0023764993 0.0026981161 0.0020847939 0.0401586824
[6,] 0.003732237 0.0025338499 0.0028776442 0.0022220297 0.0016004173
[7,] 0.006483352 0.0045523488 0.0050892276 0.0040569473 0.0029997593
[8,] 0.008401858 0.0060389695 0.0067597276 0.0053697111 0.0039575729
[9,] 0.012398528 0.0088061102 0.0098845055 0.0078132411 0.0057404116
[10,] 0.005174948 0.0035019448 0.0039719917 0.0030984806 0.0023004465
[11,] 0.002864379 0.0019502481 0.0022124945 0.0017123914 0.0012252743
[12,] 0.008450507 0.0060727695 0.0067978960 0.0053995583 0.0039793498
[13,] 0.009647954 0.0069047173 0.0077373683 0.0061342117 0.0045153648
[14,] 0.003036693 0.0020665218 0.0023448451 0.0018140834 0.0012971039
[15,] 0.006431964 0.0045069425 0.0050384741 0.0040168511 0.0029705044
[16,] 0.004274207 0.0028953870 0.0032909440 0.0025366183 0.0018582626
[17,] 0.001019824 0.0006958571 0.0007887969 0.0006115611 0.0004389276
[18,] 0.021162722 0.0148352675 0.0166928972 0.0131373023 0.0096249324
[19,] 0.001754391 0.0011966398 0.0013566479 0.0010515130 0.0007543015
[20,] 0.007120642 0.0051154636 0.0057186561 0.0045542072 0.0033625680
[,6] [,7] [,8] [,9] [,10]
[1,] 0.0037322373 0.0064833515 0.008401858 0.012398528 0.0051749484
[2,] 0.0025338499 0.0045523488 0.006038970 0.008806110 0.0035019448
[3,] 0.0028776442 0.0050892276 0.006759728 0.009884505 0.0039719917
[4,] 0.0022220297 0.0040569473 0.005369711 0.007813241 0.0030984806
[5,] 0.0016004173 0.0029997593 0.003957573 0.005740412 0.0023004465
[6,] 0.0429213432 0.0032000875 0.004223948 0.006129724 0.0024525529
[7,] 0.0032000875 0.0776962790 0.007376869 0.010839972 0.0044258747
[8,] 0.0042239481 0.0073768695 0.101469709 0.013610984 0.0058670986
[9,] 0.0061297244 0.0108399724 0.013610984 0.145720691 0.0085497394
[10,] 0.0024525529 0.0044258747 0.005867099 0.008549739 0.0603389749
[11,] 0.0013160329 0.0024584545 0.003239456 0.004693184 0.0018882346
[12,] 0.0042472268 0.0074191704 0.009478120 0.013668384 0.0058998664
[13,] 0.0048202032 0.0084603611 0.010675567 0.015081223 0.0067064091
[14,] 0.0013934264 0.0026058836 0.003434764 0.004977611 0.0020007067
[15,] 0.0031688154 0.0055617878 0.007317016 0.010747307 0.0043818549
[16,] 0.0019798777 0.0036619353 0.004839951 0.007032667 0.0028018497
[17,] 0.0004710923 0.0008759647 0.001152750 0.001667949 0.0006738797
[18,] 0.0102821064 0.0183855203 0.023526910 0.031543156 0.0143947850
[19,] 0.0008096773 0.0015067189 0.001983242 0.002870225 0.0011588027
[20,] 0.0035879141 0.0062419697 0.008119152 0.011989184 0.0049717937
[,11] [,12] [,13] [,14] [,15]
[1,] 0.0028643788 0.008450507 0.009647954 0.0030366928 0.0064319641
[2,] 0.0019502481 0.006072769 0.006904717 0.0020665218 0.0045069425
[3,] 0.0022124945 0.006797896 0.007737368 0.0023448451 0.0050384741
[4,] 0.0017123914 0.005399558 0.006134212 0.0018140834 0.0040168511
[5,] 0.0012252743 0.003979350 0.004515365 0.0012971039 0.0029705044
[6,] 0.0013160329 0.004247227 0.004820203 0.0013934264 0.0031688154
[7,] 0.0024584545 0.007419170 0.008460361 0.0026058836 0.0055617878
[8,] 0.0032394561 0.009478120 0.010675567 0.0034347640 0.0073170160
[9,] 0.0046931835 0.013668384 0.015081223 0.0049776115 0.0107473066
[10,] 0.0018882346 0.005899866 0.006706409 0.0020007067 0.0043818549
[11,] 0.0327578552 0.003257213 0.003694280 0.0010480086 0.0024346001
[12,] 0.0032572130 0.102010224 0.010724216 0.0034536096 0.0073589162
[13,] 0.0036942799 0.010724216 0.115314393 0.0039174705 0.0083902417
[14,] 0.0010480086 0.003453610 0.003917471 0.0347627729 0.0025805668
[15,] 0.0024346001 0.007358916 0.008390242 0.0025805668 0.0769701592
[16,] 0.0015276777 0.004866735 0.005525980 0.0016180460 0.0036259547
[17,] 0.0003527623 0.001159043 0.001313939 0.0003761049 0.0008675107
[18,] 0.0078606233 0.023638836 0.026393757 0.0083392295 0.0182213615
[19,] 0.0006059566 0.001994077 0.002260750 0.0006461439 0.0014921643
[20,] 0.0027542889 0.008166424 0.009329958 0.0029198540 0.0061912162
[,16] [,17] [,18] [,19] [,20]
[1,] 0.0042742071 0.0010198236 0.021162722 0.0017543911 0.007120642
[2,] 0.0028953870 0.0006958571 0.014835268 0.0011966398 0.005115464
[3,] 0.0032909440 0.0007887969 0.016692897 0.0013566479 0.005718656
[4,] 0.0025366183 0.0006115611 0.013137302 0.0010515130 0.004554207
[5,] 0.0018582626 0.0004389276 0.009624932 0.0007543015 0.003362568
[6,] 0.0019798777 0.0004710923 0.010282106 0.0008096773 0.003587914
[7,] 0.0036619353 0.0008759647 0.018385520 0.0015067189 0.006241970
[8,] 0.0048399514 0.0011527504 0.023526910 0.0019832423 0.008119152
[9,] 0.0070326670 0.0016679492 0.031543156 0.0028702253 0.011989184
[10,] 0.0028018497 0.0006738797 0.014394785 0.0011588027 0.004971794
[11,] 0.0015276777 0.0003527623 0.007860623 0.0006059566 0.002754289
[12,] 0.0048667349 0.0011590435 0.023638836 0.0019940766 0.008166424
[13,] 0.0055259804 0.0013139393 0.026393757 0.0022607503 0.009329958
[14,] 0.0016180460 0.0003761049 0.008339229 0.0006461439 0.002919854
[15,] 0.0036259547 0.0008675107 0.018221362 0.0014921643 0.006191216
[16,] 0.0493637409 0.0005460989 0.011810244 0.0009388111 0.004108154
[17,] 0.0005460989 0.0116140248 0.002790485 0.0002041539 0.000980808
[18,] 0.0118102439 0.0027904849 0.242136509 0.0048028196 0.020421365
[19,] 0.0009388111 0.0002041539 0.004802820 0.0199937150 0.001687221
[20,] 0.0041081542 0.0009808080 0.020421365 0.0016872205 0.086701381
id district population cases
35 5554 LK Borken 370196 86
83 8117 LK Goeppingen 255807 67
107 3254 LK Hildesheim 288623 85
109 6434 LK Hochtaunuskreis 226043 8
130 3457 LK Leer 165088 5
140 3355 LK Lueneburg 176445 57
157 5770 LK Minden-Luebbecke 319401 86
210 8119 LK Rems-Murr-Kreis 417131 110
219 5382 LK Rhein-Sieg-Kreis 599042 72
223 9187 LK Rosenheim 248047 67
257 1061 LK Steinburg 134664 22
273 5978 LK Unna 419353 42
290 5170 LK Wesel 474045 8
294 15091 LK Wittenberg 142906 22
324 5314 SK Bonn 316416 11
342 16051 SK Erfurt 202929 188
361 9464 SK Hof 47744 12
371 5315 SK Koeln 995397 35
418 3405 SK Wilhelmshaven 82192 17
423 5124 SK Wuppertal 356420 62
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