Introduction
In statistics, sampling is the selection of a subset of elements from a population, here defined as a complete set of subjects of interest.
Since often it is too expensive or logistically impossible to collect data for every case in a population, sampling is instead used as a cheap and fast methodology to estimate its characteristics.
Different sampling schemes exist, but they can be grouped into 2 main categories, i.e. sampling with or without replacement:
- sampling with replacement implies each element in the population may appear multiple times in one sample
- in sampling without replacement, each member of the population can be chosen only once in one sample
In this article, we show how to draw samples with
replacement from a population by using the functions
implemented in resampling.
For further information on how to draw repeated samples with replacement, see Resampling with replacement.
Setup
Seed
Then, we set a seed for the random number generation (RNG). In fact,
different R sessions have different seeds created from current time and
process ID by default, and consequently different simulation results. By
fixing a seed we ensure we will be able to reproduce the results of this
vignette. We can specify a seed by calling ?set.seed.
#Set a seed for RNG
set.seed(
#A seed
seed = 5381L, #a randomly chosen integer value
#The kind of RNG to use
kind = "Mersenne-Twister", #we make explicit the current R default value
#The kind of Normal generation
normal.kind = "Inversion" #we make explicit the current R default value
)Sampling With Replacement
As previously wrote, sampling with replacement implies that elements of a population can be chosen multiple times in one sample. There are different techniques of sampling with replacement, including:
- simple random sampling
- random sampling with unequal probabilities
- stratified sampling
- balanced sampling (a special case of stratified sampling)
The available methods can be listed through the
?listAvailableSamplingMethods function call, setting the
input argument to 'rswr'.
?listAvailableSamplingMethods returns a table with two
columns:
-
id: the id of the sampling method, to be used in the function calls -
name: the name of the sampling method
#list sampling methods
sampling.methods = listAvailableSamplingMethods(x = 'rswr')
#print in table
knitr::kable(x = sampling.methods)| id | name |
|---|---|
| rswr | random sampling with replacement |
| srswr | simple random sampling with replacement |
| stratified_rswr | stratified random sampling with replacement |
| balanced_rswr | balanced random sampling with replacement |
| bootstrap | ordinary bootstrap sampling |
The name of the sampling functions can be retrieved by calling
?listSamplingFunctionNames.
#list sampling function names
sampling.function.names = listSamplingFunctionNames(x = 'rswr')
#print in table
knitr::kable(x = sampling.function.names)| id | name |
|---|---|
| rswr | sampleWithReplacement |
| srswr | simpleRandomSampleWithReplacement |
| stratified_rswr | stratifiedSampleWithReplacement |
| balanced_rswr | balancedSampleWithReplacement |
| bootstrap | bootstrapSample |
Each function is documented. To learn more about a specific method it
is possible to use the ? operator. For example, let’s check
the function ?simpleRandomSampleWithReplacement.
#See documentation
?simpleRandomSampleWithReplacementFrom the documentation, we can see that the function accepts 2 arguments in input:
-
N: the population size -
n: the sample size
Simple Random Sampling
Simple random sampling (SRS) is the easiest form of sampling with replacement. In SRS with replacement, each element of the population has the same probability of being selected for the sample.
#Simple random sampling with replacement
simpleRandomSampleWithReplacement(
N = 10,
n = 8
)
#> [1] 1 9 10 3 4 1 9 8Random Sampling With Unequal Probability
The concept of random sampling with unequal
probability was perhaps introduced by Hansen and Hurwitz in the
context of sampling with replacement (Hansen and
Hurwitz 1943). Under this sampling design, elements of the
population have different probabilities of being selected. We can use
?sampleWithReplacement to draw our sample. For example,
let’s assume our population of interest has 10 elements, and that the
first 3 elements have an higher chance of being selected.
#Random sampling with replacement
sampleWithReplacement(
N = 10,
n = 8,
prob = c(rep(3,3), rep(1,7))
)
#> [1] 6 3 4 2 9 1 2 1Stratified Random Sampling
When a population can be partitioned into groups (i.e. strata or subpopulations) having certain properties in common, a stratified sampling approach can be used. This sampling design is adopted to ensure that subgroups of the population are represented in the taken sample.
A stratified sample with replacement can be taken by
using ?stratifiedSampleWithReplacement which accept a
strata argument in input.
#Define strata
strata = c(rep("a", 3),rep("b", 6))
#Stratified sampling with replacement
stratifiedSampleWithReplacement(
strata = strata,
n = 9
)
#> [1] 5 2 7 8 8 3 3 8 5?stratifiedSampleWithReplacement implements the
so-called “proportionate allocation”, in which the proportion of the
strata in the population is maintained in the samples.
#Define strata
strata = c(rep("a", 3),rep("b", 6))
#Check ratio
table(strata)/length(strata)
#> strata
#> a b
#> 0.3333333 0.6666667
#Stratified sampling with replacement
s = stratifiedSampleWithReplacement(
strata = strata,
n = 9
)
#Check ratio in the sample
table(strata[s])/length(strata[s])
#>
#> a b
#> 0.3333333 0.6666667Balanced Random Sampling
Balanced sampling is a special case of stratified sampling used to ensure that subgroups of the population are equally represented in the taken sample.
#Define strata
strata = c(rep("a", 3),rep("b", 6))
#Check ratio
table(strata)/length(strata)
#> strata
#> a b
#> 0.3333333 0.6666667
#Balanced sampling with replacement
s = balancedSampleWithReplacement(
strata = strata,
n = 8
)
#Check ratio in the sample
table(strata[s])/length(strata[s])
#>
#> a b
#> 0.5 0.5Bootstrap Sampling
Inspired by earlier work on the jackknife, Bradley Efron published the bootstrap method in 1979 (Efron 1979). An ordinary bootstrap sample is a special case of random sample with replacement where the sample size is equivalent to the population size.
A bootstrap sample can be taken by using
?bootstrapSample:
#Bootstrap sampling
bootstrapSample(
N = 10
)
#> [1] 10 3 3 4 3 9 8 2 2 5