In statistical surveys, when subpopulations within an overall population vary, it could be advantageous to sample each subpopulation (stratum) independently.
Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. The strata should define a partition of the population. That is, it should be collectively exhaustive and mutually exclusive: every element in the population must be assigned to one and only one stratum. Then simple random sampling is applied within each stratum. The objective is to improve the precision of the sample by reducing sampling error. It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population.
Assume that we need to estimate the average number of votes for each candidate in an election. Assume that a country has 3 towns: Town A has 1 million factory workers, Town B has 2 million office workers and Town C has 3 million retirees. We can choose to get a random sample of size 60 over the entire population but there is some chance that the resulting random sample is poorly balanced across these towns and hence is biased, causing a significant error in estimation (when the outcome of interest has a different distribution, in terms of the parameter of interest, between the towns). Instead, if we choose to take a random sample of 10, 20 and 30 from Town A, B and C respectively, then we can produce a smaller error in estimation for the same total sample size. This method is generally used when a population is not a homogeneous group.
Stratified sampling strategies
Proportionate allocation uses a sampling fraction in each of the strata that are proportional to that of the total population. For instance, if the population consists of n total individuals, m of which are male and f female (and where m + f = n), then the relative size of the two samples (x1 = m/n males, x2 = f/n females) should reflect this proportion.
Optimum allocation (or disproportionate allocation) - The sampling fraction of each stratum is proportionate to both the proportion (as above) and the standard deviation of the distribution of the variable. Larger samples are taken in the strata with the greatest variability to generate the least possible overall sampling variance.
A real-world example of using stratified sampling would be for a political survey. If the respondents needed to reflect the diversity of the population, the researcher would specifically seek to include participants of various minority groups such as race or religion, based on their proportionality to the total population as mentioned above. A stratified survey could thus claim to be more representative of the population than a survey of simple random sampling or systematic sampling.
If measurements within strata have a lower standard deviation (as compared to the overall standard deviation in the population), stratification gives a smaller error in estimation.
For many applications, measurements become more manageable and/or cheaper when the population is grouped into strata.
When it is desirable to have estimates of the population parameters for groups within the population - stratified sampling verifies we have enough samples from the strata of interest.
If the population density varies greatly within a region, stratified sampling will ensure that estimates can be made with equal accuracy in different parts of the region, and that comparisons of sub-regions can be made with equal statistical power. For example, in Ontario a survey taken throughout the province might use a larger sampling fraction in the less populated north, since the disparity in population between north and south is so great that a sampling fraction based on the provincial sample as a whole might result in the collection of only a handful of data from the north.
Disadvantages
Stratified sampling is not useful when the population cannot be exhaustively partitioned into disjoint subgroups. It would be a misapplication of the technique to make subgroups' sample sizes proportional to the amount of data available from the subgroups, rather than scaling sample sizes to subgroup sizes (or to their variances, if known to vary significantly—e.g. using an F Test). Data representing each subgroup are taken to be of equal importance if suspected variation among them warrants stratified sampling. If subgroup variances differ significantly and the data needs to be stratified by variance, it is not possible to simultaneously make each subgroup sample size proportional to subgroup size within the total population. For an efficient way to partition sampling resources among groups that vary in their means, variance and costs, see "optimum allocation". The problem of stratified sampling in the case of unknown class priors (ratio of subpopulations in the entire population) can have a deleterious effect on the performance of any analysis on the dataset, e.g. classification.[3] In that regard, minimax sampling ratio can be used to make the dataset robust with respect to uncertainty in the underlying data generating process.[3]
Combining sub-strata to ensure adequate numbers can lead to Simpson's paradox, where trends that exist in different groups of data disappear or even reverse when the groups are combined.
Mean and standard error
The mean and variance of stratified random sampling are given by:[2]
where,
number of strata
the sum of all stratum sizes
size of stratum
sample mean of stratum
number of observations in stratum
sample standard deviation of stratum
Note that the term ( − ) / ( − ), which equals 1 − , is a finite population correction and must be expressed in "sample units". Foregoing the finite population correction gives:
where the = / is the population weight of stratum .
Sample size allocation
For the proportional allocation strategy, the size of the sample in each stratum is taken in proportion to the size of the stratum. Suppose that in a company there are the following staff:[4]
male, full-time: 90
male, part-time: 18
female, full-time: 9
female, part-time: 63
total: 180
and we are asked to take a sample of 40 staff, stratified according to the above categories.
The first step is to calculate the percentage of each group of the total.
% male, full-time = 90 ÷ 180 = 50%
% male, part-time = 18 ÷ 180 = 10%
% female, full-time = 9 ÷ 180 = 5%
% female, part-time = 63 ÷ 180 = 35%
This tells us that of our sample of 40,
50% (20 individuals) should be male, full-time.
10% (4 individuals) should be male, part-time.
5% (2 individuals) should be female, full-time.
35% (14 individuals) should be female, part-time.
Another easy way without having to calculate the percentage is to multiply each group size by the sample size and divide by the total population size (size of entire staff):
^ ab"6.1 How to Use Stratified Sampling | STAT 506". onlinecourses.science.psu.edu. Retrieved 2015-07-23.
^ abShahrokh Esfahani, Mohammad; Dougherty, Edward R. (2014). "Effect of separate sampling on classification accuracy". Bioinformatics. 30 (2): 242–250. doi:10.1093/bioinformatics/btt662. PMID 24257187.
^Hunt, Neville; Tyrrell, Sidney (2001). "Stratified Sampling". Webpage at Coventry University. Archived from the original on 13 October 2013. Retrieved 12 July 2012.
Further reading
Särndal, Carl-Erik; et al. (2003). "Stratified Sampling". Model Assisted Survey Sampling. New York: Springer. pp. 100–109. ISBN0-387-40620-4.
March 04, 2023
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This article needs additional citations for verification Please help improve this article by adding citations to reliable sources Unsourced material may be challenged and removed Find sources Stratified sampling news newspapers books scholar JSTOR December 2020 Learn how and when to remove this template message In statistics stratified sampling is a method of sampling from a population which can be partitioned into subpopulations Stratified sampling example In statistical surveys when subpopulations within an overall population vary it could be advantageous to sample each subpopulation stratum independently Stratification is the process of dividing members of the population into homogeneous subgroups before sampling The strata should define a partition of the population That is it should be collectively exhaustive and mutually exclusive every element in the population must be assigned to one and only one stratum Then simple random sampling is applied within each stratum The objective is to improve the precision of the sample by reducing sampling error It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population In computational statistics stratified sampling is a method of variance reduction when Monte Carlo methods are used to estimate population statistics from a known population 1 Contents 1 Example 2 Stratified sampling strategies 3 Advantages 4 Disadvantages 5 Mean and standard error 6 Sample size allocation 7 See also 8 References 9 Further readingExample EditAssume that we need to estimate the average number of votes for each candidate in an election Assume that a country has 3 towns Town A has 1 million factory workers Town B has 2 million office workers and Town C has 3 million retirees We can choose to get a random sample of size 60 over the entire population but there is some chance that the resulting random sample is poorly balanced across these towns and hence is biased causing a significant error in estimation when the outcome of interest has a different distribution in terms of the parameter of interest between the towns Instead if we choose to take a random sample of 10 20 and 30 from Town A B and C respectively then we can produce a smaller error in estimation for the same total sample size This method is generally used when a population is not a homogeneous group Stratified sampling strategies EditProportionate allocation uses a sampling fraction in each of the strata that are proportional to that of the total population For instance if the population consists of n total individuals m of which are male and f female and where m f n then the relative size of the two samples x1 m n males x2 f n females should reflect this proportion Optimum allocation or disproportionate allocation The sampling fraction of each stratum is proportionate to both the proportion as above and the standard deviation of the distribution of the variable Larger samples are taken in the strata with the greatest variability to generate the least possible overall sampling variance A real world example of using stratified sampling would be for a political survey If the respondents needed to reflect the diversity of the population the researcher would specifically seek to include participants of various minority groups such as race or religion based on their proportionality to the total population as mentioned above A stratified survey could thus claim to be more representative of the population than a survey of simple random sampling or systematic sampling Advantages EditThe reasons to use stratified sampling rather than simple random sampling include 2 If measurements within strata have a lower standard deviation as compared to the overall standard deviation in the population stratification gives a smaller error in estimation For many applications measurements become more manageable and or cheaper when the population is grouped into strata When it is desirable to have estimates of the population parameters for groups within the population stratified sampling verifies we have enough samples from the strata of interest If the population density varies greatly within a region stratified sampling will ensure that estimates can be made with equal accuracy in different parts of the region and that comparisons of sub regions can be made with equal statistical power For example in Ontario a survey taken throughout the province might use a larger sampling fraction in the less populated north since the disparity in population between north and south is so great that a sampling fraction based on the provincial sample as a whole might result in the collection of only a handful of data from the north Disadvantages EditStratified sampling is not useful when the population cannot be exhaustively partitioned into disjoint subgroups It would be a misapplication of the technique to make subgroups sample sizes proportional to the amount of data available from the subgroups rather than scaling sample sizes to subgroup sizes or to their variances if known to vary significantly e g using an F Test Data representing each subgroup are taken to be of equal importance if suspected variation among them warrants stratified sampling If subgroup variances differ significantly and the data needs to be stratified by variance it is not possible to simultaneously make each subgroup sample size proportional to subgroup size within the total population For an efficient way to partition sampling resources among groups that vary in their means variance and costs see optimum allocation The problem of stratified sampling in the case of unknown class priors ratio of subpopulations in the entire population can have a deleterious effect on the performance of any analysis on the dataset e g classification 3 In that regard minimax sampling ratio can be used to make the dataset robust with respect to uncertainty in the underlying data generating process 3 Combining sub strata to ensure adequate numbers can lead to Simpson s paradox where trends that exist in different groups of data disappear or even reverse when the groups are combined Mean and standard error EditThe mean and variance of stratified random sampling are given by 2 x 1 N h 1 L N h x h displaystyle bar x frac 1 N sum h 1 L N h bar x h s x 2 h 1 L N h N 2 N h n h N h 1 s h 2 n h displaystyle s bar x 2 sum h 1 L left frac N h N right 2 left frac N h n h N h 1 right frac s h 2 n h where L displaystyle L number of strataN displaystyle N the sum of all stratum sizesN h displaystyle N h size of stratum h displaystyle h x h displaystyle bar x h sample mean of stratum h displaystyle h n h displaystyle n h number of observations in stratum h displaystyle h s h displaystyle s h sample standard deviation of stratum h displaystyle h Note that the term N h displaystyle N h n h displaystyle n h N h displaystyle N h 1 displaystyle 1 which equals 1 n h 1 N h 1 displaystyle frac n h 1 N h 1 is a finite population correction and N h displaystyle N h must be expressed in sample units Foregoing the finite population correction gives s x 2 h 1 L N h N 2 s h 2 n h displaystyle s bar x 2 sum h 1 L left frac N h N right 2 frac s h 2 n h where the w h displaystyle w h N h displaystyle N h N displaystyle N is the population weight of stratum h displaystyle h Sample size allocation EditFor the proportional allocation strategy the size of the sample in each stratum is taken in proportion to the size of the stratum Suppose that in a company there are the following staff 4 male full time 90 male part time 18 female full time 9 female part time 63 total 180and we are asked to take a sample of 40 staff stratified according to the above categories The first step is to calculate the percentage of each group of the total male full time 90 180 50 male part time 18 180 10 female full time 9 180 5 female part time 63 180 35 This tells us that of our sample of 40 50 20 individuals should be male full time 10 4 individuals should be male part time 5 2 individuals should be female full time 35 14 individuals should be female part time Another easy way without having to calculate the percentage is to multiply each group size by the sample size and divide by the total population size size of entire staff male full time 90 40 180 20 male part time 18 40 180 4 female full time 9 40 180 2 female part time 63 40 180 14See also Edit Mathematics portalOpinion poll Multistage sampling Statistical benchmarking Stratified sample size Stratification clinical trials References Edit Botev Z Ridder A 2017 Variance Reduction Wiley StatsRef Statistics Reference Online 1 6 doi 10 1002 9781118445112 stat07975 ISBN 9781118445112 a b 6 1 How to Use Stratified Sampling STAT 506 onlinecourses science psu edu Retrieved 2015 07 23 a b Shahrokh Esfahani Mohammad Dougherty Edward R 2014 Effect of separate sampling on classification accuracy Bioinformatics 30 2 242 250 doi 10 1093 bioinformatics btt662 PMID 24257187 Hunt Neville Tyrrell Sidney 2001 Stratified Sampling Webpage at Coventry University Archived from the original on 13 October 2013 Retrieved 12 July 2012 Further reading EditSarndal Carl Erik et al 2003 Stratified Sampling Model Assisted Survey Sampling New York Springer pp 100 109 ISBN 0 387 40620 4 Retrieved from https en wikipedia org w index php title Stratified sampling amp oldid 1141100685, wikipedia, wiki, book, books, library,
