# Creating stratified subsamples from a sample (repeated observations) - Post stratification

I have a random sample of individuals in a country (without knowing in which province they live). From this sample I want to make stratified random samples on the provincial level (let's assume there are three provinces), based on information I have about the job division in the province and the information I have of the job of each individual (and perhaps other characteristics). I found this link on Wikipedia which appears to quite well describe what I am doing.

If I add the sub-samples of these provinces back together, I will have repeated observations. Although I assume that will not affect the estimates because I am using random sampling, it will affect the standard deviations (because of the repeated observations).

Does anyone know if I have to account for this and if yes how I should account for this?

• you need to explain what you are trying to do in more detail. Create a toy data set with 10 observations, show your 3 provinces, show your job categorizations, etc. This is not a stratified sampling situation, and this is not a post-stratification situation, you have something else in mind that does not fit these definitions. (Edit your question, do not reply below.) – StasK Nov 27 '20 at 14:16

Upon reading the question, it is not clear to me that you have completely followed, what at least one researcher has described, the steps consistent with the associated high-standard ascribed to the stratified random sampling scheme.

To assist, I refer you to this dissertation work which enumerates seven critical steps. Further, the steps themselves make one fully aware of the disadvantages associated with the precise implementation of such a sampling technique.

Here are relevant portions of the article, to quote:

STEP ONE: Define the population: In our example, the population is the 10,000 students at the University of Bath....Since we are interested in all of these university students, we can say that our sampling frame is all 10,000 students. If we were only interested in female university students, for example, we would exclude all males in creating our sampling frame, which would be much less than 10,000.

STEP TWO: Choose the relevant stratification: If we wanted to look at the differences in male and female students, this would mean choosing gender as the stratification…

STEP THREE: List the population: We need to identify all 10,000 students at the University of Bath... You can read about this later in the article under Disadvantages (limitations) of stratified random sampling.

STEP FOUR: List the population according to the chosen stratification: As with the simple random sampling and systematic random sampling techniques, we need to assign a consecutive number from 1 to $$N_k$$ to each of the students in each stratum. As a result, we would end up with two lists, one detailing all male students and one detailing all female students.

STEP FIVE: Choose your sample size: Let's imagine that we choose a sample size of 100 students. The sample is expressed as n. This number was chosen because it reflects the limit of our budget and the time... However, we could have also determined the sample size we needed using a sample size calculation, which is a particularly useful statistical tool.

STEP SIX: Calculate a proportionate stratification: Imagine that of the 10,000 students, 60% of these are female and 40% male. We need to ensure that the number of units selected for the sample from each stratum is proportionate to the number of males and females in the population…

STEP SEVEN: Use a simple random or systematic sample to select your sample: Now that we have chosen to sample 40 male and 60 female students, we still need to select these students from our two lists of male and female students (see STEP FOUR above). We do this using either simple random sampling or systematic random sampling.

Listed advantages and disadvantages (linked to feasibility issues) of stratified random sampling are described after first noting that stratified random sampling is one of the 'gold standards' of sampling techniques, albeit, with many challenges. To continue quoting:

Advantages of stratified random sampling: The aim of the stratified random sample is to reduce the potential for human bias in the selection of cases to be included in the sample. As a result, the stratified random sample provides us with a sample that is highly representative of the population being studied, assuming that there is limited missing data. Since the units selected for inclusion within the sample are chosen using probabilistic methods, stratified random sampling allows us to make statistical conclusions from the data collected that will be considered to be valid. Relative to the simple random sample, the selection of units using a stratified procedure can be viewed as superior because it improves the potential for the units to be more evenly spread over the population. Furthermore, where the samples are the same size, a stratified random sample can provide greater precision than a simple random sample. Because of the greater precision of a stratified random sample compared with a simple random sample, it may be possible to use a smaller sample, which saves time and money. The stratified random sample also improves the representation of particular strata (groups) within the population, as well as ensuring that these strata are not over-represented. Together, this helps the researcher to compare strata, as well as make more valid inferences from the sample to the population.

Disadvantages (limitations) of stratified random sampling: A stratified random sample can only be carried out if a complete list of the population is available...It must also be possible for the list of the population to be clearly delineated into each stratum; that is, each unit from the population must only belong to one stratum. In our example, this would be fairly simple, since our strata are male and female students. Clearly, a student could only be classified as either male or female. No student could fit into both categories (ignoring transgender issues)...increase overall sample size required for the research, which can increase costs and time to carry out the research. Attaining a complete list of the population can be difficult for a number of reasons: Even if a list is readily available, it may be challenging to gain access to that list...There may be no single list detailing the population you are interested in. As a result, it may be difficult and time consuming to bring together numerous sub-lists to create a final list from which you want to select your sample...Indeed, it will be more complex and time consuming to prepare this list compared with simple random sampling and systematic random sampling. Many lists will not be in the public domain and their purchase may be expensive;... In terms of human populations (as opposed to other types of populations; see the article: Sampling: The basics), some of these populations will be expensive and time consuming to contact, even where a list is available... In the case of human populations, to avoid potential bias in your sample, you will also need to try and ensure that an adequate proportion of your sample takes part in the research. This may require re-contacting non-respondents,...

Note, it is statistically important to subsample in the stratums according to appropriate proportions in the parent population in order to construct the recommended stratified sample stratums. Somewhat daunting, however, are issues around sample list availability, preparation and updating for accuracy, especially for human populations, which likely can be difficult and expensive to accurately accommodate in practice.

[EDIT] Related to a comment I further quote:

Furthermore, imagine extending the sampling requirements such that we were also interested in how career goals changed depending on whether a student was an undergraduate or graduate. Since the strata must be mutually exclusive and collectively exclusive, this means that we would need to sample four strata from the population: undergraduate males, undergraduate females, graduate males, and graduate females. This will increase overall sample size required for the research, which can increase costs and time to carry out the research.

In particular, note that the constructed stratums are "mutually exclusive and collectively exclusive" so having observations repeated across strata is not in accord with a stratified sampling scheme. However, at the expense of multiplying the dimension of the sampling scheme, as cited in the case above, you may be able to proceed.

• Thank you for your post. Regretfully this is not really what I am looking for. The stratification process is not the issue. The issue is having repeated observations in the sample, as a result of the stratification process, which is not addressed in your post. If you would have any information about that I would be extremely grateful. – Tom Kisters Nov 26 '20 at 15:10
• I have provided a further extract from the cited source that may be of assistance. – AJKOER Nov 26 '20 at 15:44