sampling procedure in research paper

Sampling Methods & Strategies 101

Everything you need to know (including examples)

By: Derek Jansen (MBA) | Expert Reviewed By: Kerryn Warren (PhD) | January 2023

If you’re new to research, sooner or later you’re bound to wander into the intimidating world of sampling methods and strategies. If you find yourself on this page, chances are you’re feeling a little overwhelmed or confused. Fear not – in this post we’ll unpack sampling in straightforward language , along with loads of examples .

Overview: Sampling Methods & Strategies

• What is sampling in a research context?
• The two overarching approaches

Simple random sampling

Stratified random sampling, cluster sampling, systematic sampling, purposive sampling, convenience sampling, snowball sampling.

• How to choose the right sampling method

What (exactly) is sampling?

At the simplest level, sampling (within a research context) is the process of selecting a subset of participants from a larger group . For example, if your research involved assessing US consumers’ perceptions about a particular brand of laundry detergent, you wouldn’t be able to collect data from every single person that uses laundry detergent (good luck with that!) – but you could potentially collect data from a smaller subset of this group.

In technical terms, the larger group is referred to as the population , and the subset (the group you’ll actually engage with in your research) is called the sample . Put another way, you can look at the population as a full cake and the sample as a single slice of that cake. In an ideal world, you’d want your sample to be perfectly representative of the population, as that would allow you to generalise your findings to the entire population. In other words, you’d want to cut a perfect cross-sectional slice of cake, such that the slice reflects every layer of the cake in perfect proportion.

Achieving a truly representative sample is, unfortunately, a little trickier than slicing a cake, as there are many practical challenges and obstacles to achieving this in a real-world setting. Thankfully though, you don’t always need to have a perfectly representative sample – it all depends on the specific research aims of each study – so don’t stress yourself out about that just yet!

With the concept of sampling broadly defined, let’s look at the different approaches to sampling to get a better understanding of what it all looks like in practice.

The two overarching sampling approaches

At the highest level, there are two approaches to sampling: probability sampling and non-probability sampling . Within each of these, there are a variety of sampling methods , which we’ll explore a little later.

Probability sampling involves selecting participants (or any unit of interest) on a statistically random basis , which is why it’s also called “random sampling”. In other words, the selection of each individual participant is based on a pre-determined process (not the discretion of the researcher). As a result, this approach achieves a random sample.

Probability-based sampling methods are most commonly used in quantitative research , especially when it’s important to achieve a representative sample that allows the researcher to generalise their findings.

Non-probability sampling , on the other hand, refers to sampling methods in which the selection of participants is not statistically random . In other words, the selection of individual participants is based on the discretion and judgment of the researcher, rather than on a pre-determined process.

Non-probability sampling methods are commonly used in qualitative research , where the richness and depth of the data are more important than the generalisability of the findings.

If that all sounds a little too conceptual and fluffy, don’t worry. Let’s take a look at some actual sampling methods to make it more tangible.

Need a helping hand?

Probability-based sampling methods

First, we’ll look at four common probability-based (random) sampling methods:

Importantly, this is not a comprehensive list of all the probability sampling methods – these are just four of the most common ones. So, if you’re interested in adopting a probability-based sampling approach, be sure to explore all the options.

Simple random sampling involves selecting participants in a completely random fashion , where each participant has an equal chance of being selected. Basically, this sampling method is the equivalent of pulling names out of a hat , except that you can do it digitally. For example, if you had a list of 500 people, you could use a random number generator to draw a list of 50 numbers (each number, reflecting a participant) and then use that dataset as your sample.

Thanks to its simplicity, simple random sampling is easy to implement , and as a consequence, is typically quite cheap and efficient . Given that the selection process is completely random, the results can be generalised fairly reliably. However, this also means it can hide the impact of large subgroups within the data, which can result in minority subgroups having little representation in the results – if any at all. To address this, one needs to take a slightly different approach, which we’ll look at next.

Stratified random sampling is similar to simple random sampling, but it kicks things up a notch. As the name suggests, stratified sampling involves selecting participants randomly , but from within certain pre-defined subgroups (i.e., strata) that share a common trait . For example, you might divide the population into strata based on gender, ethnicity, age range or level of education, and then select randomly from each group.

The benefit of this sampling method is that it gives you more control over the impact of large subgroups (strata) within the population. For example, if a population comprises 80% males and 20% females, you may want to “balance” this skew out by selecting a random sample from an equal number of males and females. This would, of course, reduce the representativeness of the sample, but it would allow you to identify differences between subgroups. So, depending on your research aims, the stratified approach could work well.

Next on the list is cluster sampling. As the name suggests, this sampling method involves sampling from naturally occurring, mutually exclusive clusters within a population – for example, area codes within a city or cities within a country. Once the clusters are defined, a set of clusters are randomly selected and then a set of participants are randomly selected from each cluster.

Now, you’re probably wondering, “how is cluster sampling different from stratified random sampling?”. Well, let’s look at the previous example where each cluster reflects an area code in a given city.

With cluster sampling, you would collect data from clusters of participants in a handful of area codes (let’s say 5 neighbourhoods). Conversely, with stratified random sampling, you would need to collect data from all over the city (i.e., many more neighbourhoods). You’d still achieve the same sample size either way (let’s say 200 people, for example), but with stratified sampling, you’d need to do a lot more running around, as participants would be scattered across a vast geographic area. As a result, cluster sampling is often the more practical and economical option.

If that all sounds a little mind-bending, you can use the following general rule of thumb. If a population is relatively homogeneous , cluster sampling will often be adequate. Conversely, if a population is quite heterogeneous (i.e., diverse), stratified sampling will generally be more appropriate.

The last probability sampling method we’ll look at is systematic sampling. This method simply involves selecting participants at a set interval , starting from a random point .

For example, if you have a list of students that reflects the population of a university, you could systematically sample that population by selecting participants at an interval of 8 . In other words, you would randomly select a starting point – let’s say student number 40 – followed by student 48, 56, 64, etc.

What’s important with systematic sampling is that the population list you select from needs to be randomly ordered . If there are underlying patterns in the list (for example, if the list is ordered by gender, IQ, age, etc.), this will result in a non-random sample, which would defeat the purpose of adopting this sampling method. Of course, you could safeguard against this by “shuffling” your population list using a random number generator or similar tool.

Non-probability-based sampling methods

Right, now that we’ve looked at a few probability-based sampling methods, let’s look at three non-probability methods :

Again, this is not an exhaustive list of all possible sampling methods, so be sure to explore further if you’re interested in adopting a non-probability sampling approach.

First up, we’ve got purposive sampling – also known as judgment , selective or subjective sampling. Again, the name provides some clues, as this method involves the researcher selecting participants using his or her own judgement , based on the purpose of the study (i.e., the research aims).

For example, suppose your research aims were to understand the perceptions of hyper-loyal customers of a particular retail store. In that case, you could use your judgement to engage with frequent shoppers, as well as rare or occasional shoppers, to understand what judgements drive the two behavioural extremes .

Purposive sampling is often used in studies where the aim is to gather information from a small population (especially rare or hard-to-find populations), as it allows the researcher to target specific individuals who have unique knowledge or experience . Naturally, this sampling method is quite prone to researcher bias and judgement error, and it’s unlikely to produce generalisable results, so it’s best suited to studies where the aim is to go deep rather than broad .

Next up, we have convenience sampling. As the name suggests, with this method, participants are selected based on their availability or accessibility . In other words, the sample is selected based on how convenient it is for the researcher to access it, as opposed to using a defined and objective process.

Naturally, convenience sampling provides a quick and easy way to gather data, as the sample is selected based on the individuals who are readily available or willing to participate. This makes it an attractive option if you’re particularly tight on resources and/or time. However, as you’d expect, this sampling method is unlikely to produce a representative sample and will of course be vulnerable to researcher bias , so it’s important to approach it with caution.

Last but not least, we have the snowball sampling method. This method relies on referrals from initial participants to recruit additional participants. In other words, the initial subjects form the first (small) snowball and each additional subject recruited through referral is added to the snowball, making it larger as it rolls along .

Snowball sampling is often used in research contexts where it’s difficult to identify and access a particular population. For example, people with a rare medical condition or members of an exclusive group. It can also be useful in cases where the research topic is sensitive or taboo and people are unlikely to open up unless they’re referred by someone they trust.

Simply put, snowball sampling is ideal for research that involves reaching hard-to-access populations . But, keep in mind that, once again, it’s a sampling method that’s highly prone to researcher bias and is unlikely to produce a representative sample. So, make sure that it aligns with your research aims and questions before adopting this method.

How to choose a sampling method

Now that we’ve looked at a few popular sampling methods (both probability and non-probability based), the obvious question is, “ how do I choose the right sampling method for my study?”. When selecting a sampling method for your research project, you’ll need to consider two important factors: your research aims and your resources .

As with all research design and methodology choices, your sampling approach needs to be guided by and aligned with your research aims, objectives and research questions – in other words, your golden thread. Specifically, you need to consider whether your research aims are primarily concerned with producing generalisable findings (in which case, you’ll likely opt for a probability-based sampling method) or with achieving rich , deep insights (in which case, a non-probability-based approach could be more practical). Typically, quantitative studies lean toward the former, while qualitative studies aim for the latter, so be sure to consider your broader methodology as well.

The second factor you need to consider is your resources and, more generally, the practical constraints at play. If, for example, you have easy, free access to a large sample at your workplace or university and a healthy budget to help you attract participants, that will open up multiple options in terms of sampling methods. Conversely, if you’re cash-strapped, short on time and don’t have unfettered access to your population of interest, you may be restricted to convenience or referral-based methods.

In short, be ready for trade-offs – you won’t always be able to utilise the “perfect” sampling method for your study, and that’s okay. Much like all the other methodological choices you’ll make as part of your study, you’ll often need to compromise and accept practical trade-offs when it comes to sampling. Don’t let this get you down though – as long as your sampling choice is well explained and justified, and the limitations of your approach are clearly articulated, you’ll be on the right track.

Let’s recap…

In this post, we’ve covered the basics of sampling within the context of a typical research project.

• Sampling refers to the process of defining a subgroup (sample) from the larger group of interest (population).
• The two overarching approaches to sampling are probability sampling (random) and non-probability sampling .
• Common probability-based sampling methods include simple random sampling, stratified random sampling, cluster sampling and systematic sampling.
• Common non-probability-based sampling methods include purposive sampling, convenience sampling and snowball sampling.
• When choosing a sampling method, you need to consider your research aims , objectives and questions, as well as your resources and other practical constraints .

If you’d like to see an example of a sampling strategy in action, be sure to check out our research methodology chapter sample .

Last but not least, if you need hands-on help with your sampling (or any other aspect of your research), take a look at our 1-on-1 coaching service , where we guide you through each step of the research process, at your own pace.

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This post was based on one of our popular Research Bootcamps . If you're working on a research project, you'll definitely want to check this out ...

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Excellent and helpful. Best site to get a full understanding of Research methodology. I’m nolonger as “clueless “..😉

Excellent and helpful for junior researcher!

Grad Coach tutorials are excellent – I recommend them to everyone doing research. I will be working with a sample of imprisoned women and now have a much clearer idea concerning sampling. Thank you to all at Grad Coach for generously sharing your expertise with students.

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• Sampling Methods | Types, Techniques, & Examples

Sampling Methods | Types, Techniques, & Examples

Published on 3 May 2022 by Shona McCombes . Revised on 10 October 2022.

When you conduct research about a group of people, it’s rarely possible to collect data from every person in that group. Instead, you select a sample. The sample is the group of individuals who will actually participate in the research.

To draw valid conclusions from your results, you have to carefully decide how you will select a sample that is representative of the group as a whole. There are two types of sampling methods:

• Probability sampling involves random selection, allowing you to make strong statistical inferences about the whole group. It minimises the risk of selection bias .
• Non-probability sampling involves non-random selection based on convenience or other criteria, allowing you to easily collect data.

You should clearly explain how you selected your sample in the methodology section of your paper or thesis.

Table of contents

Population vs sample, probability sampling methods, non-probability sampling methods, frequently asked questions about sampling.

First, you need to understand the difference between a population and a sample , and identify the target population of your research.

• The population is the entire group that you want to draw conclusions about.
• The sample is the specific group of individuals that you will collect data from.

The population can be defined in terms of geographical location, age, income, and many other characteristics.

It is important to carefully define your target population according to the purpose and practicalities of your project.

If the population is very large, demographically mixed, and geographically dispersed, it might be difficult to gain access to a representative sample.

Sampling frame

The sampling frame is the actual list of individuals that the sample will be drawn from. Ideally, it should include the entire target population (and nobody who is not part of that population).

You are doing research on working conditions at Company X. Your population is all 1,000 employees of the company. Your sampling frame is the company’s HR database, which lists the names and contact details of every employee.

Sample size

The number of individuals you should include in your sample depends on various factors, including the size and variability of the population and your research design. There are different sample size calculators and formulas depending on what you want to achieve with statistical analysis .

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Probability sampling means that every member of the population has a chance of being selected. It is mainly used in quantitative research . If you want to produce results that are representative of the whole population, probability sampling techniques are the most valid choice.

There are four main types of probability sample.

1. Simple random sampling

In a simple random sample , every member of the population has an equal chance of being selected. Your sampling frame should include the whole population.

To conduct this type of sampling, you can use tools like random number generators or other techniques that are based entirely on chance.

You want to select a simple random sample of 100 employees of Company X. You assign a number to every employee in the company database from 1 to 1000, and use a random number generator to select 100 numbers.

2. Systematic sampling

Systematic sampling is similar to simple random sampling, but it is usually slightly easier to conduct. Every member of the population is listed with a number, but instead of randomly generating numbers, individuals are chosen at regular intervals.

All employees of the company are listed in alphabetical order. From the first 10 numbers, you randomly select a starting point: number 6. From number 6 onwards, every 10th person on the list is selected (6, 16, 26, 36, and so on), and you end up with a sample of 100 people.

If you use this technique, it is important to make sure that there is no hidden pattern in the list that might skew the sample. For example, if the HR database groups employees by team, and team members are listed in order of seniority, there is a risk that your interval might skip over people in junior roles, resulting in a sample that is skewed towards senior employees.

3. Stratified sampling

Stratified sampling involves dividing the population into subpopulations that may differ in important ways. It allows you draw more precise conclusions by ensuring that every subgroup is properly represented in the sample.

To use this sampling method, you divide the population into subgroups (called strata) based on the relevant characteristic (e.g., gender, age range, income bracket, job role).

Based on the overall proportions of the population, you calculate how many people should be sampled from each subgroup. Then you use random or systematic sampling to select a sample from each subgroup.

The company has 800 female employees and 200 male employees. You want to ensure that the sample reflects the gender balance of the company, so you sort the population into two strata based on gender. Then you use random sampling on each group, selecting 80 women and 20 men, which gives you a representative sample of 100 people.

4. Cluster sampling

Cluster sampling also involves dividing the population into subgroups, but each subgroup should have similar characteristics to the whole sample. Instead of sampling individuals from each subgroup, you randomly select entire subgroups.

If it is practically possible, you might include every individual from each sampled cluster. If the clusters themselves are large, you can also sample individuals from within each cluster using one of the techniques above. This is called multistage sampling .

This method is good for dealing with large and dispersed populations, but there is more risk of error in the sample, as there could be substantial differences between clusters. It’s difficult to guarantee that the sampled clusters are really representative of the whole population.

The company has offices in 10 cities across the country (all with roughly the same number of employees in similar roles). You don’t have the capacity to travel to every office to collect your data, so you use random sampling to select 3 offices – these are your clusters.

In a non-probability sample , individuals are selected based on non-random criteria, and not every individual has a chance of being included.

This type of sample is easier and cheaper to access, but it has a higher risk of sampling bias . That means the inferences you can make about the population are weaker than with probability samples, and your conclusions may be more limited. If you use a non-probability sample, you should still aim to make it as representative of the population as possible.

Non-probability sampling techniques are often used in exploratory and qualitative research . In these types of research, the aim is not to test a hypothesis about a broad population, but to develop an initial understanding of a small or under-researched population.

1. Convenience sampling

A convenience sample simply includes the individuals who happen to be most accessible to the researcher.

This is an easy and inexpensive way to gather initial data, but there is no way to tell if the sample is representative of the population, so it can’t produce generalisable results.

You are researching opinions about student support services in your university, so after each of your classes, you ask your fellow students to complete a survey on the topic. This is a convenient way to gather data, but as you only surveyed students taking the same classes as you at the same level, the sample is not representative of all the students at your university.

2. Voluntary response sampling

Similar to a convenience sample, a voluntary response sample is mainly based on ease of access. Instead of the researcher choosing participants and directly contacting them, people volunteer themselves (e.g., by responding to a public online survey).

Voluntary response samples are always at least somewhat biased, as some people will inherently be more likely to volunteer than others.

You send out the survey to all students at your university and many students decide to complete it. This can certainly give you some insight into the topic, but the people who responded are more likely to be those who have strong opinions about the student support services, so you can’t be sure that their opinions are representative of all students.

3. Purposive sampling

Purposive sampling , also known as judgement sampling, involves the researcher using their expertise to select a sample that is most useful to the purposes of the research.

It is often used in qualitative research , where the researcher wants to gain detailed knowledge about a specific phenomenon rather than make statistical inferences, or where the population is very small and specific. An effective purposive sample must have clear criteria and rationale for inclusion.

You want to know more about the opinions and experiences of students with a disability at your university, so you purposely select a number of students with different support needs in order to gather a varied range of data on their experiences with student services.

4. Snowball sampling

If the population is hard to access, snowball sampling can be used to recruit participants via other participants. The number of people you have access to ‘snowballs’ as you get in contact with more people.

You are researching experiences of homelessness in your city. Since there is no list of all homeless people in the city, probability sampling isn’t possible. You meet one person who agrees to participate in the research, and she puts you in contact with other homeless people she knows in the area.

A sample is a subset of individuals from a larger population. Sampling means selecting the group that you will actually collect data from in your research.

For example, if you are researching the opinions of students in your university, you could survey a sample of 100 students.

Statistical sampling allows you to test a hypothesis about the characteristics of a population. There are various sampling methods you can use to ensure that your sample is representative of the population as a whole.

Samples are used to make inferences about populations . Samples are easier to collect data from because they are practical, cost-effective, convenient, and manageable.

Probability sampling means that every member of the target population has a known chance of being included in the sample.

Probability sampling methods include simple random sampling , systematic sampling , stratified sampling , and cluster sampling .

In non-probability sampling , the sample is selected based on non-random criteria, and not every member of the population has a chance of being included.

Common non-probability sampling methods include convenience sampling , voluntary response sampling, purposive sampling , snowball sampling , and quota sampling .

Sampling bias occurs when some members of a population are systematically more likely to be selected in a sample than others.

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Sampling Methods: A guide for researchers

Affiliation.

• 1 Arizona School of Dentistry & Oral Health A.T. Still University, Mesa, AZ, USA [email protected].
• PMID: 37553279

Sampling is a critical element of research design. Different methods can be used for sample selection to ensure that members of the study population reflect both the source and target populations, including probability and non-probability sampling. Power and sample size are used to determine the number of subjects needed to answer the research question. Characteristics of individuals included in the sample population should be clearly defined to determine eligibility for study participation and improve power. Sample selection methods differ based on study design. The purpose of this short report is to review common sampling considerations and related errors.

Keywords: research design; sample size; sampling.

Copyright © 2023 The American Dental Hygienists’ Association.

• Research Design*
• Sample Size

• En español – ExME
• Em português – EME

What are sampling methods and how do you choose the best one?

Posted on 18th November 2020 by Mohamed Khalifa

This tutorial will introduce sampling methods and potential sampling errors to avoid when conducting medical research.

Introduction to sampling methods

Examples of different sampling methods, choosing the best sampling method.

It is important to understand why we sample the population; for example, studies are built to investigate the relationships between risk factors and disease. In other words, we want to find out if this is a true association, while still aiming for the minimum risk for errors such as: chance, bias or confounding .

However, it would not be feasible to experiment on the whole population, we would need to take a good sample and aim to reduce the risk of having errors by proper sampling technique.

What is a sampling frame?

A sampling frame is a record of the target population containing all participants of interest. In other words, it is a list from which we can extract a sample.

What makes a good sample?

A good sample should be a representative subset of the population we are interested in studying, therefore, with each participant having equal chance of being randomly selected into the study.

We could choose a sampling method based on whether we want to account for sampling bias; a random sampling method is often preferred over a non-random method for this reason. Random sampling examples include: simple, systematic, stratified, and cluster sampling. Non-random sampling methods are liable to bias, and common examples include: convenience, purposive, snowballing, and quota sampling. For the purposes of this blog we will be focusing on random sampling methods .

Example: We want to conduct an experimental trial in a small population such as: employees in a company, or students in a college. We include everyone in a list and use a random number generator to select the participants

Advantages: Generalisable results possible, random sampling, the sampling frame is the whole population, every participant has an equal probability of being selected

Disadvantages: Less precise than stratified method, less representative than the systematic method

Example: Every nth patient entering the out-patient clinic is selected and included in our sample

Advantages: More feasible than simple or stratified methods, sampling frame is not always required

Disadvantages:  Generalisability may decrease if baseline characteristics repeat across every nth participant

Example: We have a big population (a city) and we want to ensure representativeness of all groups with a pre-determined characteristic such as: age groups, ethnic origin, and gender

Advantages:  Inclusive of strata (subgroups), reliable and generalisable results

Disadvantages: Does not work well with multiple variables

Example: 10 schools have the same number of students across the county. We can randomly select 3 out of 10 schools as our clusters

Advantages: Readily doable with most budgets, does not require a sampling frame

Disadvantages: Results may not be reliable nor generalisable

How can you identify sampling errors?

Non-random selection increases the probability of sampling (selection) bias if the sample does not represent the population we want to study. We could avoid this by random sampling and ensuring representativeness of our sample with regards to sample size.

An inadequate sample size decreases the confidence in our results as we may think there is no significant difference when actually there is. This type two error results from having a small sample size, or from participants dropping out of the sample.

In medical research of disease, if we select people with certain diseases while strictly excluding participants with other co-morbidities, we run the risk of diagnostic purity bias where important sub-groups of the population are not represented.

Furthermore, measurement bias may occur during re-collection of risk factors by participants (recall bias) or assessment of outcome where people who live longer are associated with treatment success, when in fact people who died were not included in the sample or data analysis (survivors bias).

By following the steps below we could choose the best sampling method for our study in an orderly fashion.

Research objectiveness

Firstly, a refined research question and goal would help us define our population of interest. If our calculated sample size is small then it would be easier to get a random sample. If, however, the sample size is large, then we should check if our budget and resources can handle a random sampling method.

Sampling frame availability

Secondly, we need to check for availability of a sampling frame (Simple), if not, could we make a list of our own (Stratified). If neither option is possible, we could still use other random sampling methods, for instance, systematic or cluster sampling.

Study design

Moreover, we could consider the prevalence of the topic (exposure or outcome) in the population, and what would be the suitable study design. In addition, checking if our target population is widely varied in its baseline characteristics. For example, a population with large ethnic subgroups could best be studied using a stratified sampling method.

Random sampling

Finally, the best sampling method is always the one that could best answer our research question while also allowing for others to make use of our results (generalisability of results). When we cannot afford a random sampling method, we can always choose from the non-random sampling methods.

To sum up, we now understand that choosing between random or non-random sampling methods is multifactorial. We might often be tempted to choose a convenience sample from the start, but that would not only decrease precision of our results, and would make us miss out on producing research that is more robust and reliable.

References (pdf)

Mohamed Khalifa

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No Comments on What are sampling methods and how do you choose the best one?

Thank you for this overview. A concise approach for research.

really helps! am an ecology student preparing to write my lab report for sampling.

I learned a lot to the given presentation.. It’s very comprehensive… Thanks for sharing…

Very informative and useful for my study. Thank you

Oversimplified info on sampling methods. Probabilistic of the sampling and sampling of samples by chance does rest solely on the random methods. Factors such as the random visits or presentation of the potential participants at clinics or sites could be sufficiently random in nature and should be used for the sake of efficiency and feasibility. Nevertheless, this approach has to be taken only after careful thoughts. Representativeness of the study samples have to be checked at the end or during reporting by comparing it to the published larger studies or register of some kind in/from the local population.

Thank you so much Mr.mohamed very useful and informative article

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Home » Sampling Methods – Types, Techniques and Examples

Sampling Methods – Types, Techniques and Examples

Table of Contents

Sampling refers to the process of selecting a subset of data from a larger population or dataset in order to analyze or make inferences about the whole population.

In other words, sampling involves taking a representative sample of data from a larger group or dataset in order to gain insights or draw conclusions about the entire group.

Sampling Methods

Sampling methods refer to the techniques used to select a subset of individuals or units from a larger population for the purpose of conducting statistical analysis or research.

Sampling is an essential part of the Research because it allows researchers to draw conclusions about a population without having to collect data from every member of that population, which can be time-consuming, expensive, or even impossible.

Types of Sampling Methods

Sampling can be broadly categorized into two main categories:

Probability Sampling

This type of sampling is based on the principles of random selection, and it involves selecting samples in a way that every member of the population has an equal chance of being included in the sample.. Probability sampling is commonly used in scientific research and statistical analysis, as it provides a representative sample that can be generalized to the larger population.

Type of Probability Sampling :

• Simple Random Sampling: In this method, every member of the population has an equal chance of being selected for the sample. This can be done using a random number generator or by drawing names out of a hat, for example.
• Systematic Sampling: In this method, the population is first divided into a list or sequence, and then every nth member is selected for the sample. For example, if every 10th person is selected from a list of 100 people, the sample would include 10 people.
• Stratified Sampling: In this method, the population is divided into subgroups or strata based on certain characteristics, and then a random sample is taken from each stratum. This is often used to ensure that the sample is representative of the population as a whole.
• Cluster Sampling: In this method, the population is divided into clusters or groups, and then a random sample of clusters is selected. Then, all members of the selected clusters are included in the sample.
• Multi-Stage Sampling : This method combines two or more sampling techniques. For example, a researcher may use stratified sampling to select clusters, and then use simple random sampling to select members within each cluster.

Non-probability Sampling

This type of sampling does not rely on random selection, and it involves selecting samples in a way that does not give every member of the population an equal chance of being included in the sample. Non-probability sampling is often used in qualitative research, where the aim is not to generalize findings to a larger population, but to gain an in-depth understanding of a particular phenomenon or group. Non-probability sampling methods can be quicker and more cost-effective than probability sampling methods, but they may also be subject to bias and may not be representative of the larger population.

Types of Non-probability Sampling :

• Convenience Sampling: In this method, participants are chosen based on their availability or willingness to participate. This method is easy and convenient but may not be representative of the population.
• Purposive Sampling: In this method, participants are selected based on specific criteria, such as their expertise or knowledge on a particular topic. This method is often used in qualitative research, but may not be representative of the population.
• Snowball Sampling: In this method, participants are recruited through referrals from other participants. This method is often used when the population is hard to reach, but may not be representative of the population.
• Quota Sampling: In this method, a predetermined number of participants are selected based on specific criteria, such as age or gender. This method is often used in market research, but may not be representative of the population.
• Volunteer Sampling: In this method, participants volunteer to participate in the study. This method is often used in research where participants are motivated by personal interest or altruism, but may not be representative of the population.

Applications of Sampling Methods

Applications of Sampling Methods from different fields:

• Psychology : Sampling methods are used in psychology research to study various aspects of human behavior and mental processes. For example, researchers may use stratified sampling to select a sample of participants that is representative of the population based on factors such as age, gender, and ethnicity. Random sampling may also be used to select participants for experimental studies.
• Sociology : Sampling methods are commonly used in sociological research to study social phenomena and relationships between individuals and groups. For example, researchers may use cluster sampling to select a sample of neighborhoods to study the effects of economic inequality on health outcomes. Stratified sampling may also be used to select a sample of participants that is representative of the population based on factors such as income, education, and occupation.
• Social sciences: Sampling methods are commonly used in social sciences to study human behavior and attitudes. For example, researchers may use stratified sampling to select a sample of participants that is representative of the population based on factors such as age, gender, and income.
• Marketing : Sampling methods are used in marketing research to collect data on consumer preferences, behavior, and attitudes. For example, researchers may use random sampling to select a sample of consumers to participate in a survey about a new product.
• Healthcare : Sampling methods are used in healthcare research to study the prevalence of diseases and risk factors, and to evaluate interventions. For example, researchers may use cluster sampling to select a sample of health clinics to participate in a study of the effectiveness of a new treatment.
• Environmental science: Sampling methods are used in environmental science to collect data on environmental variables such as water quality, air pollution, and soil composition. For example, researchers may use systematic sampling to collect soil samples at regular intervals across a field.
• Education : Sampling methods are used in education research to study student learning and achievement. For example, researchers may use stratified sampling to select a sample of schools that is representative of the population based on factors such as demographics and academic performance.

Examples of Sampling Methods

Probability Sampling Methods Examples:

• Simple random sampling Example : A researcher randomly selects participants from the population using a random number generator or drawing names from a hat.
• Stratified random sampling Example : A researcher divides the population into subgroups (strata) based on a characteristic of interest (e.g. age or income) and then randomly selects participants from each subgroup.
• Systematic sampling Example : A researcher selects participants at regular intervals from a list of the population.

Non-probability Sampling Methods Examples:

• Convenience sampling Example: A researcher selects participants who are conveniently available, such as students in a particular class or visitors to a shopping mall.
• Purposive sampling Example : A researcher selects participants who meet specific criteria, such as individuals who have been diagnosed with a particular medical condition.
• Snowball sampling Example : A researcher selects participants who are referred to them by other participants, such as friends or acquaintances.

How to Conduct Sampling Methods

some general steps to conduct sampling methods:

• Define the population: Identify the population of interest and clearly define its boundaries.
• Choose the sampling method: Select an appropriate sampling method based on the research question, characteristics of the population, and available resources.
• Determine the sample size: Determine the desired sample size based on statistical considerations such as margin of error, confidence level, or power analysis.
• Create a sampling frame: Develop a list of all individuals or elements in the population from which the sample will be drawn. The sampling frame should be comprehensive, accurate, and up-to-date.
• Select the sample: Use the chosen sampling method to select the sample from the sampling frame. The sample should be selected randomly, or if using a non-random method, every effort should be made to minimize bias and ensure that the sample is representative of the population.
• Collect data: Once the sample has been selected, collect data from each member of the sample using appropriate research methods (e.g., surveys, interviews, observations).
• Analyze the data: Analyze the data collected from the sample to draw conclusions about the population of interest.

When to use Sampling Methods

Sampling methods are used in research when it is not feasible or practical to study the entire population of interest. Sampling allows researchers to study a smaller group of individuals, known as a sample, and use the findings from the sample to make inferences about the larger population.

Sampling methods are particularly useful when:

• The population of interest is too large to study in its entirety.
• The cost and time required to study the entire population are prohibitive.
• The population is geographically dispersed or difficult to access.
• The research question requires specialized or hard-to-find individuals.
• The data collected is quantitative and statistical analyses are used to draw conclusions.

Purpose of Sampling Methods

The main purpose of sampling methods in research is to obtain a representative sample of individuals or elements from a larger population of interest, in order to make inferences about the population as a whole. By studying a smaller group of individuals, known as a sample, researchers can gather information about the population that would be difficult or impossible to obtain from studying the entire population.

Sampling methods allow researchers to:

• Study a smaller, more manageable group of individuals, which is typically less time-consuming and less expensive than studying the entire population.
• Reduce the potential for data collection errors and improve the accuracy of the results by minimizing sampling bias.
• Make inferences about the larger population with a certain degree of confidence, using statistical analyses of the data collected from the sample.
• Improve the generalizability and external validity of the findings by ensuring that the sample is representative of the population of interest.

Characteristics of Sampling Methods

Here are some characteristics of sampling methods:

• Randomness : Probability sampling methods are based on random selection, meaning that every member of the population has an equal chance of being selected. This helps to minimize bias and ensure that the sample is representative of the population.
• Representativeness : The goal of sampling is to obtain a sample that is representative of the larger population of interest. This means that the sample should reflect the characteristics of the population in terms of key demographic, behavioral, or other relevant variables.
• Size : The size of the sample should be large enough to provide sufficient statistical power for the research question at hand. The sample size should also be appropriate for the chosen sampling method and the level of precision desired.
• Efficiency : Sampling methods should be efficient in terms of time, cost, and resources required. The method chosen should be feasible given the available resources and time constraints.
• Bias : Sampling methods should aim to minimize bias and ensure that the sample is representative of the population of interest. Bias can be introduced through non-random selection or non-response, and can affect the validity and generalizability of the findings.
• Precision : Sampling methods should be precise in terms of providing estimates of the population parameters of interest. Precision is influenced by sample size, sampling method, and level of variability in the population.
• Validity : The validity of the sampling method is important for ensuring that the results obtained from the sample are accurate and can be generalized to the population of interest. Validity can be affected by sampling method, sample size, and the representativeness of the sample.

Advantages of Sampling Methods

Sampling methods have several advantages, including:

• Cost-Effective : Sampling methods are often much cheaper and less time-consuming than studying an entire population. By studying only a small subset of the population, researchers can gather valuable data without incurring the costs associated with studying the entire population.
• Convenience : Sampling methods are often more convenient than studying an entire population. For example, if a researcher wants to study the eating habits of people in a city, it would be very difficult and time-consuming to study every single person in the city. By using sampling methods, the researcher can obtain data from a smaller subset of people, making the study more feasible.
• Accuracy: When done correctly, sampling methods can be very accurate. By using appropriate sampling techniques, researchers can obtain a sample that is representative of the entire population. This allows them to make accurate generalizations about the population as a whole based on the data collected from the sample.
• Time-Saving: Sampling methods can save a lot of time compared to studying the entire population. By studying a smaller sample, researchers can collect data much more quickly than they could if they studied every single person in the population.
• Less Bias : Sampling methods can reduce bias in a study. If a researcher were to study the entire population, it would be very difficult to eliminate all sources of bias. However, by using appropriate sampling techniques, researchers can reduce bias and obtain a sample that is more representative of the entire population.

Limitations of Sampling Methods

• Sampling Error : Sampling error is the difference between the sample statistic and the population parameter. It is the result of selecting a sample rather than the entire population. The larger the sample, the lower the sampling error. However, no matter how large the sample size, there will always be some degree of sampling error.
• Selection Bias: Selection bias occurs when the sample is not representative of the population. This can happen if the sample is not selected randomly or if some groups are underrepresented in the sample. Selection bias can lead to inaccurate conclusions about the population.
• Non-response Bias : Non-response bias occurs when some members of the sample do not respond to the survey or study. This can result in a biased sample if the non-respondents differ from the respondents in important ways.
• Time and Cost : While sampling can be cost-effective, it can still be expensive and time-consuming to select a sample that is representative of the population. Depending on the sampling method used, it may take a long time to obtain a sample that is large enough and representative enough to be useful.
• Limited Information : Sampling can only provide information about the variables that are measured. It may not provide information about other variables that are relevant to the research question but were not measured.
• Generalization : The extent to which the findings from a sample can be generalized to the population depends on the representativeness of the sample. If the sample is not representative of the population, it may not be possible to generalize the findings to the population as a whole.

About the author

Muhammad Hassan

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SYSTEMATIC REVIEW article

This article is part of the research topic.

Reviews in Gastroenterology 2023

Electrogastrography Measurement Systems and Analysis Methods Used in Clinical Practice and Research: Comprehensive Review Provisionally Accepted

• 1 VSB-Technical University of Ostrava, Czechia

The final, formatted version of the article will be published soon.

Electrogastrography (EGG) is a non-invasive method with high diagnostic potential for the prevention of gastroenterological pathologies in clinical practice. In this paper, a review of the measurement systems, procedures, and methods of analysis used in electrogastrography is presented. A critical review of historical and current literature is conducted, focusing on electrode placement, measurement apparatus, measurement procedures, and time-frequency domain methods of filtration and analysis of the non-invasively measured electrical activity of the stomach.As a result a total of 129 relevant articles with primary aim on experimental diet were reviewed in this study. Scopus, PubMed and Web of Science databases were used to search for articles in English language, according to the specific query and using PRISMA method. The research topic of electrogastrography has been continuously growing in popularity since the first measurement by professor Alvarez 100 years ago and there are many researchers and companies interested in EGG nowadays. Measurement apparatus and procedures are still being developed in both commercial and research settings. There are plenty variable electrode layouts, ranging from minimal numbers of electrodes for ambulatory measurements to very high numbers of electrodes for spatial measurements. Most authors used in their research anatomically approximated layout with 2 active electrodes in bipolar connection and commercial electrogastrograph with sampling rate of 2 or 4 Hz. Test subjects were usually healthy adults and diet was controlled. However, evaluation methods are being developed at a slower pace and usually the signals are classified only based on dominant frequency. The main review contributions include the overview of spectrum of measurement systems and procedures for electrogastrography developed by many authors, but a firm medical standard has not yet been defined. Therefore, it is not possible to use this method in clinical practice for objective diagnosis.

Keywords: electrogastrography, non-invasive method, Measurement systems, Electrode placement, Measurement apparatus, Signal processing

Received: 19 Jan 2024; Accepted: 03 Jun 2024.

Copyright: © 2024 Oczka, Augustynek, Penhaker and Kubicek. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Dr. Jan Kubicek, VSB-Technical University of Ostrava, Ostrava, 708 33, Moravian-Silesian Region, Czechia

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Purposeful sampling for qualitative data collection and analysis in mixed method implementation research

Lawrence a. palinkas.

1 School of Social Work, University of Southern California, Los Angeles, CA 90089-0411

Sarah M. Horwitz

2 Department of Child and Adolescent Psychiatry, New York University, New York, NY

Carla A. Green

3 Center for Health Research, Kaiser Permanente Northwest, Portland, OR

Jennifer P. Wisdom

4 George Washington University, Washington DC

Naihua Duan

5 New York State Neuropsychiatric Institute and Department of Psychiatry, Columbia University, New York, NY

Kimberly Hoagwood

Purposeful sampling is widely used in qualitative research for the identification and selection of information-rich cases related to the phenomenon of interest. Although there are several different purposeful sampling strategies, criterion sampling appears to be used most commonly in implementation research. However, combining sampling strategies may be more appropriate to the aims of implementation research and more consistent with recent developments in quantitative methods. This paper reviews the principles and practice of purposeful sampling in implementation research, summarizes types and categories of purposeful sampling strategies and provides a set of recommendations for use of single strategy or multistage strategy designs, particularly for state implementation research.

Recently there have been several calls for the use of mixed method designs in implementation research ( Proctor et al., 2009 ; Landsverk et al., 2012 ; Palinkas et al. 2011 ; Aarons et al., 2012). This has been precipitated by the realization that the challenges of implementing evidence-based and other innovative practices, treatments, interventions and programs are sufficiently complex that a single methodological approach is often inadequate. This is particularly true of efforts to implement evidence-based practices (EBPs) in statewide systems where relationships among key stakeholders extend both vertically (from state to local organizations) and horizontally (between organizations located in different parts of a state). As in other areas of research, mixed method designs are viewed as preferable in implementation research because they provide a better understanding of research issues than either qualitative or quantitative approaches alone ( Palinkas et al., 2011 ). In such designs, qualitative methods are used to explore and obtain depth of understanding as to the reasons for success or failure to implement evidence-based practice or to identify strategies for facilitating implementation while quantitative methods are used to test and confirm hypotheses based on an existing conceptual model and obtain breadth of understanding of predictors of successful implementation ( Teddlie & Tashakkori, 2003 ).

Sampling strategies for quantitative methods used in mixed methods designs in implementation research are generally well-established and based on probability theory. In contrast, sampling strategies for qualitative methods in implementation studies are less explicit and often less evident. Although the samples for qualitative inquiry are generally assumed to be selected purposefully to yield cases that are “information rich” (Patton, 2001), there are no clear guidelines for conducting purposeful sampling in mixed methods implementation studies, particularly when studies have more than one specific objective. Moreover, it is not entirely clear what forms of purposeful sampling are most appropriate for the challenges of using both quantitative and qualitative methods in the mixed methods designs used in implementation research. Such a consideration requires a determination of the objectives of each methodology and the potential impact of selecting one strategy to achieve one objective on the selection of other strategies to achieve additional objectives.

In this paper, we present different approaches to the use of purposeful sampling strategies in implementation research. We begin with a review of the principles and practice of purposeful sampling in implementation research, a summary of the types and categories of purposeful sampling strategies, and a set of recommendations for matching the appropriate single strategy or multistage strategy to study aims and quantitative method designs.

Principles of Purposeful Sampling

Purposeful sampling is a technique widely used in qualitative research for the identification and selection of information-rich cases for the most effective use of limited resources ( Patton, 2002 ). This involves identifying and selecting individuals or groups of individuals that are especially knowledgeable about or experienced with a phenomenon of interest ( Cresswell & Plano Clark, 2011 ). In addition to knowledge and experience, Bernard (2002) and Spradley (1979) note the importance of availability and willingness to participate, and the ability to communicate experiences and opinions in an articulate, expressive, and reflective manner. In contrast, probabilistic or random sampling is used to ensure the generalizability of findings by minimizing the potential for bias in selection and to control for the potential influence of known and unknown confounders.

As Morse and Niehaus (2009) observe, whether the methodology employed is quantitative or qualitative, sampling methods are intended to maximize efficiency and validity. Nevertheless, sampling must be consistent with the aims and assumptions inherent in the use of either method. Qualitative methods are, for the most part, intended to achieve depth of understanding while quantitative methods are intended to achieve breadth of understanding ( Patton, 2002 ). Qualitative methods place primary emphasis on saturation (i.e., obtaining a comprehensive understanding by continuing to sample until no new substantive information is acquired) ( Miles & Huberman, 1994 ). Quantitative methods place primary emphasis on generalizability (i.e., ensuring that the knowledge gained is representative of the population from which the sample was drawn). Each methodology, in turn, has different expectations and standards for determining the number of participants required to achieve its aims. Quantitative methods rely on established formulae for avoiding Type I and Type II errors, while qualitative methods often rely on precedents for determining number of participants based on type of analysis proposed (e.g., 3-6 participants interviewed multiple times in a phenomenological study versus 20-30 participants interviewed once or twice in a grounded theory study), level of detail required, and emphasis of homogeneity (requiring smaller samples) versus heterogeneity (requiring larger samples) ( Guest, Bunce & Johnson., 2006 ; Morse & Niehaus, 2009 ; Padgett, 2008 ).

Types of purposeful sampling designs

There exist numerous purposeful sampling designs. Examples include the selection of extreme or deviant (outlier) cases for the purpose of learning from an unusual manifestations of phenomena of interest; the selection of cases with maximum variation for the purpose of documenting unique or diverse variations that have emerged in adapting to different conditions, and to identify important common patterns that cut across variations; and the selection of homogeneous cases for the purpose of reducing variation, simplifying analysis, and facilitating group interviewing. A list of some of these strategies and examples of their use in implementation research is provided in Table 1 .

Purposeful sampling strategies in implementation research

Embedded in each strategy is the ability to compare and contrast, to identify similarities and differences in the phenomenon of interest. Nevertheless, some of these strategies (e.g., maximum variation sampling, extreme case sampling, intensity sampling, and purposeful random sampling) are used to identify and expand the range of variation or differences, similar to the use of quantitative measures to describe the variability or dispersion of values for a particular variable or variables, while other strategies (e.g., homogeneous sampling, typical case sampling, criterion sampling, and snowball sampling) are used to narrow the range of variation and focus on similarities. The latter are similar to the use of quantitative central tendency measures (e.g., mean, median, and mode). Moreover, certain strategies, like stratified purposeful sampling or opportunistic or emergent sampling, are designed to achieve both goals. As Patton (2002 , p. 240) explains, “the purpose of a stratified purposeful sample is to capture major variations rather than to identify a common core, although the latter may also emerge in the analysis. Each of the strata would constitute a fairly homogeneous sample.”

Challenges to use of purposeful sampling

Despite its wide use, there are numerous challenges in identifying and applying the appropriate purposeful sampling strategy in any study. For instance, the range of variation in a sample from which purposive sample is to be taken is often not really known at the outset of a study. To set as the goal the sampling of information-rich informants that cover the range of variation assumes one knows that range of variation. Consequently, an iterative approach of sampling and re-sampling to draw an appropriate sample is usually recommended to make certain the theoretical saturation occurs ( Miles & Huberman, 1994 ). However, that saturation may be determined a-priori on the basis of an existing theory or conceptual framework, or it may emerge from the data themselves, as in a grounded theory approach ( Glaser & Strauss, 1967 ). Second, there are a not insignificant number in the qualitative methods field who resist or refuse systematic sampling of any kind and reject the limiting nature of such realist, systematic, or positivist approaches. This includes critics of interventions and “bottom up” case studies and critiques. However, even those who equate purposeful sampling with systematic sampling must offer a rationale for selecting study participants that is linked with the aims of the investigation (i.e., why recruit these individuals for this particular study? What qualifies them to address the aims of the study?). While systematic sampling may be associated with a post-positivist tradition of qualitative data collection and analysis, such sampling is not inherently limited to such analyses and the need for such sampling is not inherently limited to post-positivist qualitative approaches ( Patton, 2002 ).

Purposeful Sampling in Implementation Research

Characteristics of implementation research.

In implementation research, quantitative and qualitative methods often play important roles, either simultaneously or sequentially, for the purpose of answering the same question through convergence of results from different sources, answering related questions in a complementary fashion, using one set of methods to expand or explain the results obtained from use of the other set of methods, using one set of methods to develop questionnaires or conceptual models that inform the use of the other set, and using one set of methods to identify the sample for analysis using the other set of methods ( Palinkas et al., 2011 ). A review of mixed method designs in implementation research conducted by Palinkas and colleagues (2011) revealed seven different sequential and simultaneous structural arrangements, five different functions of mixed methods, and three different ways of linking quantitative and qualitative data together. However, this review did not consider the sampling strategies involved in the types of quantitative and qualitative methods common to implementation research, nor did it consider the consequences of the sampling strategy selected for one method or set of methods on the choice of sampling strategy for the other method or set of methods. For instance, one of the most significant challenges to sampling in sequential mixed method designs lies in the limitations the initial method may place on sampling for the subsequent method. As Morse and Neihaus (2009) observe, when the initial method is qualitative, the sample selected may be too small and lack randomization necessary to fulfill the assumptions for a subsequent quantitative analysis. On the other hand, when the initial method is quantitative, the sample selected may be too large for each individual to be included in qualitative inquiry and lack purposeful selection to reduce the sample size to one more appropriate for qualitative research. The fact that potential participants were recruited and selected at random does not necessarily make them information rich.

A re-examination of the 22 studies and an additional 6 studies published since 2009 revealed that only 5 studies ( Aarons & Palinkas, 2007 ; Bachman et al., 2009 ; Palinkas et al., 2011 ; Palinkas et al., 2012 ; Slade et al., 2003) made a specific reference to purposeful sampling. An additional three studies ( Henke et al., 2008 ; Proctor et al., 2007 ; Swain et al., 2010 ) did not make explicit reference to purposeful sampling but did provide a rationale for sample selection. The remaining 20 studies provided no description of the sampling strategy used to identify participants for qualitative data collection and analysis; however, a rationale could be inferred based on a description of who were recruited and selected for participation. Of the 28 studies, 3 used more than one sampling strategy. Twenty-one of the 28 studies (75%) used some form of criterion sampling. In most instances, the criterion used is related to the individual’s role, either in the research project (i.e., trainer, team leader), or the agency (program director, clinical supervisor, clinician); in other words, criterion of inclusion in a certain category (criterion-i), in contrast to cases that are external to a specific criterion (criterion-e). For instance, in a series of studies based on the National Implementing Evidence-Based Practices Project, participants included semi-structured interviews with consultant trainers and program leaders at each study site ( Brunette et al., 2008 ; Marshall et al., 2008 ; Marty et al., 2007; Rapp et al., 2010 ; Woltmann et al., 2008 ). Six studies used some form of maximum variation sampling to ensure representativeness and diversity of organizations and individual practitioners. Two studies used intensity sampling to make contrasts. Aarons and Palinkas (2007) , for example, purposefully selected 15 child welfare case managers representing those having the most positive and those having the most negative views of SafeCare, an evidence-based prevention intervention, based on results of a web-based quantitative survey asking about the perceived value and usefulness of SafeCare. Kramer and Burns (2008) recruited and interviewed clinicians providing usual care and clinicians who dropped out of a study prior to consent to contrast with clinicians who provided the intervention under investigation. One study ( Hoagwood et al., 2007 ), used a typical case approach to identify participants for a qualitative assessment of the challenges faced in implementing a trauma-focused intervention for youth. One study ( Green & Aarons, 2011 ) used a combined snowball sampling/criterion-i strategy by asking recruited program managers to identify clinicians, administrative support staff, and consumers for project recruitment. County mental directors, agency directors, and program managers were recruited to represent the policy interests of implementation while clinicians, administrative support staff and consumers were recruited to represent the direct practice perspectives of EBP implementation.

Table 2 below provides a description of the use of different purposeful sampling strategies in mixed methods implementation studies. Criterion-i sampling was most frequently used in mixed methods implementation studies that employed a simultaneous design where the qualitative method was secondary to the quantitative method or studies that employed a simultaneous structure where the qualitative and quantitative methods were assigned equal priority. These mixed method designs were used to complement the depth of understanding afforded by the qualitative methods with the breadth of understanding afforded by the quantitative methods (n = 13), to explain or elaborate upon the findings of one set of methods (usually quantitative) with the findings from the other set of methods (n = 10), or to seek convergence through triangulation of results or quantifying qualitative data (n = 8). The process of mixing methods in the large majority (n = 18) of these studies involved embedding the qualitative study within the larger quantitative study. In one study (Goia & Dziadosz, 2008), criterion sampling was used in a simultaneous design where quantitative and qualitative data were merged together in a complementary fashion, and in two studies (Aarons et al., 2012; Zazelli et al., 2008 ), quantitative and qualitative data were connected together, one in sequential design for the purpose of developing a conceptual model ( Zazelli et al., 2008 ), and one in a simultaneous design for the purpose of complementing one another (Aarons et al., 2012). Three of the six studies that used maximum variation sampling used a simultaneous structure with quantitative methods taking priority over qualitative methods and a process of embedding the qualitative methods in a larger quantitative study ( Henke et al., 2008 ; Palinkas et al., 2010; Slade et al., 2008 ). Two of the six studies used maximum variation sampling in a sequential design ( Aarons et al., 2009 ; Zazelli et al., 2008 ) and one in a simultaneous design (Henke et al., 2010) for the purpose of development, and three used it in a simultaneous design for complementarity ( Bachman et al., 2009 ; Henke et al., 2008; Palinkas, Ell, Hansen, Cabassa, & Wells, 2011 ). The two studies relying upon intensity sampling used a simultaneous structure for the purpose of either convergence or expansion, and both studies involved a qualitative study embedded in a larger quantitative study ( Aarons & Palinkas, 2007 ; Kramer & Burns, 2008 ). The single typical case study involved a simultaneous design where the qualitative study was embedded in a larger quantitative study for the purpose of complementarity ( Hoagwood et al., 2007 ). The snowball/maximum variation study involved a sequential design where the qualitative study was merged into the quantitative data for the purpose of convergence and conceptual model development ( Green & Aarons, 2011 ). Although not used in any of the 28 implementation studies examined here, another common sequential sampling strategy is using criteria sampling of the larger quantitative sample to produce a second-stage qualitative sample in a manner similar to maximum variation sampling, except that the former narrows the range of variation while the latter expands the range.

Purposeful sampling strategies and mixed method designs in implementation research

Criterion-i sampling as a purposeful sampling strategy shares many characteristics with random probability sampling, despite having different aims and different procedures for identifying and selecting potential participants. In both instances, study participants are drawn from agencies, organizations or systems involved in the implementation process. Individuals are selected based on the assumption that they possess knowledge and experience with the phenomenon of interest (i.e., the implementation of an EBP) and thus will be able to provide information that is both detailed (depth) and generalizable (breadth). Participants for a qualitative study, usually service providers, consumers, agency directors, or state policy-makers, are drawn from the larger sample of participants in the quantitative study. They are selected from the larger sample because they meet the same criteria, in this case, playing a specific role in the organization and/or implementation process. To some extent, they are assumed to be “representative” of that role, although implementation studies rarely explain the rationale for selecting only some and not all of the available role representatives (i.e., recruiting 15 providers from an agency for semi-structured interviews out of an available sample of 25 providers). From the perspective of qualitative methodology, participants who meet or exceed a specific criterion or criteria possess intimate (or, at the very least, greater) knowledge of the phenomenon of interest by virtue of their experience, making them information-rich cases.

However, criterion sampling may not be the most appropriate strategy for implementation research because by attempting to capture both breadth and depth of understanding, it may actually be inadequate to the task of accomplishing either. Although qualitative methods are often contrasted with quantitative methods on the basis of depth versus breadth, they actually require elements of both in order to provide a comprehensive understanding of the phenomenon of interest. Ideally, the goal of achieving theoretical saturation by providing as much detail as possible involves selection of individuals or cases that can ensure all aspects of that phenomenon are included in the examination and that any one aspect is thoroughly examined. This goal, therefore, requires an approach that sequentially or simultaneously expands and narrows the field of view, respectively. By selecting only individuals who meet a specific criterion defined on the basis of their role in the implementation process or who have a specific experience (e.g., engaged only in an implementation defined as successful or only in one defined as unsuccessful), one may fail to capture the experiences or activities of other groups playing other roles in the process. For instance, a focus only on practitioners may fail to capture the insights, experiences, and activities of consumers, family members, agency directors, administrative staff, or state policy leaders in the implementation process, thus limiting the breadth of understanding of that process. On the other hand, selecting participants on the basis of whether they were a practitioner, consumer, director, staff, or any of the above, may fail to identify those with the greatest experience or most knowledgeable or most able to communicate what they know and/or have experienced, thus limiting the depth of understanding of the implementation process.

To address the potential limitations of criterion sampling, other purposeful sampling strategies should be considered and possibly adopted in implementation research ( Figure 1 ). For instance, strategies placing greater emphasis on breadth and variation such as maximum variation, extreme case, confirming and disconfirming case sampling are better suited for an examination of differences, while strategies placing greater emphasis on depth and similarity such as homogeneous, snowball, and typical case sampling are better suited for an examination of commonalities or similarities, even though both types of sampling strategies include a focus on both differences and similarities. Alternatives to criterion sampling may be more appropriate to the specific functions of mixed methods, however. For instance, using qualitative methods for the purpose of complementarity may require that a sampling strategy emphasize similarity if it is to achieve depth of understanding or explore and develop hypotheses that complement a quantitative probability sampling strategy achieving breadth of understanding and testing hypotheses ( Kemper et al., 2003 ). Similarly, mixed methods that address related questions for the purpose of expanding or explaining results or developing new measures or conceptual models may require a purposeful sampling strategy aiming for similarity that complements probability sampling aiming for variation or dispersion. A narrowly focused purposeful sampling strategy for qualitative analysis that “complements” a broader focused probability sample for quantitative analysis may help to achieve a balance between increasing inference quality/trustworthiness (internal validity) and generalizability/transferability (external validity). A single method that focuses only on a broad view may decrease internal validity at the expense of external validity ( Kemper et al., 2003 ). On the other hand, the aim of convergence (answering the same question with either method) may suggest use of a purposeful sampling strategy that aims for breadth that parallels the quantitative probability sampling strategy.

Purposeful and Random Sampling Strategies for Mixed Method Implementation Studies

• (1) Priority and sequencing of Qualitative (QUAL) and Quantitative (QUAN) can be reversed.
• (2) Refers to emphasis of sampling strategy.

Furthermore, the specific nature of implementation research suggests that a multistage purposeful sampling strategy be used. Three different multistage sampling strategies are illustrated in Figure 1 below. Several qualitative methodologists recommend sampling for variation (breadth) before sampling for commonalities (depth) ( Glaser, 1978 ; Bernard, 2002 ) (Multistage I). Also known as a “funnel approach”, this strategy is often recommended when conducting semi-structured interviews ( Spradley, 1979 ) or focus groups ( Morgan, 1997 ). This approach begins with a broad view of the topic and then proceeds to narrow down the conversation to very specific components of the topic. However, as noted earlier, the lack of a clear understanding of the nature of the range may require an iterative approach where each stage of data analysis helps to determine subsequent means of data collection and analysis ( Denzen, 1978 ; Patton, 2001) (Multistage II). Similarly, multistage purposeful sampling designs like opportunistic or emergent sampling, allow the option of adding to a sample to take advantage of unforeseen opportunities after data collection has been initiated (Patton, 2001, p. 240) (Multistage III). Multistage I models generally involve two stages, while a Multistage II model requires a minimum of 3 stages, alternating from sampling for variation to sampling for similarity. A Multistage III model begins with sampling for variation and ends with sampling for similarity, but may involve one or more intervening stages of sampling for variation or similarity as the need or opportunity arises.

Multistage purposeful sampling is also consistent with the use of hybrid designs to simultaneously examine intervention effectiveness and implementation. An extension of the concept of “practical clinical trials” ( Tunis, Stryer & Clancey, 2003 ), effectiveness-implementation hybrid designs provide benefits such as more rapid translational gains in clinical intervention uptake, more effective implementation strategies, and more useful information for researchers and decision makers ( Curran et al., 2012 ). Such designs may give equal priority to the testing of clinical treatments and implementation strategies (Hybrid Type 2) or give priority to the testing of treatment effectiveness (Hybrid Type 1) or implementation strategy (Hybrid Type 3). Curran and colleagues (2012) suggest that evaluation of the intervention’s effectiveness will require or involve use of quantitative measures while evaluation of the implementation process will require or involve use of mixed methods. When conducting a Hybrid Type 1 design (conducting a process evaluation of implementation in the context of a clinical effectiveness trial), the qualitative data could be used to inform the findings of the effectiveness trial. Thus, an effectiveness trial that finds substantial variation might purposefully select participants using a broader strategy like sampling for disconfirming cases to account for the variation. For instance, group randomized trials require knowledge of the contexts and circumstances similar and different across sites to account for inevitable site differences in interventions and assist local implementations of an intervention ( Bloom & Michalopoulos, 2013 ; Raudenbush & Liu, 2000 ). Alternatively, a narrow strategy may be used to account for the lack of variation. In either instance, the choice of a purposeful sampling strategy is determined by the outcomes of the quantitative analysis that is based on a probability sampling strategy. In Hybrid Type 2 and Type 3 designs where the implementation process is given equal or greater priority than the effectiveness trial, the purposeful sampling strategy must be first and foremost consistent with the aims of the implementation study, which may be to understand variation, central tendencies, or both. In all three instances, the sampling strategy employed for the implementation study may vary based on the priority assigned to that study relative to the effectiveness trial. For instance, purposeful sampling for a Hybrid Type 1 design may give higher priority to variation and comparison to understand the parameters of implementation processes or context as a contribution to an understanding of effectiveness outcomes (i.e., using qualitative data to expand upon or explain the results of the effectiveness trial), In effect, these process measures could be seen as modifiers of innovation/EBP outcome. In contrast, purposeful sampling for a Hybrid Type 3 design may give higher priority to similarity and depth to understand the core features of successful outcomes only.

Finally, multistage sampling strategies may be more consistent with innovations in experimental designs representing alternatives to the classic randomized controlled trial in community-based settings that have greater feasibility, acceptability, and external validity. While RCT designs provide the highest level of evidence, “in many clinical and community settings, and especially in studies with underserved populations and low resource settings, randomization may not be feasible or acceptable” ( Glasgow, et al., 2005 , p. 554). Randomized trials are also “relatively poor in assessing the benefit from complex public health or medical interventions that account for individual preferences for or against certain interventions, differential adherence or attrition, or varying dosage or tailoring of an intervention to individual needs” ( Brown et al., 2009 , p. 2). Several alternatives to the randomized design have been proposed, such as “interrupted time series,” “multiple baseline across settings” or “regression-discontinuity” designs. Optimal designs represent one such alternative to the classic RCT and are addressed in detail by Duan and colleagues (this issue) . Like purposeful sampling, optimal designs are intended to capture information-rich cases, usually identified as individuals most likely to benefit from the experimental intervention. The goal here is not to identify the typical or average patient, but patients who represent one end of the variation in an extreme case, intensity sampling, or criterion sampling strategy. Hence, a sampling strategy that begins by sampling for variation at the first stage and then sampling for homogeneity within a specific parameter of that variation (i.e., one end or the other of the distribution) at the second stage would seem the best approach for identifying an “optimal” sample for the clinical trial.

Another alternative to the classic RCT are the adaptive designs proposed by Brown and colleagues ( Brown et al, 2006 ; Brown et al., 2008 ; Brown et al., 2009 ). Adaptive designs are a sequence of trials that draw on the results of existing studies to determine the next stage of evaluation research. They use cumulative knowledge of current treatment successes or failures to change qualities of the ongoing trial. An adaptive intervention modifies what an individual subject (or community for a group-based trial) receives in response to his or her preferences or initial responses to an intervention. Consistent with multistage sampling in qualitative research, the design is somewhat iterative in nature in the sense that information gained from analysis of data collected at the first stage influences the nature of the data collected, and the way they are collected, at subsequent stages ( Denzen, 1978 ). Furthermore, many of these adaptive designs may benefit from a multistage purposeful sampling strategy at early phases of the clinical trial to identify the range of variation and core characteristics of study participants. This information can then be used for the purposes of identifying optimal dose of treatment, limiting sample size, randomizing participants into different enrollment procedures, determining who should be eligible for random assignment (as in the optimal design) to maximize treatment adherence and minimize dropout, or identifying incentives and motives that may be used to encourage participation in the trial itself.

Alternatives to the classic RCT design may also be desirable in studies that adopt a community-based participatory research framework ( Minkler & Wallerstein, 2003 ), considered to be an important tool on conducting implementation research ( Palinkas & Soydan, 2012 ). Such frameworks suggest that identification and recruitment of potential study participants will place greater emphasis on the priorities and “local knowledge” of community partners than on the need to sample for variation or uniformity. In this instance, the first stage of sampling may approximate the strategy of sampling politically important cases ( Patton, 2002 ) at the first stage, followed by other sampling strategies intended to maximize variations in stakeholder opinions or experience.

On the basis of this review, the following recommendations are offered for the use of purposeful sampling in mixed method implementation research. First, many mixed methods studies in health services research and implementation science do not clearly identify or provide a rationale for the sampling procedure for either quantitative or qualitative components of the study ( Wisdom et al., 2011 ), so a primary recommendation is for researchers to clearly describe their sampling strategies and provide the rationale for the strategy.

Second, use of a single stage strategy for purposeful sampling for qualitative portions of a mixed methods implementation study should adhere to the same general principles that govern all forms of sampling, qualitative or quantitative. Kemper and colleagues (2003) identify seven such principles: 1) the sampling strategy should stem logically from the conceptual framework as well as the research questions being addressed by the study; 2) the sample should be able to generate a thorough database on the type of phenomenon under study; 3) the sample should at least allow the possibility of drawing clear inferences and credible explanations from the data; 4) the sampling strategy must be ethical; 5) the sampling plan should be feasible; 6) the sampling plan should allow the researcher to transfer/generalize the conclusions of the study to other settings or populations; and 7) the sampling scheme should be as efficient as practical.

Third, the field of implementation research is at a stage itself where qualitative methods are intended primarily to explore the barriers and facilitators of EBP implementation and to develop new conceptual models of implementation process and outcomes. This is especially important in state implementation research, where fiscal necessities are driving policy reforms for which knowledge about EBP implementation barriers and facilitators are urgently needed. Thus a multistage strategy for purposeful sampling should begin first with a broader view with an emphasis on variation or dispersion and move to a narrow view with an emphasis on similarity or central tendencies. Such a strategy is necessary for the task of finding the optimal balance between internal and external validity.

Fourth, if we assume that probability sampling will be the preferred strategy for the quantitative components of most implementation research, the selection of a single or multistage purposeful sampling strategy should be based, in part, on how it relates to the probability sample, either for the purpose of answering the same question (in which case a strategy emphasizing variation and dispersion is preferred) or the for answering related questions (in which case, a strategy emphasizing similarity and central tendencies is preferred).

Fifth, it should be kept in mind that all sampling procedures, whether purposeful or probability, are designed to capture elements of both similarity and differences, of both centrality and dispersion, because both elements are essential to the task of generating new knowledge through the processes of comparison and contrast. Selecting a strategy that gives emphasis to one does not mean that it cannot be used for the other. Having said that, our analysis has assumed at least some degree of concordance between breadth of understanding associated with quantitative probability sampling and purposeful sampling strategies that emphasize variation on the one hand, and between the depth of understanding and purposeful sampling strategies that emphasize similarity on the other hand. While there may be some merit to that assumption, depth of understanding requires both an understanding of variation and common elements.

Finally, it should also be kept in mind that quantitative data can be generated from a purposeful sampling strategy and qualitative data can be generated from a probability sampling strategy. Each set of data is suited to a specific objective and each must adhere to a specific set of assumptions and requirements. Nevertheless, the promise of mixed methods, like the promise of implementation science, lies in its ability to move beyond the confines of existing methodological approaches and develop innovative solutions to important and complex problems. For states engaged in EBP implementation, the need for these solutions is urgent.

Multistage Purposeful Sampling Strategies

Acknowledgments

This study was funded through a grant from the National Institute of Mental Health (P30-MH090322: K. Hoagwood, PI).

The Transformed MG-Extended Exponential Distribution: Properties and Applications

• Research Article
• Open access
• Published: 05 June 2024

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• Addisalem Assaye Menberu   ORCID: orcid.org/0009-0001-9635-6081 1 &
• Ayele Taye Goshu   ORCID: orcid.org/0000-0002-5056-0132 1  

Our research paper introduces a newly developed probability distribution called the transformed MG-extended exponential (TMGEE) distribution. This distribution is derived from the exponential distribution using the modified Frechet approach, but it has a more adaptable hazard function and unique features that we have explained in detail. We conducted simulation studies using two methods: rejection sampling and inverse transform sampling, to produce summaries and show distributional properties. Moreover, we applied the TMGEE distribution to three real datasets from the health area to demonstrate its applicability. We used the maximum likelihood estimation technique to estimate the distribution’s parameters. Our results indicate that the TMGEE distribution provides a better fit for the three sets of data as compared to nine other commonly used probability distributions, including Weibull, exponential, and lognormal distributions.

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1 Introduction

The exponential probability distribution is a commonly used model for various real-world situations in fields such as engineering, business, economics, medicine, and biology. In modeling Poisson processes, the inter-arrival duration can typically be determined using the exponential distribution. However, in real-life situations, the assumption of constant hazard rates and constant event occurrence rates may not hold, making the model less adaptable. To address this issue, a new probability distribution called the Transformed MG-Extended Exponential Distribution has been developed. The distribution described here modifies the exponential distribution by raising its cumulative distribution to a power determined by an additional parameter. This parameter influences the shape of the distribution and allows for the representation of certain characteristics in real-world phenomena that were not accounted for by the original exponential distribution. The method involves a transformation that enhances the flexibility of the distribution.

New probability distributions have been introduced by adding new parameter(s) to existing distributions, making them more adaptable to various scenarios through transformation methods.

Several distributions have been studied in the literature that are derived from the exponential distribution. For example, Gupta and Kundu [ 1 ] introduced a new parameter to derive the exponentiated exponential distribution. Merovci [ 2 ] used a quadratic rank transformation map to obtain the transmuted exponentiated exponential distribution from the exponentiated exponential distribution. Similarly, Oguntunde and Adejumo [ 3 ] generalized the exponential distribution to a two-parameter model using the same transformation technique. Hussian [ 4 ] examined the transmuted exponentiated gamma distribution, which is a generalization of the exponentiated gamma distribution. Enahoro et al. [ 5 ] applied the performance rating of the transmuted exponential distribution to a real-life dataset. Nadarajah and Kotz [ 6 ] studied the beta exponential distribution, which is generated from the logit of the exponential distribution.

Cordeiro and Castro [ 7 ] studied a new family of generalized distributions called Kumaraswamy distributions, which includes Weibull, gamma, normal, Gumbel, and inverse Gaussian distributions. Mahdavi and Kundu [ 8 ] developed a new alpha power transformation method by adding a new parameter to the exponential distribution, producing new probability distributions. Recently, Khalil et al. [ 9 ] developed a new Modified Frechet distribution by adding a shape parameter to the Frechet distribution. In their work, [ 10 ] introduced a new Modified Frechet-Rayleigh distribution. They used the Rayleigh distribution as a base distribution and added a shape parameter to their derivation.

Alzaatreh et al. [ 11 ] combined the T-X method with the probability density function of the exponential distribution to introduce a novel technique for generating new probability distributions. Marshal and Olkin [ 12 ] proposed a new technique by adding a parameter to a family of distributions using the Weibull distribution as a base distribution. Tahir et al. [ 13 ] explored a novel Weibull G-family distribution and its characteristics by using the Weibull distribution as a base distribution. When analyzing lifetime or failure time data, the Weibull and gamma distributions with two and three parameters are commonly used. However, these models have a limitation in that their hazard functions are monotonic, i.e., they either increase or decrease with time. This may not be suitable for all scenarios. For example, as pointed out by [ 14 ], the hazard function of the Weibull distribution grows from zero to infinity as its shape parameter increases. This makes it unsuitable for modeling lifetime data in survival analysis, where some events have increasing risks over time and constant risks after a certain point.

The paper is organized into several sections for ease of navigation. The introduction can be found in Sect.  1 . The transformed MG-extended exponential distribution and its associated density, cumulative distribution function, survival, and hazard function graphs are detailed in Sects.  2 and  3 . The definitions of quintiles, mean, moments, and moment-generating functions are provided in Sects.  4 through  6 . Section  7 explains the definition of order statistics, while Sect.  8 covers parameter estimation and the characteristics of estimators. The paper also includes a simulation study in Sect.  9 and applications discussed in Sect.  10 . Finally, the conclusion can be found in Sect.  11 .

2 The Transformed MG-Extended Exponential Distribution

A study conducted by [ 9 ] introduced a modified Frechet approach for generating probability distributions. The study presented the cumulative distribution function and the density function, which are defined by Eqs.  1 and  2 , respectively. These functions were generated based on the base or induced distribution function, F ( x ), of a random variable X . The equation for the distribution function of the modefied Frechet is provided below.

The probability density function is subsequently specified to be

where \(\alpha\) is the distribution’s extra parameter.

The aforementioned approach is employed in this study to develop a new probability distribution based on the exponential probability distribution, named the Transformed MG-Extended Exponential (TMGEE) distribution. The TMGEE distribution is introduced to improve time-to-event data models for survival analysis and to increase the flexibility of the distribution for modeling real-world problems. Equations  3 and  4 define the cumulative distribution and density functions of the exponential distribution, respectively.

where \(\lambda\) is the distribution’s rate parameter, often indicating the rate at which events occur in a Poisson point process with a mean of \(1/\lambda\) and variance 1/ \(\lambda ^2\) .

The new proposed TMGEE cumulative distribution function is defined in Eq.  5 , for the shape parameter \(\alpha\) and scale parameter \(\lambda\) by substituting the cumulative distribution function defined in Eq.  3 to Eq.  1 .

By differentiating the cumulative distribution with respect to x , we determine the probability density function. Equation  6 thus defines the probability density function of the TMGEE distribution.

Proposition 1

The TMGEE distribution, \(f_{TMGEE}(x;\alpha ,\lambda )\) is a legitimate probability density function.

\(f_{TMGEE}(x;\alpha ,\lambda )\) is obviously non-negative for all \(x \ge 0\) , and

Substituting \(u = (1-e^{-\lambda x})^{\alpha}\) , and \(du = \alpha \lambda e^{-\lambda x} (1-e^{- \lambda x})^{\alpha -1}dx\) , we have 

Note that the TMGEE probability density function has a decreasing curve with a form akin to the exponential distribution’s probability density function for \(0<\alpha \le 1\) . The exponential distribution, however, is not a special case of it. The probability density function and cumulative distribution function of the TMGEE distribution are displayed in Fig.  1 for a few selected \(\alpha\) and \(\lambda\) values. \(\square\)

The density function and distribution function of the TMGEE distribution for some chosen values of \(\alpha\) and \(\lambda\)

3 Survival and Hazard Functions of the TMGEE Distribution

The survival and hazard functions of the TMGEE distribution are defined, respectively, by Eqs.   7 and  8 below.

For some chosen values of \(\alpha\) and \(\lambda\) , Fig.  2 displays the survival and hazard function of the TMGEE distribution. It’s worth noting that the hazard function has distinct characteristics depending on the selected parameters:

When \(0 < \alpha \le 1\) , the curve continuously decreases until settling to the value of \(\lambda\) .

When \(\alpha > 1\) and \(\alpha > \lambda\) , the curve rises until stabilizing at the value of \(\lambda\) .

Finally, when \(\alpha > 1\) and \(\lambda \ge \alpha\) , the curve reaches its highest value and then slightly decreases before settling to the value of \(\lambda\) .

The survival and hazard functions of the TMGEE distribution for some chosen values of \(\alpha\) and \(\lambda\)

Proposition 2

The function \(f_{TMGEE}(x;\alpha ,\lambda )\) is convex for \(0<\alpha \le 1\) and concave for \(\alpha >1.\)

Proposition 2 can be verified by differentiating the log of the TMGEE probability density function with respect to x . The results indicate that \(f_{TMGEE}(x;\alpha ,\lambda )\) is decreasing for \(0<\alpha \le 1\) and unimodal for \(\alpha >1\) . Equation  9 can be used to get the modal value.

4 Quantiles of the TMGEE Distribution

To obtain the pth quantile value for the TMGEE distribution, where X is a random variable such that X \(\sim\) TMGEE( \(\alpha\) , \(\lambda\) ), follow these steps:

Find the inverse function, \(F^{-1}(.)\) , based on the cumulative distribution function Eq.  5 .

Generate a random variable U , such that \(U \sim U(0,1)\) .

Then, use the formula \(X_{p} = F^{-1}(U)\) to obtain the pth quantile value \(X_{p}\) .

From Eq.  5 , \(F_{TMGEE}(x)=\frac{e^{-{(1-e^{-\lambda x})}^{\alpha }}-1}{e^{-1}-1}\) , and \(F(X_{p}) = u\) implies

Consequently, the quantile function of the TMGEE distribution is given by Eq.  10 as follows:

The median is defined in Eq.  11 and is found by substituting \(u=0.5\) in Eq.  10 .

Equations  12 and  13 give the following definitions for the first and third quartiles, \(Q_{1}\) and \(Q_{3}\) , respectively.

5 Mean and Moments of the TMGEE Distribution

5.1 the mean.

If X is a random variable such that X \(\sim\) TMGEE( \(\alpha\) , \(\lambda\) ), then mean, \(\mu = E(X)\) , can be derived as follows:

If \(y = (1-e^{-\lambda x})\) , and \(e^{-\lambda x} = 1-y\) , then dy \(=\) \(\lambda (1-y) dx\) . As a result, the mean can be written as

If \(z=y^{\alpha }\) , the mean can be evaluated as follows:

By using the series expansion of \(log(1-z^{\frac{1}{\alpha }}) = -\sum _{k=1}^{\infty }\frac{z^{\frac{k}{\alpha }}}{k}\) , for \(|{z^{\frac{1}{\alpha }}}| < 1\) ,

Therefore, using the infinite series representation of the incomplete gamma integral, Eq.  14 can be used to define the mean of the random variable following the TMGEE distribution as:

where the discussion of the upper incomplete gamma function \(\Gamma \left( .,.\right)\) can be found in the study by [ 15 ].

5.2 Moments of the TMGEE Distribution

5.2.1 moments.

The r -th moment, denoted by \(\mu _r\) , of the TMGEE random variable can be defined as follows:

For \(y = 1-e^{-\lambda x}, e^{-\lambda x} = 1-y\) and \(dx = \frac{dy}{\lambda(1-y)}\) , \(\mu _{r}^{/}\) can be written as

When \(z=y^\alpha , \mu _{r}^{/}\) can be expressed as

For \(|z^{\frac{1}{\alpha }}| <1\) , the rth moment can be simplified by substituting the series expansion

in Eq.  15 as:

The \(r^{th}\) moment of the TMGEE distribution can be found by expanding the summation part of the equation, which is located just above, and then integrating each term that results from the expansion. Equation  16 defines the final result. The incomplete gamma function is used in this derivation.

The first moment defined in Eq.  16 is equivalent to the mean of the random variable following the TMGEE distribution given in Eq.  14 .

5.2.2 Central Moments

The rth central moment of a random variable X is defined as follows.

For a random variable that follows the TMGEE distribution, Eq.  17 can be utilized to formulate the rth central moment.

The zero moment and central moments are equal to one, i.e., \(\mu _{0}^{/} = \mu _{0} =1\) and the first central moment, \(\mu _{1}\) is zero.

The variance of the random variable X , following the TMGEE distribution, can be determined using Eq.  17 .

where \(\mu\) is the mean defined in Eqs.  14 and   16 can be used to find the second moment \(\mu _{2}^{/}\) . The standard deviation can be determined using the square root of the variance defined in Eq.  18 .

5.3 Skewness and Kurtosis of the TMGEE Distribution

To determine the coefficients of skewness and kurtosis for the random variable \(X\) \(\sim\) \(TMGEE(\alpha ,\lambda )\) , we apply the methods of [ 16 , 17 ]. Equations  19 and  20 define the results.

where \(CS_{TMGEE}\) and \(CK_{TMGEE}\) represent the coefficients of skewness and kurtosis for the TMGEE distribution, respectively. The standard deviation is denoted by \(\sigma\) , and the third and fourth-order central moments are represented by \(\mu _{3}\) and \(\mu _{4}\) , respectively.

The inverse transform algorithm based on Eq.  10 with specified values of \(\alpha\) and \(\lambda\) is used to replicate eight random samples of size 100,000 from the Transformed MG-Extended Exponential probability distribution to better familiarize with it.

The summary statistics for each sample are shown in Table  1 including mean, standard deviation (Sd), skewness, and kurtosis. The table also shows how, depending on the value of \(\alpha\) , the new distribution alters the base distribution. For instance, when \(\lambda = 2\) , the base distribution’s mean and standard deviation are both \(\frac{1}{\lambda }\) and equal to 0.5. The mean and standard deviation of the simulated data, however, are approximately 0.22 and 0.35, respectively.

6 The Moment Generating Function of the TMGEE Distribution

The moment-generating function (MGF), \(M_{x}(t)\) , of the TMGEE random variable X is derived as follows:

By setting \(u = 1-e^{-\lambda x}, du = \lambda e^{-\lambda x} dx\) and \(e^{x} = (1-u)^{-\frac{1}{\lambda }}\) , the MGF can be written as:

For \(z = u^\alpha , u=z^{\frac{1}{\alpha }}\) and \(dz = \alpha u^{\alpha -1} du\) , we have

We use the following series expansion to formulate the moment-generating function:

For \(|z^{\frac{1}{\alpha }}| <1\) , \(M_{z}(t)\) can be expressed as:

Consequently, the MGF is defined by Eq.  21 .

7 The Order Statistics of the TMGEE Distribution

If \(X_{1}, X_{2},..., X_{n}\) are samples drawn at random from the TMGEE distribution and \(X_{(1)}, X_{(2)},..., X_{(n)}\) are the order statistics, then Eq.  22 defines the probability density function of the ith order statistics \(X_{i:n}\) as follows:

By substituting the density function f ( x ) and the cumulative distribution function F ( x ) of the random variable X in Eq.  22 , the ith order statistics, \(f_{i:n}(x)\) can be written as

Consequently, Eq.  23 defines the ith order statistics of the TMGEE distribution.

Equations  24 and  25 define the first order statistics, \(f_{1:n}(x)\) , and nth order statistics, \(f_{n:n} (x)\) .

8 Parameter Estimation: The Maximum Likelihood Method

Let the respective realization be \(\varvec{x} = (x_{1}, x_{2},x_{3},...,x_{n})\) for \(X_{1}, X_{2}, X_{3},..., X_{n}\) independent and identically distributed ( iid ) random samples of size n from the TMGEE distribution. We estimate the distribution’s parameters using the maximum likelihood estimation approach.

The probability density function \(f_{TMGEE}(\varvec{x};\alpha ,\lambda )\) can be used to express the likelihood function, where \(\alpha\) and \(\lambda\) are unknown parameters.

Equation  26 can be used to define the likelihood function pf the distribution’s parameters.

The log-likelihood function, which is defined by Eq.  27 , is obtained by utilizing Eq.  26 .

We obtain the maximum likelihood estimates (MLEs) by differentiating the log-likelihood function with respect to the parameters \(\alpha\) and \(\lambda\) . Equating the results to zero gives the score function in Eqs.  28 and  29 .

Equations  30 and  31 give the second derivatives of the likelihood function specified in Eq.  27 with respect to \(\alpha\) and \(\lambda\) , respectively.

Since the score function defined in Eqs.  28 and  29 cannot be solved in closed form, numerical methods can be employed to estimate the parameters. Equation  32 defines the observed Fisher information matrix of the random variable X .

The estimated variances of the parameters, which are specified on the diagonal of the variance-covariance matrix defined in Eq.  33 ), can be derived by inverting the Fisher information matrix defined in Eq. ( 32 ). By taking the square roots of the estimated variances, the estimated standard errors \(\hat{\sigma }(\hat{\alpha })\) and \(\hat{\sigma }(\hat{\lambda })\) are produced. Meanwhile, the off-diagonal elements represent the estimated covariances between the parameter estimates.

8.1 Asymptotic Distribution of MLEs

The MLEs are intrinsically random variables that depend on the sample size. [ 16 ] and [ 18 ] addressed the asymptotic distribution of MLEs and the properties of the estimators. The ML estimates consistently converge in probability to the true values. Owing to the general asymptotic theory of the MLEs, the sampling distribution of \((\hat{\alpha }-\alpha )/ \sqrt{\hat{\sigma }^2(\hat{\alpha })}\) and \((\hat{\lambda }-\lambda )/ \sqrt{\hat{\sigma }^2(\hat{\lambda })}\) can be approximately represented by the standard normal distribution. The asymptotic confidence intervals of the parameter estimates \(\hat{\alpha }\) and \(\hat{\lambda }\) can be determined using Eq.  34 .

where \(\sqrt{\hat{\sigma }^2(\hat{\alpha })}\) and \(\sqrt{\hat{\sigma }^2(\hat{\lambda })}\) , respectively, are the estimated standard errors of the estimates \(\hat{\alpha }\) and \(\hat{\lambda }\) . \(z_{t/2}\) is the 100 t upper percentage point of the standard normal distribution.

9 Simulation Studies

9.1 rejection sampling method.

In our first simulation study, we used the rejection sampling method, also known as acceptance/rejection sampling, to generate samples from a target distribution with a density function of f ( x ). This method allows us to simulate a random variable, X , without directly sampling from the distribution. We achieve this by using two independent random variables, U and X , where U is a uniform random variable between 0 and 1, and X is a random variable with a proposal density of g ( x ). The goal of this method is to ensure that f ( x ) is less than or equal to c times g ( x ) for every value of x , where g ( x ) is an arbitrary proposal density and c is a finite constant.

According to [ 19 ], the rejection sampling technique is versatile enough to derive values from g ( x ) even without complete information about the specification of f ( x ). Here is a step-by-step guide of the rejection sampling algorithm on how to generate a random variable X that follows the probability density function f ( x ).

Generate a candidate X randomly from a distribution g ( x ).

Compute the acceptance probability \(\alpha = \frac{1}{c} \cdot \frac{f(X)}{g(X)}\) , where c is a constant such that \(cg(x) \ge f(x)\) for all x .

Generate a random number u from a uniform distribution on the interval  (0, 1).

If \(u < \alpha\) , accept X as a sample from f ( x ) and return X .

If \(u \ge \alpha\) , reject X and go back to step 1.

By following steps 1–4 repeatedly, we were able to obtain a set of samples that adhered to the probability density function f ( x ).

We used the rejection sampling method and inverse transform of the Monte Carlo methods to generate a random sample of observations for a random variable X , which followed the TMGEE distribution.

To illustrate this, we simulated random samples for two probability density functions: \(f_{TMGEE}(1.5,1.5)\) and \(f_{TMGEE}(5,3)\) , using both the aforementioned methods. For the first distribution, we used the exponential distribution \(f_{exp}(0.8)\) as a proposal density, while for the second distribution, we employed the Weibull distribution \(f_{weib}(1.5,0.8)\) . In each simulation, we randomly selected 10,000 samples. The R codes can be found in the appendix section of  A and  B .

To sample from \(f_{TMGEE}(1.5,1.5)\) , we used the rejection sampling method and obtained 10,000 observations from the \(f_{exp}(0.8)\) distribution. Out of these samples, 7182 were accepted as draws from \(f_{TMGEE}(1.5,1.5)\) , which resulted in an acceptance rate of 71.82%. The histogram in Fig.  3 a shows the accepted draws from the rejection sampling method, while the density plot represents the target distribution.

Similarly, we used the rejection sampling method to obtain 10,000 observations from the \(f_{weib}(1.5,0.8)\) distribution. Out of the 10,000 samples, 6679 were accepted as draws from \(f_{TMGEE}(5,3)\) , resulting in an acceptance rate of 66.79%. The accepted draws are displayed in the histogram in Fig.  3 b, and the density plot in the figure represents the target distribution.

It has been confirmed that the proposal distribution used in each of the cases has a remarkable agreement with the results obtained using the rejection sampling method.

a Histogram created with accepted draws of rejection sampling using exponential proposal density and density plot of TMGEE (1.5, 1.5) distribution using inverse transform sampling. b Histogram created with accepted draws of rejection sampling using Weibull proposal density and density plot of TMGEE (5, 3) distribution using inverse transform sampling

9.2 Inverse Transform-Based Sampling Method

To examine the MLEs of \(\hat{\theta } \in (\hat{\alpha }, \hat{\lambda })\) for \(\theta\) , we conducted a Monte Carlo simulation study utilizing the quantile function of the TMGEE distribution as defined by Eq.  10 . We examined the precision of the estimators using bias and mean squared error (MSE) and observed their characteristics. We generated samples of sizes 50, 100, 150, 300, and 1000 using the quantile function of the TMGEE distribution and performed simulations with R = 1,000. Following that, we computed the bias, MSE, variance, and MLEs. To simulate and estimate, we followed these steps:

Define the likelihood function of the model parameters.

Obtain the MLEs by minimizing the negative log-likelihood function using the Optim approach of [ 20 ].

Repeat the estimation for the R simulations.

Compute the bias, MSE, and variance of the estimates.

We assume that the distribution’s \(\alpha\) values are 0.5, 2, 3, 5, and 8 and its \(\lambda\) values are 0.5, 3, 5, and 10 to run the simulations. There are 20 different parameter combinations for each of the generated sample sizes. Bias and MSE are calculated using Eqs.  35 and  36 , respectively.

where \(\theta \in (\alpha ,\lambda )\) and \(MSE(\hat{\theta }) = (Bias(\hat{\theta }))^2+Var(\hat{\theta })\)

The Monte Carlo simulation was conducted for each parameter setting to estimate the random variable’s bias, MSE, and variance. The results of the parameter estimates are presented in Tables  2 , 3 , 4 ,  5 , and  6 .

As part of our analysis, we ran R = 10,000 simulations and generated samples of size n = 1,000 for each simulation. This helped us evaluate the asymptotic distribution of the TMGEE distribution’s parameter estimates. To do this, we utilized the inverse transform method with the parameter values of \(\alpha =1.5\) , \(\lambda =2\) , and \(\alpha =5\) , \(\lambda =3\) . We’ve presented the outcomes of the respective estimates in Figs.  4 and  5 .

Based on the simulation studies carried out, using the varying sample sizes, samll (n = 50) to large (n = 1000), we can draw the following conclusions: As we increase the sample size, the MSE, the variance of the parameter estimates decreases, and the estimates of the parameters converge towards their true values. This means that the MLEs of the TMGEE parameters are unbiased and consistent. Upon examining the histograms in Figs.  4 and  5 and considering the large sample properties of the MLEs, it has been established that the asymptotic distribution of the MLEs of the TMGEE distribution parameters is normal. Furthermore, the simulation study has revealed that there is a positive correlation between the MLEs of the TMGEE distribution’s parameters for some parameter settings (refer to Fig.  6 ).

The results of 10,000 simulations for sample sizes of 1000 in each, with corresponding histogram and density plots for \(\alpha = 1.5\) in a and for \(\lambda = 2\) in b

The results of 10.000 simulations for sample sizes of 1.000 in each, with corresponding histogram and density plots for \(\alpha = 5\) in a and for \(\lambda = 3\) in b

The results of estimated alpha, estimated lambda, and Pearson correlations for different parameter settings from 1,000 simulations for sample sizes of 1000 each

10 Applications

In this section, we compare the TMGEE distribution with various probability distributions, including the exponential (Exp) distribution, the Weibull (WE) distribution, the lognormal (LN) distribution, and the alpha power exponential (APE) and alpha power Weibull (APW) distributions studied by [ 8 ]. We also compare the exponentiated Weibull distribution (EW) introduced by [ 21 ], the exponentiated Kumaraswamy G family distribution where the baseline distribution is exponential (EKG-E), as studied by [ 11 ], and the Kumaraswamy G family distributions (KG-W and KG-G) researched by [ 7 ], where the baseline distributions are Weibull and gamma, respectively. We define the probability density functions and cumulative distribution functions of some of these distributions below.

Exponentiated Weibull distribution(EW)

\(f(x;\beta , c,\alpha ) = \alpha \beta c (\beta x)^{c-1} (1-e^{-(\beta x)^c})^{\alpha -1}e^{-(\beta x)^c}\)

\(F(x;\beta , c,\alpha ) = (1-e^{-(\beta x)^c})^{\alpha }, x>0, c>0, \alpha >0\) and \(\beta >0\)

Alpha power exponential distribution(APE)

\(f(x;\alpha ,\lambda ) = \frac{log(\alpha )\lambda e^{-\lambda x} \alpha ^{1-e^{-\lambda x}}}{\alpha -1}\)

\(F(x,\alpha ,\lambda ) = \frac{\alpha ^{1-e^{-\lambda x}}-1}{\alpha -1}, x>0, \alpha >0, \alpha \ne 1\) and \(\lambda >0\)

The alpha power Weibull distribution (APW)

\(f(x;\alpha ,\lambda , \beta ) = \frac{log(\alpha )\lambda \beta e^{-\lambda x^\beta } x^{\beta -1} \alpha ^{1-e^{-\lambda x^\beta }}}{\alpha -1}\)

\(F(x;\alpha ,\lambda ,\beta ) = \frac{1-\alpha ^{1-e^{-\lambda x^\beta }}}{1-\alpha },x>0, \alpha >0, \alpha \ne 1\) and \(\lambda>0,\beta >0\)

Exponentiated Kumaraswamy G family distributions (EKG-E)

\(f(x;a,b,c) = abcg(x)G(x)^{\alpha -1} (1-G(x)^a)^{b-1}(1-(1-G(x)^a)^b)^{c-1}\)

\(F(x;a,b,c) = (1-(1-G(x)^a)^b)^c, x>0, a>0, b>0,c>0\)

where g ( x ) and G ( x ) are, respectively, the pdf and CDF of the exponential distribution.

Kumaraswamy G family distributions (KG-W and KG-G)

\(f(x;a,b) = abg(x)G(x)^{\alpha -1} (1-G(x)^a)^{b-1}(1-G(x)^a)^{b-1}\)

\(F(x;a,b) = 1-(1-G(x)^a)^b, x>0, a>0, b>0,c>0\)

where the Weibull and gamma distributions’ respective pdf and CDF are denoted as g ( x ) and G ( x ).

Three real datasets were used to illustrate the fitting of distributions. One of the datasets used in this study was obtained from [ 22 ]. The dataset consists of the frailty term assessed in a study of the recurrence time of infections in 38 patients receiving kidney dialysis. The data set includes 76 observations. [ 23 ] explained that each person has a distinct level of frailty or an observed heterogeneity term, which determines their risk of death in proportional hazard models. Another dataset was obtained from the systematic review and meta-analysis conducted by [ 24 ]. This dataset shows overall mortality rates among people who injected drugs. The third dataset consists of 128 individuals with bladder cancer, and it shows the duration of each patient’s remission in months. This dataset was used by [ 25 ] to compare the fits of the five-parameter beta-exponentiated Pareto distribution. Tables  7 ,  8 , and  9 display these datasets.

The empirical and density plots for the data sets that are shown in Tables  7 ,  8 , and  9

MLEs and model fitting statistics were obtained by fitting the models with numerical techniques adapted from [ 26 ] and [ 17 ]. We provided the MLEs and standard errors of the model parameters for three datasets in Tables  10 ,  11 , and  12 . To compare the fit of the models, we used various information criteria, such as Akaike information criteria (AIC), corrected Akaike information criteria (AICc), Bayesian information criteria (BIC), Hannan-Quinn information criterion (HQIC), Kolmogorov-Smirnov test statistic (K-S), and its p-value. These information criteria, as explained in [ 27 , 28 ], are useful in comparing model fits in various applications. However, AIC can be vulnerable to overfitting when sample sizes are small. Therefore, to correct this, we added a bias-correction term of second order to AIC to obtain AICc, which performs better in small samples. The bias-correction term raises the penalty on the number of parameters compared to AIC. As the sample size increases, the term asymptotically approaches zero, and AICc moves closer to AIC. In large samples, the HQIC penalizes complex models less than the BIC. However, individual AIC and AICc values are not interpretable [ 27 ], so they need to be rescaled to \(\Delta _i\) , which is defined as follows.

Let \(\Delta _i\) denote the difference between \(AIC_i\) and \(AIC_{min}\) , where \(AIC_{min}\) is the minimum value of AIC among all models. According to [ 27 ], the best model has \(\Delta _i=0\) . A smaller value of the information criteria indicates a better fit, regardless of the specific criteria used. The equations defined from  37 to  40 for k and \(l(\hat{\theta })\) can be used to compute some of the model fitting statistics, where k is the number of parameters to be estimated and \(l(\hat{\theta })\) is a likelihood.

The values of the model-fitting statistics are summarized in Tables  13 ,  14 , and  15 . Goodness-of-fit plots were produced for three datasets (shown in Tables  7 ,  8 , and  9 ) using specific distributions (TMGEE, exponential, Weibull, and lognormal) with methods proposed by [ 26 ]. The results of these plots can be seen in Figs.  7 ,  8 , and  9 . Our analysis shows that the newly proposed TMGEE probability distribution provides a better fit than the previously considered distributions when applied to the three datasets.

Q–Q plots (TMGEE, exponential, Weibull, and lognormal distributions)

P–P plots (TMGEE, exponential, weibull, and Lognormal distributions)

11 Conclusions

In this research, a new probability distribution called the Transformed MG-Extended Exponential Distribution (TMGEE) has been developed from the exponential probability distribution. The distribution’s detailed features have been explored and derived, and the approach of maximum likelihood has been used to estimate the parameters. We obtained unbiased and consistent estimates. Two simulation experiments have been conducted using the rejection sampling and inverse-transform sampling techniques. The usefulness of the new distribution has been evaluated using three different real datasets.

To evaluate the maximum likelihood estimates, we have used different statistical tools such as the Kolmogorov-Smirnov test statistic, the Hannan-Quinn information criteria, the corrected Akaike information criteria, the Bayesian information criteria, and the Akaike information criteria. The newly introduced TMGEE distribution has been found to fit the three sets of data far better than some of the most commonly used probability distributions, including the Weibull, exponential, and lognormal distributions.

We recommend further studies on this probability distribution in statistical theory, such as the Bayesian parameter estimation method and its application to other datasets involving lifetime and time-to-event processes.

Data availability

We have used secondary data. All data sets are available in the citations given.

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Addisalem Assaye Menberu & Ayele Taye Goshu

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The idea and conceptualization of this study was conceived by AAM and ATG. AAM derived the model and its properties. ATG verified the computations. AAM drafted the paper. Both AAM and ATG edited the draft and the final manuscript. ATG supervised the whole study.

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Appendix a: ar sampling with exponential distribution as proposal density.

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Menberu, A.A., Goshu, A.T. The Transformed MG-Extended Exponential Distribution: Properties and Applications. J Stat Theory Appl (2024). https://doi.org/10.1007/s44199-024-00078-8

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1. Introduction

3. results and discussion, 4. conclusions and outlook, 5. data availability, supporting information.

research papers \(\def\hfill{\hskip 5em}\def\hfil{\hskip 3em}\def\eqno#1{\hfil {#1}}\)

What shapes template-matching performance in cryogenic electron tomography in situ ?

a European Molecular Biology Laboratory Hamburg, Notkestrasse 85, 22607 Hamburg, Germany, b Centre for Structural Systems Biology (CSSB), Notkestrasse 85, 22607 Hamburg, Germany, and c Structural and Computational Biology Unit, European Molecular Biology Laboratory, Meyerhofstrasse 1, 69117 Heidelberg, Germany * Correspondence e-mail: [email protected]

The detection of specific biological macromolecules in cryogenic electron tomography data is frequently approached by applying cross-correlation-based 3D template matching. To reduce computational cost and noise, high binning is used to aggregate voxels before template matching. This remains a prevalent practice in both practical applications and methods development. Here, the relation between template size, shape and angular sampling is systematically evaluated to identify ribosomes in a ground-truth annotated data set. It is shown that at the commonly used binning, a detailed subtomogram average, a sphere and a heart emoji result in near-identical performance. These findings indicate that with current template-matching practices macromolecules can only be detected with high precision if their shape and size are sufficiently different from the background. Using theoretical considerations, the experimental results are rationalized and it is discussed why primarily low-frequency information remains at high binning and that template matching fails to be accurate because similarly shaped and sized macromolecules have similar low-frequency spectra. These challenges are discussed and potential enhancements for future template-matching methodologies are proposed.

Keywords: cryo-electron tomography ; template matching ; computer vision ; particle picking .

3.1. Shape and size are the major determinants for template-matching precision

To further confirm this finding, we also ran control experiments using an Influenza A virus hemagglutinin (HA) template, which has a markedly different shape to a ribosome ( Supplementary Fig. S1 a ). HA is a trimer with a total molecular weight of 180 kDa that has an approximately cylindrical shape with a length of ∼17 nm and a width of ∼6 nm. We scaled the radius analogous to the previous structures and calculated the precision with respect to the ground-truth data. The precision was near 0% for sizes up to 10 voxels, and only for larger radii did the precision increase as the structure further approaches the shape and size of the ribosome ( Supplementary Fig. S1 b ). This further underscores the observation that at this level of binning, template matching is less dependent on the structure and overall focuses on shape and size.

3.2. Angular sampling does not improve precision

3.3. ribosome and fatty-acid synthase are not discernible with conventional template matching, 3.4. theory.

We now aim to rationalize our empirical observations by examining the analytical form of the Fourier transforms of several geometric shapes and discussing them in the context of cross-correlation calculation. Based on our assessment, we conclude that template matching on the typically used 4–8 times binning is primarily driven by shape and size and list the associated implications.

where k is the wavenumber in the Fourier domain and t is the position vector in the real domain. From this, it becomes apparent that a shift in the spatial domain corresponds to a frequency-dependent phase shift in the Fourier domain. Since |exp(− i 2 π kn )| = 1, the magnitude of the Fourier transform is independent of the phase shift. The cross-correlation in the real domain can be obtained by inverse Fourier transform of the element-wise product of amplitudes A and the sum of phases ψ ,

A sphere of radius R centered around the origin can be defined in real space as

where j 1 ( x ) is the spherical Bessel function of first kind and order defined as

A one-dimensional rectangle, i.e. a box function, can be defined in real space as

where w is the width of the box function. The Fourier transform of the one-dimensional box function g ( r ) is

The definition of the box-function Fourier transform can be used to synthesize the Fourier transform of three-dimensional rectangles with width a , b and c as

where k x , k y and k z are the wavenumbers corresponding to the spatial dimensions x , y and z , respectively.

The cylinder is essentially a combination of a circle and a box function and can be defined as

where R is the radius of the circle and h is the width of the box function. We can make use of the cylindrical symmetry and the separability of the Fourier transform to derive the closed form of the cylinder Fourier transform as follows:

These theoretical considerations have three important implications for template matching. (i) Since template matching at this binning is primarily about matching object size, macromolecules of similar size to the macromolecule of interest will be identified as false positives. (ii) Small macromolecules would mainly be represented through high frequencies, which overlap with noise in the data. This relation makes template-matching small macromolecules at this binning near-impossible. (iii) More accurate templates are unlikely to improve the template-matching performance because high-resolution information cannot be accurately represented at the typically used 4–8 times binning.

Lastly, we suggest broadening benchmark entities beyond large and highly abundant globular structures such as the ribosome when evaluating new template-matching algorithms. In particular, providing test sets of particles that have similar low-frequency information is necessary to determine the discriminatory power of novel template-matching methods, score functions or processing approaches. Novel methods should also be validated against the simple geometric shapes considered here to ensure that they perform better and justify the higher computational cost.

The tomogram and ground-truth picks are freely available from EMPIAR (EMPIAR-10988, TS_037). The scaled maps for the ribosome (EMDB entry EMD-3228 ), sphere, emoji and HA at various radii, the resulting picks, the raw data for plots and the scripts used are freely available on GitHub at https://github.com/maurerv/ribosomeTemplateMatching .

Supplementary Figure S1. DOI: https://doi.org/10.1107/S2059798324004303/vo5016sup1.pdf

Acknowledgements

We thank Lukas Grunwald [Max Planck Institute for the Structure and Dynamics of Matter, Center for Free-Electron Laser Science (CFEL), Hamburg, Germany] for critical reading and editing of the theory section and Julia Mahamid (EMBL Heidelberg, Germany) for critical reading of, and feedback on, the manuscript. We thank the EMBL IT and HPC resources for providing essential computational infrastructure. Author contributions were as follows. VM and MS conceived the study, performed the research, wrote the initial draft and edited the manuscript. JK provided feedback and edited the manuscript. The authors declare no conflicts of interest. Open access funding enabled and organized by Projekt DEAL.

Funding information

VM and JK acknowledge funding from the CSSB flagship project Plasmofraction. MS acknowledges support from a research fellowship from the EMBL Interdisciplinary Postdoc (EIPOD) Programme under Marie Curie Cofund Actions MSCA-COFUND-FP (grant agreement No. 847543).

This is an open-access article distributed under the terms of the Creative Commons Attribution (CC-BY) Licence , which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are cited.