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Research Methodology – Types, Examples and writing Guide

Table of Contents

Research Methodology

Research Methodology

Definition:

Research Methodology refers to the systematic and scientific approach used to conduct research, investigate problems, and gather data and information for a specific purpose. It involves the techniques and procedures used to identify, collect , analyze , and interpret data to answer research questions or solve research problems . Moreover, They are philosophical and theoretical frameworks that guide the research process.

Structure of Research Methodology

Research methodology formats can vary depending on the specific requirements of the research project, but the following is a basic example of a structure for a research methodology section:

I. Introduction

  • Provide an overview of the research problem and the need for a research methodology section
  • Outline the main research questions and objectives

II. Research Design

  • Explain the research design chosen and why it is appropriate for the research question(s) and objectives
  • Discuss any alternative research designs considered and why they were not chosen
  • Describe the research setting and participants (if applicable)

III. Data Collection Methods

  • Describe the methods used to collect data (e.g., surveys, interviews, observations)
  • Explain how the data collection methods were chosen and why they are appropriate for the research question(s) and objectives
  • Detail any procedures or instruments used for data collection

IV. Data Analysis Methods

  • Describe the methods used to analyze the data (e.g., statistical analysis, content analysis )
  • Explain how the data analysis methods were chosen and why they are appropriate for the research question(s) and objectives
  • Detail any procedures or software used for data analysis

V. Ethical Considerations

  • Discuss any ethical issues that may arise from the research and how they were addressed
  • Explain how informed consent was obtained (if applicable)
  • Detail any measures taken to ensure confidentiality and anonymity

VI. Limitations

  • Identify any potential limitations of the research methodology and how they may impact the results and conclusions

VII. Conclusion

  • Summarize the key aspects of the research methodology section
  • Explain how the research methodology addresses the research question(s) and objectives

Research Methodology Types

Types of Research Methodology are as follows:

Quantitative Research Methodology

This is a research methodology that involves the collection and analysis of numerical data using statistical methods. This type of research is often used to study cause-and-effect relationships and to make predictions.

Qualitative Research Methodology

This is a research methodology that involves the collection and analysis of non-numerical data such as words, images, and observations. This type of research is often used to explore complex phenomena, to gain an in-depth understanding of a particular topic, and to generate hypotheses.

Mixed-Methods Research Methodology

This is a research methodology that combines elements of both quantitative and qualitative research. This approach can be particularly useful for studies that aim to explore complex phenomena and to provide a more comprehensive understanding of a particular topic.

Case Study Research Methodology

This is a research methodology that involves in-depth examination of a single case or a small number of cases. Case studies are often used in psychology, sociology, and anthropology to gain a detailed understanding of a particular individual or group.

Action Research Methodology

This is a research methodology that involves a collaborative process between researchers and practitioners to identify and solve real-world problems. Action research is often used in education, healthcare, and social work.

Experimental Research Methodology

This is a research methodology that involves the manipulation of one or more independent variables to observe their effects on a dependent variable. Experimental research is often used to study cause-and-effect relationships and to make predictions.

Survey Research Methodology

This is a research methodology that involves the collection of data from a sample of individuals using questionnaires or interviews. Survey research is often used to study attitudes, opinions, and behaviors.

Grounded Theory Research Methodology

This is a research methodology that involves the development of theories based on the data collected during the research process. Grounded theory is often used in sociology and anthropology to generate theories about social phenomena.

Research Methodology Example

An Example of Research Methodology could be the following:

Research Methodology for Investigating the Effectiveness of Cognitive Behavioral Therapy in Reducing Symptoms of Depression in Adults

Introduction:

The aim of this research is to investigate the effectiveness of cognitive-behavioral therapy (CBT) in reducing symptoms of depression in adults. To achieve this objective, a randomized controlled trial (RCT) will be conducted using a mixed-methods approach.

Research Design:

The study will follow a pre-test and post-test design with two groups: an experimental group receiving CBT and a control group receiving no intervention. The study will also include a qualitative component, in which semi-structured interviews will be conducted with a subset of participants to explore their experiences of receiving CBT.

Participants:

Participants will be recruited from community mental health clinics in the local area. The sample will consist of 100 adults aged 18-65 years old who meet the diagnostic criteria for major depressive disorder. Participants will be randomly assigned to either the experimental group or the control group.

Intervention :

The experimental group will receive 12 weekly sessions of CBT, each lasting 60 minutes. The intervention will be delivered by licensed mental health professionals who have been trained in CBT. The control group will receive no intervention during the study period.

Data Collection:

Quantitative data will be collected through the use of standardized measures such as the Beck Depression Inventory-II (BDI-II) and the Generalized Anxiety Disorder-7 (GAD-7). Data will be collected at baseline, immediately after the intervention, and at a 3-month follow-up. Qualitative data will be collected through semi-structured interviews with a subset of participants from the experimental group. The interviews will be conducted at the end of the intervention period, and will explore participants’ experiences of receiving CBT.

Data Analysis:

Quantitative data will be analyzed using descriptive statistics, t-tests, and mixed-model analyses of variance (ANOVA) to assess the effectiveness of the intervention. Qualitative data will be analyzed using thematic analysis to identify common themes and patterns in participants’ experiences of receiving CBT.

Ethical Considerations:

This study will comply with ethical guidelines for research involving human subjects. Participants will provide informed consent before participating in the study, and their privacy and confidentiality will be protected throughout the study. Any adverse events or reactions will be reported and managed appropriately.

Data Management:

All data collected will be kept confidential and stored securely using password-protected databases. Identifying information will be removed from qualitative data transcripts to ensure participants’ anonymity.

Limitations:

One potential limitation of this study is that it only focuses on one type of psychotherapy, CBT, and may not generalize to other types of therapy or interventions. Another limitation is that the study will only include participants from community mental health clinics, which may not be representative of the general population.

Conclusion:

This research aims to investigate the effectiveness of CBT in reducing symptoms of depression in adults. By using a randomized controlled trial and a mixed-methods approach, the study will provide valuable insights into the mechanisms underlying the relationship between CBT and depression. The results of this study will have important implications for the development of effective treatments for depression in clinical settings.

How to Write Research Methodology

Writing a research methodology involves explaining the methods and techniques you used to conduct research, collect data, and analyze results. It’s an essential section of any research paper or thesis, as it helps readers understand the validity and reliability of your findings. Here are the steps to write a research methodology:

  • Start by explaining your research question: Begin the methodology section by restating your research question and explaining why it’s important. This helps readers understand the purpose of your research and the rationale behind your methods.
  • Describe your research design: Explain the overall approach you used to conduct research. This could be a qualitative or quantitative research design, experimental or non-experimental, case study or survey, etc. Discuss the advantages and limitations of the chosen design.
  • Discuss your sample: Describe the participants or subjects you included in your study. Include details such as their demographics, sampling method, sample size, and any exclusion criteria used.
  • Describe your data collection methods : Explain how you collected data from your participants. This could include surveys, interviews, observations, questionnaires, or experiments. Include details on how you obtained informed consent, how you administered the tools, and how you minimized the risk of bias.
  • Explain your data analysis techniques: Describe the methods you used to analyze the data you collected. This could include statistical analysis, content analysis, thematic analysis, or discourse analysis. Explain how you dealt with missing data, outliers, and any other issues that arose during the analysis.
  • Discuss the validity and reliability of your research : Explain how you ensured the validity and reliability of your study. This could include measures such as triangulation, member checking, peer review, or inter-coder reliability.
  • Acknowledge any limitations of your research: Discuss any limitations of your study, including any potential threats to validity or generalizability. This helps readers understand the scope of your findings and how they might apply to other contexts.
  • Provide a summary: End the methodology section by summarizing the methods and techniques you used to conduct your research. This provides a clear overview of your research methodology and helps readers understand the process you followed to arrive at your findings.

When to Write Research Methodology

Research methodology is typically written after the research proposal has been approved and before the actual research is conducted. It should be written prior to data collection and analysis, as it provides a clear roadmap for the research project.

The research methodology is an important section of any research paper or thesis, as it describes the methods and procedures that will be used to conduct the research. It should include details about the research design, data collection methods, data analysis techniques, and any ethical considerations.

The methodology should be written in a clear and concise manner, and it should be based on established research practices and standards. It is important to provide enough detail so that the reader can understand how the research was conducted and evaluate the validity of the results.

Applications of Research Methodology

Here are some of the applications of research methodology:

  • To identify the research problem: Research methodology is used to identify the research problem, which is the first step in conducting any research.
  • To design the research: Research methodology helps in designing the research by selecting the appropriate research method, research design, and sampling technique.
  • To collect data: Research methodology provides a systematic approach to collect data from primary and secondary sources.
  • To analyze data: Research methodology helps in analyzing the collected data using various statistical and non-statistical techniques.
  • To test hypotheses: Research methodology provides a framework for testing hypotheses and drawing conclusions based on the analysis of data.
  • To generalize findings: Research methodology helps in generalizing the findings of the research to the target population.
  • To develop theories : Research methodology is used to develop new theories and modify existing theories based on the findings of the research.
  • To evaluate programs and policies : Research methodology is used to evaluate the effectiveness of programs and policies by collecting data and analyzing it.
  • To improve decision-making: Research methodology helps in making informed decisions by providing reliable and valid data.

Purpose of Research Methodology

Research methodology serves several important purposes, including:

  • To guide the research process: Research methodology provides a systematic framework for conducting research. It helps researchers to plan their research, define their research questions, and select appropriate methods and techniques for collecting and analyzing data.
  • To ensure research quality: Research methodology helps researchers to ensure that their research is rigorous, reliable, and valid. It provides guidelines for minimizing bias and error in data collection and analysis, and for ensuring that research findings are accurate and trustworthy.
  • To replicate research: Research methodology provides a clear and detailed account of the research process, making it possible for other researchers to replicate the study and verify its findings.
  • To advance knowledge: Research methodology enables researchers to generate new knowledge and to contribute to the body of knowledge in their field. It provides a means for testing hypotheses, exploring new ideas, and discovering new insights.
  • To inform decision-making: Research methodology provides evidence-based information that can inform policy and decision-making in a variety of fields, including medicine, public health, education, and business.

Advantages of Research Methodology

Research methodology has several advantages that make it a valuable tool for conducting research in various fields. Here are some of the key advantages of research methodology:

  • Systematic and structured approach : Research methodology provides a systematic and structured approach to conducting research, which ensures that the research is conducted in a rigorous and comprehensive manner.
  • Objectivity : Research methodology aims to ensure objectivity in the research process, which means that the research findings are based on evidence and not influenced by personal bias or subjective opinions.
  • Replicability : Research methodology ensures that research can be replicated by other researchers, which is essential for validating research findings and ensuring their accuracy.
  • Reliability : Research methodology aims to ensure that the research findings are reliable, which means that they are consistent and can be depended upon.
  • Validity : Research methodology ensures that the research findings are valid, which means that they accurately reflect the research question or hypothesis being tested.
  • Efficiency : Research methodology provides a structured and efficient way of conducting research, which helps to save time and resources.
  • Flexibility : Research methodology allows researchers to choose the most appropriate research methods and techniques based on the research question, data availability, and other relevant factors.
  • Scope for innovation: Research methodology provides scope for innovation and creativity in designing research studies and developing new research techniques.

Research Methodology Vs Research Methods

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Here's What You Need to Understand About Research Methodology

Deeptanshu D

Table of Contents

Research methodology involves a systematic and well-structured approach to conducting scholarly or scientific inquiries. Knowing the significance of research methodology and its different components is crucial as it serves as the basis for any study.

Typically, your research topic will start as a broad idea you want to investigate more thoroughly. Once you’ve identified a research problem and created research questions , you must choose the appropriate methodology and frameworks to address those questions effectively.

What is the definition of a research methodology?

Research methodology is the process or the way you intend to execute your study. The methodology section of a research paper outlines how you plan to conduct your study. It covers various steps such as collecting data, statistical analysis, observing participants, and other procedures involved in the research process

The methods section should give a description of the process that will convert your idea into a study. Additionally, the outcomes of your process must provide valid and reliable results resonant with the aims and objectives of your research. This thumb rule holds complete validity, no matter whether your paper has inclinations for qualitative or quantitative usage.

Studying research methods used in related studies can provide helpful insights and direction for your own research. Now easily discover papers related to your topic on SciSpace and utilize our AI research assistant, Copilot , to quickly review the methodologies applied in different papers.

Analyze and understand research methodologies faster with SciSpace Copilot

The need for a good research methodology

While deciding on your approach towards your research, the reason or factors you weighed in choosing a particular problem and formulating a research topic need to be validated and explained. A research methodology helps you do exactly that. Moreover, a good research methodology lets you build your argument to validate your research work performed through various data collection methods, analytical methods, and other essential points.

Just imagine it as a strategy documented to provide an overview of what you intend to do.

While undertaking any research writing or performing the research itself, you may get drifted in not something of much importance. In such a case, a research methodology helps you to get back to your outlined work methodology.

A research methodology helps in keeping you accountable for your work. Additionally, it can help you evaluate whether your work is in sync with your original aims and objectives or not. Besides, a good research methodology enables you to navigate your research process smoothly and swiftly while providing effective planning to achieve your desired results.

What is the basic structure of a research methodology?

Usually, you must ensure to include the following stated aspects while deciding over the basic structure of your research methodology:

1. Your research procedure

Explain what research methods you’re going to use. Whether you intend to proceed with quantitative or qualitative, or a composite of both approaches, you need to state that explicitly. The option among the three depends on your research’s aim, objectives, and scope.

2. Provide the rationality behind your chosen approach

Based on logic and reason, let your readers know why you have chosen said research methodologies. Additionally, you have to build strong arguments supporting why your chosen research method is the best way to achieve the desired outcome.

3. Explain your mechanism

The mechanism encompasses the research methods or instruments you will use to develop your research methodology. It usually refers to your data collection methods. You can use interviews, surveys, physical questionnaires, etc., of the many available mechanisms as research methodology instruments. The data collection method is determined by the type of research and whether the data is quantitative data(includes numerical data) or qualitative data (perception, morale, etc.) Moreover, you need to put logical reasoning behind choosing a particular instrument.

4. Significance of outcomes

The results will be available once you have finished experimenting. However, you should also explain how you plan to use the data to interpret the findings. This section also aids in understanding the problem from within, breaking it down into pieces, and viewing the research problem from various perspectives.

5. Reader’s advice

Anything that you feel must be explained to spread more awareness among readers and focus groups must be included and described in detail. You should not just specify your research methodology on the assumption that a reader is aware of the topic.  

All the relevant information that explains and simplifies your research paper must be included in the methodology section. If you are conducting your research in a non-traditional manner, give a logical justification and list its benefits.

6. Explain your sample space

Include information about the sample and sample space in the methodology section. The term "sample" refers to a smaller set of data that a researcher selects or chooses from a larger group of people or focus groups using a predetermined selection method. Let your readers know how you are going to distinguish between relevant and non-relevant samples. How you figured out those exact numbers to back your research methodology, i.e. the sample spacing of instruments, must be discussed thoroughly.

For example, if you are going to conduct a survey or interview, then by what procedure will you select the interviewees (or sample size in case of surveys), and how exactly will the interview or survey be conducted.

7. Challenges and limitations

This part, which is frequently assumed to be unnecessary, is actually very important. The challenges and limitations that your chosen strategy inherently possesses must be specified while you are conducting different types of research.

The importance of a good research methodology

You must have observed that all research papers, dissertations, or theses carry a chapter entirely dedicated to research methodology. This section helps maintain your credibility as a better interpreter of results rather than a manipulator.

A good research methodology always explains the procedure, data collection methods and techniques, aim, and scope of the research. In a research study, it leads to a well-organized, rationality-based approach, while the paper lacking it is often observed as messy or disorganized.

You should pay special attention to validating your chosen way towards the research methodology. This becomes extremely important in case you select an unconventional or a distinct method of execution.

Curating and developing a strong, effective research methodology can assist you in addressing a variety of situations, such as:

  • When someone tries to duplicate or expand upon your research after few years.
  • If a contradiction or conflict of facts occurs at a later time. This gives you the security you need to deal with these contradictions while still being able to defend your approach.
  • Gaining a tactical approach in getting your research completed in time. Just ensure you are using the right approach while drafting your research methodology, and it can help you achieve your desired outcomes. Additionally, it provides a better explanation and understanding of the research question itself.
  • Documenting the results so that the final outcome of the research stays as you intended it to be while starting.

Instruments you could use while writing a good research methodology

As a researcher, you must choose which tools or data collection methods that fit best in terms of the relevance of your research. This decision has to be wise.

There exists many research equipments or tools that you can use to carry out your research process. These are classified as:

a. Interviews (One-on-One or a Group)

An interview aimed to get your desired research outcomes can be undertaken in many different ways. For example, you can design your interview as structured, semi-structured, or unstructured. What sets them apart is the degree of formality in the questions. On the other hand, in a group interview, your aim should be to collect more opinions and group perceptions from the focus groups on a certain topic rather than looking out for some formal answers.

In surveys, you are in better control if you specifically draft the questions you seek the response for. For example, you may choose to include free-style questions that can be answered descriptively, or you may provide a multiple-choice type response for questions. Besides, you can also opt to choose both ways, deciding what suits your research process and purpose better.

c. Sample Groups

Similar to the group interviews, here, you can select a group of individuals and assign them a topic to discuss or freely express their opinions over that. You can simultaneously note down the answers and later draft them appropriately, deciding on the relevance of every response.

d. Observations

If your research domain is humanities or sociology, observations are the best-proven method to draw your research methodology. Of course, you can always include studying the spontaneous response of the participants towards a situation or conducting the same but in a more structured manner. A structured observation means putting the participants in a situation at a previously decided time and then studying their responses.

Of all the tools described above, it is you who should wisely choose the instruments and decide what’s the best fit for your research. You must not restrict yourself from multiple methods or a combination of a few instruments if appropriate in drafting a good research methodology.

Types of research methodology

A research methodology exists in various forms. Depending upon their approach, whether centered around words, numbers, or both, methodologies are distinguished as qualitative, quantitative, or an amalgamation of both.

1. Qualitative research methodology

When a research methodology primarily focuses on words and textual data, then it is generally referred to as qualitative research methodology. This type is usually preferred among researchers when the aim and scope of the research are mainly theoretical and explanatory.

The instruments used are observations, interviews, and sample groups. You can use this methodology if you are trying to study human behavior or response in some situations. Generally, qualitative research methodology is widely used in sociology, psychology, and other related domains.

2. Quantitative research methodology

If your research is majorly centered on data, figures, and stats, then analyzing these numerical data is often referred to as quantitative research methodology. You can use quantitative research methodology if your research requires you to validate or justify the obtained results.

In quantitative methods, surveys, tests, experiments, and evaluations of current databases can be advantageously used as instruments If your research involves testing some hypothesis, then use this methodology.

3. Amalgam methodology

As the name suggests, the amalgam methodology uses both quantitative and qualitative approaches. This methodology is used when a part of the research requires you to verify the facts and figures, whereas the other part demands you to discover the theoretical and explanatory nature of the research question.

The instruments for the amalgam methodology require you to conduct interviews and surveys, including tests and experiments. The outcome of this methodology can be insightful and valuable as it provides precise test results in line with theoretical explanations and reasoning.

The amalgam method, makes your work both factual and rational at the same time.

Final words: How to decide which is the best research methodology?

If you have kept your sincerity and awareness intact with the aims and scope of research well enough, you must have got an idea of which research methodology suits your work best.

Before deciding which research methodology answers your research question, you must invest significant time in reading and doing your homework for that. Taking references that yield relevant results should be your first approach to establishing a research methodology.

Moreover, you should never refrain from exploring other options. Before setting your work in stone, you must try all the available options as it explains why the choice of research methodology that you finally make is more appropriate than the other available options.

You should always go for a quantitative research methodology if your research requires gathering large amounts of data, figures, and statistics. This research methodology will provide you with results if your research paper involves the validation of some hypothesis.

Whereas, if  you are looking for more explanations, reasons, opinions, and public perceptions around a theory, you must use qualitative research methodology.The choice of an appropriate research methodology ultimately depends on what you want to achieve through your research.

Frequently Asked Questions (FAQs) about Research Methodology

1. how to write a research methodology.

You can always provide a separate section for research methodology where you should specify details about the methods and instruments used during the research, discussions on result analysis, including insights into the background information, and conveying the research limitations.

2. What are the types of research methodology?

There generally exists four types of research methodology i.e.

  • Observation
  • Experimental
  • Derivational

3. What is the true meaning of research methodology?

The set of techniques or procedures followed to discover and analyze the information gathered to validate or justify a research outcome is generally called Research Methodology.

4. Where lies the importance of research methodology?

Your research methodology directly reflects the validity of your research outcomes and how well-informed your research work is. Moreover, it can help future researchers cite or refer to your research if they plan to use a similar research methodology.

sample research methodology paper

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How To Write The Methodology Chapter

The what, why & how explained simply (with examples).

By: Jenna Crossley (PhD) | Reviewed By: Dr. Eunice Rautenbach | September 2021 (Updated April 2023)

So, you’ve pinned down your research topic and undertaken a review of the literature – now it’s time to write up the methodology section of your dissertation, thesis or research paper . But what exactly is the methodology chapter all about – and how do you go about writing one? In this post, we’ll unpack the topic, step by step .

Overview: The Methodology Chapter

  • The purpose  of the methodology chapter
  • Why you need to craft this chapter (really) well
  • How to write and structure the chapter
  • Methodology chapter example
  • Essential takeaways

What (exactly) is the methodology chapter?

The methodology chapter is where you outline the philosophical underpinnings of your research and outline the specific methodological choices you’ve made. The point of the methodology chapter is to tell the reader exactly how you designed your study and, just as importantly, why you did it this way.

Importantly, this chapter should comprehensively describe and justify all the methodological choices you made in your study. For example, the approach you took to your research (i.e., qualitative, quantitative or mixed), who  you collected data from (i.e., your sampling strategy), how you collected your data and, of course, how you analysed it. If that sounds a little intimidating, don’t worry – we’ll explain all these methodological choices in this post .

Free Webinar: Research Methodology 101

Why is the methodology chapter important?

The methodology chapter plays two important roles in your dissertation or thesis:

Firstly, it demonstrates your understanding of research theory, which is what earns you marks. A flawed research design or methodology would mean flawed results. So, this chapter is vital as it allows you to show the marker that you know what you’re doing and that your results are credible .

Secondly, the methodology chapter is what helps to make your study replicable. In other words, it allows other researchers to undertake your study using the same methodological approach, and compare their findings to yours. This is very important within academic research, as each study builds on previous studies.

The methodology chapter is also important in that it allows you to identify and discuss any methodological issues or problems you encountered (i.e., research limitations ), and to explain how you mitigated the impacts of these. Every research project has its limitations , so it’s important to acknowledge these openly and highlight your study’s value despite its limitations . Doing so demonstrates your understanding of research design, which will earn you marks. We’ll discuss limitations in a bit more detail later in this post, so stay tuned!

Need a helping hand?

sample research methodology paper

How to write up the methodology chapter

First off, it’s worth noting that the exact structure and contents of the methodology chapter will vary depending on the field of research (e.g., humanities, chemistry or engineering) as well as the university . So, be sure to always check the guidelines provided by your institution for clarity and, if possible, review past dissertations from your university. Here we’re going to discuss a generic structure for a methodology chapter typically found in the sciences.

Before you start writing, it’s always a good idea to draw up a rough outline to guide your writing. Don’t just start writing without knowing what you’ll discuss where. If you do, you’ll likely end up with a disjointed, ill-flowing narrative . You’ll then waste a lot of time rewriting in an attempt to try to stitch all the pieces together. Do yourself a favour and start with the end in mind .

Section 1 – Introduction

As with all chapters in your dissertation or thesis, the methodology chapter should have a brief introduction. In this section, you should remind your readers what the focus of your study is, especially the research aims . As we’ve discussed many times on the blog, your methodology needs to align with your research aims, objectives and research questions. Therefore, it’s useful to frontload this component to remind the reader (and yourself!) what you’re trying to achieve.

In this section, you can also briefly mention how you’ll structure the chapter. This will help orient the reader and provide a bit of a roadmap so that they know what to expect. You don’t need a lot of detail here – just a brief outline will do.

The intro provides a roadmap to your methodology chapter

Section 2 – The Methodology

The next section of your chapter is where you’ll present the actual methodology. In this section, you need to detail and justify the key methodological choices you’ve made in a logical, intuitive fashion. Importantly, this is the heart of your methodology chapter, so you need to get specific – don’t hold back on the details here. This is not one of those “less is more” situations.

Let’s take a look at the most common components you’ll likely need to cover. 

Methodological Choice #1 – Research Philosophy

Research philosophy refers to the underlying beliefs (i.e., the worldview) regarding how data about a phenomenon should be gathered , analysed and used . The research philosophy will serve as the core of your study and underpin all of the other research design choices, so it’s critically important that you understand which philosophy you’ll adopt and why you made that choice. If you’re not clear on this, take the time to get clarity before you make any further methodological choices.

While several research philosophies exist, two commonly adopted ones are positivism and interpretivism . These two sit roughly on opposite sides of the research philosophy spectrum.

Positivism states that the researcher can observe reality objectively and that there is only one reality, which exists independently of the observer. As a consequence, it is quite commonly the underlying research philosophy in quantitative studies and is oftentimes the assumed philosophy in the physical sciences.

Contrasted with this, interpretivism , which is often the underlying research philosophy in qualitative studies, assumes that the researcher performs a role in observing the world around them and that reality is unique to each observer . In other words, reality is observed subjectively .

These are just two philosophies (there are many more), but they demonstrate significantly different approaches to research and have a significant impact on all the methodological choices. Therefore, it’s vital that you clearly outline and justify your research philosophy at the beginning of your methodology chapter, as it sets the scene for everything that follows.

The research philosophy is at the core of the methodology chapter

Methodological Choice #2 – Research Type

The next thing you would typically discuss in your methodology section is the research type. The starting point for this is to indicate whether the research you conducted is inductive or deductive .

Inductive research takes a bottom-up approach , where the researcher begins with specific observations or data and then draws general conclusions or theories from those observations. Therefore these studies tend to be exploratory in terms of approach.

Conversely , d eductive research takes a top-down approach , where the researcher starts with a theory or hypothesis and then tests it using specific observations or data. Therefore these studies tend to be confirmatory in approach.

Related to this, you’ll need to indicate whether your study adopts a qualitative, quantitative or mixed  approach. As we’ve mentioned, there’s a strong link between this choice and your research philosophy, so make sure that your choices are tightly aligned . When you write this section up, remember to clearly justify your choices, as they form the foundation of your study.

Methodological Choice #3 – Research Strategy

Next, you’ll need to discuss your research strategy (also referred to as a research design ). This methodological choice refers to the broader strategy in terms of how you’ll conduct your research, based on the aims of your study.

Several research strategies exist, including experimental , case studies , ethnography , grounded theory, action research , and phenomenology . Let’s take a look at two of these, experimental and ethnographic, to see how they contrast.

Experimental research makes use of the scientific method , where one group is the control group (in which no variables are manipulated ) and another is the experimental group (in which a specific variable is manipulated). This type of research is undertaken under strict conditions in a controlled, artificial environment (e.g., a laboratory). By having firm control over the environment, experimental research typically allows the researcher to establish causation between variables. Therefore, it can be a good choice if you have research aims that involve identifying causal relationships.

Ethnographic research , on the other hand, involves observing and capturing the experiences and perceptions of participants in their natural environment (for example, at home or in the office). In other words, in an uncontrolled environment.  Naturally, this means that this research strategy would be far less suitable if your research aims involve identifying causation, but it would be very valuable if you’re looking to explore and examine a group culture, for example.

As you can see, the right research strategy will depend largely on your research aims and research questions – in other words, what you’re trying to figure out. Therefore, as with every other methodological choice, it’s essential to justify why you chose the research strategy you did.

Methodological Choice #4 – Time Horizon

The next thing you’ll need to detail in your methodology chapter is the time horizon. There are two options here: cross-sectional and longitudinal . In other words, whether the data for your study were all collected at one point in time (cross-sectional) or at multiple points in time (longitudinal).

The choice you make here depends again on your research aims, objectives and research questions. If, for example, you aim to assess how a specific group of people’s perspectives regarding a topic change over time , you’d likely adopt a longitudinal time horizon.

Another important factor to consider is simply whether you have the time necessary to adopt a longitudinal approach (which could involve collecting data over multiple months or even years). Oftentimes, the time pressures of your degree program will force your hand into adopting a cross-sectional time horizon, so keep this in mind.

Methodological Choice #5 – Sampling Strategy

Next, you’ll need to discuss your sampling strategy . There are two main categories of sampling, probability and non-probability sampling.

Probability sampling involves a random (and therefore representative) selection of participants from a population, whereas non-probability sampling entails selecting participants in a non-random  (and therefore non-representative) manner. For example, selecting participants based on ease of access (this is called a convenience sample).

The right sampling approach depends largely on what you’re trying to achieve in your study. Specifically, whether you trying to develop findings that are generalisable to a population or not. Practicalities and resource constraints also play a large role here, as it can oftentimes be challenging to gain access to a truly random sample. In the video below, we explore some of the most common sampling strategies.

Methodological Choice #6 – Data Collection Method

Next up, you’ll need to explain how you’ll go about collecting the necessary data for your study. Your data collection method (or methods) will depend on the type of data that you plan to collect – in other words, qualitative or quantitative data.

Typically, quantitative research relies on surveys , data generated by lab equipment, analytics software or existing datasets. Qualitative research, on the other hand, often makes use of collection methods such as interviews , focus groups , participant observations, and ethnography.

So, as you can see, there is a tight link between this section and the design choices you outlined in earlier sections. Strong alignment between these sections, as well as your research aims and questions is therefore very important.

Methodological Choice #7 – Data Analysis Methods/Techniques

The final major methodological choice that you need to address is that of analysis techniques . In other words, how you’ll go about analysing your date once you’ve collected it. Here it’s important to be very specific about your analysis methods and/or techniques – don’t leave any room for interpretation. Also, as with all choices in this chapter, you need to justify each choice you make.

What exactly you discuss here will depend largely on the type of study you’re conducting (i.e., qualitative, quantitative, or mixed methods). For qualitative studies, common analysis methods include content analysis , thematic analysis and discourse analysis . In the video below, we explain each of these in plain language.

For quantitative studies, you’ll almost always make use of descriptive statistics , and in many cases, you’ll also use inferential statistical techniques (e.g., correlation and regression analysis). In the video below, we unpack some of the core concepts involved in descriptive and inferential statistics.

In this section of your methodology chapter, it’s also important to discuss how you prepared your data for analysis, and what software you used (if any). For example, quantitative data will often require some initial preparation such as removing duplicates or incomplete responses . Similarly, qualitative data will often require transcription and perhaps even translation. As always, remember to state both what you did and why you did it.

Section 3 – The Methodological Limitations

With the key methodological choices outlined and justified, the next step is to discuss the limitations of your design. No research methodology is perfect – there will always be trade-offs between the “ideal” methodology and what’s practical and viable, given your constraints. Therefore, this section of your methodology chapter is where you’ll discuss the trade-offs you had to make, and why these were justified given the context.

Methodological limitations can vary greatly from study to study, ranging from common issues such as time and budget constraints to issues of sample or selection bias . For example, you may find that you didn’t manage to draw in enough respondents to achieve the desired sample size (and therefore, statistically significant results), or your sample may be skewed heavily towards a certain demographic, thereby negatively impacting representativeness .

In this section, it’s important to be critical of the shortcomings of your study. There’s no use trying to hide them (your marker will be aware of them regardless). By being critical, you’ll demonstrate to your marker that you have a strong understanding of research theory, so don’t be shy here. At the same time, don’t beat your study to death . State the limitations, why these were justified, how you mitigated their impacts to the best degree possible, and how your study still provides value despite these limitations .

Section 4 – Concluding Summary

Finally, it’s time to wrap up the methodology chapter with a brief concluding summary. In this section, you’ll want to concisely summarise what you’ve presented in the chapter. Here, it can be a good idea to use a figure to summarise the key decisions, especially if your university recommends using a specific model (for example, Saunders’ Research Onion ).

Importantly, this section needs to be brief – a paragraph or two maximum (it’s a summary, after all). Also, make sure that when you write up your concluding summary, you include only what you’ve already discussed in your chapter; don’t add any new information.

Keep it simple

Methodology Chapter Example

In the video below, we walk you through an example of a high-quality research methodology chapter from a dissertation. We also unpack our free methodology chapter template so that you can see how best to structure your chapter.

Wrapping Up

And there you have it – the methodology chapter in a nutshell. As we’ve mentioned, the exact contents and structure of this chapter can vary between universities , so be sure to check in with your institution before you start writing. If possible, try to find dissertations or theses from former students of your specific degree program – this will give you a strong indication of the expectations and norms when it comes to the methodology chapter (and all the other chapters!).

Also, remember the golden rule of the methodology chapter – justify every choice ! Make sure that you clearly explain the “why” for every “what”, and reference credible methodology textbooks or academic sources to back up your justifications.

If you need a helping hand with your research methodology (or any other component of your research), be sure to check out our private coaching service , where we hold your hand through every step of the research journey. Until next time, good luck!

sample research methodology paper

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What is Research Methodology? Definition, Types, and Examples

sample research methodology paper

Research methodology 1,2 is a structured and scientific approach used to collect, analyze, and interpret quantitative or qualitative data to answer research questions or test hypotheses. A research methodology is like a plan for carrying out research and helps keep researchers on track by limiting the scope of the research. Several aspects must be considered before selecting an appropriate research methodology, such as research limitations and ethical concerns that may affect your research.

The research methodology section in a scientific paper describes the different methodological choices made, such as the data collection and analysis methods, and why these choices were selected. The reasons should explain why the methods chosen are the most appropriate to answer the research question. A good research methodology also helps ensure the reliability and validity of the research findings. There are three types of research methodology—quantitative, qualitative, and mixed-method, which can be chosen based on the research objectives.

What is research methodology ?

A research methodology describes the techniques and procedures used to identify and analyze information regarding a specific research topic. It is a process by which researchers design their study so that they can achieve their objectives using the selected research instruments. It includes all the important aspects of research, including research design, data collection methods, data analysis methods, and the overall framework within which the research is conducted. While these points can help you understand what is research methodology, you also need to know why it is important to pick the right methodology.

Why is research methodology important?

Having a good research methodology in place has the following advantages: 3

  • Helps other researchers who may want to replicate your research; the explanations will be of benefit to them.
  • You can easily answer any questions about your research if they arise at a later stage.
  • A research methodology provides a framework and guidelines for researchers to clearly define research questions, hypotheses, and objectives.
  • It helps researchers identify the most appropriate research design, sampling technique, and data collection and analysis methods.
  • A sound research methodology helps researchers ensure that their findings are valid and reliable and free from biases and errors.
  • It also helps ensure that ethical guidelines are followed while conducting research.
  • A good research methodology helps researchers in planning their research efficiently, by ensuring optimum usage of their time and resources.

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Types of research methodology.

There are three types of research methodology based on the type of research and the data required. 1

  • Quantitative research methodology focuses on measuring and testing numerical data. This approach is good for reaching a large number of people in a short amount of time. This type of research helps in testing the causal relationships between variables, making predictions, and generalizing results to wider populations.
  • Qualitative research methodology examines the opinions, behaviors, and experiences of people. It collects and analyzes words and textual data. This research methodology requires fewer participants but is still more time consuming because the time spent per participant is quite large. This method is used in exploratory research where the research problem being investigated is not clearly defined.
  • Mixed-method research methodology uses the characteristics of both quantitative and qualitative research methodologies in the same study. This method allows researchers to validate their findings, verify if the results observed using both methods are complementary, and explain any unexpected results obtained from one method by using the other method.

What are the types of sampling designs in research methodology?

Sampling 4 is an important part of a research methodology and involves selecting a representative sample of the population to conduct the study, making statistical inferences about them, and estimating the characteristics of the whole population based on these inferences. There are two types of sampling designs in research methodology—probability and nonprobability.

  • Probability sampling

In this type of sampling design, a sample is chosen from a larger population using some form of random selection, that is, every member of the population has an equal chance of being selected. The different types of probability sampling are:

  • Systematic —sample members are chosen at regular intervals. It requires selecting a starting point for the sample and sample size determination that can be repeated at regular intervals. This type of sampling method has a predefined range; hence, it is the least time consuming.
  • Stratified —researchers divide the population into smaller groups that don’t overlap but represent the entire population. While sampling, these groups can be organized, and then a sample can be drawn from each group separately.
  • Cluster —the population is divided into clusters based on demographic parameters like age, sex, location, etc.
  • Convenience —selects participants who are most easily accessible to researchers due to geographical proximity, availability at a particular time, etc.
  • Purposive —participants are selected at the researcher’s discretion. Researchers consider the purpose of the study and the understanding of the target audience.
  • Snowball —already selected participants use their social networks to refer the researcher to other potential participants.
  • Quota —while designing the study, the researchers decide how many people with which characteristics to include as participants. The characteristics help in choosing people most likely to provide insights into the subject.

What are data collection methods?

During research, data are collected using various methods depending on the research methodology being followed and the research methods being undertaken. Both qualitative and quantitative research have different data collection methods, as listed below.

Qualitative research 5

  • One-on-one interviews: Helps the interviewers understand a respondent’s subjective opinion and experience pertaining to a specific topic or event
  • Document study/literature review/record keeping: Researchers’ review of already existing written materials such as archives, annual reports, research articles, guidelines, policy documents, etc.
  • Focus groups: Constructive discussions that usually include a small sample of about 6-10 people and a moderator, to understand the participants’ opinion on a given topic.
  • Qualitative observation : Researchers collect data using their five senses (sight, smell, touch, taste, and hearing).

Quantitative research 6

  • Sampling: The most common type is probability sampling.
  • Interviews: Commonly telephonic or done in-person.
  • Observations: Structured observations are most commonly used in quantitative research. In this method, researchers make observations about specific behaviors of individuals in a structured setting.
  • Document review: Reviewing existing research or documents to collect evidence for supporting the research.
  • Surveys and questionnaires. Surveys can be administered both online and offline depending on the requirement and sample size.

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What are data analysis methods.

The data collected using the various methods for qualitative and quantitative research need to be analyzed to generate meaningful conclusions. These data analysis methods 7 also differ between quantitative and qualitative research.

Quantitative research involves a deductive method for data analysis where hypotheses are developed at the beginning of the research and precise measurement is required. The methods include statistical analysis applications to analyze numerical data and are grouped into two categories—descriptive and inferential.

Descriptive analysis is used to describe the basic features of different types of data to present it in a way that ensures the patterns become meaningful. The different types of descriptive analysis methods are:

  • Measures of frequency (count, percent, frequency)
  • Measures of central tendency (mean, median, mode)
  • Measures of dispersion or variation (range, variance, standard deviation)
  • Measure of position (percentile ranks, quartile ranks)

Inferential analysis is used to make predictions about a larger population based on the analysis of the data collected from a smaller population. This analysis is used to study the relationships between different variables. Some commonly used inferential data analysis methods are:

  • Correlation: To understand the relationship between two or more variables.
  • Cross-tabulation: Analyze the relationship between multiple variables.
  • Regression analysis: Study the impact of independent variables on the dependent variable.
  • Frequency tables: To understand the frequency of data.
  • Analysis of variance: To test the degree to which two or more variables differ in an experiment.

Qualitative research involves an inductive method for data analysis where hypotheses are developed after data collection. The methods include:

  • Content analysis: For analyzing documented information from text and images by determining the presence of certain words or concepts in texts.
  • Narrative analysis: For analyzing content obtained from sources such as interviews, field observations, and surveys. The stories and opinions shared by people are used to answer research questions.
  • Discourse analysis: For analyzing interactions with people considering the social context, that is, the lifestyle and environment, under which the interaction occurs.
  • Grounded theory: Involves hypothesis creation by data collection and analysis to explain why a phenomenon occurred.
  • Thematic analysis: To identify important themes or patterns in data and use these to address an issue.

How to choose a research methodology?

Here are some important factors to consider when choosing a research methodology: 8

  • Research objectives, aims, and questions —these would help structure the research design.
  • Review existing literature to identify any gaps in knowledge.
  • Check the statistical requirements —if data-driven or statistical results are needed then quantitative research is the best. If the research questions can be answered based on people’s opinions and perceptions, then qualitative research is most suitable.
  • Sample size —sample size can often determine the feasibility of a research methodology. For a large sample, less effort- and time-intensive methods are appropriate.
  • Constraints —constraints of time, geography, and resources can help define the appropriate methodology.

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How to write a research methodology .

A research methodology should include the following components: 3,9

  • Research design —should be selected based on the research question and the data required. Common research designs include experimental, quasi-experimental, correlational, descriptive, and exploratory.
  • Research method —this can be quantitative, qualitative, or mixed-method.
  • Reason for selecting a specific methodology —explain why this methodology is the most suitable to answer your research problem.
  • Research instruments —explain the research instruments you plan to use, mainly referring to the data collection methods such as interviews, surveys, etc. Here as well, a reason should be mentioned for selecting the particular instrument.
  • Sampling —this involves selecting a representative subset of the population being studied.
  • Data collection —involves gathering data using several data collection methods, such as surveys, interviews, etc.
  • Data analysis —describe the data analysis methods you will use once you’ve collected the data.
  • Research limitations —mention any limitations you foresee while conducting your research.
  • Validity and reliability —validity helps identify the accuracy and truthfulness of the findings; reliability refers to the consistency and stability of the results over time and across different conditions.
  • Ethical considerations —research should be conducted ethically. The considerations include obtaining consent from participants, maintaining confidentiality, and addressing conflicts of interest.

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Frequently Asked Questions

Q1. What are the key components of research methodology?

A1. A good research methodology has the following key components:

  • Research design
  • Data collection procedures
  • Data analysis methods
  • Ethical considerations

Q2. Why is ethical consideration important in research methodology?

A2. Ethical consideration is important in research methodology to ensure the readers of the reliability and validity of the study. Researchers must clearly mention the ethical norms and standards followed during the conduct of the research and also mention if the research has been cleared by any institutional board. The following 10 points are the important principles related to ethical considerations: 10

  • Participants should not be subjected to harm.
  • Respect for the dignity of participants should be prioritized.
  • Full consent should be obtained from participants before the study.
  • Participants’ privacy should be ensured.
  • Confidentiality of the research data should be ensured.
  • Anonymity of individuals and organizations participating in the research should be maintained.
  • The aims and objectives of the research should not be exaggerated.
  • Affiliations, sources of funding, and any possible conflicts of interest should be declared.
  • Communication in relation to the research should be honest and transparent.
  • Misleading information and biased representation of primary data findings should be avoided.

Q3. What is the difference between methodology and method?

A3. Research methodology is different from a research method, although both terms are often confused. Research methods are the tools used to gather data, while the research methodology provides a framework for how research is planned, conducted, and analyzed. The latter guides researchers in making decisions about the most appropriate methods for their research. Research methods refer to the specific techniques, procedures, and tools used by researchers to collect, analyze, and interpret data, for instance surveys, questionnaires, interviews, etc.

Research methodology is, thus, an integral part of a research study. It helps ensure that you stay on track to meet your research objectives and answer your research questions using the most appropriate data collection and analysis tools based on your research design.

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  • Research methodologies. Pfeiffer Library website. Accessed August 15, 2023. https://library.tiffin.edu/researchmethodologies/whatareresearchmethodologies
  • Types of research methodology. Eduvoice website. Accessed August 16, 2023. https://eduvoice.in/types-research-methodology/
  • The basics of research methodology: A key to quality research. Voxco. Accessed August 16, 2023. https://www.voxco.com/blog/what-is-research-methodology/
  • Sampling methods: Types with examples. QuestionPro website. Accessed August 16, 2023. https://www.questionpro.com/blog/types-of-sampling-for-social-research/
  • What is qualitative research? Methods, types, approaches, examples. Researcher.Life blog. Accessed August 15, 2023. https://researcher.life/blog/article/what-is-qualitative-research-methods-types-examples/
  • What is quantitative research? Definition, methods, types, and examples. Researcher.Life blog. Accessed August 15, 2023. https://researcher.life/blog/article/what-is-quantitative-research-types-and-examples/
  • Data analysis in research: Types & methods. QuestionPro website. Accessed August 16, 2023. https://www.questionpro.com/blog/data-analysis-in-research/#Data_analysis_in_qualitative_research
  • Factors to consider while choosing the right research methodology. PhD Monster website. Accessed August 17, 2023. https://www.phdmonster.com/factors-to-consider-while-choosing-the-right-research-methodology/
  • What is research methodology? Research and writing guides. Accessed August 14, 2023. https://paperpile.com/g/what-is-research-methodology/
  • Ethical considerations. Business research methodology website. Accessed August 17, 2023. https://research-methodology.net/research-methodology/ethical-considerations/

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15 Research Methodology Examples

research methodologies examples, explained below

Research methodologies can roughly be categorized into three group: quantitative, qualitative, and mixed-methods.

  • Qualitative Research : This methodology is based on obtaining deep, contextualized, non-numerical data. It can occur, for example, through open-ended questioning of research particiapnts in order to understand human behavior. It’s all about describing and analyzing subjective phenomena such as emotions or experiences.
  • Quantitative Research: This methodology is rationally-based and relies heavily on numerical analysis of empirical data . With quantitative research, you aim for objectivity by creating hypotheses and testing them through experiments or surveys, which allow for statistical analyses.
  • Mixed-Methods Research: Mixed-methods research combines both previous types into one project. We have more flexibility when designing our research study with mixed methods since we can use multiple approaches depending on our needs at each time. Using mixed methods can help us validate our results and offer greater predictability than just either type of methodology alone could provide.

Below are research methodologies that fit into each category.

chris

Qualitative Research Methodologies

1. case study.

Conducts an in-depth examination of a specific case, individual, or event to understand a phenomenon.

Instead of examining a whole population for numerical trend data, case study researchers seek in-depth explanations of one event.

The benefit of case study research is its ability to elucidate overlooked details of interesting cases of a phenomenon (Busetto, Wick & Gumbinger, 2020). It offers deep insights for empathetic, reflective, and thoughtful understandings of that phenomenon.

However, case study findings aren’t transferrable to new contexts or for population-wide predictions. Instead, they inform practitioner understandings for nuanced, deep approaches to future instances (Liamputtong, 2020).

2. Grounded Theory

Grounded theory involves generating hypotheses and theories through the collection and interpretation of data (Faggiolani, n.d.). Its distinguishing features is that it doesn’t test a hypothesis generated prior to analysis, but rather generates a hypothesis or ‘theory’ that emerges from the data.

It also involves the application of inductive reasoning and is often contrasted with the hypothetico-deductive model of scientific research. This research methodology was developed by Barney Glaser and Anselm Strauss in the 1960s (Glaser & Strauss, 2009). 

The basic difference between traditional scientific approaches to research and grounded theory is that the latter begins with a question, then collects data, and the theoretical framework is said to emerge later from this data.

By contrast, scientists usually begin with an existing theoretical framework , develop hypotheses, and only then start collecting data to verify or falsify the hypotheses.

3. Ethnography

In ethnographic research , the researcher immerses themselves within the group they are studying, often for long periods of time.

This type of research aims to understand the shared beliefs, practices, and values of a particular community by immersing the researcher within the cultural group.

Although ethnographic research cannot predict or identify trends in an entire population, it can create detailed explanations of cultural practices and comparisons between social and cultural groups.

When a person conducts an ethnographic study of themselves or their own culture, it can be considered autoethnography .

Its strength lies in producing comprehensive accounts of groups of people and their interactions.

Common methods researchers use during an ethnographic study include participant observation , thick description, unstructured interviews, and field notes vignettes. These methods can provide detailed and contextualized descriptions of their subjects.

Example Study

Liquidated: An Ethnography of Wall Street by Karen Ho involves an anthropologist who embeds herself with Wall Street firms to study the culture of Wall Street bankers and how this culture affects the broader economy and world.

4. Phenomenology

Phenomenology to understand and describe individuals’ lived experiences concerning a specific phenomenon.

As a research methodology typically used in the social sciences , phenomenology involves the study of social reality as a product of intersubjectivity (the intersection of people’s cognitive perspectives) (Zahavi & Overgaard, n.d.).

This philosophical approach was first developed by Edmund Husserl.

5. Narrative Research

Narrative research explores personal stories and experiences to understand their meanings and interpretations.

It is also known as narrative inquiry and narrative analysis(Riessman, 1993).

This approach to research uses qualitative material like journals, field notes, letters, interviews, texts, photos, etc., as its data.

It is aimed at understanding the way people create meaning through narratives (Clandinin & Connelly, 2004).

6. Discourse Analysis

A discourse analysis examines the structure, patterns, and functions of language in context to understand how the text produces social constructs.

This methodology is common in critical theory , poststructuralism , and postmodernism. Its aim is to understand how language constructs discourses (roughly interpreted as “ways of thinking and constructing knowledge”).

As a qualitative methodology , its focus is on developing themes through close textual analysis rather than using numerical methods. Common methods for extracting data include semiotics and linguistic analysis.

7. Action Research

Action research involves researchers working collaboratively with stakeholders to address problems, develop interventions, and evaluate effectiveness.

Action research is a methodology and philosophy of research that is common in the social sciences.

The term was first coined in 1944 by Kurt Lewin, a German-American psychologist who also introduced applied research and group communication (Altrichter & Gstettner, 1993).

Lewin originally defined action research as involving two primary processes: taking action and doing research (Lewin, 1946).

Action research involves planning, action, and information-seeking about the result of the action.

Since Lewin’s original formulation, many different theoretical approaches to action research have been developed. These include action science, participatory action research, cooperative inquiry, and living educational theory among others.

Using Digital Sandbox Gaming to Improve Creativity Within Boys’ Writing (Ellison & Drew, 2019) is a study conducted by a school teacher who used video games to help teach his students English. It involved action research, where he interviewed his students to see if the use of games as stimuli for storytelling helped draw them into the learning experience, and iterated on his teaching style based on their feedback (disclaimer: I am the second author of this study).

See More: Examples of Qualitative Research

Quantitative Research Methodologies

8. experimental design.

As the name suggests, this type of research is based on testing hypotheses in experimental settings by manipulating variables and observing their effects on other variables.

The main benefit lies in its ability to manipulate specific variables to determine their effect on outcomes which is a great method for those looking for causational links in their research.

This is common, for example, in high-school science labs, where students are asked to introduce a variable into a setting in order to examine its effect.

9. Non-Experimental Design

Non-experimental design observes and measures associations between variables without manipulating them.

It can take, for example, the form of a ‘fly on the wall’ observation of a phenomenon, allowing researchers to examine authentic settings and changes that occur naturally in the environment.

10. Cross-Sectional Design

Cross-sectional design involves analyzing variables pertaining to a specific time period and at that exact moment.

This approach allows for an extensive examination and comparison of distinct and independent subjects, thereby offering advantages over qualitative methodologies such as case studies or surveys.

While cross-sectional design can be extremely useful in taking a ‘snapshot in time’, as a standalone method, it is not useful for examining changes in subjects after an intervention. The next methodology addresses this issue.

The prime example of this type of study is a census. A population census is mailed out to every house in the country, and each household must complete the census on the same evening. This allows the government to gather a snapshot of the nation’s demographics, beliefs, religion, and so on.

11. Longitudinal Design

Longitudinal research gathers data from the same subjects over an extended period to analyze changes and development.

In contrast to cross-sectional tactics, longitudinal designs examine variables more than once, over a pre-determined time span, allowing for multiple data points to be taken at different times.

A cross-sectional design is also useful for examining cohort effects , by comparing differences or changes in multiple different generations’ beliefs over time.

With multiple data points collected over extended periods ,it’s possible to examine continuous changes within things like population dynamics or consumer behavior. This makes detailed analysis of change possible.

12. Quasi-Experimental Design

Quasi-experimental design involves manipulating variables for analysis, but uses pre-existing groups of subjects rather than random groups.

Because the groups of research participants already exist, they cannot be randomly assigned to a cohort as with a true experimental design study. This makes inferring a causal relationship more difficult, but is nonetheless often more feasible in real-life settings.

Quasi-experimental designs are generally considered inferior to true experimental designs.

13. Correlational Research

Correlational research examines the relationships between two or more variables, determining the strength and direction of their association.

Similar to quasi-experimental methods, this type of research focuses on relationship differences between variables.

This approach provides a fast and easy way to make initial hypotheses based on either positive or negative correlation trends that can be observed within dataset.

Methods used for data analysis may include statistic correlations such as Pearson’s or Spearman’s.

Mixed-Methods Research Methodologies

14. sequential explanatory design (quan→qual).

This methodology involves conducting quantitative analysis first, then supplementing it with a qualitative study.

It begins by collecting quantitative data that is then analyzed to determine any significant patterns or trends.

Secondly, qualitative methods are employed. Their intent is to help interpret and expand the quantitative results.

This offers greater depth into understanding both large and smaller aspects of research questions being addressed.

The rationale behind this approach is to ensure that your data collection generates richer context for gaining insight into the particular issue across different levels, integrating in one study, qualitative exploration as well as statistical procedures.

15. Sequential Exploratory Design (QUAL→QUAN)

This methodology goes in the other direction, starting with qualitative analysis and ending with quantitative analysis.

It starts with qualitative research that delves deeps into complex areas and gathers rich information through interviewing or observing participants.

After this stage of exploration comes to an end, quantitative techniques are used to analyze the collected data through inferential statistics.

The idea is that a qualitative study can arm the researchers with a strong hypothesis testing framework, which they can then apply to a larger sample size using qualitative methods.

When I first took research classes, I had a lot of trouble distinguishing between methodologies and methods.

The key is to remember that the methodology sets the direction, while the methods are the specific tools to be used. A good analogy is transport: first you need to choose a mode (public transport, private transport, motorized transit, non-motorized transit), then you can choose a tool (bus, car, bike, on foot).

While research methodologies can be split into three types, each type has many different nuanced methodologies that can be chosen, before you then choose the methods – or tools – to use in the study. Each has its own strengths and weaknesses, so choose wisely!

Altrichter, H., & Gstettner, P. (1993). Action Research: A closed chapter in the history of German social science? Educational Action Research , 1 (3), 329–360. https://doi.org/10.1080/0965079930010302

Audi, R. (1999). The Cambridge dictionary of philosophy . Cambridge ; New York : Cambridge University Press. http://archive.org/details/cambridgediction00audi

Clandinin, D. J., & Connelly, F. M. (2004). Narrative Inquiry: Experience and Story in Qualitative Research . John Wiley & Sons.

Creswell, J. W. (2008). Educational Research: Planning, Conducting, and Evaluating Quantitative and Qualitative Research . Pearson/Merrill Prentice Hall.

Faggiolani, C. (n.d.). Perceived Identity: Applying Grounded Theory in Libraries . https://doi.org/10.4403/jlis.it-4592

Gauch, H. G. (2002). Scientific Method in Practice . Cambridge University Press.

Glaser, B. G., & Strauss, A. L. (2009). The Discovery of Grounded Theory: Strategies for Qualitative Research . Transaction Publishers.

Kothari, C. R. (2004). Research Methodology: Methods and Techniques . New Age International.

Kuada, J. (2012). Research Methodology: A Project Guide for University Students . Samfundslitteratur.

Lewin, K. (1946). Action research and minority problems. Journal of Social Issues , 2,  4 , 34–46. https://doi.org/10.1111/j.1540-4560.1946.tb02295.x

Mills, J., Bonner, A., & Francis, K. (2006). The Development of Constructivist Grounded Theory. International Journal of Qualitative Methods , 5 (1), 25–35. https://doi.org/10.1177/160940690600500103

Mingers, J., & Willcocks, L. (2017). An integrative semiotic methodology for IS research. Information and Organization , 27 (1), 17–36. https://doi.org/10.1016/j.infoandorg.2016.12.001

OECD. (2015). Frascati Manual 2015: Guidelines for Collecting and Reporting Data on Research and Experimental Development . Organisation for Economic Co-operation and Development. https://www.oecd-ilibrary.org/science-and-technology/frascati-manual-2015_9789264239012-en

Peirce, C. S. (1992). The Essential Peirce, Volume 1: Selected Philosophical Writings (1867–1893) . Indiana University Press.

Reese, W. L. (1980). Dictionary of Philosophy and Religion: Eastern and Western Thought . Humanities Press.

Riessman, C. K. (1993). Narrative analysis . Sage Publications, Inc.

Saussure, F. de, & Riedlinger, A. (1959). Course in General Linguistics . Philosophical Library.

Thomas, C. G. (2021). Research Methodology and Scientific Writing . Springer Nature.

Zahavi, D., & Overgaard, S. (n.d.). Phenomenological Sociology—The Subjectivity of Everyday Life .

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Crafting a comprehensive research paper can be daunting. Understanding diverse citation styles and various subject areas presents a challenge for many.

Without clear examples, students often feel lost and overwhelmed, unsure of how to start or which style fits their subject.

Explore our collection of expertly written research paper examples. We’ve covered various citation styles and a diverse range of subjects.

So, read on!

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Research Paper Example for Different Formats

Following a specific formatting style is essential while writing a research paper . Knowing the conventions and guidelines for each format can help you in creating a perfect paper. Here we have gathered examples of research paper for most commonly applied citation styles :

Social Media and Social Media Marketing: A Literature Review

APA Research Paper Example

APA (American Psychological Association) style is commonly used in social sciences, psychology, and education. This format is recognized for its clear and concise writing, emphasis on proper citations, and orderly presentation of ideas.

Here are some research paper examples in APA style:

Research Paper Example APA 7th Edition

Research Paper Example MLA

MLA (Modern Language Association) style is frequently employed in humanities disciplines, including literature, languages, and cultural studies. An MLA research paper might explore literature analysis, linguistic studies, or historical research within the humanities. 

Here is an example:

Found Voices: Carl Sagan

Research Paper Example Chicago

Chicago style is utilized in various fields like history, arts, and social sciences. Research papers in Chicago style could delve into historical events, artistic analyses, or social science inquiries. 

Here is a research paper formatted in Chicago style:

Chicago Research Paper Sample

Research Paper Example Harvard

Harvard style is widely used in business, management, and some social sciences. Research papers in Harvard style might address business strategies, case studies, or social policies.

View this sample Harvard style paper here:

Harvard Research Paper Sample

Examples for Different Research Paper Parts

A research paper has different parts. Each part is important for the overall success of the paper. Chapters in a research paper must be written correctly, using a certain format and structure.

The following are examples of how different sections of the research paper can be written.

Research Proposal

The research proposal acts as a detailed plan or roadmap for your study, outlining the focus of your research and its significance. It's essential as it not only guides your research but also persuades others about the value of your study.

Example of Research Proposal

An abstract serves as a concise overview of your entire research paper. It provides a quick insight into the main elements of your study. It summarizes your research's purpose, methods, findings, and conclusions in a brief format.

Research Paper Example Abstract

Literature Review 

A literature review summarizes the existing research on your study's topic, showcasing what has already been explored. This section adds credibility to your own research by analyzing and summarizing prior studies related to your topic.

Literature Review Research Paper Example

Methodology

The methodology section functions as a detailed explanation of how you conducted your research. This part covers the tools, techniques, and steps used to collect and analyze data for your study.

Methods Section of Research Paper Example

How to Write the Methods Section of a Research Paper

The conclusion summarizes your findings, their significance and the impact of your research. This section outlines the key takeaways and the broader implications of your study's results.

Research Paper Conclusion Example

Research Paper Examples for Different Fields

Research papers can be about any subject that needs a detailed study. The following examples show research papers for different subjects.

History Research Paper Sample

Preparing a history research paper involves investigating and presenting information about past events. This may include exploring perspectives, analyzing sources, and constructing a narrative that explains the significance of historical events.

View this history research paper sample:

Many Faces of Generalissimo Fransisco Franco

Sociology Research Paper Sample

In sociology research, statistics and data are harnessed to explore societal issues within a particular region or group. These findings are thoroughly analyzed to gain an understanding of the structure and dynamics present within these communities. 

Here is a sample:

A Descriptive Statistical Analysis within the State of Virginia

Science Fair Research Paper Sample

A science research paper involves explaining a scientific experiment or project. It includes outlining the purpose, procedures, observations, and results of the experiment in a clear, logical manner.

Here are some examples:

Science Fair Paper Format

What Do I Need To Do For The Science Fair?

Psychology Research Paper Sample

Writing a psychology research paper involves studying human behavior and mental processes. This process includes conducting experiments, gathering data, and analyzing results to understand the human mind, emotions, and behavior.

Here is an example psychology paper:

The Effects of Food Deprivation on Concentration and Perseverance

Art History Research Paper Sample

Studying art history includes examining artworks, understanding their historical context, and learning about the artists. This helps analyze and interpret how art has evolved over various periods and regions.

Check out this sample paper analyzing European art and impacts:

European Art History: A Primer

Research Paper Example Outline

Before you plan on writing a well-researched paper, make a rough draft. An outline can be a great help when it comes to organizing vast amounts of research material for your paper.

Here is an outline of a research paper example:

Here is a downloadable sample of a standard research paper outline:

Research Paper Outline

Want to create the perfect outline for your paper? Check out this in-depth guide on creating a research paper outline for a structured paper!

Good Research Paper Examples for Students

Here are some more samples of research paper for students to learn from:

Fiscal Research Center - Action Plan

Qualitative Research Paper Example

Research Paper Example Introduction

How to Write a Research Paper Example

Research Paper Example for High School

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  • How to Write a Research Proposal | Examples & Templates

How to Write a Research Proposal | Examples & Templates

Published on October 12, 2022 by Shona McCombes and Tegan George. Revised on November 21, 2023.

Structure of a research proposal

A research proposal describes what you will investigate, why it’s important, and how you will conduct your research.

The format of a research proposal varies between fields, but most proposals will contain at least these elements:

Introduction

Literature review.

  • Research design

Reference list

While the sections may vary, the overall objective is always the same. A research proposal serves as a blueprint and guide for your research plan, helping you get organized and feel confident in the path forward you choose to take.

Table of contents

Research proposal purpose, research proposal examples, research design and methods, contribution to knowledge, research schedule, other interesting articles, frequently asked questions about research proposals.

Academics often have to write research proposals to get funding for their projects. As a student, you might have to write a research proposal as part of a grad school application , or prior to starting your thesis or dissertation .

In addition to helping you figure out what your research can look like, a proposal can also serve to demonstrate why your project is worth pursuing to a funder, educational institution, or supervisor.

Research proposal length

The length of a research proposal can vary quite a bit. A bachelor’s or master’s thesis proposal can be just a few pages, while proposals for PhD dissertations or research funding are usually much longer and more detailed. Your supervisor can help you determine the best length for your work.

One trick to get started is to think of your proposal’s structure as a shorter version of your thesis or dissertation , only without the results , conclusion and discussion sections.

Download our research proposal template

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Writing a research proposal can be quite challenging, but a good starting point could be to look at some examples. We’ve included a few for you below.

  • Example research proposal #1: “A Conceptual Framework for Scheduling Constraint Management”
  • Example research proposal #2: “Medical Students as Mediators of Change in Tobacco Use”

Like your dissertation or thesis, the proposal will usually have a title page that includes:

  • The proposed title of your project
  • Your supervisor’s name
  • Your institution and department

The first part of your proposal is the initial pitch for your project. Make sure it succinctly explains what you want to do and why.

Your introduction should:

  • Introduce your topic
  • Give necessary background and context
  • Outline your  problem statement  and research questions

To guide your introduction , include information about:

  • Who could have an interest in the topic (e.g., scientists, policymakers)
  • How much is already known about the topic
  • What is missing from this current knowledge
  • What new insights your research will contribute
  • Why you believe this research is worth doing

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

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sample research methodology paper

As you get started, it’s important to demonstrate that you’re familiar with the most important research on your topic. A strong literature review  shows your reader that your project has a solid foundation in existing knowledge or theory. It also shows that you’re not simply repeating what other people have already done or said, but rather using existing research as a jumping-off point for your own.

In this section, share exactly how your project will contribute to ongoing conversations in the field by:

  • Comparing and contrasting the main theories, methods, and debates
  • Examining the strengths and weaknesses of different approaches
  • Explaining how will you build on, challenge, or synthesize prior scholarship

Following the literature review, restate your main  objectives . This brings the focus back to your own project. Next, your research design or methodology section will describe your overall approach, and the practical steps you will take to answer your research questions.

To finish your proposal on a strong note, explore the potential implications of your research for your field. Emphasize again what you aim to contribute and why it matters.

For example, your results might have implications for:

  • Improving best practices
  • Informing policymaking decisions
  • Strengthening a theory or model
  • Challenging popular or scientific beliefs
  • Creating a basis for future research

Last but not least, your research proposal must include correct citations for every source you have used, compiled in a reference list . To create citations quickly and easily, you can use our free APA citation generator .

Some institutions or funders require a detailed timeline of the project, asking you to forecast what you will do at each stage and how long it may take. While not always required, be sure to check the requirements of your project.

Here’s an example schedule to help you get started. You can also download a template at the button below.

Download our research schedule template

If you are applying for research funding, chances are you will have to include a detailed budget. This shows your estimates of how much each part of your project will cost.

Make sure to check what type of costs the funding body will agree to cover. For each item, include:

  • Cost : exactly how much money do you need?
  • Justification : why is this cost necessary to complete the research?
  • Source : how did you calculate the amount?

To determine your budget, think about:

  • Travel costs : do you need to go somewhere to collect your data? How will you get there, and how much time will you need? What will you do there (e.g., interviews, archival research)?
  • Materials : do you need access to any tools or technologies?
  • Help : do you need to hire any research assistants for the project? What will they do, and how much will you pay them?

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Methodology

  • Sampling methods
  • Simple random sampling
  • Stratified sampling
  • Cluster sampling
  • Likert scales
  • Reproducibility

 Statistics

  • Null hypothesis
  • Statistical power
  • Probability distribution
  • Effect size
  • Poisson distribution

Research bias

  • Optimism bias
  • Cognitive bias
  • Implicit bias
  • Hawthorne effect
  • Anchoring bias
  • Explicit bias

Once you’ve decided on your research objectives , you need to explain them in your paper, at the end of your problem statement .

Keep your research objectives clear and concise, and use appropriate verbs to accurately convey the work that you will carry out for each one.

I will compare …

A research aim is a broad statement indicating the general purpose of your research project. It should appear in your introduction at the end of your problem statement , before your research objectives.

Research objectives are more specific than your research aim. They indicate the specific ways you’ll address the overarching aim.

A PhD, which is short for philosophiae doctor (doctor of philosophy in Latin), is the highest university degree that can be obtained. In a PhD, students spend 3–5 years writing a dissertation , which aims to make a significant, original contribution to current knowledge.

A PhD is intended to prepare students for a career as a researcher, whether that be in academia, the public sector, or the private sector.

A master’s is a 1- or 2-year graduate degree that can prepare you for a variety of careers.

All master’s involve graduate-level coursework. Some are research-intensive and intend to prepare students for further study in a PhD; these usually require their students to write a master’s thesis . Others focus on professional training for a specific career.

Critical thinking refers to the ability to evaluate information and to be aware of biases or assumptions, including your own.

Like information literacy , it involves evaluating arguments, identifying and solving problems in an objective and systematic way, and clearly communicating your ideas.

The best way to remember the difference between a research plan and a research proposal is that they have fundamentally different audiences. A research plan helps you, the researcher, organize your thoughts. On the other hand, a dissertation proposal or research proposal aims to convince others (e.g., a supervisor, a funding body, or a dissertation committee) that your research topic is relevant and worthy of being conducted.

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  • Published: 01 March 2024

A Bayesian Bernoulli-Exponential joint model for binary longitudinal outcomes and informative time with applications to bladder cancer recurrence data

  • Michael Safo Oduro 1 , 2  

BMC Medical Research Methodology volume  24 , Article number:  54 ( 2024 ) Cite this article

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A variety of methods exist for the analysis of longitudinal data, many of which are characterized with the assumption of fixed visit time points for study individuals. This, however is not always a tenable assumption. Phenomenon that alter subject visit patterns such as adverse events due to investigative treatment administered, travel or any other emergencies may result in unbalanced data and varying individual visit time points. Visit times can be considered informative, because subsequent or current subject outcomes can change or be adapted due to previous subject outcomes.

In this paper, a Bayesian Bernoulli-Exponential model for analyzing joint binary outcomes and exponentially distributed informative visit times is developed. Via statistical simulations, the influence of controlled variations in visit patterns, prior and sample size schemes on model performance is assessed. As an application example, the proposed model is applied to a Bladder Cancer Recurrence data.

Results and conclusions

Results from the simulation analysis indicated that the Bayesian Bernoulli-Exponential joint model converged in stationarity, and performed relatively better for small to medium sample size scenarios with less varying time sequences regardless of the choice of prior. In larger samples, the model performed better for less varying time sequences. This model’s application to the bladder cancer data showed a statistically significant effect of prior tumor recurrence on the probability of subsequent recurrences.

Peer Review reports

Introduction

Longitudinal data entail observations collected repeatedly on subjects over time. In medical research, the collection of correlated, longitudinal data is a common phenomenon. Ranging from the assessment of response changes and trends over time to understanding disease progression, the benefits longitudinal approaches are enormous [ 1 , 2 ]. A defining feature of longitudinal data is the dependency that characterizes observations extending over time, the type of outcome measured and sometimes, the assumption of fixed time measurements for subjects [ 3 , 4 , 5 ]. The broad assumption of fixed time measurements, predetermined by study design, however is not always a tenable assumption. For instance, in a clinical trial, there is the potential for different visit mechanisms. Study subjects are likely to miss scheduled visits, and a proportion of them are prone to adverse events from investigative treatments. Also, due to poor health conditions, individuals may self elect to visit the investigative site or hospital more intensely than their study counterparts. These occurrences may result not just in unbalanced data for subjects, but also varying visit profiles. Thus, the time structure adopted for the study can be considered informative. In a broad sense, this indicates that outcomes measured at subsequent time points are influenced or can be adapted based on outcomes measured in current time. This necessitates the use of advanced methods that address the informative time structure rather than standard, traditional approaches, which are limited by the assumption of fixed time. To handle such scenarios, Bronsert [ 6 ] developed a classical joint model, involving Gaussian outcomes and exponentially distributed informative time. Later, Alomair [ 7 ] extended Bronsert’s model to include time dependent covariates. Classical informative time joint models have also been developed by Seo [ 8 ], involving longitudinal outcomes from the exponential families and exponentially distributed informative time. These joint models used the maximum likelihood estimation approach for estimating model parameters, and the authors broadly discussed associated computational complexities.

A Bayesian technique for modeling joint longitudinal outcomes and informative time points has been developed by Zaagan [ 9 ] but only for Gaussian distributed outcomes. The objectives of this research paper are twofold. First, we develop a Bayesian joint model for analyzing binary longitudinal outcomes and informative times. Then, via statistical simulations, we examine the influence of controlled variations in subject visit patterns, different prior specifications and sample size schemes on the proposed model. This proceeds with model convergence assessment and model evaluation. The proposed Bayesian-Exponential joint model is applied to a Bladder cancer recurrence data resulting from a clinical trial involving patients with bladder cancer conducted by the Veterans Administration Co-operative Urological Research Group (VACURG) [ 10 , 11 ].

Data and methods

The bayesian bernoulli-exponential joint model formulation and likelihood specification.

The exponential family of distributions covers a broad range of response distributions including Gaussian and Non-Gaussian distributions [ 12 , 13 ]. For example, the Normal, Gamma, Poisson, Bernoulli, and Beta distributions are a part of the parametric set of distributions included in the family. Suppose the observations \(y_{1},y_{2},y_{3},\cdots , y_{n}\) are independent observations of a response variable, the exponential family of distributions from which the independent observations are sampled, can be specified as

\(\theta _{i}\) represents the canonical parameter.

\(\phi\) is a scale parameter and \(m_{i}(\cdot ), s(\cdot )\) , and \(r(\cdot )\) are known functions which relates to the variances of distributions in the exponential family.

\(m_{i}(\phi )\) can be specified as \(m_{i}(\phi )=\frac{\phi }{u_{i}}\) , and \(u_{i}\) ’s are predetermined weights.

The canonical or location parameter characterizes a so called canonical link function, and relates to the means of the distributions in the exponential family.

Assume we have a set of n participants enrolled in a clinical trial, have to visit an investigative site over time and are followed over an interval from \((0,\tau ]\) . A response observation for the i th participant measured at the k th visit time point can be specified as \(y_{ik}\) . We can further specify vectors of individual responses and their associated visit time points as

Here, the subscript \(n_{i}\) allows for varying participant visit times. We can thus specify the joint distribution of recorded responses and time points as

where \(\Theta\) is a vector of unknown parameters to be estimated. Using these ideas, and in line with Seo [ 8 ] we can further specify a model that incorporates the joint distribution of responses and time points \(\varvec{y}_{ik}\) and \(\varvec{t}_{i_{n}}\) with the underlying assumption that the current response depends on the one-step prior response \(\left( y_{ik-1}\right)\) , and current visit time point \(\left( t_{i k}\right)\) . It is important to note, however, that subsequent responses, \(y_{i k}\) will not be solely conditioned on observation time, \(t_{i k}\) but also on the most recent prior response, \(y_{ik-1}\) and observation time. This distribution can be specified as;

This formulation forms the premise for specifying the joint model with response observations sampled from the Bernoulli distribution. Time is considered informative and assumed to be exponentially distributed. The joint distribution for binary longitudinal outcomes and informative time given the underlying assumption of a one step dependency can be specified as;

Note that, \(\mu _{ik}=E\left( Y_{ik}\right) =P\left( Y_{ik}=1\right)\) .

More specifically for the Bernoulli distribution the link function can be specified as a logit link

which in the context of this study can be expressed as;

Furthermore, the specified mean function for the initial value for the i th participant and that after the initial value can be expressed as

respectively. Hence, our final model specification for the parametric joint Bernoulli-Exponential model can be expressed as;

\(\varvec{\alpha }\) is a vector of regression parameters denoting the effect of covariates on observed responses.

\(\psi\) represents the effect of the prior responses on average current responses.

\(\vartheta\) represents the effect of current response time on the mean responses,

\(\xi\) is a constant parameter associated with time

\(\gamma\) characterizes the effect of previous response on mean time and \(\varvec{X}\) is the design matrix.

The resulting likelihood function, a product of the density functions for s subjects, can be specified as,

It is further important to clarify, that one key underlying assumption of this model, following Lin and Ying [ 14 ],Lin, Scharfstein, and Rosenheck [ 15 ], Liang, Wenbin and Zhiliang [ 16 ] and Sun, Sun, and Liu [ 17 ], is that censoring time, \(Z_i\) in this study is noninformative in the sense that given covariates \(\left( X_i\right) , Z_i\) is independent of the observation times \(\left\{ t_{i k}\right.\) , \(k \ge 1\}\) and longitudinal outcomes \(Y_i(\cdot )\) . This basically means that given the covariate history up to time k , the distribution of the future covariate path up to any time \(t>k\) is independent of whether or not there is an observation on \(X_i\) at time k .

Specification of priors

After the likelihood function of the Bernoulli-Exponential joint model distribution has been specified, the next step in the Bayesian model specification is the identification of a suitable prior. In this study, informative and non-informative priors are considered. Both priors serve important roles in Bayesian analysis, and the choice between them depends on the specific goals and available information in a given analysis [ 18 ]. Non-informative priors, also known as weak,vague or diffuse priors, are designed to have minimal influence on the posterior distribution. They can make Bayesian analysis robust to situations where there is little prior information or when prior beliefs are uncertain. They prevent strong prior assumptions from biasing results when there is limited prior knowledge [ 19 ]. One of the primary benefits of informative priors, on the other hand, is that they allow to incorporate expert domain knowledge and prior information into the analysis [ 20 , 21 ]. This is invaluable when experts have insights that can improve parameter estimation, and, in situations with limited or noisy data, informative priors can lead to more stable and accurate parameter estimates. Finally, informative priors explicitly quantify prior beliefs and uncertainty, which allows to integrate these beliefs with observed data. In this study, for both informative and non-informative prior scenarios, we consider the vector of mean parameters \((\varvec{\alpha })\) as having a multivariate normal distribution [ 19 , 22 , 23 , 24 ]. This is specified as;

Furthermore, we consider the parameters associated with time or visit to similarly follow a Gaussian distribution;

Note that the prior distributions of our joint model parameters are considered independent and thus,

For the informative prior setting, fixed values for the prior means, \((\varvec{\mu }_{\alpha },{\mu }_{\vartheta },{\mu }_{\psi },{\mu }_{\xi },{\mu }_{\omega })\) and their corresponding variances \((\varvec{\Sigma }_{\alpha },\nu _{\vartheta },\nu _{\psi },\nu _{\xi },\nu _{\omega })\) are adopted, since we do not have expert or historical estimates yet for these kind of studies. More specifically, we can denote the mean vector of \(\varvec{\alpha }\) , \(\varvec{\mu }_{\alpha }\) with a prior mean vector and corresponding covariance matrix as;

where \(\text{I}_{s}\) represents an identity matrix whose dimension depends on s individuals and \(\phi\) . More broadly, we set predetermined prior mean values for the visit parameters as;

and their corresponding prior variances as

Regarding the non-informative prior setting, two approaches are considered. First, Gaussian non-informative priors are adopted for all mean and variance parameters of both the response and time parameters. More broadly, to express prior ignorance, the prior means \(({\mu }_{\alpha },{\mu }_{\vartheta },{\mu }_{\psi },{\mu }_{\xi },{\mu }_{\omega })\) are set to zero and the variance-covariance for \(\phi \varvec{\Sigma }_{\alpha }\) can be set as a diagonal matrix with large variance. Similarly the corresponding prior variances for the other parameters are set very large to express prior ignorance. Thus, the non-informative priors are set up as,

For the second case of non-informative prior, we consider the Jeffreys prior [ 25 ] an appealing reference prior widely used in Bayesian inference. This prior is considered for the response/outcome parameters and Gaussian non-informative priors are still considered in this study for visit parameters. The Jeffreys prior is obtained by applying the Jeffreys rule which defines the prior density to be directly proportional to the square root of the determinant of the Fisher information matrix. That is, for a set of parameters \(\varvec{\theta }=\left( \theta _{1}, \ldots , \theta _{n}\right)\) , the Jeffreys prior is given by,

The Fisher information matrix is defined by,

and L is the likelihood function that specifies the probability for data y given the parameters \(\theta\) . It is appropriate so far as \(\textbf{I}(\theta )\) is positive definite. Aside its geometric interpretation, one of the appealing reasons for its usage is the concept of parameterization invariance [ 26 ]. This means that the prior is invariant with regards to one-to-one transformations. The principle can be extended for multidimensional parameters. To establish ideas for the Jeffreys prior for response parameters, which result from the exponential family of distributions, the likelihood functions of the distributions and associated score vectors need to be specified.

Let \(\phi _{i}\) ’s be known and \(\varvec{X}^{\prime }\) assume a rank q . Also let, \(\theta _{i}=z\left( \varvec{x}_{i}^{\prime } \varvec{\alpha }\right)\) and \(m^{-1}\left( \phi _{i}\right) =\phi ^{-1}w\) . The likelihood function for Generalized linear models with responses from the exponential family of distributions can generally be specified as;

The score vector is represented by;

The resulting Fisher information matrix is specified as;

\(\varvec{P}=\text {Diag}\left( m^{-1}\left( \phi _{i}\right) ,\cdots , m^{-1}\left( \phi _{n}\right) \right)\) which is an \(n \times n\) diagonal matrix of the weights \(w_{i}\) .

\(\varvec{V}(\varvec{\alpha })=\text {Diag}\left( s^{\prime \prime }\left( \left( \varvec{x}_{1}^{\prime } \varvec{\alpha }\right) \right) , \cdots , s^{\prime \prime }\left( \left( \varvec{x}_{n}^{\prime } \varvec{\alpha }\right) \right) \right)\) which reflects an \(n \times n\) diagonal matrix of \(v_{i}=\) \(\frac{\partial ^{2} s\left( \theta _{i}\right) }{\partial \theta _{i}^{2}}\) .

\(\varvec{\Delta }(\varvec{\alpha })=\text {Diag}\left( s^{\prime }\left( \varvec{x}_{1}^{T} \varvec{\alpha }\right) , \cdots , s^{\prime }\left( \varvec{x}_{n}^{T} \varvec{\alpha }\right) \right)\) is an a \(n \times n\) diagonal matrix of \(\delta _{i}=\frac{\partial s\left( \theta _{i}\right) }{\partial \eta _{i}}\) and is an adjustment for the link function.

The Jeffreys prior thus for \(\varvec{\alpha }\) assuming \(\phi\) is known, is specified as

Based on this derivation, Jeffreys non-informative prior considered for response parameters and Gaussian non-informative priors maintained for the visit parameters can be specified as;

Posterior distribution specification and Bayesian joint parameter estimation

The next step in the Bayesian model development is the specification of the posterior distribution, which has a directly proportional relationship with the model likelihood and the priors specified. For the scenario where Gaussian priors are considered for both the response and visit parameters and also for both informative and non informative settings, the resulting Bayesian Bernoulli-Exponential joint model posterior specification can be obtained as;

Also for the scenario where Jeffreys priors are considered for the parameters of the Bernoulli response and Gaussian priors for the visit parameters (non informative settings), the resulting Bayesian Bernoulli-Exponential joint model can be parameterized as;

Here, \(\varvec{V}(\varvec{\alpha })=\text {diag}\left( v_{1},v_{2} \ldots , v_{n}\right)\) and \(v_{i}=\mu _{ik}(1-\mu _{ik})\) . Note that,

The next goal is to obtain posterior summary estimates for inference. Analytical calculations of the posterior distributions are possible, but often untenable due to laborious calculations involving the integration constant. Integral approximation methods can be adopted but only if few parameters are involved [ 19 , 24 ]. In situations such as this study involving many parameters to be estimated, one can resort to Markov Chain Monte Carlo Methods (MCMC) [ 27 ]. The MCMC methods are viable simulation approaches for sampling from posterior distributions and computing posterior summary measures. They are premised on a Markov Chain construction that subsequently converges to a so-called target distribution. The two most popular MCMC methods are the Gibbs sampling and the Metropolis-Hastings algorithm [ 27 , 28 , 29 ]. In this study, we adopt the Gibbs sampling procedure for generating samples from the joint posterior distributions of the unknown parameters in our model. It is important to clarify, however, that the Gibbs sampler, performs iterative draws from posterior conditional distributions instead of directly sampling from the joint posterior distribution. This approach enhances the utility of the Gibbs Sampler, especially when dealing with complex joint posteriors that can be challenging to handle directly. Then, posterior summaries can be computed. In each step of the algorithm, random values are generated from unidimensional distributions [ 30 ]. A brief summary of the Gibbs sampling algorithm is as follows;

Predetermined initial values \(\varvec{\theta }^{(0)}\) need to be specified.

For \(t=1, \ldots , T\) iterations,

Set \(\varvec{\theta }=\varvec{\theta }^{(t-1)}\) .

For \(k=1, \ldots , r\) , we can update \(\theta _{k}\) from \(\theta _{k} \sim p\left( \theta _{k} \mid \theta _{1}, \ldots , \theta _{k-1}, \theta _{k+1}, \ldots , \theta _{r}\right)\) .

Now, if the current state of the chain \(\theta\) is \(\theta ^{(t)}=\left( \theta _{1}^{(t)} \ldots , \theta _{r}\right)\) , then we can generate the new parameters by,

The distributions, \(p\left( \theta _{k} \mid \theta _{1}^{(t)}, \theta _{2}^{(t)}, \ldots , \theta _{k-1}^{(t)}, \theta _{k+1}^{(t-1)}, \ldots , \theta _{q}^{(t-1)}, \varvec{y}\right)\) are known as the full, complete or conditional distributions. Summarily, the Gibbs sampling algorithm helps to iteratively generate samples from our posterior distribution based on prespecified starting values. Initial portions of the Markov chains are discarded in an attempt to mask the influence of initial values. This is called the burn-in part. Resulting posterior summary measures such as the posterior mean, posterior standard deviation and Bayesian credible intervals are obtained from the MCMC output. Furthermore, we assess convergence of the Markov chains via the diagnosis of ergodic mean plots of estimated parameters and the Heidelberger and Welch diagnostic test which is a more formal convergence diagnostic method [ 31 ].

Model evaluation

To assess the Bayesian Bernoulli-Exponential joint model, the Bayesian model evaluation criteria called the Deviance Information Criterion (DIC) is used [ 32 ]. The DIC measure comprises a “goodness of fit” and “complexity” term and is obtained as;

where \(\hat{D}(\varvec{\theta })\) is the deviance calculated at the posterior mean of the parameters and \(p_{\text{D}}\) characterizes the “effective” number of parameters relating the complexity of the models. \(p_{\text{D}}\) is the difference between the posterior mean deviance, \(\overline{D(\varvec{\theta })}\) and deviance calculated at the posterior mean of the parameters, \(\hat{D}(\varvec{\theta })\) . Smaller values of DIC justify a better fit of the model. In line with this derivation, the DIC measure for the Bayesian Bernoulli-Exponential model is specified as;

Simulation study

In order to assess the Bayesian Bernoulli-Exponential model in terms of how it can be influenced by controlled variations in sample size, visit schema and types of prior distributions on the parameter estimates we present in this subsection, a simulation study. More precisely, the simulation study helps establish the validity of the joint model in random scenarios via data generation and parameter estimation. It is important to clarify, however, that this present study is an extension of the studies of Bronsert [ 6 ], Lin [ 33 ], Seo [ 8 ] and Zaagan [ 9 ] and thus for computational convenience, an abundant level of consistency is maintained in terms of simulation conditions. All simulations are performed in R software via the R2OpenBugs package. This package provides a means to program Bayesian models in R via an OpenBugs software [ 34 , 35 ]. To develop the Bayesian joint model, the structure of the data to be simulated is clearly defined. We simulate data involving two categorical variables, each having three levels, and two continuous variables. The longitudinal responses are simulated from a Bernoulli distribution. The first response is simulated from the distribution, and then the subsequent response is computed based on the relationship between the prior outcome and the prior time for predicting the average response based on starting parameter values in Table  1 . It is important to clarify, however that during the simulation exercise, only “plausible” starting values from the range of starting values in Table  1 are utilized. It is not the intent of this study to analyze the impact of all four range of starting values. The visit times for each of the corresponding responses are simulated from an exponential distribution.

Furthermore, we simulate design structures that consider varying visit schemes and sample sizes to effectively study trends or patterns associated with the model. In this study, three varying sample sizes with four sub design visit structures entailing both balanced and unbalanced visit structures are considered and shown in Table  2 . Also, three prior schemes are considered, that is Gaussian informative, Gaussian non-informative and Jeffreys non-informative priors.

Thus, the simulation matrix involves three varying sample size designs, three varying prior schemes and three visit design structures. To further clarify the visit structure, as an example to signal an unbalanced visit pattern, when the sample size is 180 and the number of observations is 20 & 6 , this exemplifies 90 participants having 20 recorded observations and another 90 subjects have 6 measured outcomes each. This simulation design scheme results in 27 differing designs for the simulation analysis of the Bayesian Bernoulli-Exponential joint model.

After data generation, the simulation analysis involves estimating the joint model parameters via the package R2Openbugs in R software. It commences by first “sinking” in generated parameter values which that serve as initial values for the MCMC estimation process. Then, the likelihood of the Bayesian joint model is calculated based on the design structures and priors specified. Parameter estimation proceeds with the Gibbs Sampling approach, which has earlier been discussed. This generates dependent Markov chains for our model parameters by drawing samples from the posterior distribution using initial parameter values that were embedded in the simulation design. Markov chains are run iteratively 30,000 times, and the first 10,000 iterations are discarded to serve as burn-in, effectively mitigating the influence of the initial values. Thinning intervals of three iterations are considered to monitor autocorrelations of the generated values. Subsequently, to monitor convergence of Markov chains and their associated posterior parameters, the Heidelberger and Welch convergence tests are computed. Then, posterior summaries such as the mean, standard deviation, and credible interval limits are presented. It is instructive to note that the simulations were replicated a 1000 times and inferences were premised on the averaged estimates and associated credible intervals. Finally, inferences via comparisons for different specification of the prior distribution and their sample size and visit design schemes for the model are made along with Deviance Information Criterion measures.

Simulation results: model convergence assessment of the Bayesian Bernoulli-Exponential joint model

To evaluate convergence of the Markov chains of the model parameters, a formal diagnostic test, called the Heidelberger and Welch test [ 31 ] is used. It is expected that after the burn-in period, the Gibbs Sampling algorithm produces samples from the posterior distribution that attains a stationary distribution. The Heidelberger and Welch test constitutes a stationary and half-width test and calculates a test statistic to accept or reject the null hypothesis that the Markov chains are from a stationary distribution. The half-width test is based on a computed \(95\%\) confidence interval for the mean, using the chain that earlier passed the stationarity test. The resulting ratio of the interval half-width and the mean compared with a threshold ( \(\varepsilon =0.1\) ) determines whether the half-width test is passed or not. More precisely, the test passes if the ratio between the half-width and the mean is lesser than \(\varepsilon\) . Selected convergence results based on the Heidelberger and Welch test are presented for the Bayesian Bernoulli-Exponential joint model across select scenarios and shown in the Tables  3 , 4  and 5 . These results cut across all prior scenarios (informative, non-informative, Jeffreys non-informative Prior), sample sizes (18, 54, 180) and visit patterns (10, balanced), (5 &3, Unbalanced), (20 & 6 ,Unbalanced). Inferring from the Heidelberger and Welch tests conducted across the broad range of scenarios selected, no issues were observed with the convergence of the MCMC chains for the Bayesian Bernoulli-Exponential Joint model. More precisely, the p -values resulting from the stationarity test for all estimated model parameters, regardless of prior, sample size or visit schemes were statistically insignificant. This suggests that the sampled values for parameters are from a stationary process. A further indication is that our model parameter estimation can be implemented with precision because MCMC chains are in a stationary distribution.

Simulation results: parameter estimation and evaluation of the Bayesian Bernoulli-Exponential model

In this section, the influence of controlled variations in sample size, visit sequences and type of prior distributions on the estimated parameters of the Bayesian Bernoulli-Exponential model are examined. Consistency in the direction of these estimates and their associated credible intervals are checked. For ease of reporting, we present a select number of results from the various simulation scenarios. Posterior means, standard deviations and credible intervals of select scenarios are presented in Tables  6 , 7 , 8 , 9  and 10 .

Fixing sample sizes and priors across scenarios and examining the effect of varying sequences on parameter estimates, a consistent trend in magnitude and direction of the estimates and their log-transformation were observed across all scenarios. For example, the parameter estimates of results obtained from the model when sample size and time sequence 54(10) and \(54(20 \& 6)\) , 18(10) and \(18(5 \& 3)\) , 180(10) and \(180(5 \& 3)\) under informative prior scheme were not markedly different in terms of their magnitude and direction. As an example, the posterior means and standard deviations obtained for the model scenario, sample size and visit scheme 180(10) under informative prior scheme from were \(\alpha _{1}:0.100(0.183)\) , \(\alpha _{2}:0.065(0.194)\) , \(\alpha _{3}:0.106(0.069)\) , \(\alpha _{4}:-0.036(0.137)\) , \(\alpha _{5}:0.226(0.138)\) , \(\alpha _{6}:0.348(0.0.058)\) , \(\alpha _{7}:0.724(0.068)\) , \(~\gamma :-0.023(0.0560)\) , \(\psi :-0.916(0.120)\) , \(\vartheta :-0.116(0.026)\) , \(\xi :-0.972(0.0.048)\) . These estimates are not markedly different in magnitude and direction from when the time sequence changed to \(20 \& 6\) under the same scenario where the resulting estimates obtained were \(\alpha _{1}:0.200(0.133),\alpha _{2}:0.122(0.124),\alpha _{3}:0.335(0.125),\alpha _{4}:0.156(0.128),\alpha _{5}:0.149 (0.128), \alpha _{6}:0.302 (0.055), \alpha _{7}:0.779 (0.063), \gamma :-0.058 (0.050), \psi :-1.052 (0.114), \vartheta :-0.105 (0.024),\xi :-0.957 (0.044)\) . This pattern was similarly observed across the other scenarios, fixing sample sizes, priors and varying the time-sequences and broadly demonstrates a consistency in estimation performance. This further indicates that varying time sequences do not considerably affect the resulting estimates. Examining the credible interval(CI) widths under the different schemes reveal an interesting trend. As the sample sizes across all scenarios increased, albeit keeping priors and time sequences constant, the CI widths were increasingly narrow, implying that when our proposed model is applied to datasets of increasing sample sizes, the resulting estimates are obtained with higher precision. For instance, as an example, we compare parameter estimates and their CI widths under a select Gaussian non-informative prior scenario for these model scenarios 18(10), 54(10) and 180(10) (see Table  11 ). The trend observed from the presented estimates are quite obvious; increasing sample sizes applied to the proposed Bayesian Bernoulli-Exponential model increases precision of the model estimates. This broadly cuts across all scenarios.

Simulation results: evaluation of the Bayesian Bernoulli-Exponential model

Finally, model performance is evaluated under the various simulation scenarios via the Deviance Information Criterion (DIC). Since there are a lot of DIC values computed for varying scenarios, they are presented graphically for ease of evaluation and clarity. The DIC plots of the selected simulation scenarios applied to the model are presented in Figs.  1 , 2 , 3 , 4 , 5 , 6 , 7  and 8 . First, we fix sample sizes and compare how the model performs across the type of prior and visit sequence. Regardless of the kind of prior chosen for the model parameters, it is observed in Fig.  1 that in the smallest sample considered, 18, the model performs better for the time sequence \(5 \& 3\) , reflected by lower DIC values across all prior scenarios. This is followed by the balanced time sequence, 10. In fact, there’s no marked difference between the DIC value of the time sequence \(5 \& 3 (599.8)\) and 10(628.4) when considering the Jeffreys prior and fixing the sample size at 18. This trend is consistently observed, even when the sample sizes are fixed at 54 and 180 (see Figs.  2 and  3 ). The model still performs better for the time sequence \(5 \& 3\) followed by 10. The next step in the model evaluation process involved fixing priors and comparing the models across competing sample sizes and sequences. For both Gaussian informative and non-informative priors, the DIC’s are very large for the time sequence \(20 \& 6\) and sample size 180 signaling that the model may not be robust for scenarios where visit sequences of individuals vary significantly. When Jeffreys prior is considered, yet again DIC’s obtained for the model in small sample size 18 and sequence type \(5 \& 3\) are very low indicating better performance followed closely by sample size 54, time sequence \(5 \& 3\) . This model scenario performs better across all samples and sequences than the considered Informative and Non-Informative Prior Scenario. The DIC values were at par in samples 54 and 180 for time sequence 10 and \(5 \& 3\) when the Jeffreys prior was considered. Finally, an observation of model performance across sample size and prior schemes while keeping the time sequence fixed is made. Across time sequence \(5 \& 3\) , the model performs better overall for sample size 18 and 54 regardless of prior chosen. No marked differences are observed however when the Jeffreys prior is used for sample size 18 and 54 as evidenced by Fig.  5 . Furthermore, model performance does not broadly vary for the sample size 180, regardless of the prior chosen for sequence \(5 \& 3\) and \(20 \& 6\) . The results for the visit sequence 10 are quite consistent with \(5 \& 3\) when compared. Models perform better in small sample size 18 scenarios as reflected by their lower DIC values, followed by 54.

The DIC values for sample size 54 and 180, however are close when the Jeffreys prior is considered for time sequence 10. Overall, model evaluation of the Bayesian Bernoulli-Exponential Model suggest a relatively better fit for small and medium sample size scenarios (18 and 54) with less varying time sequences (5 &3) and (10), regardless of prior choice. For larger samples (180), the models performs fairly well for less varying time sequences (5 &3) but not significantly so for time sequences (20& 6) regardless of the choice of prior.

figure 1

Deviance information criterion plot for keeping sample sizes fixed at 18 and examining influence across priors and design schemes

figure 2

Deviance information criterion plot for keeping sample sizes fixed at 54 and examining influence across priors and design schemes

figure 3

Deviance information criterion plot for keeping sample sizes fixed at 180 and examining influence across priors and design schemes

figure 4

DIC plot for keeping prior fixed at gaussian non-informative and examining influence across sample size and design schemes

figure 5

Deviance information criterion plot for keeping visit sequence fixed at \(5 \& 3\) and examining influence across sample size and prior schemes

figure 6

DIC plot for keeping visit sequence fixed at 10 and examining influence across sample size and prior schemes

figure 7

Deviance information criterion plot for keeping prior fixed at gaussian informative and examining influence across sample size and design schemes

figure 8

DIC plot for keeping prior fixed at jeffreys non-informative and examining influence across sample size and design schemes

A model application to bladder cancer recurrence data

In this section, the proposed Bayesian Joint Bernoulli-Exponential model is applied to a real-world dataset, called the Bladder Cancer Data. This data is openly available in \(\text{R}\) software, specifically in the “Survival” package [ 36 ] and results from a clinical trial on patients with bladder cancer conducted by the Veterans Administration Co-operative Urological Research Group (VACURG) [ 10 , 11 ]. The bladder cancer dataset in \(\text{R}\) software comprises information on 85 subjects, measured four times, with randomly assigned treatments of only thiotepa or a placebo. 38 patients are assigned to the placebo group and 47 to the treatment(thiotepa) group. Data on patient experienced number of recurrences are collected including the number of initial tumours present pre-trial randomization. Other variables include “stop”, which measures the time interval in months since the last visit. The next scheduled visit is dependent on bladder tumor recurrence at the time of measurement, indicating that time can be considered informative, and that subsequent visits are likely be influenced by previous visits. Also, the intensity of visits depend on tumor recurrences. Furthermore, there is an “event” variable, which is a binary variable representing the recurrence of tumor(1) or (0) for non-recurrence attributable to reasons like death. The variables along with their description are given in Table  12 below.

This data is analyzed with the following objectives in mind. Is there an effect of treatment type, size in centimeters(cm) of the largest initial tumor, initial number of tumors on the likelihood of tumor recurrence? Furthermore, is there an effect of prior recurrences(outcomes) on the likelihood of current recurrence? To answer these research questions, our proposed Bayesian Bernoulli-Exponential Joint model is fitted to the data. The binary “event” variable is used as the response and the predictors included in the model are treatment type, size in cm of the largest initial tumor, initial number of tumors and other time variables. Just as previously discussed in the Data and methods section, the Bayesian model involves the specification of a joint likelihood, priors and then the posterior distribution.

Here, three types of priors are considered and compared across the models. In this regard, the non-informative Gaussian priors considered for this model is,

The Gaussian Informative priors considered for this model is,

Furthermore, we consider Jeffreys non-informative priors for the \(\alpha\) parameters and Gaussian non-informative priors for the visit parameters. The resulting posterior distribution of the Bayesian Bernoulli-Exponential Joint model for the bladder cancer data, for the instance where the Jeffreys prior considered for the parameters of the Bernoulli response process and Gaussian priors for the visit parameters in non informative settings is considered is;

Here, \(\varvec{V}(\varvec{\alpha })=\text {diag}\left( v_{1},v_{2} \ldots , v_{n}\right)\) and \(v_{i}=\mu _{ik}(1-\mu _{ik})\) . and,

\(\alpha _{s}\) are regression parameters representing the effect of the predictors; treatment type( \(x_{2}\) ), initial number of tumors, ( \(x_{3}\) ) and size in (cm)( \(x_{4}\) ) of the largest initial tumor on the likelihood of tumor recurrence.

\(\psi\) represents the effect of the prior recurrence on the mean response of the current recurrence and \(\vartheta\) characterizes the effect of current recurrence time on the mean recurrence,

\(\xi\) is a constant parameter associated with time and \(\gamma\) is the effect of the previous recurrence on the mean time.

Other components are already explained thoroughly in the Data and methods section. Note that the posterior distribution changes when the priors change in the Gaussian and non-Gaussian settings considered for all parameters. Then, after the posterior specification, we proceed with the joint parameter estimation with the Gibbs sampling approach in R software. For each of the three prior scenarios considered, the Markov chains are run iteratively 30,000 times, and the first 10,000 iterations are discarded to serve as burn-in. Convergence of the markov chains and associated posterior parameters are monitored via the Heidelberger and Welch tests. Then, posterior summaries are computed. Parameter significance is inferred via credible intervals and the models are compared with the Deviance Information Criteria Measure. Results of the Heidelberg and Welch convergence tests from the application to the bladder cancer data with the different prior scenarios are presented in Table  13 . Inferring from the tests conducted, no issues were observed with the convergence of the MCMC chains. Overall, we can proceed with posterior summary inference with precision since the MCMC chains are in a stationary distribution.

After convergence assessment of the model, inference based on the posterior summary measures is the next step. Posterior means, standard deviations and associated credible intervals of the prior scenarios are presented in Table  14 along with their corresponding DIC’s. The best model is chosen based on the least DIC value. Observing the results, the model under the Jeffreys non-informative prior, yielded the least DIC (1108) value. Ergo, parameter inference is based on the Bayesian Bernoulli-Exponential model with Jeffreys prior specified. The results demonstrate that the effect of treatment type is statistically significant on the likelihood of cancer recurrence inferring from its credible interval \(\alpha _{2}=0.216\) (0.232, 0.411). The initial number of tumors have a significant effect \(\alpha _{3}=0.036\) (0.001, 0.108) on the likelihood of cancer recurrence and hence a significant prognostic factor. Furthermore, the size in cm of the largest tumor has a significant marker on the likelihood of cancer recurrence. Afterwards, the time parameters are observed. The effect of prior tumor recurrence on the mean response of current tumor recurrence, represented by \(\psi\) is statistically significant \(-0.408(-1.009,-0.135)\) , indicating that previous tumor recurrences influence the probability of subsequent recurrences. Additionally, the effect of current recurrence time( \(\vartheta\) ) is significant on average recurrence, reflected by the estimated probability (0.157)(0.018, 0.337).

Discussions and conclusions

Broad assumptions underlie the usage of longitudinal analysis approaches, ranging from univariate designs to the even the most complex conditional and marginal modeling approaches. One of the common assumptions, albeit implausible in certain scenarios, is the supposition that time is always fixed and predetermined by statistical design. Phenomenons may alter the time trajectory of study subjects, like sickness or adverse events in clinical trials, which may result in not only irregular time points for subjects, but also imbalanced data and differing visit intensities. This implies current visit outcomes being informative to subsequent ones. It is also important to emphasize that the issue of informative censoring may be less problematic in the context of an informative time/schedule designs, given the assumed observation schedule protocols. In simpler terms, individuals with more severe conditions requiring early interventions or treatments, which could lead to informative censoring, would also have shorter observation schedules and, consequently, more “frequent” measurements. This assumption underlies the simulation design for this study. In this article, we have developed a Bayesian joint model for longitudinal outcomes from the exponential family of distributions with particular emphasis on Bernoulli distributed longitudinal outcomes and exponentially distributed informative time points. An assessment of the influence of controlled sample size scenarios, visit and prior specification schemes on the estimated parameters of the proposed Bayesian Bernoulli-Exponential joint model was performed via simulations and was evaluated based on Deviance Information Criteria.

The methods commenced with specifying likelihoods for the joint outcome and time distributions, specification of priors, and then a discussion on the Markov Chain Monte Carlo Approach for estimating posterior parameters. The priors considered were Gaussian informative priors, Gaussian non-informative priors and Jeffreys non-informative priors. Convergence analysis was performed with the Heilderberg and Welch Test. Once the models converged, posterior inference followed and models were evaluated based on Deviance Information Criteria. Inference from the Heidelberger and Welch Tests conducted across selected simulation scenarios for the Bayesian Bernoulli-Exponential broadly suggested no pertinent issues with the convergence or stationarity of MCMC chains for estimated parameters irrespective of prior specified, sample size or visit schemes. Fixing sample sizes and priors across selected scenarios of the model and examining effect of varying sequences on parameter estimates, a consistent trend in magnitude and direction of the estimates and their transformations were observed.

As sample sizes increased, albeit keeping priors and time sequences constant, credible interval widths were increasingly narrow, indicating that when the proposed model is applied to datasets of increasing sample sizes, resulting estimates are obtained with higher precision. Overall, evaluation made for the Bayesian Bernoulli-Exponential model indicated better performance for the less intense visit sequence \(5 \& 3\) scenario, reflected by lower DIC values, followed by the balanced visit sequence 10 regardless of sample size or prior type. Sample sizes across various simulation scenarios performed similarly well, only that the difference in performance was largely attributable to the sequence of individual visits. Finally, the proposed model has been applied to a bladder cancer recurrence data to serve as an application example.

Availability of data and materials

Beyond the simulation analysis, the data that support the findings of this study and for the model application is openly available in R software, called ‘bladder’ specifically in the “Survival” package [ 36 ] and results from a clinical trial on patients with bladder cancer conducted by the Veterans Administration Co-operative Urological Research Group (VACURG) [ 10 , 11 ].

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Acknowledgements

The author acknowledges the support of Dr. Khalil Shafie, Dr. Han Yu and Dr. Khaledi Bahaedin at the University of Northern Colorado for their insightful comments on this project during the author’s Ph.D.’s dissertation.

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Oduro, M. A Bayesian Bernoulli-Exponential joint model for binary longitudinal outcomes and informative time with applications to bladder cancer recurrence data. BMC Med Res Methodol 24 , 54 (2024). https://doi.org/10.1186/s12874-024-02160-2

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