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Mathematics and Statistics Theses and Dissertations

Theses/dissertations from 2023 2023.

Classification of Finite Topological Quandles and Shelves via Posets , Hitakshi Lahrani

Applied Analysis for Learning Architectures , Himanshu Singh

Rational Functions of Degree Five That Permute the Projective Line Over a Finite Field , Christopher Sze

Theses/Dissertations from 2022 2022

New Developments in Statistical Optimal Designs for Physical and Computer Experiments , Damola M. Akinlana

Advances and Applications of Optimal Polynomial Approximants , Raymond Centner

Data-Driven Analytical Predictive Modeling for Pancreatic Cancer, Financial & Social Systems , Aditya Chakraborty

On Simultaneous Similarity of d-tuples of Commuting Square Matrices , Corey Connelly

Symbolic Computation of Lump Solutions to a Combined (2+1)-dimensional Nonlinear Evolution Equation , Jingwei He

Boundary behavior of analytic functions and Approximation Theory , Spyros Pasias

Stability Analysis of Delay-Driven Coupled Cantilevers Using the Lambert W-Function , Daniel Siebel-Cortopassi

A Functional Optimization Approach to Stochastic Process Sampling , Ryan Matthew Thurman

Theses/Dissertations from 2021 2021

Riemann-Hilbert Problems for Nonlocal Reverse-Time Nonlinear Second-order and Fourth-order AKNS Systems of Multiple Components and Exact Soliton Solutions , Alle Adjiri

Zeros of Harmonic Polynomials and Related Applications , Azizah Alrajhi

Combination of Time Series Analysis and Sentiment Analysis for Stock Market Forecasting , Hsiao-Chuan Chou

Uncertainty Quantification in Deep and Statistical Learning with applications in Bio-Medical Image Analysis , K. Ruwani M. Fernando

Data-Driven Analytical Modeling of Multiple Myeloma Cancer, U.S. Crop Production and Monitoring Process , Lohuwa Mamudu

Long-time Asymptotics for mKdV Type Reduced Equations of the AKNS Hierarchy in Weighted L 2 Sobolev Spaces , Fudong Wang

Online and Adjusted Human Activities Recognition with Statistical Learning , Yanjia Zhang

Theses/Dissertations from 2020 2020

Bayesian Reliability Analysis of The Power Law Process and Statistical Modeling of Computer and Network Vulnerabilities with Cybersecurity Application , Freeh N. Alenezi

Discrete Models and Algorithms for Analyzing DNA Rearrangements , Jasper Braun

Bayesian Reliability Analysis for Optical Media Using Accelerated Degradation Test Data , Kun Bu

On the p(x)-Laplace equation in Carnot groups , Robert D. Freeman

Clustering methods for gene expression data of Oxytricha trifallax , Kyle Houfek

Gradient Boosting for Survival Analysis with Applications in Oncology , Nam Phuong Nguyen

Global and Stochastic Dynamics of Diffusive Hindmarsh-Rose Equations in Neurodynamics , Chi Phan

Restricted Isometric Projections for Differentiable Manifolds and Applications , Vasile Pop

On Some Problems on Polynomial Interpolation in Several Variables , Brian Jon Tuesink

Numerical Study of Gap Distributions in Determinantal Point Process on Low Dimensional Spheres: L -Ensemble of O ( n ) Model Type for n = 2 and n = 3 , Xiankui Yang

Non-Associative Algebraic Structures in Knot Theory , Emanuele Zappala

Theses/Dissertations from 2019 2019

Field Quantization for Radiative Decay of Plasmons in Finite and Infinite Geometries , Maryam Bagherian

Probabilistic Modeling of Democracy, Corruption, Hemophilia A and Prediabetes Data , A. K. M. Raquibul Bashar

Generalized Derivations of Ternary Lie Algebras and n-BiHom-Lie Algebras , Amine Ben Abdeljelil

Fractional Random Weighted Bootstrapping for Classification on Imbalanced Data with Ensemble Decision Tree Methods , Sean Charles Carter

Hierarchical Self-Assembly and Substitution Rules , Daniel Alejandro Cruz

Statistical Learning of Biomedical Non-Stationary Signals and Quality of Life Modeling , Mahdi Goudarzi

Probabilistic and Statistical Prediction Models for Alzheimer’s Disease and Statistical Analysis of Global Warming , Maryam Ibrahim Habadi

Essays on Time Series and Machine Learning Techniques for Risk Management , Michael Kotarinos

The Systems of Post and Post Algebras: A Demonstration of an Obvious Fact , Daviel Leyva

Reconstruction of Radar Images by Using Spherical Mean and Regular Radon Transforms , Ozan Pirbudak

Analyses of Unorthodox Overlapping Gene Segments in Oxytricha Trifallax , Shannon Stich

An Optimal Medium-Strength Regularity Algorithm for 3-uniform Hypergraphs , John Theado

Power Graphs of Quasigroups , DayVon L. Walker

Theses/Dissertations from 2018 2018

Groups Generated by Automata Arising from Transformations of the Boundaries of Rooted Trees , Elsayed Ahmed

Non-equilibrium Phase Transitions in Interacting Diffusions , Wael Al-Sawai

A Hybrid Dynamic Modeling of Time-to-event Processes and Applications , Emmanuel A. Appiah

Lump Solutions and Riemann-Hilbert Approach to Soliton Equations , Sumayah A. Batwa

Developing a Model to Predict Prevalence of Compulsive Behavior in Individuals with OCD , Lindsay D. Fields

Generalizations of Quandles and their cohomologies , Matthew J. Green

Hamiltonian structures and Riemann-Hilbert problems of integrable systems , Xiang Gu

Optimal Latin Hypercube Designs for Computer Experiments Based on Multiple Objectives , Ruizhe Hou

Human Activity Recognition Based on Transfer Learning , Jinyong Pang

Signal Detection of Adverse Drug Reaction using the Adverse Event Reporting System: Literature Review and Novel Methods , Minh H. Pham

Statistical Analysis and Modeling of Cyber Security and Health Sciences , Nawa Raj Pokhrel

Machine Learning Methods for Network Intrusion Detection and Intrusion Prevention Systems , Zheni Svetoslavova Stefanova

Orthogonal Polynomials With Respect to the Measure Supported Over the Whole Complex Plane , Meng Yang

Theses/Dissertations from 2017 2017

Modeling in Finance and Insurance With Levy-It'o Driven Dynamic Processes under Semi Markov-type Switching Regimes and Time Domains , Patrick Armand Assonken Tonfack

Prevalence of Typical Images in High School Geometry Textbooks , Megan N. Cannon

On Extending Hansel's Theorem to Hypergraphs , Gregory Sutton Churchill

Contributions to Quandle Theory: A Study of f-Quandles, Extensions, and Cohomology , Indu Rasika U. Churchill

Linear Extremal Problems in the Hardy Space H p for 0 p , Robert Christopher Connelly

Statistical Analysis and Modeling of Ovarian and Breast Cancer , Muditha V. Devamitta Perera

Statistical Analysis and Modeling of Stomach Cancer Data , Chao Gao

Structural Analysis of Poloidal and Toroidal Plasmons and Fields of Multilayer Nanorings , Kumar Vijay Garapati

Dynamics of Multicultural Social Networks , Kristina B. Hilton

Cybersecurity: Stochastic Analysis and Modelling of Vulnerabilities to Determine the Network Security and Attackers Behavior , Pubudu Kalpani Kaluarachchi

Generalized D-Kaup-Newell integrable systems and their integrable couplings and Darboux transformations , Morgan Ashley McAnally

Patterns in Words Related to DNA Rearrangements , Lukas Nabergall

Time Series Online Empirical Bayesian Kernel Density Segmentation: Applications in Real Time Activity Recognition Using Smartphone Accelerometer , Shuang Na

Schreier Graphs of Thompson's Group T , Allen Pennington

Cybersecurity: Probabilistic Behavior of Vulnerability and Life Cycle , Sasith Maduranga Rajasooriya

Bayesian Artificial Neural Networks in Health and Cybersecurity , Hansapani Sarasepa Rodrigo

Real-time Classification of Biomedical Signals, Parkinson’s Analytical Model , Abolfazl Saghafi

Lump, complexiton and algebro-geometric solutions to soliton equations , Yuan Zhou

Theses/Dissertations from 2016 2016

A Statistical Analysis of Hurricanes in the Atlantic Basin and Sinkholes in Florida , Joy Marie D'andrea

Statistical Analysis of a Risk Factor in Finance and Environmental Models for Belize , Sherlene Enriquez-Savery

Putnam's Inequality and Analytic Content in the Bergman Space , Matthew Fleeman

On the Number of Colors in Quandle Knot Colorings , Jeremy William Kerr

Statistical Modeling of Carbon Dioxide and Cluster Analysis of Time Dependent Information: Lag Target Time Series Clustering, Multi-Factor Time Series Clustering, and Multi-Level Time Series Clustering , Doo Young Kim

Some Results Concerning Permutation Polynomials over Finite Fields , Stephen Lappano

Hamiltonian Formulations and Symmetry Constraints of Soliton Hierarchies of (1+1)-Dimensional Nonlinear Evolution Equations , Solomon Manukure

Modeling and Survival Analysis of Breast Cancer: A Statistical, Artificial Neural Network, and Decision Tree Approach , Venkateswara Rao Mudunuru

Generalized Phase Retrieval: Isometries in Vector Spaces , Josiah Park

Leonard Systems and their Friends , Jonathan Spiewak

Resonant Solutions to (3+1)-dimensional Bilinear Differential Equations , Yue Sun

Statistical Analysis and Modeling Health Data: A Longitudinal Study , Bhikhari Prasad Tharu

Global Attractors and Random Attractors of Reaction-Diffusion Systems , Junyi Tu

Time Dependent Kernel Density Estimation: A New Parameter Estimation Algorithm, Applications in Time Series Classification and Clustering , Xing Wang

On Spectral Properties of Single Layer Potentials , Seyed Zoalroshd

Theses/Dissertations from 2015 2015

Analysis of Rheumatoid Arthritis Data using Logistic Regression and Penalized Approach , Wei Chen

Active Tile Self-assembly and Simulations of Computational Systems , Daria Karpenko

Nearest Neighbor Foreign Exchange Rate Forecasting with Mahalanobis Distance , Vindya Kumari Pathirana

Statistical Learning with Artificial Neural Network Applied to Health and Environmental Data , Taysseer Sharaf

Radial Versus Othogonal and Minimal Projections onto Hyperplanes in l_4^3 , Richard Alan Warner

Ensemble Learning Method on Machine Maintenance Data , Xiaochuang Zhao

Theses/Dissertations from 2014 2014

Properties of Graphs Used to Model DNA Recombination , Ryan Arredondo

Recursive Methods in Number Theory, Combinatorial Graph Theory, and Probability , Jonathan Burns

On the Classification of Groups Generated by Automata with 4 States over a 2-Letter Alphabet , Louis Caponi

Statistical Analysis, Modeling, and Algorithms for Pharmaceutical and Cancer Systems , Bong-Jin Choi

Topological Data Analysis of Properties of Four-Regular Rigid Vertex Graphs , Grant Mcneil Conine

Trend Analysis and Modeling of Health and Environmental Data: Joinpoint and Functional Approach , Ram C. Kafle

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• Master's Thesis

As an integral component of the Master of Science in Statistical Science program, you can submit and defend a Master's Thesis. Your Master's Committee administers this oral examination. If you choose to defend a thesis, it is advisable to commence your research early, ideally during your second semester or the summer following your first year in the program. It's essential to allocate sufficient time for the thesis writing process. Your thesis advisor, who also serves as the committee chair, must approve both your thesis title and proposal. The final thesis work necessitates approval from all committee members and must adhere to the  Master's thesis requirements  set forth by the Duke University Graduate School.

Master’s BEST Award 

Each second-year Duke Master’s of Statistical Science (MSS) student defending their MSS thesis may be eligible for the  Master’s BEST Award . The Statistical Science faculty BEST Award Committee selects the awardee based on the submitted thesis of MSS thesis students, and the award is presented at the departmental graduation ceremony. 

Thesis Proposal

All second-year students choosing to do a thesis must submit a proposal (not more than two pages) approved by their thesis advisor to the Master's Director via Qualtrics by November 10th.  The thesis proposal should include a title,  the thesis advisor, committee members, and a description of your work. The description must introduce the research topic, outline its main objectives, and emphasize the significance of the research and its implications while identifying gaps in existing statistical literature. In addition, it can include some of the preliminary results. 

Committee members

MSS Students will have a thesis committee, which includes three faculty members - two must be departmental primary faculty, and the third could be from an external department in an applied area of the student’s interest, which must be a  Term Graduate Faculty through the Graduate School or have a secondary appointment with the Department of Statistical Science. All Committee members must be familiar with the Student’s work.  The department coordinates Committee approval. The thesis defense committee must be approved at least 30 days before the defense date.

Thesis Timeline and  Departmental Process:

Before defense:.

Intent to Graduate: Students must file an Intent to Graduate in ACES, specifying "Thesis Defense" during the application. For graduation deadlines, please refer to https://gradschool.duke.edu/academics/preparing-graduate .

Scheduling Thesis Defense: The student collaborates with the committee to set the date and time for the defense and communicates this information to the department, along with the thesis title. The defense must be scheduled during regular class sessions. Be sure to review the thesis defense and submission deadlines at https://gradschool.duke.edu/academics/theses-and-dissertations/

Room Reservations: The department arranges room reservations and sends confirmation details to the student, who informs committee members of the location.

Defense Announcement: The department prepares a defense announcement, providing a copy to the student and chair. After approval, it is signed by the Master's Director and submitted to the Graduate School. Copies are also posted on department bulletin boards.

Initial Thesis Submission: Two weeks before the defense, the student submits the initial thesis to the committee and the Graduate School. Detailed thesis formatting guidelines can be found at https://gradschool.duke.edu/academics/theses-and-dissertations.

Advisor Notification: The student requests that the advisor email [email protected] , confirming the candidate's readiness for defense. This step should be completed before the exam card appointment.

Format Check Appointment: One week before the defense, the Graduate School contacts the student to schedule a format check appointment. Upon approval, the Graduate School provides the Student Master’s Exam Card, which enables the student to send a revised thesis copy to committee members.

MSS Annual Report Form: The department provides the student with the MSS Annual Report Form to be presented at the defense.

Post Defense:

Communication of Defense Outcome: The committee chair conveys the defense results to the student, including any necessary follow-up actions in case of an unsuccessful defense.

In Case of Failure: If a student does not pass the thesis defense, the committee's decision to fail the student must be accompanied by explicit and clear comments from the chair, specifying deficiencies and areas that require attention for improvement.

Documentation: The student should ensure that the committee signs the Title Page, Abstract Page, and Exam Card.

Annual Report Form: The committee chair completes the Annual Report Form.

Master's Director Approval: The Master's director must provide their approval by signing the Exam Card.

Form Submission: Lastly, the committee chair is responsible for returning all completed and signed forms to the Department.

Final Thesis Submission: The student must meet the Graduate School requirement by submitting the final version of their Thesis to the Graduate School via ProQuest before the specified deadline. For detailed information, visit https://gradschool.duke.edu/academics/preparinggraduate .

• The Stochastic Proximal Distance Algorithm
• Logistic-tree Normal Mixture for Clustering Microbiome Compositions
• Inference for Dynamic Treatment Regimes using Overlapping Sampling Splitting
• Bayesian Modeling for Identifying Selection in B Cell Maturation
• Differentially Private Verification with Survey Weights
• Stable Variable Selection for Sparse Linear Regression in a Non-uniqueness Regime  
• A Cost-Sensitive, Semi-Supervised, and Active Learning Approach for Priority Outlier Investigation
• Bayesian Decoupling: A Decision Theory-Based Approach to Bayesian Variable Selection
• A Differentially Private Bayesian Approach to Replication Analysis
• Numerical Approximation of Gaussian-Smoothed Optimal Transport
• Computational Challenges to Bayesian Density Discontinuity Regression
• Hierarchical Signal Propagation for Household Level Sales in Bayesian Dynamic Models
• Logistic Tree Gaussian Processes (LoTgGaP) for Microbiome Dynamics and Treatment Effects
• Bayesian Inference on Ratios Subject to Differentially Private Noise
• Multiple Imputation Inferences for Count Data
• An Euler Characteristic Curve Based Representation of 3D Shapes in Statistical Analysis
• An Investigation Into the Bias & Variance of Almost Matching Exactly Methods
• Comparison of Bayesian Inference Methods for Probit Network Models
• Differentially Private Counts with Additive Constraints
• Multi-Scale Graph Principal Component Analysis for Connectomics
• MCMC Sampling Geospatial Partitions for Linear Models
• Bayesian Dynamic Network Modeling with Censored Flow Data  
• An Application of Graph Diffusion for Gesture Classification
• Easy and Efficient Bayesian Infinite Factor Analysis
• Analyzing Amazon CD Reviews with Bayesian Monitoring and Machine Learning Methods
• Missing Data Imputation for Voter Turnout Using Auxiliary Margins
• Generalized and Scalable Optimal Sparse Decision Trees
• Construction of Objective Bayesian Prior from Bertrand’s Paradox and the Principle of Indifference
• Rethinking Non-Linear Instrumental Variables
• Clustering-Enhanced Stochastic Gradient MCMC for Hidden Markov Models
• Optimal Sparse Decision Trees
• Bayesian Density Regression with a Jump Discontinuity at a Given Threshold
• Forecasting the Term Structure of Interest Rates: A Bayesian Dynamic Graphical Modeling Approach
• Testing Between Different Types of Poisson Mixtures with Applications to Neuroscience
• Multiple Imputation of Missing Covariates in Randomized Controlled Trials
• A Bayesian Strategy to the 20 Question Game with Applications to Recommender Systems
• Applied Factor Dynamic Analysis for Macroeconomic Forecasting
• A Theory of Statistical Inference for Ensuring the Robustness of Scientific Results
• Bayesian Inference Via Partitioning Under Differential Privacy
• A Bayesian Forward Simulation Approach to Establishing a Realistic Prior Model for Complex Geometrical Objects
• Two Applications of Summary Statistics: Integrating Information Across Genes and Confidence Intervals with Missing Data
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Department of Statistics – Academic Commons Link to Recent Ph.D. Dissertations (2011 – present)

2022 Ph.D. Dissertations

Andrew Davison

Statistical Perspectives on Modern Network Embedding Methods

Sponsor: Tian Zheng

Nabarun Deb

Blessing of Dependence and Distribution-Freeness in Statistical Hypothesis Testing

Sponsor: Bodhisattva Sen / Co-Sponsor: Sumit Mukherjee

Elliot Gordon Rodriguez

Advances in Machine Learning for Compositional Data

Sponsor: John Cunningham

Charles Christopher Margossian

Modernizing Markov Chains Monte Carlo for Scientific and Bayesian Modeling

Sponsor: Andrew Gelman

Alejandra Quintos Lima

Dissertation TBA

Sponsor: Philip Protter

Bridgette Lynn Ratcliffe

Statistical approach to tagging stellar birth groups in the Milky Way

Sponsor: Bodhisattva Sen

Chengliang Tang

Latent Variable Models for Events on Social Networks

On Recovering the Best Rank-? Approximation from Few Entries

Sponsor: Ming Yuan

Sponsor: Sumit Mukherjee

2021 Ph.D. Dissertations

On the Construction of Minimax Optimal Nonparametric Tests with Kernel Embedding Methods

Sponsor: Liam Paninski

Advances in Statistical Machine Learning Methods for Neural Data Science

Milad Bakhshizadeh

Phase retrieval in the high-dimensional regime

Chi Wing Chu

Semiparametric Inference of Censored Data with Time-dependent Covariates

Miguel Angel Garrido Garcia

Characterization of the Fluctuations in a Symmetric Ensemble of Rank-Based Interacting Particles

Sponsor: Ioannis Karatzas

Rishabh Dudeja

High-dimensional Asymptotics for Phase Retrieval with Structured Sensing Matrices

Sponsor: Arian Maleki

Statistical Learning for Process Data

Sponsor: Jingchen Liu

Toward a scalable Bayesian workflow

2020 Ph.D. Dissertations

Jonathan Auerbach

Some Statistical Models for Prediction

Sponsor: Shaw-Hwa Lo

Adji Bousso Dieng

Deep Probabilistic Graphical Modeling

Sponsor: David Blei

Guanhua Fang

Latent Variable Models in Measurement: Theory and Application

Sponsor: Zhiliang Ying

Promit Ghosal

Time Evolution of the Kardar-Parisi-Zhang Equation

Sponsor: Ivan Corwin

Partition-based Model Representation Learning

Sihan Huang

Community Detection in Social Networks: Multilayer Networks and Pairwise Covariates

Peter JinHyung Lee

Spike Sorting for Large-scale Multi-electrode Array Recordings in Primate Retina

Statistical Analysis of Complex Data in Survival and Event History Analysis

Multiple Causal Inference with Bayesian Factor Models

New perspectives in cross-validation

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Department of Statistics

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What do senior theses in Statistics look like?

This is a brief overview of thesis writing; for more information, please see our  complete guide here . Senior theses in Statistics cover a wide range of topics, across the spectrum from applied to theoretical. Typically, senior theses are expected to have one of the following three flavors:                                                                                                            

1. Novel statistical theory or methodology, supported by extensive mathematical and/or simulation results, along with a clear account of how the research extends or relates to previous related work.

2. An analysis of a complex data set that advances understanding in a related field, such as public health, economics, government, or genetics. Such a thesis may rely entirely on existing methods, but should give useful results and insights into an interesting applied problem.                                                                                 

3. An analysis of a complex data set in which new methods or modifications of published methods are required. While the thesis does not necessarily contain an extensive mathematical study of the new methods, it should contain strong plausibility arguments or simulations supporting the use of the new methods.

A good thesis is clear, readable, and well-motivated, justifying the applicability of the methods used rather than, for example, mechanically running regressions without discussing the assumptions (and whether they are plausible), performing diagnostics, and checking whether the conclusions make sense. 

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Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

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• HCA Healthc J Med
• v.1(2); 2020
• PMC10324782

Introduction to Research Statistical Analysis: An Overview of the Basics

Christian vandever.

1 HCA Healthcare Graduate Medical Education

Description

This article covers many statistical ideas essential to research statistical analysis. Sample size is explained through the concepts of statistical significance level and power. Variable types and definitions are included to clarify necessities for how the analysis will be interpreted. Categorical and quantitative variable types are defined, as well as response and predictor variables. Statistical tests described include t-tests, ANOVA and chi-square tests. Multiple regression is also explored for both logistic and linear regression. Finally, the most common statistics produced by these methods are explored.

Introduction

Statistical analysis is necessary for any research project seeking to make quantitative conclusions. The following is a primer for research-based statistical analysis. It is intended to be a high-level overview of appropriate statistical testing, while not diving too deep into any specific methodology. Some of the information is more applicable to retrospective projects, where analysis is performed on data that has already been collected, but most of it will be suitable to any type of research. This primer will help the reader understand research results in coordination with a statistician, not to perform the actual analysis. Analysis is commonly performed using statistical programming software such as R, SAS or SPSS. These allow for analysis to be replicated while minimizing the risk for an error. Resources are listed later for those working on analysis without a statistician.

After coming up with a hypothesis for a study, including any variables to be used, one of the first steps is to think about the patient population to apply the question. Results are only relevant to the population that the underlying data represents. Since it is impractical to include everyone with a certain condition, a subset of the population of interest should be taken. This subset should be large enough to have power, which means there is enough data to deliver significant results and accurately reflect the study’s population.

The first statistics of interest are related to significance level and power, alpha and beta. Alpha (α) is the significance level and probability of a type I error, the rejection of the null hypothesis when it is true. The null hypothesis is generally that there is no difference between the groups compared. A type I error is also known as a false positive. An example would be an analysis that finds one medication statistically better than another, when in reality there is no difference in efficacy between the two. Beta (β) is the probability of a type II error, the failure to reject the null hypothesis when it is actually false. A type II error is also known as a false negative. This occurs when the analysis finds there is no difference in two medications when in reality one works better than the other. Power is defined as 1-β and should be calculated prior to running any sort of statistical testing. Ideally, alpha should be as small as possible while power should be as large as possible. Power generally increases with a larger sample size, but so does cost and the effect of any bias in the study design. Additionally, as the sample size gets bigger, the chance for a statistically significant result goes up even though these results can be small differences that do not matter practically. Power calculators include the magnitude of the effect in order to combat the potential for exaggeration and only give significant results that have an actual impact. The calculators take inputs like the mean, effect size and desired power, and output the required minimum sample size for analysis. Effect size is calculated using statistical information on the variables of interest. If that information is not available, most tests have commonly used values for small, medium or large effect sizes.

When the desired patient population is decided, the next step is to define the variables previously chosen to be included. Variables come in different types that determine which statistical methods are appropriate and useful. One way variables can be split is into categorical and quantitative variables. ( Table 1 ) Categorical variables place patients into groups, such as gender, race and smoking status. Quantitative variables measure or count some quantity of interest. Common quantitative variables in research include age and weight. An important note is that there can often be a choice for whether to treat a variable as quantitative or categorical. For example, in a study looking at body mass index (BMI), BMI could be defined as a quantitative variable or as a categorical variable, with each patient’s BMI listed as a category (underweight, normal, overweight, and obese) rather than the discrete value. The decision whether a variable is quantitative or categorical will affect what conclusions can be made when interpreting results from statistical tests. Keep in mind that since quantitative variables are treated on a continuous scale it would be inappropriate to transform a variable like which medication was given into a quantitative variable with values 1, 2 and 3.

Categorical vs. Quantitative Variables

Both of these types of variables can also be split into response and predictor variables. ( Table 2 ) Predictor variables are explanatory, or independent, variables that help explain changes in a response variable. Conversely, response variables are outcome, or dependent, variables whose changes can be partially explained by the predictor variables.

Response vs. Predictor Variables

Choosing the correct statistical test depends on the types of variables defined and the question being answered. The appropriate test is determined by the variables being compared. Some common statistical tests include t-tests, ANOVA and chi-square tests.

T-tests compare whether there are differences in a quantitative variable between two values of a categorical variable. For example, a t-test could be useful to compare the length of stay for knee replacement surgery patients between those that took apixaban and those that took rivaroxaban. A t-test could examine whether there is a statistically significant difference in the length of stay between the two groups. The t-test will output a p-value, a number between zero and one, which represents the probability that the two groups could be as different as they are in the data, if they were actually the same. A value closer to zero suggests that the difference, in this case for length of stay, is more statistically significant than a number closer to one. Prior to collecting the data, set a significance level, the previously defined alpha. Alpha is typically set at 0.05, but is commonly reduced in order to limit the chance of a type I error, or false positive. Going back to the example above, if alpha is set at 0.05 and the analysis gives a p-value of 0.039, then a statistically significant difference in length of stay is observed between apixaban and rivaroxaban patients. If the analysis gives a p-value of 0.91, then there was no statistical evidence of a difference in length of stay between the two medications. Other statistical summaries or methods examine how big of a difference that might be. These other summaries are known as post-hoc analysis since they are performed after the original test to provide additional context to the results.

Analysis of variance, or ANOVA, tests can observe mean differences in a quantitative variable between values of a categorical variable, typically with three or more values to distinguish from a t-test. ANOVA could add patients given dabigatran to the previous population and evaluate whether the length of stay was significantly different across the three medications. If the p-value is lower than the designated significance level then the hypothesis that length of stay was the same across the three medications is rejected. Summaries and post-hoc tests also could be performed to look at the differences between length of stay and which individual medications may have observed statistically significant differences in length of stay from the other medications. A chi-square test examines the association between two categorical variables. An example would be to consider whether the rate of having a post-operative bleed is the same across patients provided with apixaban, rivaroxaban and dabigatran. A chi-square test can compute a p-value determining whether the bleeding rates were significantly different or not. Post-hoc tests could then give the bleeding rate for each medication, as well as a breakdown as to which specific medications may have a significantly different bleeding rate from each other.

A slightly more advanced way of examining a question can come through multiple regression. Regression allows more predictor variables to be analyzed and can act as a control when looking at associations between variables. Common control variables are age, sex and any comorbidities likely to affect the outcome variable that are not closely related to the other explanatory variables. Control variables can be especially important in reducing the effect of bias in a retrospective population. Since retrospective data was not built with the research question in mind, it is important to eliminate threats to the validity of the analysis. Testing that controls for confounding variables, such as regression, is often more valuable with retrospective data because it can ease these concerns. The two main types of regression are linear and logistic. Linear regression is used to predict differences in a quantitative, continuous response variable, such as length of stay. Logistic regression predicts differences in a dichotomous, categorical response variable, such as 90-day readmission. So whether the outcome variable is categorical or quantitative, regression can be appropriate. An example for each of these types could be found in two similar cases. For both examples define the predictor variables as age, gender and anticoagulant usage. In the first, use the predictor variables in a linear regression to evaluate their individual effects on length of stay, a quantitative variable. For the second, use the same predictor variables in a logistic regression to evaluate their individual effects on whether the patient had a 90-day readmission, a dichotomous categorical variable. Analysis can compute a p-value for each included predictor variable to determine whether they are significantly associated. The statistical tests in this article generate an associated test statistic which determines the probability the results could be acquired given that there is no association between the compared variables. These results often come with coefficients which can give the degree of the association and the degree to which one variable changes with another. Most tests, including all listed in this article, also have confidence intervals, which give a range for the correlation with a specified level of confidence. Even if these tests do not give statistically significant results, the results are still important. Not reporting statistically insignificant findings creates a bias in research. Ideas can be repeated enough times that eventually statistically significant results are reached, even though there is no true significance. In some cases with very large sample sizes, p-values will almost always be significant. In this case the effect size is critical as even the smallest, meaningless differences can be found to be statistically significant.

These variables and tests are just some things to keep in mind before, during and after the analysis process in order to make sure that the statistical reports are supporting the questions being answered. The patient population, types of variables and statistical tests are all important things to consider in the process of statistical analysis. Any results are only as useful as the process used to obtain them. This primer can be used as a reference to help ensure appropriate statistical analysis.

Funding Statement

This research was supported (in whole or in part) by HCA Healthcare and/or an HCA Healthcare affiliated entity.

Conflicts of Interest

The author declares he has no conflicts of interest.

Christian Vandever is an employee of HCA Healthcare Graduate Medical Education, an organization affiliated with the journal’s publisher.

This research was supported (in whole or in part) by HCA Healthcare and/or an HCA Healthcare affiliated entity. The views expressed in this publication represent those of the author(s) and do not necessarily represent the official views of HCA Healthcare or any of its affiliated entities.

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A Grand Journey of Statistical Hierarchical Modelling 

Advances in empirical bayes modeling and bayesian computation , advances in statistical network modeling and nonlinear time series modeling , advances in the normal-normal hierarchical model , analysis, modeling, and optimal experimental design under uncertainty: from carbon nano-structures to 3d printing , bayesian biclustering on discrete data: variable selection methods , bayesian learning of relationships , a bayesian perspective on factorial experiments using potential outcomes , building interpretable models: from bayesian networks to neural networks , causal inference under network interference: a framework for experiments on social networks , complications in causal inference: incorporating information observed after treatment is assigned , diagnostic tools in missing data and causal inference on time series , dilemmas in design: from neyman and fisher to 3d printing , distributed and multiphase inference in theory and practice: principles, modeling, and computation for high-throughput science , essays in causal inference and public policy , expediting scientific discoveries with bayesian statistical methods , exploring objective causal inference in case-noncase studies under the rubin causal model , exploring the role of randomization in causal inference , extensions of randomization-based methods for causal inference , g-squared statistic for detecting dependence, additive modeling, and calibration concordance for astrophysical data .

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Dissertations & Theses

The following is a list of recent statistics and biostatistics PhD Dissertations and Masters Theses.

Jeffrey Gory (2017) PhD Dissertation (Statistics): Marginally Interpretable Generalized Linear Mixed Models Advisors: Peter Craigmile & Steven MacEachern

Yi Lu (2017) PhD Dissertation (Statistics): Function Registration from a Bayesian Perspective Advisors: Radu Herbei & Sebastian Kurtek

Michael Matthews (2017) PhD Dissertation (Statistics):  Extending Ranked Sampling in Inferential Procedures Advisor: Douglas Wolfe

Anna Smith (2017) PhD Dissertation (Statistics):  Statistical Methodology for Multiple Networks Advisor: Catherine Calder

Weiyi Xie (2017) PhD Dissertation (Statistics): A Geometric Approach to Visualization of Variability in Univariate and Multivariate Functional Data Advisor: Sebastian Kurtek

Jingying Zeng (2017) Masters Thesis (Statistics): Latent Factor Models for Recommender Systems and Market Segmentation Through Clustering Advisors: Matthew Pratola & Laura Kubatko

Han Zhang (2017) PhD Dissertation (Statistics): Detecting Rare Haplotype-Environmental Interaction and Nonlinear Effects of Rare Haplotypes using Bayesian LASSO on Quantitative Traits Advisor: Shili Lin

Mark Burch (2016) PhD Dissertation (Biostatistics): Statistical Methods for Network Epidemic Models Advisor: Grzegorz Rempala

Po-hsu Chen (2016) PhD Dissertation (Statistics):  Modeling Multivariate Simulator Outputs with Applications to Prediction and Sequential Pareto Minimization Advisors: Thomas Santner & Angela Dean

Yanan Jia (2016) PhD Dissertation (Statistics): Generalized Bilinear Mixed-Effects Models for Multi-Indexed Multivariate Data Advisor: Catherine Calder

Rong Lu (2016) PhD Dissertation (Biostatistics): Statistical Methods for Functional Genomics Studies Using Observational Data Advisor: Grzegorz Rempala (Public Health)

Junyan Wang (2016) PhD Dissertation (Statistics): Empirical Bayes Model Averaging in the Presence of Model Misfit Advisors: Mario Peruggia & Christopher Hans

Ran Wei (2016) PhD Dissertation (Statistics):  On Estimation Problems in Network Sampling Advisors: David Sivakoff & Elizabeth Stasny

Hui Yang (2016) PhD Dissertation (Statistics):  Adjusting for Bounding and Time-in-Sample Eects in the National Crime Victimization Survey (NCVS) Property Crime Rate Estimation Advisors: Elizabeth Stasny & Asuman Turkmen

Matthew Brems (2015) Masters Thesis (Statistis): The Rare Disease Assumption: The Good, The Bad, and The Ugly Advisor: Shili Lin

Linchao Chen (2015) PhD Dissertation (Statistics):  Predictive Modeling of Spatio-Temporal Datasets in High Dimensions Advisors: Mark Berliner & Christopher Hans

Casey Davis (2015) PhD Dissertation (Statistics):  A Bayesian Approach to Prediction and Variable Selection Using Nonstationary Gaussian Processes Advisors: Christopher Hans & Thomas Santner

Victor Gendre (2015) Masters Thesis (Statistics): Predicting short term exchange rates with Bayesian autoregressive state space models: an investigation of the Metropolis Hastings algorithm forecasting efficiency Advisor: Radu Herbei

Zhengyu Hu (2015) PhD Dissertation (Statistics):  Initializing the EM Algorithm for Data Clustering and Sub-population Detection Advisors: Steven MacEachern & Joseph Verducci

David Kline (2015) PhD Dissertation (Biostatistics): Systematically Missing Subject-Level Data in Longitudinal Research Synthesis Advisors: Eloise Kaizar, Rebecca Andridge (Public Health)

Andrew Landgraf (2015) PhD Dissertation (Statistics): Generalized Principal Component Analysis: Dimensionality Reduction through the Projection of Natural Parameters Advisor: Yoonkyung Lee

Andrew Olsen (2015) PhD Dissertation (Statistics):  When Infinity is Too Long to Wait: On the Convergence of Markov Chain Monte Carlo Methods Advisor: Radu Herbei

Elizabeth   Petraglia (2015) PhD Dissertation (Statistics):  Estimating County-Level Aggravated Assault Rates by Combining Data from the National Crime Victimization Survey (NCVS) and the National Incident-Based Reporting System (NIBRS) Advisor: Elizabeth Stasny

Mark   Risser (2015) PhD Dissertation (Statistics):  Spatially-Varying Covariance Functions for Nonstationary Spatial Process Modeling Advisor: Catherine Calder

John Stettler (2015) PhD Dissertation (Statistics):  The Discrete Threshold Regression Model Advisor: Mario Peruggia

Zachary   Thomas (2015) PhD Dissertation (Statistics):  Bayesian Hierarchical Space-Time Clustering Methods Advisor: Mark Berliner

Sivaranjani   Vaidyanathan (2015) PhD Dissertation (Statistics):  Bayesian Models for Computer Model Calibration and Prediction Advisor: Mark Berliner

Xiaomu Wang (2015) PhD Dissertation (Statistics): Robust Bayes in Hierarchical Modeling and Empirical Bayes Analysis in Multivariate Estimation Advisor: Mark Berliner

Staci White (2015) PhD Dissertation (Statistics):  Quantifying Model Error in Bayesian Parameter Estimation Advisor: Radu Herbei

Jiaqi Zaetz (2015) PhD Dissertation (Statistics): A Riemannian Framework for Shape Analysis of Annotated 3D Objects Advisor: Sebastian Kurtek

Fangyuan Zhang (2015) PhD Dissertation (Biostatistics): Detecting genomic imprinting and maternal effects in family-based association studies Advisor: Shili Lin

How To Write The Results/Findings Chapter

For quantitative studies (dissertations & theses).

By: Derek Jansen (MBA) | Expert Reviewed By: Kerryn Warren (PhD) | July 2021

So, you’ve completed your quantitative data analysis and it’s time to report on your findings. But where do you start? In this post, we’ll walk you through the results chapter (also called the findings or analysis chapter), step by step, so that you can craft this section of your dissertation or thesis with confidence. If you’re looking for information regarding the results chapter for qualitative studies, you can find that here .

Overview: Quantitative Results Chapter

• What exactly the results chapter is
• What you need to include in your chapter
• How to structure the chapter
• Tips and tricks for writing a top-notch chapter
• Free results chapter template

What exactly is the results chapter?

The results chapter (also referred to as the findings or analysis chapter) is one of the most important chapters of your dissertation or thesis because it shows the reader what you’ve found in terms of the quantitative data you’ve collected. It presents the data using a clear text narrative, supported by tables, graphs and charts. In doing so, it also highlights any potential issues (such as outliers or unusual findings) you’ve come across.

But how’s that different from the discussion chapter?

Well, in the results chapter, you only present your statistical findings. Only the numbers, so to speak – no more, no less. Contrasted to this, in the discussion chapter , you interpret your findings and link them to prior research (i.e. your literature review), as well as your research objectives and research questions . In other words, the results chapter presents and describes the data, while the discussion chapter interprets the data.

Let’s look at an example.

In your results chapter, you may have a plot that shows how respondents to a survey  responded: the numbers of respondents per category, for instance. You may also state whether this supports a hypothesis by using a p-value from a statistical test. But it is only in the discussion chapter where you will say why this is relevant or how it compares with the literature or the broader picture. So, in your results chapter, make sure that you don’t present anything other than the hard facts – this is not the place for subjectivity.

It’s worth mentioning that some universities prefer you to combine the results and discussion chapters. Even so, it is good practice to separate the results and discussion elements within the chapter, as this ensures your findings are fully described. Typically, though, the results and discussion chapters are split up in quantitative studies. If you’re unsure, chat with your research supervisor or chair to find out what their preference is.

What should you include in the results chapter?

Following your analysis, it’s likely you’ll have far more data than are necessary to include in your chapter. In all likelihood, you’ll have a mountain of SPSS or R output data, and it’s your job to decide what’s most relevant. You’ll need to cut through the noise and focus on the data that matters.

This doesn’t mean that those analyses were a waste of time – on the contrary, those analyses ensure that you have a good understanding of your dataset and how to interpret it. However, that doesn’t mean your reader or examiner needs to see the 165 histograms you created! Relevance is key.

How do I decide what’s relevant?

At this point, it can be difficult to strike a balance between what is and isn’t important. But the most important thing is to ensure your results reflect and align with the purpose of your study .  So, you need to revisit your research aims, objectives and research questions and use these as a litmus test for relevance. Make sure that you refer back to these constantly when writing up your chapter so that you stay on track.

As a general guide, your results chapter will typically include the following:

• Some demographic data about your sample
• Reliability tests (if you used measurement scales)
• Descriptive statistics
• Inferential statistics (if your research objectives and questions require these)
• Hypothesis tests (again, if your research objectives and questions require these)

We’ll discuss each of these points in more detail in the next section.

Importantly, your results chapter needs to lay the foundation for your discussion chapter . This means that, in your results chapter, you need to include all the data that you will use as the basis for your interpretation in the discussion chapter.

For example, if you plan to highlight the strong relationship between Variable X and Variable Y in your discussion chapter, you need to present the respective analysis in your results chapter – perhaps a correlation or regression analysis.

Need a helping hand?

How do I write the results chapter?

There are multiple steps involved in writing up the results chapter for your quantitative research. The exact number of steps applicable to you will vary from study to study and will depend on the nature of the research aims, objectives and research questions . However, we’ll outline the generic steps below.

Step 1 – Revisit your research questions

The first step in writing your results chapter is to revisit your research objectives and research questions . These will be (or at least, should be!) the driving force behind your results and discussion chapters, so you need to review them and then ask yourself which statistical analyses and tests (from your mountain of data) would specifically help you address these . For each research objective and research question, list the specific piece (or pieces) of analysis that address it.

At this stage, it’s also useful to think about the key points that you want to raise in your discussion chapter and note these down so that you have a clear reminder of which data points and analyses you want to highlight in the results chapter. Again, list your points and then list the specific piece of analysis that addresses each point. 

Next, you should draw up a rough outline of how you plan to structure your chapter . Which analyses and statistical tests will you present and in what order? We’ll discuss the “standard structure” in more detail later, but it’s worth mentioning now that it’s always useful to draw up a rough outline before you start writing (this advice applies to any chapter).

Step 2 – Craft an overview introduction

As with all chapters in your dissertation or thesis, you should start your quantitative results chapter by providing a brief overview of what you’ll do in the chapter and why . For example, you’d explain that you will start by presenting demographic data to understand the representativeness of the sample, before moving onto X, Y and Z.

This section shouldn’t be lengthy – a paragraph or two maximum. Also, it’s a good idea to weave the research questions into this section so that there’s a golden thread that runs through the document.

Step 3 – Present the sample demographic data

The first set of data that you’ll present is an overview of the sample demographics – in other words, the demographics of your respondents.

For example:

• What age range are they?
• How is gender distributed?
• How is ethnicity distributed?
• What areas do the participants live in?

The purpose of this is to assess how representative the sample is of the broader population. This is important for the sake of the generalisability of the results. If your sample is not representative of the population, you will not be able to generalise your findings. This is not necessarily the end of the world, but it is a limitation you’ll need to acknowledge.

Of course, to make this representativeness assessment, you’ll need to have a clear view of the demographics of the population. So, make sure that you design your survey to capture the correct demographic information that you will compare your sample to.

But what if I’m not interested in generalisability?

Well, even if your purpose is not necessarily to extrapolate your findings to the broader population, understanding your sample will allow you to interpret your findings appropriately, considering who responded. In other words, it will help you contextualise your findings . For example, if 80% of your sample was aged over 65, this may be a significant contextual factor to consider when interpreting the data. Therefore, it’s important to understand and present the demographic data.

 Step 4 – Review composite measures and the data “shape”.

Before you undertake any statistical analysis, you’ll need to do some checks to ensure that your data are suitable for the analysis methods and techniques you plan to use. If you try to analyse data that doesn’t meet the assumptions of a specific statistical technique, your results will be largely meaningless. Therefore, you may need to show that the methods and techniques you’ll use are “allowed”.

Most commonly, there are two areas you need to pay attention to:

#1: Composite measures

The first is when you have multiple scale-based measures that combine to capture one construct – this is called a composite measure .  For example, you may have four Likert scale-based measures that (should) all measure the same thing, but in different ways. In other words, in a survey, these four scales should all receive similar ratings. This is called “ internal consistency ”.

Internal consistency is not guaranteed though (especially if you developed the measures yourself), so you need to assess the reliability of each composite measure using a test. Typically, Cronbach’s Alpha is a common test used to assess internal consistency – i.e., to show that the items you’re combining are more or less saying the same thing. A high alpha score means that your measure is internally consistent. A low alpha score means you may need to consider scrapping one or more of the measures.

#2: Data shape

The second matter that you should address early on in your results chapter is data shape. In other words, you need to assess whether the data in your set are symmetrical (i.e. normally distributed) or not, as this will directly impact what type of analyses you can use. For many common inferential tests such as T-tests or ANOVAs (we’ll discuss these a bit later), your data needs to be normally distributed. If it’s not, you’ll need to adjust your strategy and use alternative tests.

To assess the shape of the data, you’ll usually assess a variety of descriptive statistics (such as the mean, median and skewness), which is what we’ll look at next.

Step 5 – Present the descriptive statistics

Now that you’ve laid the foundation by discussing the representativeness of your sample, as well as the reliability of your measures and the shape of your data, you can get started with the actual statistical analysis. The first step is to present the descriptive statistics for your variables.

For scaled data, this usually includes statistics such as:

• The mean – this is simply the mathematical average of a range of numbers.
• The median – this is the midpoint in a range of numbers when the numbers are arranged in order.
• The mode – this is the most commonly repeated number in the data set.
• Standard deviation – this metric indicates how dispersed a range of numbers is. In other words, how close all the numbers are to the mean (the average).
• Skewness – this indicates how symmetrical a range of numbers is. In other words, do they tend to cluster into a smooth bell curve shape in the middle of the graph (this is called a normal or parametric distribution), or do they lean to the left or right (this is called a non-normal or non-parametric distribution).
• Kurtosis – this metric indicates whether the data are heavily or lightly-tailed, relative to the normal distribution. In other words, how peaked or flat the distribution is.

A large table that indicates all the above for multiple variables can be a very effective way to present your data economically. You can also use colour coding to help make the data more easily digestible.

For categorical data, where you show the percentage of people who chose or fit into a category, for instance, you can either just plain describe the percentages or numbers of people who responded to something or use graphs and charts (such as bar graphs and pie charts) to present your data in this section of the chapter.

When using figures, make sure that you label them simply and clearly , so that your reader can easily understand them. There’s nothing more frustrating than a graph that’s missing axis labels! Keep in mind that although you’ll be presenting charts and graphs, your text content needs to present a clear narrative that can stand on its own. In other words, don’t rely purely on your figures and tables to convey your key points: highlight the crucial trends and values in the text. Figures and tables should complement the writing, not carry it .

Depending on your research aims, objectives and research questions, you may stop your analysis at this point (i.e. descriptive statistics). However, if your study requires inferential statistics, then it’s time to deep dive into those .

Step 6 – Present the inferential statistics

Inferential statistics are used to make generalisations about a population , whereas descriptive statistics focus purely on the sample . Inferential statistical techniques, broadly speaking, can be broken down into two groups .

First, there are those that compare measurements between groups , such as t-tests (which measure differences between two groups) and ANOVAs (which measure differences between multiple groups). Second, there are techniques that assess the relationships between variables , such as correlation analysis and regression analysis. Within each of these, some tests can be used for normally distributed (parametric) data and some tests are designed specifically for use on non-parametric data.

There are a seemingly endless number of tests that you can use to crunch your data, so it’s easy to run down a rabbit hole and end up with piles of test data. Ultimately, the most important thing is to make sure that you adopt the tests and techniques that allow you to achieve your research objectives and answer your research questions .

In this section of the results chapter, you should try to make use of figures and visual components as effectively as possible. For example, if you present a correlation table, use colour coding to highlight the significance of the correlation values, or scatterplots to visually demonstrate what the trend is. The easier you make it for your reader to digest your findings, the more effectively you’ll be able to make your arguments in the next chapter.

Step 7 – Test your hypotheses

If your study requires it, the next stage is hypothesis testing. A hypothesis is a statement , often indicating a difference between groups or relationship between variables, that can be supported or rejected by a statistical test. However, not all studies will involve hypotheses (again, it depends on the research objectives), so don’t feel like you “must” present and test hypotheses just because you’re undertaking quantitative research.

The basic process for hypothesis testing is as follows:

• Specify your null hypothesis (for example, “The chemical psilocybin has no effect on time perception).
• Specify your alternative hypothesis (e.g., “The chemical psilocybin has an effect on time perception)
• Set your significance level (this is usually 0.05)
• Calculate your statistics and find your p-value (e.g., p=0.01)
• Draw your conclusions (e.g., “The chemical psilocybin does have an effect on time perception”)

Finally, if the aim of your study is to develop and test a conceptual framework , this is the time to present it, following the testing of your hypotheses. While you don’t need to develop or discuss these findings further in the results chapter, indicating whether the tests (and their p-values) support or reject the hypotheses is crucial.

Step 8 – Provide a chapter summary

To wrap up your results chapter and transition to the discussion chapter, you should provide a brief summary of the key findings . “Brief” is the keyword here – much like the chapter introduction, this shouldn’t be lengthy – a paragraph or two maximum. Highlight the findings most relevant to your research objectives and research questions, and wrap it up.

Some final thoughts, tips and tricks

Now that you’ve got the essentials down, here are a few tips and tricks to make your quantitative results chapter shine:

• When writing your results chapter, report your findings in the past tense . You’re talking about what you’ve found in your data, not what you are currently looking for or trying to find.
• Structure your results chapter systematically and sequentially . If you had two experiments where findings from the one generated inputs into the other, report on them in order.
• Make your own tables and graphs rather than copying and pasting them from statistical analysis programmes like SPSS. Check out the DataIsBeautiful reddit for some inspiration.
• Once you’re done writing, review your work to make sure that you have provided enough information to answer your research questions , but also that you didn’t include superfluous information.

If you’ve got any questions about writing up the quantitative results chapter, please leave a comment below. If you’d like 1-on-1 assistance with your quantitative analysis and discussion, check out our hands-on coaching service , or book a free consultation with a friendly coach.

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Thank you. I will try my best to write my results.

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this was great explaination

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Home > Statistics > Dissertations, Theses, and Student Work

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Department of statistics: dissertations, theses, and student work.

Examining the Effect of Word Embeddings and Preprocessing Methods on Fake News Detection , Jessica Hauschild

Exploring Experimental Design and Multivariate Analysis Techniques for Evaluating Community Structure of Bacteria in Microbiome Data , Kelsey Karnik

Human Perception of Exponentially Increasing Data Displayed on a Log Scale Evaluated Through Experimental Graphics Tasks , Emily Robinson

Factors Influencing Student Outcomes in a Large, Online Simulation-Based Introductory Statistics Course , Ella M. Burnham

Comparing Machine Learning Techniques with State-of-the-Art Parametric Prediction Models for Predicting Soybean Traits , Susweta Ray

Using Stability to Select a Shrinkage Method , Dean Dustin

Statistical Methodology to Establish a Benchmark for Evaluating Antimicrobial Resistance Genes through Real Time PCR assay , Enakshy Dutta

Group Testing Identification: Objective Functions, Implementation, and Multiplex Assays , Brianna D. Hitt

Community Impact on the Home Advantage within NCAA Men's Basketball , Erin O'Donnell

Optimal Design for a Causal Structure , Zaher Kmail

Role of Misclassification Estimates in Estimating Disease Prevalence and a Non-Linear Approach to Study Synchrony Using Heart Rate Variability in Chickens , Dola Pathak

A Characterization of a Value Added Model and a New Multi-Stage Model For Estimating Teacher Effects Within Small School Systems , Julie M. Garai

Methods to Account for Breed Composition in a Bayesian GWAS Method which Utilizes Haplotype Clusters , Danielle F. Wilson-Wells

Beta-Binomial Kriging: A New Approach to Modeling Spatially Correlated Proportions , Aimee Schwab

Simulations of a New Response-Adaptive Biased Coin Design , Aleksandra Stein

MODELING THE DYNAMIC PROCESSES OF CHALLENGE AND RECOVERY (STRESS AND STRAIN) OVER TIME , Fan Yang

A New Approach to Modeling Multivariate Time Series on Multiple Temporal Scales , Tucker Zeleny

A Reduced Bias Method of Estimating Variance Components in Generalized Linear Mixed Models , Elizabeth A. Claassen

NEW STATISTICAL METHODS FOR ANALYSIS OF HISTORICAL DATA FROM WILDLIFE POPULATIONS , Trevor Hefley

Informative Retesting for Hierarchical Group Testing , Michael S. Black

A Test for Detecting Changes in Closed Networks Based on the Number of Communications Between Nodes , Christopher S. Wichman

GROUP TESTING REGRESSION MODELS , Boan Zhang

A Comparison of Spatial Prediction Techniques Using Both Hard and Soft Data , Megan L. Liedtke Tesar

STUDYING THE HANDLING OF HEAT STRESSED CATTLE USING THE ADDITIVE BI-LOGISTIC MODEL TO FIT BODY TEMPERATURE , Fan Yang

Estimating Teacher Effects Using Value-Added Models , Jennifer L. Green

SEQUENCE COMPARISON AND STOCHASTIC MODEL BASED ON MULTI-ORDER MARKOV MODELS , Xiang Fang

DETECTING DIFFERENTIALLY EXPRESSED GENES WHILE CONTROLLING THE FALSE DISCOVERY RATE FOR MICROARRAY DATA , SHUO JIAO

Spatial Clustering Using the Likelihood Function , April Kerby

FULLY EXPONENTIAL LAPLACE APPROXIMATION EM ALGORITHM FOR NONLINEAR MIXED EFFECTS MODELS , Meijian Zhou

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Statistical Methods in Theses: Guidelines and Explanations

Signed August 2018 Naseem Al-Aidroos, PhD, Christopher Fiacconi, PhD Deborah Powell, PhD, Harvey Marmurek, PhD, Ian Newby-Clark, PhD, Jeffrey Spence, PhD, David Stanley, PhD, Lana Trick, PhD

Version:  2.00

This document is an organizational aid, and workbook, for students. We encourage students to take this document to meetings with their advisor and committee. This guide should enhance a committee’s ability to assess key areas of a student’s work. 

In recent years a number of well-known and apparently well-established findings have  failed to replicate , resulting in what is commonly referred to as the replication crisis. The APA Publication Manual 6 th Edition notes that “The essence of the scientific method involves observations that can be repeated and verified by others.” (p. 12). However, a systematic investigation of the replicability of psychology findings published in  Science  revealed that over half of psychology findings do not replicate (see a related commentary in  Nature ). Even more disturbing, a  Bayesian reanalysis of the reproducibility project  showed that 64% of studies had sample sizes so small that strong evidence for or against the null or alternative hypotheses did not exist. Indeed, Morey and Lakens (2016) concluded that most of psychology is statistically unfalsifiable due to small sample sizes and correspondingly low power (see  article ). Our discipline’s reputation is suffering. News of the replication crisis has reached the popular press (e.g.,  The Atlantic ,   The Economist ,   Slate , Last Week Tonight ).

An increasing number of psychologists have responded by promoting new research standards that involve open science and the elimination of  Questionable Research Practices . The open science perspective is made manifest in the  Transparency and Openness Promotion (TOP) guidelines  for journal publications. These guidelines were adopted some time ago by the  Association for Psychological Science . More recently, the guidelines were adopted by American Psychological Association journals ( see details ) and journals published by Elsevier ( see details ). It appears likely that, in the very near future, most journals in psychology will be using an open science approach. We strongly advise readers to take a moment to inspect the  TOP Guidelines Summary Table . 

A key aspect of open science and the TOP guidelines is the sharing of data associated with published research (with respect to medical research, see point #35 in the  World Medical Association Declaration of Helsinki ). This practice is viewed widely as highly important. Indeed, open science is recommended by  all G7 science ministers . All Tri-Agency grants must include a data-management plan that includes plans for sharing: “ research data resulting from agency funding should normally be preserved in a publicly accessible, secure and curated repository or other platform for discovery and reuse by others.”  Moreover, a 2017 editorial published in the  New England Journal of Medicine announced that the  International Committee of Medical Journal Editors believes there is  “an ethical obligation to responsibly share data.”  As of this writing,  60% of highly ranked psychology journals require or encourage data sharing .

The increasing importance of demonstrating that findings are replicable is reflected in calls to make replication a requirement for the promotion of faculty (see details in  Nature ) and experts in open science are now refereeing applications for tenure and promotion (see details at the  Center for Open Science  and  this article ). Most dramatically, in one instance, a paper resulting from a dissertation was retracted due to misleading findings attributable to Questionable Research Practices. Subsequent to the retraction, the Ohio State University’s Board of Trustees unanimously revoked the PhD of the graduate student who wrote the dissertation ( see details ). Thus, the academic environment is changing and it is important to work toward using new best practices in lieu of older practices—many of which are synonymous with Questionable Research Practices. Doing so should help you avoid later career regrets and subsequent  public mea culpas . One way to achieve your research objectives in this new academic environment is  to incorporate replications into your research . Replications are becoming more common and there are even websites dedicated to helping students conduct replications (e.g.,  Psychology Science Accelerator ) and indexing the success of replications (e.g., Curate Science ). You might even consider conducting a replication for your thesis (subject to committee approval).

As early-career researchers, it is important to be aware of the changing academic environment. Senior principal investigators may be  reluctant to engage in open science  (see this student perspective in a  blog post  and  podcast ) and research on resistance to data sharing indicates that one of the barriers to sharing data is that researchers do not feel that they have knowledge of  how to share data online . This document is an educational aid and resource to provide students with introductory knowledge of how to participate in open science and online data sharing to start their education on these subjects. 

Guidelines and Explanations

In light of the changes in psychology, faculty members who teach statistics/methods have reviewed the literature and generated this guide for graduate students. The guide is intended to enhance the quality of student theses by facilitating their engagement in open and transparent research practices and by helping them avoid Questionable Research Practices, many of which are now deemed unethical and covered in the ethics section of textbooks.

This document is an informational tool.

How to Start

In order to follow best practices, some first steps need to be followed. Here is a list of things to do:

• Get an Open Science account. Registration at  osf.io  is easy!
• If conducting confirmatory hypothesis testing for your thesis, pre-register your hypotheses (see Section 1-Hypothesizing). The Open Science Foundation website has helpful  tutorials  and  guides  to get you going.
• Also, pre-register your data analysis plan. Pre-registration typically includes how and when you will stop collecting data, how you will deal with violations of statistical assumptions and points of influence (“outliers”), the specific measures you will use, and the analyses you will use to test each hypothesis, possibly including the analysis script. Again, there is a lot of help available for this. 

Exploratory and Confirmatory Research Are Both of Value, But Do Not Confuse the Two

We note that this document largely concerns confirmatory research (i.e., testing hypotheses). We by no means intend to devalue exploratory research. Indeed, it is one of the primary ways that hypotheses are generated for (possible) confirmation. Instead, we emphasize that it is important that you clearly indicate what of your research is exploratory and what is confirmatory. Be clear in your writing and in your preregistration plan. You should explicitly indicate which of your analyses are exploratory and which are confirmatory. Please note also that if you are engaged in exploratory research, then Null Hypothesis Significance Testing (NHST) should probably be avoided (see rationale in  Gigerenzer  (2004) and  Wagenmakers et al., (2012) ). 

This document is structured around the stages of thesis work:  hypothesizing, design, data collection, analyses, and reporting – consistent with the headings used by Wicherts et al. (2016). We also list the Questionable Research Practices associated with each stage and provide suggestions for avoiding them. We strongly advise going through all of these sections during thesis/dissertation proposal meetings because a priori decisions need to be made prior to data collection (including analysis decisions). 

To help to ensure that the student has informed the committee about key decisions at each stage, there are check boxes at the end of each section.

How to Use This Document in a Proposal Meeting

• Print off a copy of this document and take it to the proposal meeting.
• During the meeting, use the document to seek assistance from faculty to address potential problems.
• Revisit responses to issues raised by this document (especially the Analysis and Reporting Stages) when you are seeking approval to proceed to defense.

Consultation and Help Line

Note that the Center for Open Science now has a help line (for individual researchers and labs) you can call for help with open science issues. They also have training workshops. Please see their  website  for details.

• Hypothesizing
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Statistics PhD theses

2015 onwards.

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Writing with Descriptive Statistics

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Usually there is no good way to write a statistic. It rarely sounds good, and often interrupts the structure or flow of your writing. Oftentimes the best way to write descriptive statistics is to be direct. If you are citing several statistics about the same topic, it may be best to include them all in the same paragraph or section.

The mean of exam two is 77.7. The median is 75, and the mode is 79. Exam two had a standard deviation of 11.6.

Overall the company had another excellent year. We shipped 14.3 tons of fertilizer for the year, and averaged 1.7 tons of fertilizer during the summer months. This is an increase over last year, where we shipped only 13.1 tons of fertilizer, and averaged only 1.4 tons during the summer months. (Standard deviations were as followed: this summer .3 tons, last summer .4 tons).

Some fields prefer to put means and standard deviations in parentheses like this:

If you have lots of statistics to report, you should strongly consider presenting them in tables or some other visual form. You would then highlight statistics of interest in your text, but would not report all of the statistics. See the section on statistics and visuals for more details.

If you have a data set that you are using (such as all the scores from an exam) it would be unusual to include all of the scores in a paper or article. One of the reasons to use statistics is to condense large amounts of information into more manageable chunks; presenting your entire data set defeats this purpose.

At the bare minimum, if you are presenting statistics on a data set, it should include the mean and probably the standard deviation. This is the minimum information needed to get an idea of what the distribution of your data set might look like. How much additional information you include is entirely up to you. In general, don't include information if it is irrelevant to your argument or purpose. If you include statistics that many of your readers would not understand, consider adding the statistics in a footnote or appendix that explains it in more detail.

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