13 – Assumptions of parametric tests

Introduction

Chapters 8, 10 and 12 were concerned primarily with tests of means among groups of treatments. ANOVA, t-tests, linear models, all involve estimation of parameters, qualities of populations. Although we have included assumptions about these statistics along the way, this chapter provides a summary about assumptions needed to be met in order to correctly interpret results of these statistical tests. Assumptions of parametric tests include how the data are presumed to be distributed (eg, normality) and about the variability within groups (eg, we assume equal or “homogeneity” of variances).

One important caveat: you can always estimate regardless of whether or not the assumptions are met. And, certainly, R and other statistical software will allow you to perform these calculations without warning. However, to the extent one or more assumptions do not hold, your conclusions, eg, p-value and Type I error, will be influenced. That’s what we mean by statistical thinking — knowing when your conclusions are valid.

This is the classic approach — provide tests of assumptions to justify use of ANOVA, etc. The modern approach, perhaps even the best practice approach, is instead to use more powerful statistical modeling approach, eg, generalized linear model (GLS) to model for correlations among residuals (lack of independence assumption) or heteroscedastic variances (equal residual variances).

Quizzes in this chapter

A total of 29 questions among the several subchapters, a mix of true or false and multiple choice question format.

Chapter 13 contents

2 – Introduction

Why biostatistics, not “biometry”?

Chapter 1 – Getting started, we presented a brief introduction to statistical thinking, a systematic approach to how we ask questions about the world from data. The chief learning objective of this eBook, then, is to provide the reader with a framework and practical computing skills needed for statistical reasoning in the biological sciences. Chapter 1 – Getting started also offered a justification for why undergraduate biology students should (are required) learn (bio)statistics. In my day, most of us took statistics as part of our graduate training. The curriculum for science students has accelerated now — it is now assumed that as part of undergraduate career students gain experience working with data and developing quantitative reasoning skills. Biostatistics courses are designed to help you achieve this understanding. We expand on that point in Ch2.1 – Why biostatistics?.

Note 1: I took a business statistics class as an undergraduate student at the University of Washington in the early 1980s.  The textbook was the 6th edition of John E. Freund‘s Modern Elementary Statistics. Biostatistics course was not an option for a biology undergraduate student at the time. During my first year of graduate school at the University of Wisconsin, textbooks were the 2nd edition of Biometry by Robert R. Sokal and F. James Rohlf) and the 8th edition of Statistical Methods by G.W. Snedecor and W.G. Cochran.

Biostatistics generally refers to the application of statistical methods to biological research in the context of medicine and public health. The emphasis of biostatistics is on experimental design and hypothesis testing in clinical and epidemiological contexts. The term biostatistics first appears in the literature in the 1940s (Pubmed). On the other hand, biometry, sensu the title of Sokal and Rohlf‘s legendary textbook, is a broader term, associated with the application of mathematical and statistical techniques to biological problems, especially in fields like ecology, genetics, and evolution. Our eBook provides real world problems as examples from across the biological field, including agriculture, ecology, evolution, genetics as well as health topics, thus the subject of this text is more in line with the spirit of biometry. However, I titled the book “Mike’s Biostatistics Book,” not “Mike’s Biometry Book,” because the term biometry is now conflated with biometrics (Oxford English Dictionary), which refers to the technology used to identify individuals based on biological characteristics. 

Chapter 2 begins my sales pitch: Why biostatistics? We’ll also provide a selective history of biostatistics and how statistical thinking informs the design of experiments. Chapter 2 concludes with how statistical reasoning fits in the scientific method — the systematic approach to acquiring knowledge through observation, experimentation, and reasoning (Chapter 2.5) — and introduces project-based learning approaches used in the biostatistics course at Chaminade University.

Quizzes in this chapter

A total of 50 questions among the several subchapters, a mix of true or false and multiple choice question format.

Homework to go with this topic.

Homework 1: Project data life cycle in Mike’s Workbook for Biostatistics.

Chapter 2 contents

 

 

1 – Getting started

What this book is about.

The primary theme of this applied statistics book is to foster statistical thinking. By developing statistical thinking approaches, biology students gain a way to understand and approach biology problems using statistical concepts and reasoning. Thus, my task in writing chapters was to focus on a problem-solving approach by using real-world data and examples, introducing and expanding on conceptual understanding, and modeling the process of statistical investigation. Our goal is to help students achieve competence in the foundations of biostatistics. Adopting a statistical thinking philosophy is crucial for professionals across medical and scientific disciplines because it provides the analytical tools needed to draw valid, evidence-based conclusions from complex biological and health data.

How to use this book.

This eBook is intended to accompany and support students at Chaminade University of Honolulu enrolled in BI-311 Biostatistics, a one semester introduction to biostatistics. Like all textbooks, the intent is to provide the reader with a guided and interactive presentation about the subject. However, the text should not be taken as the only voice — there are plenty of good textbooks, many free, to help you learn statistics. You are encouraged to seek additional help with the material. 

Homework and projects.

Mike’s Biostatistics Book is a stand alone product, but the purpose of the book is to provide content for my biostatistics course. Homework and projects designed to build confidence with the material are provided in a separate workbook, also available as a free eBook as Mike’s Workbook for Biostatistics, provides tutorials on R code and project management, homework inspired by chapter contents. The workbook also provides guidance for student-directed projects that enable students to learn biostatistics firsthand.

The websites serve the course,

Mike’s Biostatistics Book is hosted at biostatistics.letgen.org. Organization at the site is facilitated by the WordPress wp-gitbook theme by Tom J. Nowell at https://github.com/tomjn/wp-gitbook. Thank you, Tom!

The course, BI-311 Biostatistics, is a CANVAS CMS website is accessible through chaminade.edu. The course website helps me to organize and support the course. BI-311 is web-enhanced course, not a blended or hybrid course; that is, online materials are presented to supplement your work in class and do not replace “face-to-face” time. During — and since — the Covid-19 pandemic I produced a number of videos based on lecture materials for BI-311; many of these are still available at my YouTube channel for the course — they all lack, shall we say, a polished approach.

Students of BI-311 enrolled in CANVAS will find lecture slides, help with your computer, help with R statistical programming language and R Commander, a basic statistics GUI that works with R, and an extensive glossary covering statistics and data science terminology. BI-311 students submit their work to CANVAS course site hosted by Chaminade University. Online quizzes provide rapid feedback and suggestions for further study. 

Material on CANVAS course site  is organized to follow the table of contents from Mike’s Biostatistics book.  For a 16-week semester, the course would be divided into three parts

Part 1. Chapter 1 — 7 → Exam 1

Part 2. Chapters 8 – 14 → Exam 2

Part 3. Chapters 15 — 18 → Exam 3

In addition, students are expected to complete a self-directed student research project, typically a small meta-analysis, but can also be original research based on experiments. I typically do not permit student-led original survey work — surveys are deceptively challenging to do well, require approval from IRB (which is doable, but again, challenging given the time constraitns of the semester), and may detract from the purpose of the project, which is to support learning biostatistics.

Book conventions.

Chapters are divided into main subject sections. Subheadings within sections indicate that important concepts follow, concepts you will be expected to understand and demonstrate that understanding on BI-311 quizzes and exams.

Figures and tables in each chapter start with Figure 1 and Table 1. If a figure from chapter 3 is referred to in chapter 5, then the reference to that figure will be Figure 3.1, with Figure 1 referring to the first figure in Chapter 5. Additional notes about the book were provided in the Preface.

Equations were written using LaTeX; in some cases, equations are presented as images.

Each section also include questions and additional readings, particularly where there are opinions or interpretations of statistical concepts, references are provided. For example, we spend considerable time discussing what the “p value” means. Additional readings either extend the discussion or provide context to the topic.

Like any new subject, a key to your success will be to learn the language. Terms in the text requiring definitions and your attention are in bold. Take care with definitions in statistics: while the words are recognizable, their meanings are often distinctly different from common usage.

On many pages I include parenthetical notes, which are intended to provide additional information or tangential information while minimizing disruption to the reader’s attention to the main text. Parenthetical notes are enclosed in green box, like this one.

Note 1: Parenthetical notes will be enclosed in green box (eg, this note about statistical terms on this page). 

R code.

Throughout the text we will also include relevant R code. R code you type will appear in a sentence as a code block, like help.start(), or as a preformatted shaded text block, like so

help.start()

R code you type will be in blue; any output from R will be displayed as red type. 

Note 2: Text formatting according to the blue/red principle is not consistent — a work in progress.

Each section includes worked examples that are presented to illustrate concepts, demonstrate how R can be used, and include interpretations as appropriate. 

Questions for the reader.

I have added questions at the end of most book sections to emphasize important concepts or to have you explore more about a subject. Questions range from extensions of concepts introduced in the section to exercises and problems to solve. Questions in the text are generally short answer or require a numerical solution.

New as of Fall 2025. Most chapters include about 10 questions based on the readings, a total of 928 as of December. Quiz questions are multiple choice or True/False; click on a response and get immediate feedback. Questions in the book are intended to provide the student with an outline of topics likely to show up on quizzes and exams. On the other hand, the accompanying workbook, Mike’s Workbook for Biostatistics, provides students with opportunities to conduct more detailed data analysis and to learn about the R language.

Note 3: Like a lot of folks, I’m playing with large language AI models — ChatGPT in particular. Some of the book quiz questions were drafted with AI assistance. None are word-for-word AI, however. 

I also now routinely use ChatGPT Work to draft R code snippets.

 

Chapter 1 contents