## Introduction

# Acknowledgements

The authors would like to thank Alex Nones for proofreading the manuscript during its various stages. Also, thanks to Karl Broman for contributing the “Plots to Avoid” section and to Stephanie Hicks for designing some of the exercises.

This book was conceived during the teaching of several HarvardX courses, coordinated by Heather Sternshein. We are also grateful to all the students whose questions and comments helped us improve the book. The courses were partially funded by NIH grant R25GM114818. We are very grateful to the National Institute of Health for its support.

A special thanks goes to all those that edited the book via GitHub pull requests: vjcitn, yeredh, ste-fan, molx, kern3020, josemrecio, hcorrada, neerajt, massie, jmgore75, molecules, lzamparo, eronisko, and obicke.

Cover image credit: this photograph is La Mina Falls, El Yunque National Forest, Puerto Rico, taken by Ron Kroetz https://www.flickr.com/photos/ronkroetz/14779273923 Attribution-NoDerivs 2.0 Generic (CC BY-ND 2.0)

# Introduction

The unprecedented advance in digital technology during the second half of the 20th century has produced a measurement revolution that is transforming science. In the life sciences, data analysis is now part of practically every research project. Genomics, in particular, is being driven by new measurement technologies that permit us to observe certain molecular entities for the first time. These observations are leading to discoveries analogous to identifying microorganisms and other breakthroughs permitted by the invention of the microscope. Choice examples of these technologies are microarrays and next generation sequencing.

Scientific fields that have traditionally relied upon simple data analysis techniques have been turned on their heads by these technologies. In the past, for example, researchers would measure the transcription levels of a single gene of interest. Today, it is possible to measure all 20,000+ human genes at once. Advances such as these have brought about a shift from hypothesis to discovery-driven research. However, interpreting information extracted from these massive and complex datasets requires sophisticated statistical skills as one can easily be fooled by patterns arising by chance. This has greatly elevated the importance of statistics and data analysis in the life sciences.

## What Does This Book Cover?

This book will cover several of the statistical concepts and data analytic skills needed to succeed in data-driven life science research. We go from relatively basic concepts related to computing p-values to advanced topics related to analyzing high-throughput data.

We start with one of the most important topics in statistics and in
the life sciences: statistical inference. Inference is the use of
probability to learn population characteristic from data. A typical example
is deciphering if two groups (for example, cases versus controls) are
different on average. Specific topics covered include the t-test,
confidence intervals, association tests, Monte Carlo methods,
permutation tests and statistical power. We make use of approximations
made possible by mathematical theory, such as the Central Limit
Theorem, as well as techniques made possible by modern computing. We
will learn how to compute p-values and confidence intervals and
implement basic data analyses. Throughout the book we will describe
visualization techniques in the statistical computer language *R* that
are useful for exploring new data sets. For example, we will use these
to learn when to apply robust statistical techniques.

We will then move on to an introduction to linear models and matrix algebra. We will explain why it is beneficial to use linear models to analyze differences across groups, and why matrices are useful to represent and implement linear models. We continue with a review of matrix algebra, including matrix notation and how to multiply matrices (both on paper and in R). We will then apply what we covered on matrix algebra to linear models. We will learn how to fit linear models in R, how to test the significance of differences, and how the standard errors for differences are estimated. Furthermore, we will review some practical issues with fitting linear models, including collinearity and confounding. Finally, we will learn how to fit complex models, including interaction terms, how to contrast multiple terms in R, and the powerful technique which the functions in R actually use to stably fit linear models: the QR decomposition.

In the third part of the book we cover topics related to high-dimensional data. Specifically, we describe multiple testing, error rate controlling procedures, exploratory data analysis for high-throughput data, p-value corrections and the false discovery rate. From here we move on to covering statistical modeling. In particular, we will discuss parametric distributions, including binomial and gamma distributions. Next, we will cover maximum likelihood estimation. Finally, we will discuss hierarchical models and empirical Bayes techniques and how they are applied in genomics.

We then cover the concepts of distance and dimension reduction. We will introduce the mathematical definition of distance and use this to motivate the singular value decomposition (SVD) for dimension reduction and multi-dimensional scaling. Once we learn this, we will be ready to cover hierarchical and k-means clustering. We will follow this with a basic introduction to machine learning.

We end by learning about batch effects and how component and factor analysis are used to deal with this challenge. In particular, we will examine confounding, show examples of batch effects, make the connection to factor analysis, and describe surrogate variable analysis.

## How Is This Book Different?

While statistics textbooks focus on mathematics, this book focuses on using a computer to perform data analysis. This book follows the approach of Stat Labs, by Deborah Nolan and Terry Speed. Instead of explaining the mathematics and theory, and then showing examples, we start by stating a practical data-related challenge. This book also includes the computer code that provides a solution to the problem and helps illustrate the concepts behind the solution. By running the code yourself, and seeing data generation and analysis happen live, you will get a better intuition for the concepts, the mathematics, and the theory.

We focus on the practical challenges faced by data analysts in the
life sciences and introduce mathematics as a tool that can help us
achieve scientific goals. Furthermore, throughout the book we show the
R code that performs this analysis and connect the lines of code to
the statistical and mathematical concepts we explain. All sections of
this book are reproducible as they were made using *R markdown*
documents that include R code used to produce the figures, tables and
results shown in the book. In order to distinguish it, the code is
shown in the following font:

```
x <- 2
y <- 3
print(x+y)
```

and the results in different colors, preceded by two hash
characters (*##*):

```
## [1] 5
```

We will provide links that will give you access to the raw R markdown code so you can easily follow along with the book by programming in R.

At the beginning of each chapter you will see the sentence:

The R markdown document for this section is available here.

The word “here” will be a hyperlink to the R markdown file. The best way to read this book is with a computer in front of you, scrolling through that file, and running the R code that produces the results included in the book section you are reading.