Chapter 1

Biostatistics 101

IN THIS CHAPTER

Bullet Getting up to speed on the prerequisites for biostatistics

Bullet Understanding the human research environment

Bullet Surveying the specific procedures used to analyze biological data

Bullet Estimating how many participants you need

Bullet Working with distributions

Biostatistics deals with the design and execution of scientific studies involving biology, the acquisition and analysis of data from those studies, and the interpretation and presentation of the results of those analyses. This book is meant to be a useful and easy-to-understand companion to the more formal textbooks used in graduate-level biostatistics courses. Because most of these courses teach how to analyze data from epidemiologic studies and clinical trials, this book focuses on that as well. In this first chapter, we introduce you to the fundamentals of biostatistics.

Brushing Up on Math and Stats Basics

Chapters 2 and 3 are designed to bring you up to speed on the basic math and statistical background that’s needed to understand biostatistics and give you supplementary information or context that you may find useful while reading the rest of this book.

  • Many people feel unsure of themselves when it comes to understanding mathematical formulas and equations. Although this book contains fewer formulas than many statistics books, we include them when they help illustrate a concept or describe a calculation that’s simple enough to do by hand. But if you’re a real mathophobe, you probably dread looking at any chapter that has a math expression anywhere in it. That’s why we include Chapter 2, “Overcoming Mathophobia” to show you how to read and understand the basic mathematical notation we use in this book. We cover everything from basic mathematical operations to functions and beyond.
  • If you’re in a graduate-level biostatistics course, you’ve probably already taken one or two introductory statistics courses. But that may have been a while ago, and you may feel unsure of your knowledge of the basic statistical concepts. Or you may have little or no formal statistical training but now find yourself in a work situation where you interact with clinical researchers, participate in the design of research projects, or work with the results from biological research. If so, read Chapter 3, which provides an overview of the fundamental concepts and terminology of statistics. There, you get the scoop on topics such as probability, randomness, populations, samples, statistical inference, accuracy, precision, hypothesis testing, nonparametric statistics, and simulation techniques.

Doing Calculations with the Greatest of Ease

For instructional purposes, some chapters in this book include step-by-step instructions for performing statistical tests and analyses by hand. We include such instruction only to illustrate the concepts that are involved in the procedure or to demonstrate calculations that are simple to do manually.

However, we demonstrate many of the statistical functions we talk about in this book using R, which is a free, open-source software package. If you are in a class and assigned a particular software package to use, you will have to use that software for the course, which may be commercial software associated with a fee. However, if you are learning on your own, you may choose to use open-source software, which is free. Chapter 4 provides guidance on both commercial and free software.

Concentrating on Epidemiologic Research

Remember This book covers topics that are applicable to all areas of biostatistics, concentrating on methods that are especially relevant to epidemiologic research — studies involving people. This includes clinical trials, which are experiments done to develop therapeutic interventions such as drugs. Because policy in healthcare is often based on the results from clinical trials, if you make mistake analyzing clinical trial data, it can have disastrous and wide-ranging human and financial consequences. Even if you don’t expect to ever work in a domain that relies heavily on clinical trials (such as drug development research), ensuring that you have a working knowledge of how to manage the statistical issues seen in clinical trials is critical.

Three chapters discuss clinical trials:

  • Chapter 5 describes the statistical aspects of clinical trials as three phases. First, it covers the design phase, where a study protocol is written. Next, it describes the execution phase, where data are collected, and efforts are made to prevent invalid or missing data. In the final phase, data from the study are analyzed and interpreted to answer the hypotheses.
  • Chapter 7 presents epidemiologic study designs and explains the importance of the clinical trial as a study design.
  • Chapter 20 explains the role well-designed clinical trials play in accruing evidence of causal inference in biostatistics.

Much of the work in biostatistics is using data from samples to make inferences about the background population from which the sample was drawn. Now that we have large databases, it is possible to easily take samples of data. Chapter 6 provides guidance on different ways to take samples of larger populations so you can make valid population-based estimates from these samples. Sampling is especially important when doing observational studies. While clinical trials covered are experiments, where participants are assigned interventions, in observational studies, participants are merely observed, with data collected and statistics performed to make inferences. Chapter 7 describes these observational study designs, and the statistical issues that need to be considered when analyzing data arising from such studies.

Data used in biostatistics are often collected in online databases, but some data are still collected on paper. Regardless of the source of the data, they must be put into electronic format and arranged in a certain way to be able to be analyzed using statistical software. Chapter 8 is devoted to describing how to get your data into the computer and arrange it properly so it can be analyzed correctly. It also describes how to collect and validate your data. Then in Chapter 9, we show you how to summarize each type of data and display it graphically. We explain how to make bar charts, box-and-whiskers charts, and more.

Drawing Conclusions from Your Data

Most statistical analysis involves inferring, or drawing conclusions about the population at large based on your observations of a sample drawn from that population. The theory of statistical inference is often divided into two broad sub-theories: estimation theory and decision theory.

Statistical estimation theory

Chapter 10 deals with statistical estimation theory, which addresses the question of how accurately and precisely you can estimate a population parameter from the values you observe in your sample. For example, you may want to estimate the mean blood hemoglobin concentration in adults with Type II diabetes, or the true correlation coefficient between body weight and height in certain pediatric populations. Chapter 10 describes how to estimate these parameters by constructing a confidence interval around your estimate. The confidence interval is the range that is likely to include the true population parameter, which provides an idea of the precision of your estimate.

Statistical decision theory

Much of the rest of this book deals with statistical decision theory, which is how to decide whether some effect you’ve observed in your data reflects a real difference or association in the background population or is merely the result of random fluctuations in your data or sampling. If you measure the mean blood hemoglobin concentration in two different samples of adults with Type II diabetes, you will likely get a different number. But does this difference reflect a real difference between the groups in terms of blood hemoglobin concentration? Or is this difference a result of random fluctuations? Statistical decision theory helps you decide.

In Part 4, we cover statistical decision theory in terms of comparing means and proportions between groups, as well as understanding the relationship between two or more variables.

Comparing groups

In Part 4, we show you different ways to compare groups statistically.

  • In Chapter 11, you see how to compare average values between two or more groups by using t tests and ANOVAs. We also describe their nonparametric counterparts that can be used with skewed or other non-normally distributed data.
  • Chapter 12 shows how to compare proportions between two or more groups, such as the proportions of patients responding to two different drugs, using the chi-square and Fisher Exact tests on cross-tabulated (cross-tab) data.
  • Chapter 13 focuses on one specific kind of cross-tab called the fourfold table, which has exactly two rows and two columns. Because the fourfold table provides the opportunity for some particularly insightful calculations, it’s worth a chapter of its own.
  • In Chapter 14, you discover how the terminology used in epidemiologic studies is applied to specifically formatted fourfold tables to calculate incidence and prevalence rates.

Looking for relationships between variables

Epidemiology and biostatistics are interested in causal inference, which means trying to figure out what causes particular outcomes in biological research. While it is possible to look at the relationship between two variables in a bivariate analysis, regression analysis is the part of statistics that enables you to explore the relationship between multiple variables and one outcome in the same model so you can evaluate their relative cause of the outcome. Here are some use-cases for regression:

  • You may want to know whether there’s a statistically significant association between one or more variables and an outcome, even if there are other variables in the model. You may ask: Does being overweight increase the likelihood of getting liver cancer? Or: Is exercising fewer hours per week associated with higher blood pressure measurements? In answering both of those questions, you may want to control other variables known to influence the outcome.
  • You may want to develop a formula for predicting the value of a variable from the observed values of one or more other variables. For example, you may want to predict how long a newly diagnosed cancer patient may survive based on their age, obesity status, and medical history.
  • You may be fitting a theoretical formula to some data to estimate one of the parameters appearing in that formula. An example of such a problem is determining how fast the kidneys can remove a drug from the body, which is called a terminal elimination rate constant. This can be estimated from measurements of drug concentration in the blood taken at various times after taking a dose of the drug.

Regression analysis can manage all these tasks and many more. Regression is so important in biological research that all the chapters in Part 5 are focused on some aspect of regression.

Tip If you have never learned correlation and regression analysis, read Chapter 15, which introduces these topics. We cover simple straight-line regression in Chapter 16, which includes one predictor variable. We extend that to cover multiple regression with more than one predictor variable in Chapter 17. These three chapters deal with ordinary linear regression, where you’re trying to predict the value of a numerical outcome variable from one or more other variables. An example would be trying to predict mean blood hemoglobin concentration using variables like age, blood pressure level, and Type II diabetes status. Ordinary linear regression uses a formula that’s a simple summation of terms, each of which consists of a predictor variable multiplied by a regression coefficient.

But in real-world biological and epidemiologic research, you encounter more complicated relationships. Chapter 18 describes logistic regression, where the outcome is the occurrence or non-occurrence of an event (such as being diagnosed with Type II diabetes), and you want to predict the probability that the event will occur. You also find out about several other kinds of regression in Chapter 19:

  • Poisson regression, where the outcome is the number of events that occur in an interval of time
  • Nonlinear least-squares regression, where the relationship between the predictors and numerical outcome can be more complicated than a simple summation of terms in a linear model
  • LOWESS curve-fitting, where you fit a custom function to describe your data

Finally, Part 5 ends with Chapter 20, which provides guidance on the mechanics of regression modeling, including how to develop a modeling plan, and how to choose variables to include in models.

A Matter of Life and Death: Working with Survival Data

Sooner or later, everyone dies, and in biological research, it becomes especially important to characterize that sooner-or-later part as accurately as possible using survival analysis techniques. But characterizing survival can get tricky. It’s possible to say that patients may live an average of 5.3 years after they are diagnosed with a particular disease. But what is the exact survival experience? Imagine you do a study with patients who have this disease. You may ask: Do all patients tend to live around five or six years, or do half the patients die within the first few months, and the other half survive ten years or more? And what if some patients live longer than the observational period of your study? How do you include them in your analysis? And what about participants who stopped returning calls from your study staff? You do not know if these dropouts went on to live or die. How do you include their data in your analysis?

Remember The need to study survival with data like these led to the development of survival analysis techniques. But survival analysis is not only intended to study the outcome of death. You can use survival analysis to study the time to the first occurrence of non-death events as well, like remission or recurrence of cancer, the diagnosis of a particular condition, or the resolution of a particular condition. Survival analysis techniques are presented in Part 6.

Getting to Know Statistical Distributions

Statistics books always contain tables, so why should this one be any different? Back in the not-so-good old days, when analysts had to do statistical calculations by hand, they needed to use tables of the common statistical distributions to complete the calculation of the significance test. They needed tables for the normal distribution, Student t, chi-square, Fisher F, and others. Now, software does all this for you, including calculating exact p values, so these printed tables aren’t necessary anymore.

But you should still be familiar with the common statistical distributions that may describe the fluctuations in your data, or that may be referenced in the course of performing a statistical calculation. Chapter 24 contains a list of commonly used distribution functions, with explanations of where you can expect to encounter those distributions and what they look like. We also include a description of some of their properties and how they’re related to other distributions. Some of them are accompanied by a small table of critical values, corresponding to statistical significance at α = 0.05.

Figuring Out How Many Participants You Need

Of all the statistical challenges a researcher may encounter, none seems to instill as much apprehension and insecurity as having to estimate the number of participants needed for a study. While smaller sample sizes mean less data collection work, you want to make sure your target sample size is large enough so that in the end, your study has sufficient power. You want to conduct a study with a high probability of yielding a statistically significant result if the hypothesized effect is truly present in the population.

Tip Because sample-size estimation is such an important part of the design of any research project, this book shows you how to make those estimates for the situations you’re likely to encounter when doing biological research. As we describe each statistical test in Parts 4, 5, 6, and 7, we explain how to estimate the number of participants needed to provide sufficient power for that test. In addition, Chapter 25 describes ten simple rules for getting a “quick and dirty” estimate of the required sample size.