Intuitive Biostatistics: Survival Curves

This is chapter preface and Introduction of Intuitive Biostatistics (ISBN 0-19-508607-4) by Harvey Motulsky. Copyright © 1995 by Oxford University Press Inc. All rights reserved. You may order the book from GraphPad Software with a software purchase, from any academic bookstore, or from amazon.com.

APPROACH

This book provides a nonmathematical introduction to statistics for medical students, physicians, graduate students, and researchers in the health science. So do plenty of other books, but this one has a unique approach.

Explanations rather than mathematical proofs. To those with appropriate training and inclination, mathematical notation is a wonderful way to say things clearly and concisely. Why read a page of explanation when you can read two equations? But for many (perhaps most) students and scientists, mathematical notation is confusing and threatening. This book explains concepts in words, with few mathematical proof or equations (except for those provided in reference sections marked with a # symbol.
Emphasis on interpreting results rather than analyzing data. Statistical methods will only be used by those who collect data. Statistical results will be interpreted by everyone who reads published papers. This book emphasizes the interpretation of published results, although it also explains how to analyze data. For example, I explain how to interpret P values long before I present any tests that compute P values. In some cases, I discuss statistical methods without showing the equations needed to analyze data with those methods.
Emphasis on confidence intervals. Statistical calculations generate both P values and confidence intervals. While most books emphasize the calculation of P values, I emphasize confidence intervals. Although P values and confidence intervals are related and are calculated together, I present confidence intervals first, and delay presenting P values until Chapter 10.
Examples from the clinical literature. In a few places, I've included simple examples with fake data. But most of the examples are from recent medical literature. To focus the discussion on basic statistical understanding, I have sometimes simplified the findings a bit, without (I hope) obscuring the essence of the results. These papers were not selected because they are particularly good or particularly bad, They are just a sampling of papers from good journals that I happened to stumble across when searching for examples (so the authors don't need to be particularly proud or embarrassed ((3 see their work included).
Explanation of Bayesian thinking. Bayesian thinking helps one interpret P values, lab results, genetic counseling, and linkage analysis. Whereas most introductory books ignore Bayesian analysis, I discuss it in reasonable detail in Part IV.

TOPICS COVERED

In choosing topics to include in this book I've chosen breadth over depth. This is because so many statistical methods are commonly used in the biomedical literature. Flip through any medical or scientific journal and you'll soon find use of a statistical technique not mentioned in most introductory books. To guide those who read those papers, I included many topics omitted from other books: relative risk and odds ratios, prediction intervals, nonparametric tests, survival curves, multiple comparisons, the design of clinical trials, computing the power of a test, nonlinear regression, interpretation of lab tests (sensitivity, specificity, etc.). I also briefly introduce multiple regression, logistic regression, proportional hazards regression, randomization tests, and lod scores. Analysis of variance is given less emphasis than usual.

CHAPTERS TO SKIP

As statistics books go, this one is pretty short. But realistically that it still is more than most people want to read about statistics. If you just want to learn the main ideas of statistics, with no detail, read Chapters I through 5, 10 through 13, and 19.

This book is for anyone who reads papers in the biomedical literature, not just for people who read clinical studies. Basic scientists may want to skip Chapters 6, 9, 20, 21, 32, and 33 which deal with topics uncommonly encountered in basic research. The other chapters are applicable to both clinicians and basic scientists.

ANALYZING DATA WITH COMPUTER PROGRAMS

We are lucky to live in an era where personal computers are readily available. Although this book gives the equations for many statistical tests, most people will rely on a computer program instead. Unfortunately, most statistics programs are designed for statisticians and are too complicated and too expensive for the average student or scientist. That's why my company, GraphPad Software, created GraphPad InStat, an inexpensive and extremely easy statistical program available for DOS and Macintosh computers (a Windows version is coming soon). Although this book shows sample output from InStat, you do not need InStat to follow the examples in this book or to work the problems.

Although spreadsheet programs were originally developed to perform financial calculations, current versions are very versatile and adept at statistical computation. See Appendix 3 to learn how to use Microsoft Excel to perform statistical calculations.

REFERENCES AND ACKNOWLEDGMENTS

I have organized this book in a unique way, but none of the ideas are particularly original. All of the statistical methods are standard, and have been discussed in many textbooks. Rather than give the original reference for each method, I have listed text book references in Appendix 1.

I would like to thank everyone who reviewed various sections of the book in draft form and gave valuable comments, including Jan Agosti, Cedric Garland, Ed Jackson, Arno Motulsky, Paige Searle, and Christopher Sempos. I especially want to thank Harry Frank, whose lengthy comments improved this book considerably. This book would be very different if it weren't for his repeated lengthy reviews. I also want to thank all the students who helped me shape this book over the last five years. Of course, any errors are my own responsibility. Please email comments and suggestions to HMotulsky@graphpad.com.

Introduction to Statistics

There is something fascinating about science. One gets such a wholesale return of conjecture out of a trifling investment of fact. – Mark Twain (Life on the Mississippi, 1850)

This is a book for "consumers" of statistics. The goals are to teach you enough statistics to:

Understand the statistical portions of most articles in medical journals.
Avoid being bamboozled by statistical nonsense.
Do simple statistical calculations yourself, especially those that help you interpret published literature.
Use a simple statistics computer program to analyze data.
Be able to refer to a more advanced statistics text or communicate with a statistical consultant (without an interpreter').

Many statistical books read like cookbooks; they contain the recipes for many statistical tests, and their goal (often unstated) is to train "statistical chefs," able to whip up a P value on a moments notice. This book is based on the assumption that statistical tests are best calculated by computer programs or by experts. This book, therefore, will not teach you to be a chef, but rather to become an educated connoisseur or critic who can appreciate and criticize what the chef has created. But just as you must learn a bit about the differences between broiling, boiling, baking, and basting to become a connoisseur of fine food, you must learn a bit about probability distributions and null hypotheses to become an educated consumer of the biomedical literature. Hopefully this book will make it relatively painless.

WHY DO WE NEED STATISTICAL CALCULATIONS?

When analyzing data, your goal is simple: You wish to make the strongest possible conclusions from limited amounts of data. To do this, you need to overcome two problems:

Important differences are often obscured by biological variability and/or experimental imprecision, making it difficult to distinguish real differences from random variation.
The human brain excels at finding patterns and relationships, but tends to over generalize, For example, a 3 -year-old girl recently told her buddy, "You can't become a doctor; only girls can become doctors." To her this made sense, as the only three doctors she knew were women. This inclination to over generalize does not seem to go away as you get older, and scientists have the same urge. Statistical rigor prevents you from making this kind of error.

MANY KINDS OF DATA CAN BE ANALYZED WITHOUT STATISTICAL ANALYSIS

Statistical calculations are most helpful when you are looking for fairly small differences in the face of considerable biological variability and imprecise measurements. Basic scientists asking fundamental questions can often reduce biological variability by using inbred animals or cloned cells in controlled environments. Even so, there will still be scatter among replicate data points. If you only care about differences that are large compared with the scatter, the conclusions from such studies can he obvious without statistical analysis. In such experimental systems, effects small enough to require statistical analysis are often not interesting enough to pursue.

If you are lucky enough to be studying such a system, you may heed the following aphorisms:

If you need statistics to analyze your experiment, then you've done the wrong experiment.

If your data speak for themselves, don't interrupt!

Most scientists are not so lucky. In many areas of biology, and especially in clinical research, the investigator is faced with enormous biological variability, is not able to control all relevant variables, and is interested in small effects (say 20% change). With such data, it is difficult to distinguish the signal you are looking for from the noise created by biological variability and imprecise measurements. Statistical calculations are necessary to make sense out of such data .

STATISTICAL CALCULATIONS EXTRAPOLATE FROM SAMPLE TO POPULATION

Statistical calculations allow you to make general conclusions from limited amounts of data, You can extrapolate from your data to a more general case. Statisticians say that you extrapolate from a sample to a population. The distinction between sample and population is key to understanding much of statistics. Here are four different contexts where the terms are used.

Quality control. The terms sample and population make the most sense in the context of quality control where the sample is randomly selected from the overall population. For example, a factory makes lots of items (the population), but randomly selects a few items to test (the sample). These results obtained from the sample are used to make inferences about the entire population.
Political polls. A random sample of voters (the sample) is polled, and the results, are used to make conclusions about the entire population of voters.
Clinical studies. The sample of patients studied is rarely a random sample of the larger population. However, the patients included in the study are representative of other similar patients, and the extrapolation from sample to population is still useful. There is often room for disagreement about the precise definition of the population. Is the population all such patients that come to that particular medical center, or all that come to a big city teaching hospital, or all such patients, in the country, or all such patients in the world? While the population may be defined rather vaguely, it still is clear we wish to use the sample data to make conclusions about a larger group.
Laboratory experiments. Extending the terms sample and population to laboratory experiments is a bit awkward. The data from the experiment(s) you actually performed is the sample. If you were to repeat the experiment, you'd have a different sample. The data from all the experiments you could have performed is the population. From the sample data you want to make inferences about the ideal situation.

In biomedical research, we usually assume that the population is infinite, or at least very large compared with our sample. All the methods in this book are based on that assumption. If the population has a defined size, and you have sampled a substantial fraction of the population (>10% or so), then you need to use special methods that are not presented in this book.

WHAT STATISTICAL CALCULATIONS CAN DO

Statistical reasoning uses three general approaches:

Statistical Estimation

The simplest example is calculating the mean of a sample. Although the calculation is exact, the mean you calculate from a sample is only an estimate of the population mean. This is called a point estimate. How good is the estimate? As we will see in Chapter 5, it depends on the sample size and scatter. Statistical calculations combine these to generate an interval estimate (a range of values), known as a confidence interval for the population mean. If you assume that your sample is randomly selected from (or at least representative of) the entire population, then you can be 95% sure that the mean of the population lies somewhere within the 95% confidence interval, and you can be 99% sure that the mean lies within the 99% confidence interval. Similarly, it is possible to calculate confidence intervals for proportions, for the difference or ratio of two proportions or two means, and for many other values.

Statistical Hypothesis Testing

Statistical hypothesis testing helps you decide whether an observed difference is likely to be caused by chance. Various techniques can be used to answer this question: If there is no difference between two (or more) populations, what is the probability of randomly selecting samples with a difference as large or larger than actually observed? The answer is a probability termed the P value. If the P value is small, you conclude that the difference is statistically significant and unlikely to be due to chance.

Statistical Modeling

Statistical modeling tests how well experimental data fit a mathematical model constructed from physical, chemical, genetic, or physiological principles. The most common form of statistical modeling is linear regression. These calculations determine "the best" straight line through a particular set of data points, More sophisticated modeling methods can fit curves through data points.

WHAT STATISTICAL CALCULATIONS CANNOT DO

In theory, here is how you should apply statistical analysis to a simple experiment:

Define a population you are interested in.
Randomly select a sample of subjects to study.
Randomly select half the subjects to receive one treatment, and give the other half another treatment.
Measure a single variable in each subject.
From the data you have measured in the samples, use statistical techniques to make inferences about the distribution of the variable in the population and about the effect of the treatment.

When applying statistical analysis to real data, scientists confront several problems that limit the validity of statistical reasoning. For example, consider how you would design a study to test whether a new drug is effective in treating patients infected with the human immunodeficiency virus (HIV).

The population you really care about is all patients in the world, now and in the future, who are infected with HIV. Because you can't access that population, you choose to study a more limited population: HIV patients aged 20 to 40 living in San Francisco who come to a university clinic. You may also exclude from the population patients who are too sick, who are taking other experimental drugs, who have taken experimental vaccines, or who are unable to cooperate with the experimental protocol. Even though the population you are working with is defined narrowly, you hope to extrapolate your findings to the wider population of HIV-infected patients.

Randomly sampling patients from the defined population is not practical, so instead you simply attempt to enroll all patients who come to morning clinic during two particular months. This is termed a convenience sample. The validity of statistical calculations depends on the assumption that the results obtained from this convenience sample are similar to those you would have obtained had you randomly sampled subjects from the population.

The variable you really want to measure is survival time, so you can ask whether the drug increases life span. But HIV kills slowly, so it will take a long time to accumulate enough data. As an alternative (or first step), you choose to measure the number of helper (CD4) lymphocytes. Patients infected with the HIV have low numbers of CD4 lymphocytes, so you can ask whether the drug increases CD4 cell number (or delays the reduction in CD4 cell count). To save time and expense, you have switched from an important variable (survival) to a proxy variable (CD4 cell count).

Statistical calculations are based on the assumption that the measurements are made correctly. In our HIV example, statistical calculations would not be helpful if the antibody used to identify CD4 cells was not really selective for those cells.

Statistical calculations are most often used to analyze one variable measured in a single experiment, or a series of similar experiments. But scientists usually draw general conclusions by combining evidence generated by different kinds of experiments. To assess the effectiveness of a drug to combat HIV, you might want to look at several measures of effectiveness: reduction in CD4 cell count, prolongation of life, increased quality of life, and reduction in medical costs. In addition to measuring how well the drug works, you also want to quantify the number and severity of side effects. Although your conclusion must be based on all these data, statistical methods are not very helpful in blending different kinds of data. You must use clinical or scientific judgment, as well as common sense.

In summary, statistical reasoning can not help you overcome these common problems:

The population you really care about is more diverse than the population from which your data were sampled.
You collect data from a "convenience sample" rather than a random sample.
The measured variable is a proxy for another variable you really care about.
Your measurements may be made or recorded incorrectly, and assays may not always measure exactly the right thing.

You need to combine different kinds of measurements to reach an overall conclusion.
You must use scientific and clinical judgment, common sense, and sometimes a leap of faith to overcome these problems. Statistical calculations are an important part of data analysis, but interpreting data also requires a great deal of judgment. That's what makes research challenging. This is a book about statistics, so we will focus on the statistical analysis of data. Understanding the statistical calculations is only a small part of evaluating clinical and biological research.

WHY IS IT HARD TO LEARN STATISTICS?

Five factors make it difficult for many students to learn statistics:

The terminology is deceptive. Statistics gives special meaning to many ordinary words. To understand statistics, you have to understand that the statistical meaning of terms such as sigiiificai7t, error, and h3,pothesis are distinct from the ordinary uses of these words. As you read this book, pay special attention to the statistical terms that sound like words you already know.
Many people seem to believe that statistical calculations are magical and can reach conclusions that are much stronger than is actually possible. The phrase statistically significant is seductive and is often misinterpreted.
Statistics requires mastering abstract concepts. It is not easy to think about theoretical concepts such as populations, probability distributions, and null hypotheses.
Statistics is at the interface of mathematics and science. To really grasp the concepts of statistics, you need to be able to think about it from both angles. This book emphasizes the scientific angle and avoids math. If you think like a mathematician, you may prefer a text that uses a mathematical approach.
The derivation of many statistical tests involves difficult math. Unless you study more advanced books, you must take much of statistics on faith. However, you can learn to use statistical tests and interpret the results even if you don't fully understand how they work. This situation is common in science, as few scientists really understand all the tools they use. You can interpret results from a pH meter (measures acidity) or a scintillation counter (measures radioactivity), even if you don't understand exactly how they work. You only need to know enough about how the instruments work so that you can avoid using them in inappropriate situations. Similarly, you can calculate statistical tests and interpret the results even if you don't understand how the equations were derived, as long as you know enough to use the statistical tests appropriately.

ARRANGEMENT OF THIS BOOK

Parts I through V present the basic principles of statistics. To make it easier to learn, I have separated the chapters that explain confidence intervals from those that explain P values. In practice, the two approaches are used in parallel. Basic scientists who don't care to learn about clinical studies may skip Chapters 6 (survival curves) and 9 (case-control studies) without loss of continuity.

Part VI describes the design of clinical studies and discusses how to determine sample size. Basic scientists who don't care to learn about clinical studies can skip this entire part. However, Chapter 22 (sample size) is of interest to all. Part VII explains the most common statistical tests. Even if you use a computer program to calculate the tests, reading these chapters will help you understand how the tests work. The tests mentioned in this section are described in detail.

Part VIII gives an overview of more advanced statistical tests. These tests are not described in detail, but the chapters provide enough information so that you can be an intelligent consumer of papers that use these tests. The chapters in this section do not follow a logical sequence, so you can pick and choose the topics that interest you. The only exception is that you should read Chapter 31 (multiple regression) before Chapters 32 (logistic regression) or the parts of Chapter 33 (comparing survival curves) dealing with proportional hazards regression.

The statistical principles and tests discussed in this book are widely used, and I do not give detailed references. For more information, refer to the general textbook references listed in Appendix 1.

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