Intuitive biostatistics pdf download

Comparing Two Means: The Randomization and Mann-Whitney Tests, Calculating the exact randomizationtest, Large samples:The approximaterandomizationtest, The relationship between the randomizationtest and the t test, Mann-Whitney test,22l Performing the Mann-Whitney test, Assumptions of the Mann-Whitney test, When to use nonparametrictests, Problems.

Comparing Observed and Expected Counts, Analyzing counted data, The Yates' continuity correction, Where does the equation come from? Comparing Two Proportions, Fisher's exact test, Chi-squaretest for 2 X 2 contingency tables, How to calculate the chi-squaretest for a 2 x 2 contingency table, Assumptions, Choosing between chi-squareand Fisher's test, Calculatingpower, Problems.

Further Analyses of Contingency Tables, McNemar's chi-squaretest for paired observations, Chi-squaretest with large tables more than two rows or columns , Chi-squaretest for trend, Mantel-Haenszelchi-squaretest, Multiple Regression, The uses of multiple regression, The multiple regressionmodel Assumptions of multiple regression, Interpreting the results of multiple regression, Choosing which X variablesto include in a modeI, The term multivariate statistics, Logistic Regression, Introduction to logistic regression, How logistic regressionworks, Assumptionsof logistic regression, Interpreting results from logistic regression, Comparing Survival Curves, Comparing two survival curves, Assumptions of the log-rank test, A potential trap: Comparing survival of respondersversus nonresponders, Will Rogers' phenomenon, Multiple regressionwith survival data: Proportional hazardsregression,2T4 How proportional hazardsregressionworks, Interpreting the results of proportional hazardsregression,2T6 Using Nonlinear Regressionto Fit Curves, The goals of curve fitting, An example, What's wrong with transforming curved data into straight lines?

Adjustingfor Confounding What is a confounding vanable? Choosing a Test, Review of available statisticaltests, Review of nonparametrictests, Choosing between parametric and nonparametrictests: The easycases, Choosing between parametric and nonparametrictests: The hardcases, Choosing between parametric and nonparametrictests: Doesit matter?

The Big Picture, Look at the data! References, Appendix2. Answersto Problems,J16 Appendix5. StatisticalTables, Index. One gets such a whore sar ereturn "tffil;'iT:':"I ;i"' Ttr,il;:ilffi:, TJl'; This is a book for "consumers" of statistics. The goals are to teach you enough statisticsto l. Understandthe statisticalportions of most articles in medical journals. Avoid being bamboozledby statisticalnonsense. Do simple statisticalcalculationsyourself, especially those that help you interpret published literature.

Use a simple statisticscomputer program to analyze data. Be able to refer to a more advancedstatisticstext or communicatewith 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 moment's notice. This book is based on the assumptionthat statistical tests are best calculated by computer programs or by experts.

This book, therefore,will not teachyou to be a chef, but rather to becomean educatedconnoisseur or critic who can appreciateand criticize what the chef has created. But just as you must learn a bit about the differencesbetween broiling, boiling, baking, and basting to becomea connoisseurof fine food, you must learn a bit aboutprobability distributions and null hypothesesto become an educatedconsumer of the biomedical literature.

Hopefully this book will make it relatively painless. When analyzing data, your goal is simple: You wish to make the strongestpossible conclusions from limited amounts of data. This inclination to overgeneraLize does not seem to go away as you get older, and scientistshave the sameurge.

Statistical rigor prevents you from making this kind of error. Basic scientistsasking fundamentalquestionscan often reducebiological 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 comparedwith the scatter,the conclusionsfrom such studiescan be obvious without statistical analysis. In such experimental systems, effects small enough to require statisticalanalysis are often not interestingenough to pursue. If you are lucky enough to be studying such a system,you may heed the following aphorisms: If youneedstatistics to analyzeyourexperiment, thenyou'vedonethewrongexperiment.

If your dataspeakfor themselves, Most scientists are not so lucky. In many areas of biology, and especially in clinical research,the investigatoris faced with enormousbiological variability, is not able to control all relevantvariables,and is interestedin small effects say ochange.

With suchdata,it is difficult to distinguishthe signal you are looking for from the noise Statisticalcalculationsare createdbybiological variability and imprecisemeasurements. You can extrapolatefrom your data to a more general case. Statisticianssay that you extrapolatefrom 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. Quatity control. The terms sample andpopulation makethe most sensein the context of quality control where the sample is randomly selectedfrom the overall population. For example, a factory makes lots of items the population , but randomly selectsa few items to test the sample. These results obtained from the sample are used to make inferences about the entire population.

Politicat polts. A random sample of voters the sample is polled, and the results are used to make conclusionsabout 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 representativeof other similar patients,and the extrapolationfrom sampleto population is still useful. There is often room for disagreementabout the precise definition of the population.

Is the populatio,n 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 usethe sampledata to make conclusionsabout a larger group. Laboratory experiments. Extending the terms sample and population to laboratory experimentsis a bit awkward.

The data from the experiment s you actually performed is the sample. If you were to repeatthe 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 inferencesabout the ideal situation. In biomedical research,we usually assumethat the population is infinite, or at least very large comparedwith our sample. All the methodsin this book are basedon that assumption. Although the calculation is exact, the mean you calculate from a sample is only an estimateof the population mean. This is called a point estimate. How good is the estimate? As we will see in Chapter 5, it dependson the sample size and scatter.

Statisticalcalculationscombine these to generate an interval estimate a range of values , known as a confidence interval for the population mean. If you assumethat your sampleis randomly selected from or at least representativeof the entire population, then you can be 95Vo sure that the mean of the population lies somewhere within the 95Vo confidence interval, and you can be o sure that the mean lies within the 99Vo confidence interval.

Similarly, it is possibleto calculateconfidenceintervals for proportions,for the difference or ratio of two proportions or two means,and for many other values. Statistical Hypothesis Testing Statisticalhypothesistesting helps you decide whether an observeddifference is likely to be causedby chance. Various techniquescan be used to answer this question: If there is no difference between two or more populations,what is the probability of randomly selectingsampleswith a differenceas 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 signfficant and,unlikely to be due to chance. 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 sophisticatedmodeling methodscan fit curves through data points. Define a population you are interestedin. Randomly select a sample of subjectsto study. Randomly selecthalf the subjectsto receive one treatment,and give the other half another treatment.

Measure a single variable in each subject. From the data you have measuredin the samples,use statisticaltechniquesto make inferences about the distribution of the variable in the population and about the effect of the treatment.

When applying statisticalanalysisto real data,scientistsconfront severalproblems that limit the validity of statisticalreasoning. For example, consider how you would design a study to test whether a new drug is effective in treating patientsinfected with the human immunodeficiencyvirus HIV.

The population you really care about is all patients in the world, now and in the future, who are infected with HIV. Becauseyou can't accessthat population, you chooseto study a more limited population: HIV patients aged2O to 40 living in San Franciscowho 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 experimentalvaccines,or who areunableto cooperatewith the experimentalprotocol.

Even though the population you are working with is defined narrowly, you hope to extrapolateyour findings to the wider population of HlV-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 conveniencesample. The validity of statistical calculations dependson the assumptionthat the results obtainedfrom this convenience sample are similar to those you would have obtained had you randomly sampled subjectsfrom the population.

The variable you really want to measure is survival time, so you can ask whether the drug increaseslife span. But HIV kills slowly, so it wiil take a long time to accumulate enough data. As an alternative or first step , you choose to measurethe 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. Statisticalcalculationsare basedon the assumptionthat the measurementsare made correctly. In our HIV example, statistical calculations would not be helpful if the antibody used to identify CD4 cells was not really selectivefor those cells.

Statistical calculations are most often used to analyzeone variable measuredin a single experiment, or a series of similar experiments. But scientistsusually draw general conclusionsby combining evidencegeneratedby different kinds of experiments. To assessthe effectivenessof a drug to combat HIV, you might want to look at several measuresof effectiveness:reduction in CD4 cell count, prolongation of life, increasedquality of life, and reductionin 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 conclusionmust be basedon all thesedata, statisticalmethods 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 diversethan the population from which your data were sampled.

You must use scientific and clinical judgment, common sense,and sometimesa leap of faith to overcometheseproblems. Statisticalcalculationsare an important part of data analysis,but interpreting data also requires a greatdeal of judgment. That's what makes researchchallenging. This is a book about statistics,so we will focus on the statisticalanalysisof data. Understandingthe statisticalcalculationsis only a small part of evaluating clinical and biological research.

Five factors make it difficult for many students to learn statistics: ' The terminology is deceptive. Statistics gives special meaning to many ordinary words. To understandstatistics,you have to understandthat the statisticalmeaning of terms such as signfficant, error, and hypothesis aredistinct from the ordinary uses of these words. As you read this book, pay special attention to the statisticalterms that sound like words you already know. The phrase statistically significanr is seductiveand is often misinterpreted.

It is not easyto think about theoretical conceptssuch as populations,probability distributions, and null hypotheses. To really grasp the concepts of statistics, you need to be able to think about it from both angles. If you think like a mathematician, you may prefer a text that uses a mathematical approach.

Unless you study more advancedbooks, you must take much of statistics on faith. However, you can learn to use statisticaltestsand interpretthe resultseven if you don't fully understand how they work. This situation is common in science,asfew scientistsreally understand all the tools they use. You can interpret results from a pH meter measuresacidity or a scintillation counter measuresradioactivity , 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 statisticaltests 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.

To make it easierto learn, I have separatedthe chaptersthat explain confidence intervals from those that explain P values. In practice, the two approachesare used in parallel. Basic scientistswho don't care to learn about clinical studiesmay skip Chapters6 survival curves and 9 case-controlstudies without loss of continuity. Part VI describesthe design of clinical studies and discusseshow to determine sample size. Basic scientistswho 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 chapterswill help you understandhow the tests work. The tests mentioned in this section are described in detail.

Part VIII gives an overview of more advancedstatistical tests. These tests are not described in detail, but the chaptersprovide enough information so that you can be an intelligent consumerof papersthat use thesetests.

The chaptersin this section do not follow a logical sequence,so you can pick and choosethe topics that interestyou. The only exceptionis that you should read Chapter31 multiple regression before Chapters 32 logistic regression or the parts of Chapter 33 comparing survival curves dealing with proportional hazards regression. The statisticalprinciples and tests discussedin this book are widely used, and I do not give detailed references.

For more information, refer to the general textbook referenceslisted in Appendix l. The two give complementaryinformation and are often calculated in tandem.

For the purposesof clarity and simplicity, this book presentsconfidence intervals first and then presentsP values. Confidenceintervals let you statea result with margin of eruor. This sectionexplains what this meansand how to calculate confidenceintervals. In this chapter we will consider only results expressedas a proportion or fraction. Here are some examples: the proportion of patients who become infected after a procedure, the proportion of patientswith myocardialinfarction who developheartfailure, the proportion of students who pass a course,the proportion of voters who vote for a particular candidate.

Later we will discussother kinds of variables,including measurementsand survival times. This meansthat, in the long run, a coin will land on headsabout as often as it lands on tails. But in any particular series of tosses,you may not seethe coin land on headsexactly half the time. You may even see all headsor all tails. Mathematicianshave developedequations,known as the binomial distribution. Using the binomial distribution, you can answer questionssuch as these: ' If you flip a coin 10 times, what is the probability of getting exactly 7 heads?

Perhapsyou've seenthe equationsthat help you answer thesekinds of questions,and recall that there are lots of factorials. If you're interested,the equation is presentedat the end of this chapter. The theory startswith a known probability i. When analyzing data, we need to work in the opposite direction. We don't know the overall probability. That's what we are trying to find out. We do know the proportion observedin a single sampleand wish to make inferencesabout the overall probability.

The binomial distribution can still be useful. I show you how to do this at the end of the chapter. For now, acceptthe fact that it can be done and concentrateon interpreting the results. The proportion is ,which equals0. What can you say about the probability of complications in the entire population of patients who will be treated with this drug?

There are two issuesto think about. First, you must think about whether the 14 patients are representativeof the entire population of patients who will receive the drug. Perhapsthese patients were selectedin such a way as to make them more or less likely than other patientsto develop the side effect. Statisticalcalculationscan't help you answerthat question,and we'll assumethat the sampleadequatelyrepresents the population.

The secondissueis random sampling,sometimesreferredto as margin of error. Just by chance,your sample of 14 patientsmay have had an especiallyhigh or an especially low rate of side effects. The overall proportion of side effects in the population is unlikely to equal exactly 0. Here is a secondexample. You polled randomly selectedvoters just before an election, and only 33 said they would vote for your candidate.

What can you say about the proportion of all voters who will vote for your candidate? Again, there are two issuesto deal with. First, you need to think about whether your sample is really representativeof the population of voters, and whether people tell the pollsters the truth about how they will vote. Statistical calculationscannot help you grapple with those issues.

We'lI assumethat the sampleis perfectly representativeof the population of voters and that every person will vote as they said they would on the poll. Second, you need to think about sampling error. Just by chance,your sample may contain a smaller or larger fraction of people voting for your candidate than does the overall population. Since we only know the proportion in one sample, there is no way to be sure about the proportion in the population.

The best we can do is calculate a range of values that bracket the true populationproportion. How wide does this range of values have to be? In the overall population, the fraction of patients with side effects could be as low as 0. Those values are exceedingly unlikely but not absolutely impossible. If you want to be osure that your range includes the true population value, the range has to include these possibilities.

Such a wide range is not helpful. To create a narrower and more useful range,you must acceptthe possibility that the interval will not include the true population value. It makessensethat the margin of error dependson the samplesize,so that the confidence interval is wider in the first example 14 subjects than in the second subjects. Before continuing, you should think about these two examplesand write down your intuitive estimate of the 95Vo CIs.

Do it now, before reading the answer in the next paragraph. Later in this chapter you'll learn how to calculate the confidenceinterval.

But it is easier to use an appropriatecomputer program to calculate the o CIs instantly. All examplesin this book were calculatedwith the simple program GraphPadInStat see Appendix 2 , but many other programs can perform these calculations. Here are the results. For the first example,the 95VoCI extendsfrom 0. For the second example, the 95VoCI extendsfrom 0. How good were your guesses? Many people tend to imagine that the interval is narrower than it actually is.

What does this mean? Assuming that our sampleswere randomly chosen from the entire populations,we can be 95Vo surethat the range of values includes the true population proportion. Note that there is no uncertainty about what we observed in the sample.

We are absolutely sure that 2l. Calculation of a confidenceinterval cannot overcomeany mistakesthat were made in tabulatingthosenumbers. What we don't know is the proportionin the entirepopulation. However, we can be 95Vosure that it lies within the calculatedinterval.

The term confidenceinterval, abbreviatedCI, refers to the range of values. The two ends of the CI are called the confidencelimits. When you only have measuredone sample,you don't know the value of the population proportion. It either lies within the 95VoCI you calculatedor it doesn't.

There is no way for you to know. If you were to calculate a 95VoCI from many samples,the population proportion will be included in the CI in 95Vo of the samples,but will be outside of the CI the other 5Voof the time.

More precisely, you can be 95Vo certain that the o CI calculated from your sample includes the population proportion. Figure 2. Here we assumethat the proportion of voters in the overall population who will vote for your candidateequals 0. We created 50 samples,each with subjects and calculated the 95VoCI for each. Each 95VoCI is shown as a vertical line extending from the lower confidencelimit to the upperconfidencelimit.

The value of the observed proportion in each sample is shown as a small hatch mark in the middle of each CI. The first line on the left correspondsto our example. The other 49 lines represent results that could have been obtained by random sampling from the samepopulation. Why are they surprising? You also need to consider the experimental context, previous data,and theory. Bayesian logic allows you to integrate the current experimentaldata with what you knew before the experiment.

Since Bayesian logic can be difficult to understandin the context of interpreting P values, I first present the use of Bayesian logic in interpreting the results of clinical laboratory tests in Chapter Then in Chapter 16 I explain how Bayesianlogic is used in interpretinggeneticdata. This chapter setsthe stagefor the discussion in the next two chapters. What do laboratory tests have to do with P values? Understandinghow to interpret "positive" and "negative" lab testswill help you understandhow to interpret ,,significant" and "not significant" statisticaltests.

Results can be tabulatedon a two by two contingencytable Table The rows represenr the outcome of the test positive or negative , and the columns indicate whether the diseaseis presentor absent basedupon someother method that is completely accurate, perhapsthe test of time. If the test is "positive," it may be true poiitiu.

Tp , or it may be a false positive FP test in a person without the condition being tested for. If the test is "negative," it may be a true negative TN or it may be a false negative FN test in a person who does have the condition.

How accurateis the test? It is impossibleto expressthe accuracyin one number. It takes at least two: sensitivity and specificity.

An ideal test has very high sensitivity and very high specificity: ' The sensitivityis the fraction of all those with the diseasewho get a positive testresult. Sensitivity measures how well the test identifies those with the disease,that is, how sensitive it is. If a test has a high sensitivity, it will pick up nearly everyonewith the disease.

If a test has a very high specificity, it won't mistakenly give a positive result to many people without the disease. Rememberthe meaningof specificity with this acronym: SpPln. If a test has high q,pecificity,a positive test rules in the disorder relatively few positive tests are false positive. For the purposesof this chapter, we will simplify things so that it reports either o'normal" or "abnormal. In Figure Every individual whose value is below that value the left of the dotted line does not have the condition, and every individual whose test value is above that threshold has it.

Figure Again, the solid curve representspatientswithout the condition and the dashed curve representspatientswith the condition. Whereveryou set the cutoff, somepatients will be misclassified. The dark shadedareashowsfalse positives,thosepatientsclassified as positive even though they don't have the disease.

The lighter shadedareashows false negatives,those patientsclassifiedas negativeseven though they really do have the disease. Choosing a threshold requires you to make a tradeoff between sensitivity and specificity. If you increasethe threshold move the dotted line to the right , you will increasethe specificity but decreasethe sensitivity. You have fewer false positives but more false negatives. The book provides a modern look at introductory Biostatistical concepts and the associated computational tools using the latest developments in computation and visualization in the R language environment.

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