8.1 – The null and alternative hypotheses

Introduction

Classical statistical parametric tests — t-test (one sample t-test, independent sample-t-test), analysis of variance (ANOVA), correlation, and linear regression— and nonparametric tests like  (chi-square: goodness of fit and contingency table),  share several features that we need to understand. It’s natural to see all the details as if they are specific to each test, but there’s a theme that binds all of the classical statistical inference in order to make  claim of “statistical significance.”

• a calculated test statistic
• degrees of freedom associated with the calculation of the test statistic
• a probability value or p-value which is associated with the test statistic, assuming a null hypothesis is “true” in the population from which we sample.
  • Recall from our previous discussion (Chapter 8.2) that this is not strictly the interpretation of p-value, but a short-hand for how likely the data fit the null hypothesis. P-value alone can’t tell us about “truth.”
• in the event we reject the null hypothesis, we provisionally accept the alternative hypothesis.

Statistical Inference in the NHST Framework

By inference we mean to imply some formal process by which a conclusion is reached from data analysis of outcomes of an experiment. The process at its best leads to conclusions based on evidence. In statistics, evidence comes about from the careful and reasoned application of statistical procedures and the evaluation of probability (Abelson 1995).

Formally, statistics is rich in inference process (Bzdok et al, 2018). We begin by defining the classical frequentist, aka Neyman-Pearson approach, to inference, which involves the pairing of two kinds of statistical hypotheses: the null hypothesis (H_O) and the alternative hypothesis (H_A). The null hypothesis assumes that any observed difference between the groups are the results of random chance (introduced in Ch 2.3 – A brief history of (bio)statistics ), where “random chance” implies that outcomes occur without order or pattern (Eagle 2021). Whether we reject or accept the null hypothesis — a short-hand for the more appropriate “provisionally reject” in light of the humility that our results may reflect a rare sampling event — based on the outcome of our statistical inference, the inference is evaluated against a decision criterion (cf discussion in Christiansen 2005), a fixed statistical significance level (Lehmann 1992). Significance level refers to the setting of a p-value threshold before testing is done. The threshold is often set to Type I error of 5% (Cowles & Davis 1982), but researchers should always consider whether this threshold is appropriate for their work (Benjamin et al 2017).

This inference process is referred to as Null Hypothesis Significance Testing, NHST. Additionally, a probability value will be obtained for the test outcome or test statistic value. In the Fisherian likelihood tradition, the magnitude of this statistic value can be associated with a probability value, the p-value, of how likely the result is given the null hypothesis is “true.” (Again, keep in mind that this is not strictly the interpretation of p-value, it’s a short-hand for how likely the data fit the null hypothesis. P-value alone can’t tell us about “truth”, per our discussion, Chapter 8.2.)

-1*log(0.000134,10)
[1] 3.872895

> 10^(-1*3.872895)
[1] 0.0001340001

NHST Workflow

We presented in the introduction to Chapter 8 without discussion a simple flow chart to illustrate the process of decision (Fig 1, Chapter 8). Here, we repeat the flow chart diagram and follow with descriptions of the elements.

Figure 1. Flow chart of inductive statistical reasoning.

What’s missing from the flow chart is the very necessary caveat that interpretation of the null hypothesis is associated with two kinds of error, Type I error and Type II error. These points and others are discussed in the following sections.

We start with the hypothesis statements. For illustration we discuss hypotheses in terms of comparisons involving just two groups, also called two sample tests. One sample tests in contrast refer to scenarios where you compare a sample statistic to a population value. Extending these concepts to more than two samples is straightforward, but we leave that discussion to Chapters 12 – 18.

Null hypothesis

By far the most common application of the null hypothesis testing paradigm involves the comparisons of different treatment groups on some outcome variable. These kinds of null hypotheses are the subject of Chapters 8 through 12.

The Null hypothesis (H_O) is a statement about the comparisons, eg, between a sample statistic and the population, or between two treatment groups. The former is referred to as a one tailed test whereas the latter is called a two-tailed test. The null hypothesis is typically “no statistical difference” between the comparisons.

For example, a one sample, two tailed null hypothesis.

and we read it as “there is no statistical difference between our sample mean and the population mean.” For the more likely case in which no population mean is available, we provide another example, a two sample, two tailed null hypothesis.

Here, we read the statement as “there is no difference between our two sample means.” Equivalently, we interpret the statement as both sample means estimate the same population mean.

Under the Neyman-Pearson approach to inference we have two hypotheses: the null hypothesis and the alternative hypothesis. The hull hypothesis was defined above.

Alternative hypothesis

Alternative hypothesis (H_A): If we conclude that the null hypothesis is false, or rather and more precisely, we find that we provisionally fail to reject the null hypothesis, then we provisionally accept the alternative hypothesis. The view then is that something other than random chance has influenced the sample observations. Note that the pairing of null and alternative hypotheses covers all possible outcomes. We do not, however, say that we have evidence for the alternative hypothesis under this statistical regimen (Abelson 1995). We tested the null hypothesis, not the alternative hypothesis. Thus, it is incorrect to write that, having found a statistical difference between two drug treatments, say aspirin and acetaminophen for relief of migraine symptoms, it is not correct to conclude that we have proven the case that acetaminophen improves improves symptoms of migraine sufferers.

For the one sample, two tailed null hypothesis, the alternative hypothesis is

and we read it as “there is a statistical difference between our sample mean and the population mean.” For the two sample, two tailed null hypothesis, the alternative hypothesis would be

and we read it as “there is a statistical difference between our two sample means.”

Alternative hypothesis often may be the research hypothesis.

It may be helpful to distinguish between technical hypotheses, scientific hypothesis, or the equality of different kinds of treatments. Tests of technical hypotheses include the testing of statistical assumptions like normality assumption (see Chapter 13.3) and homogeneity of variances (Chapter 13.4). The results of inferences about technical hypotheses are used by the statistician to justify selection of parametric statistical tests (Chapter 13). The testing of some scientific hypothesis like whether or not there is a positive link between lifespan and insulin-like growth factor levels in humans (Fontana et al 2008), like the link between lifespan and IGFs in other organisms (Holtzenberger et al 2003), can be further advanced by considering multiple hypotheses and a test of nested hypotheses and evaluated either in Bayesian or likelihood approaches (Chapter 16 and Chapter 17). The advantage of the Bayesian approach is that it returns evidence for or against this hypothesis: if the posterior probability is greater than the prior probability, we can we say that we have evidence for the hypothesis. This is very appealing — the challenge is that the prior probability is inherently subjective (Chapter 6.4), limited by the amount and quality of existing information pertaining to the hypothesis (Prelec 2004).

How to interpret the results of a statistical test.

Any number of statistical tests may be used to calculate the value of the test statistic. For example, a one sample t-test may be used to evaluate the difference between the sample mean and the population mean (Chapter 8.5) or the independent sample t-test may be used to evaluate the difference between means of the control group and the treatment group (Chapter 10). The test statistic is the particular value of the outcome of our evaluation of the hypothesis and it is associated with the p-value. In other words, given the assumption of a particular probability distribution, in this case the t-distribution, we can associate a probability, the p-value, that we observed the particular value of the test statistic and the null hypothesis is true in the reference population.

By convention, we determine statistical significance (Cox 1982; Whitley & Ball 2002) by assigning ahead of time a decision probability called the Type I error rate, often given the symbol α (alpha). The practice is to look up the critical value that corresponds to the outcome of the test with degrees of freedom like your experiment and at the Type I error rate that you selected. The Degrees of Freedom (DF, df, or sometimes noted by the symbol v), are the number of independent pieces of information available to you. Knowing the degrees of freedom is a crucial piece of information for making the correct tests. Each statistical test has a specific formula for obtaining the independent information available for the statistical test. We first were introduced to DF when we calculated the sample variance with the Bessel correction, n – 1, instead of dividing through by n. With the df in hand, the value of the test statistic is compared to the critical value for our null hypothesis. If the test statistic is smaller than the critical value, we fail to reject the null hypothesis. If, however, the test statistic is greater than the critical value, then we provisionally reject the null hypothesis. This critical value comes from a probability distribution appropriate for the kind of sampling and properties of the measurement we are using. In other words, the rejection criterion for the null hypothesis is set to a critical value, which corresponds to a known probability, the Type I error rate.

Before proceeding with yet another interpretation, and hopefully less technical discussion about test statistics and critical values, we need to discuss the two types of statistical errors. The Type I error rate is the statistical error assigned to the probability that we may reject a null hypothesis as a result of our evaluation of our data when in fact in the reference population, the null hypothesis is, in fact, true. In Biology we generally use Type I error α = 0.05 level of significance. We say that the probability of obtaining the observed value AND H_O is true is 1 in 20 (5%) if α = 0.05. Put another way, we are willing to reject the Null Hypothesis when there is only a 5% chance that the observations could occur and the Null hypothesis is still true. Our test statistic is associated with the p-value; the critical value is associated with the Type I error rate. If and only if the test statistic value equals the critical value will the p-value equal the Type I error rate.

The second error type associated with hypothesis testing is, β, the Type II statistical error rate. This is the case where we accept or fail to reject a null hypothesis based on our data, but in the reference population, the situation is that indeed, the null hypothesis is actually false. We must also keep in mind that, as we evaluate the p-value relative to the magnitude of the Type I error rate, and if we reject the null hypothesis, we still do not say that we have evidence for the alternative hypothesis.

Thus, we end with a concept that may take you a while to come to terms with — there are four, not two possible outcomes of an experiment.

Outcomes of an experiment

What are the possible outcomes of a comparative experiment\? We have two treatments, one in which subjects are given a treatment and the other, subjects receive a placebo. Subjects are followed and an outcome is measured. We calculate the descriptive statistics aka summary statistics, means, standard deviations, and perhaps other statistics, and then ask whether there is a difference between the statistics for the groups. So, two possible outcomes of the experiment, correct\? If the treatment has no effect, then we would expect the two groups to have roughly the same values for means, etc., in other words, any difference between the groups is due to chance fluctuations in the measurements and not because of any systematic effect due to the treatment received. Conversely, then if there is a difference due to the treatment, we expect to see a large enough difference in the statistics so that we would notice the systematic effect due to the treatment.

Actually, there are four, not two, possible outcomes of an experiment, just as there were four and not two conclusions about the results of a clinical assay. The four possible outcomes of a test of a statistical null hypothesis are illustrated in Table 1.

Table 1. When conducting hypothesis testing, four outcomes are possible.

In the actual population, a thing happens or it doesn’t. The null hypothesis is either true or it is not. But we don’t have access to the reference population, we don’t have a census. In other words, there is truth, but we don’t have access to the truth. We can weight, assigned as a probability or p-value,  our decisions by how likely our results are given the assumption that the truth is indeed “no difference.”

If you recall, we’ve seen a table like Table 1 before in our discussion of conditional probability and risk analysis (Chapter 7.3). We made the point that statistical inference and the interpretation of clinical tests are similar (Browner and Newman 1987). From the perspective of ordering a diagnostic test, the proper null hypothesis would be the patient does not have the disease. For your review, here’s that table (Table 2).

Table 2. Interpretations of results of a diagnostic or clinical test.

Thus, a positive diagnostic test result is interpreted as rejecting the null hypothesis. If the person actually does not have the disease, then the positive diagnostic test is a false positive.

Questions

1. Match the corresponding entries in the two tables. For example, which outcome from the inference/hypothesis table matches specificity of the test \?
2. Find three sources on the web for definitions of the p-value. Write out these definitions in your notes and compare them.
3. In your own words distinguish between the test statistic and the critical value.
4. All of our discussions have been about testing the null hypothesis, about accepting or rejecting, provisionally, the null hypothesis.
5. What are the p-values for -logP of 5, 6, 7, 8, 9, and 10 \? Complete the p-value -logP transform table.
6. Instead of log₁₀ transform, create a similar table but for negative natural log transform. Which is more convenient? Hint: log(x, base=exp(1))

Quiz Chapter 8.1

The null and alternate hypothesis

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Chapter 8 contents

• Introduction
• The null and alternative hypotheses
• The controversy over proper hypothesis testing
• Sampling distribution and hypothesis testing
• Tails of a test
• One sample t-test
• Confidence limits for the estimate of population mean
• References and suggested readings

2.3 – A brief history of (bio)statistics

Foundations of biostatistics: key definitions.

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Before discussing history and landmark moments in biostatistics, let’s start with basic definitions.

Bioinformatics is loosely defined as a discipline of biology primarily concerned with work involving large data sets (eg, databases), but a bioinformatician would not primarily be a statistician necessarily. Rather, a bioinformatician in addition to foundation in statistical and mathematical training would likely be fluent in at least one programming language and confident in use and design of databases.

Biostatistics, or biometry, then, refers to use of statistics in biology. Biostatistics encompasses application of statistical approaches to design, analyze, and interpret biological data collected through observation of use of experimentation. In turn, there are many broad disciplines or fields of specialty that trained biostatisticians may work.

Chance, the likelihood that a particular event will occur.

Data scientist, a very general label, is a person likely to work on “big data.” Big data may be loosely and inconsistently identified as access to large detailed and unstructured data sets such as visits and behavior within websites of tens of millions of Internet “hits” to a website like Amazon® or Google®. The data scientist would then be involved in extracting meaning from volumes of this data in a process called data mining. In the context of biology web sites like allele frequency aggregator at NCBI, which houses allele frequency information collected on human populations, the 1000 genome project, or any of the databases accessible at National Center for Biotechnology Information would constitute sources of big data for biological researchers.

Epidemiology refers to the statistics of patterns of and risk of disease in populations, particularly of humans and thus, an epidemiologist would also be considered to be a biostatistician. The statistics of epidemiology include all of the materials we will cover in this course, but perhaps if any particular analytical approach characterizes epidemiology, it would be survival analysis (review in Clark et al 2003) — in competition with compartment models to predict spread of communicable disease in populations (Brauer 2017).

Event, an outcome to which a probability is assigned.

Likelihood, the probable chance of occurrence of an event that has already occurred. Likelihood assesses how well a particular model or set of parameters explains observed data.

Probability, the chance that an event will occur in the future, given a known model or set of parameters.

Random or random chance. This is a good place to share a warning about vocabulary; statistics, like most of science, uses familiar words, but with refined and sometimes different meanings than our everyday usage. Consider our everyday use of random: “without definite aim, direction, rule, or method – subjects chosen at random” (Merriam-Webster online dictionary). In statistics, however, random refers to “an assignment of a numerical value to each possible outcome of an event” (Wikipedia, Eagle 2021). Thus, in statistics, random dictates a method of determining how likely a subject is to be included: If represents the size of the population, then random sampling implies that each individual had chance of being selected. Thus, if , then each individual has a 1% () chance of selection. This is quite different from Merriam-Webster’s definition, in which no method is assumed. To a statistician, then, random as used in everyday conversation would imply haphazard sampling or convenience sampling from a population.

Statistics may be defined as the science of collecting, organizing, and interpreting data. Statistics is a branch of applied mathematics. Note that the word statistic is also used, but refers to a calculated quantity like the mean or standard deviation. A little confusing, but the context in which statistics or statistic is appropriate is usually not a major issue.

From dice games to data science: A brief history of statistics.

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Figure 1. Soldiers playing dice, painting by Michiel Sweerts (1618–1664). Public domain image, https://commons.wikimedia.org/

The concepts of chance and probability, so crucial to statistical reasoning — because it allows us to interpret data when outcomes are inconsistent — were realized rather late in the history of mathematics. While people have been writing about applied and theoretical math for thousands of years, probability as a topic of interest by (Western) scholars seems to date only back to the late 17th century, beginning with letters written between Pierre de Fermat (1601-1665) and Blaise Pascal (1623 – 1662) and the substantial work on probability by Pierre-Simon Laplace (1749-1827). Often, research on probability developed under the watchful eyes of rich patrons more interested in gaming (sensu Fig 1), than to scientific applications. Work on permutations and combinations, essential for an understanding of probability, trace to India prior to Pascal’s work (Raju 2011).

The history of statistics goes back further if you allow for the dual use of the term statistics, both as a descriptor of the act of collecting data and as a systematic approach to the analysis of data. Prior to the 1700s, statistics was used in the sense of collection of data for use by the governments. It is not until the latter part of the 19th century that we see scholarship on statistical analytical techniques. Many of the statistical approaches we teach and use today were developed in the decades between 1880s and the 1930s. Work by Francis Galton (1822-1911), Karl Pearson (1857-1936), RA Fisher (1890-1962), Sewall Wright (1889-1988), Jerzy Neyman (1894-1981), Egon Pearson (1895-1980, Karl Pearson’s son), George EP Box (1919-2013) remains highly influential today.

Development of randomized controlled (clinical) trials, RCT, the “gold standard,” by the 1950s provides the strongest study design to evaluate the effectiveness of clinical or other kinds of interventions. The general features of RCT are (1) comparison among groups, with at least one control group, (2) randomization, and (3) blinding, where treatment options are concealed from the subjects (single-blind) and from those charged to assess the outcomes (double-blind). Sir Austin Bradford Hill (1887-1991) is generally credited with developing the RCT (Collier 2009, Matthews 2016), but we owe RA Fisher for randomization in experimental design. Hill (1965) is also known for Hill’s criteria of causality — nine principles that can be used to establish causal relationship between epidemiological (observational) evidence and disease. Chief among these nine principles are strength of the association (effect size) and consistent association (reproducibility), elements we return to often in our introduction to biostatistics.  

Since the 1950s, there has been an explosion of developments in statistics, particularly as related to power of computer. These include use of resampling, simulation, and Monte Carlo methods (Harris 2010). Resampling — the creation of new samples based on a set of observed data — is a key innovation in statistics. Its use led to a number of innovative ways to estimate the precision of an estimate (see Chapter 3.4 and Chapter 19). Monte Carlo methods, or MCM, which involves resampling from a probability distribution, is used to repeat (simulate) an experiment over and over again (Kroese et al 2014). Computers have so influenced statistics that some now define statistics as “…the study of algorithms for data analysis” (p 175, Beran 2003). So much so, the role of computers and coding to process and model data, we now have the discipline of Data Science. For more on the history of statistics, see Anderson (1992), Fienberg (1992), and Freedman 1999; for excellent, conversational books read Salsburg (2002) and McGrayne (2011). For influential women in early development of statisticians, see Anderson (1992).

Numbers and prejudice: A hidden history of statistics.

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Many statistical methods in use today, including regression and analysis of variance methods, can trace their origins to the late 1800’s and early 1900’s (Kevles 1998). Many of these early statisticians developed statistical methods to, at least in part, further their interests in understanding differences between racial groups of humans. Sir Francis Galton, who developed regression and correlation concepts (the details and extensions of which were the works of Karl Pearson), coined the term Eugenics, the “science” of improving humans through selective breeding. In an editorial, the journal Nature recently published an apology for it’s complicity providing “a platform to [eugenicists] views” (p 875, Nature 2022). Sir RA Fisher is rightfully remembered as the inventor of analysis of variance and maximum likelihood techniques (Rao 1992; Bodmer et al 2021), and perhaps more important, developed the concepts of sampling from populations, degrees of freedom, among other contributions. Moreover, work of Fisher, Wright, and others led to the realization that populations of the same species differed at many locations in the genome, thus completely refuting any notion of “racial purity” (Bodmer et al 2021). However, Fisher was an active promoter of eugenics (Bodmer et al 2021), and an award given by The Society for the Study of Evolution in his honor, the RA Fisher Prize, was renamed in 2020.

Eugenics is still with us (Liscum and Garcia 2022; see also Legacies of Eugenics: An Introduction Los Angeles Review of Books; Veit et al 2021), but the era of “Three generations of imbeciles are enough” (Buck v. Bell), and the forced sterilization eugenic programs of the Nazi (Kelves 1999), were thoroughly discredited on scientific grounds many times (click here for Eugenics Archive website). Do keep in mind that the times were different, and elements of eugenics thinking remains with us (cf discussion in Veil et al 2021 and references therein), it is interesting nevertheless to learn about the murky history of statistics and the objectives of some of the very bright people responsible for many of the statistical analyses we use today (see Stephen J. Gould’s “The Mismeasure of Man“). Several books tackle aspects of scientific racism, including 2017 Is science racist? by Jonathan Marks and 2019 Superior: The return of of race science, by Angela Saini, Cathy O’Neil’s 2017 accessible book Weapons of math destruction: How big data increases inequality and threatens democracy, and a host of articles and books listed at Wikipedia.

Keep in mind also that geneticists and statisticians were instrumental in showing why Eugenics was unscientific, at best. See partial listing in Bodmer et al (2021). Also, see  2021 review by Rebecca Sear, Demography and the rise, apparent fall, and resurgence of eugenics, in Population studies: A journal of demography. While the algorithms of statistics should be unbiased — and we often believe that they are — clearly, humans, and the interpretations of data may be biased (Celiktutan et al 2024). Statistical reasoning absolutely requires considering sources of bias and their impacts on our conclusions.

Landmark epidemiology studies.

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John Snow (1813-1858) is credited by some as the “Father of Epidemiology”(Ramsay 2006). During a London outbreak — the sudden increase of disease events — of cholera in 1854, Snow conducted work to establish cholera mortality with source and quality of drinking water. At the time, the prevailing explanation for cholera was that it was an airborne infection. Snow’s map of cholera mortality in the Golden Square district of London in relation to a water pump on Broad Street is shown in Fig 2, Snow’s theory of contaminated water was not accepted as an explanation for cholera until after his death.

Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854, drawn and lithographed by Charles Cheffins.

Figure 2. Original map by John Snow showing the clusters of cholera cases in the London epidemic of 1854, drawn and lithographed by Charles Cheffins. Image Public Domain, from Wikipedia.

Snow’s work and dataset can be viewed and thanks to Paul Lindman and others, the work can be expanded: for example, define areas around pumps by walking distance (Fig 3). The R package is cholera. Figure 2 shows a plot like Snow’s annotated map.cholera.

Figure 3. Plot of Snow’s London using R cholera package. Triangles marked with p1-p13 represent public water pumps. Red dots represent cholera cases.

R code to make the plot was

snowMap()

Snow’s ideas about cholera were not accepted in his time and you should recognize that by itself, a cluster map supports both the airborne and waterborne theories. The cholera package contains additional data to help visualize the area, including setting regions by walking distance (Fig 4).

Figure 4. Plot of Snow’s London with walking areas drawn about the 13 water pumps. R cholera package.

R code to make the plot was

plot(neighborhoodWalking(case.set = "expected"), "area.polygons")

Epidemiology of cancer.

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This is an undeveloped section in my book. For now, I’ve listed a number of key landmark studies — a selective and incomplete list — in the history of epidemiology and cancer risk — these would serve well as case studies:

• Tobacco as a carcinogen
  • Doll R, Hill AB. Smoking and carcinoma of the lung; preliminary report. Br Med J 1950; 2: 739–748
  • Fontham ET, Correa P, Wu-Williams A, Reynolds P, Greenberg RS, Buffler PA, Chen VW, Boyd P, Alterman T, Austin DF, Liff J, Greenberg SD. Lung cancer in non-smoking women: a multicenter case-control study. Cancer Epidemiol Biomarkers Prev 1991;1:35-43
• Diet and cancer risk
  • Kolonel LN, Hankin JH, Lee J, Chu SY, Nomura AM, Hinds MW. Nutrient intakes in relation to cancer incidence in Hawaii. Br J Cancer 1981;44:332-339.
• Obesity, exercise, and cancer risk
  • Calle EE, Rodriguez C, Walker-Thurmond K, Thun MJ. Overweight, Obesity, and Mortality from Cancer in a Prospectively Studied Cohort of U.S. Adults. N Engl J Med 2003;348:1625-1638.
• Hormones and cancer risk
  • Marchbanks PA, McDonald JA, Wilson HG, Folger SG, Mandel MG, Daling JR, Bernstein L, Malone KE, Ursin G, Strom BL, Norman SA, Wingo PA, Burkman RT, Berlin JA, Simon MS, Spirtas R, Weiss LK. Oral contraceptives and the risk of breast cancer. New Engl J Med 2002;346(26):2025-2032.
• Cancer risk and occupations
  • absence of cervical cancer but high incidence of breast cancer in nuns (Ramazzini, 1713)
  • cancer of the scrotums among chimney sweeps of London (Pott, 1775)
  • Silverman DT, Hoover RN, Mason TJ, Swanson GM. Motor exhaust-related occupations and bladder cancer. Cancer Res 1986;46:2113-2116.

Please see review by Greenwald and Dunn (2009).

Questions

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help.
2. Expand my example of landmark studies (also called pivotal or seminal studies) in epidemiology of cancer to a subject of your own interest. For example, vaccination and communicable diseases like polio.

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Quiz Chapter 2.3

A history of (bio)statistics

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Chapter 2 contents

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• Introduction
• Why biostatistics?
• Why do we use R Software?
• A brief history of (bio)statistics
• Experimental Design and rise of statistics in medical research
• Scientific method and where statistics fits
• Statistical reasoning
• Chapter 2 – References and suggested reading