8.2 – The controversy over proper hypothesis testing

Introduction

Over the next several chapters we will introduce and develop an approach to statistical inference, which has been given the title “Null Hypothesis Significance Testing” or NHST.

In outline, NHST proceeds with

• statements of two hypotheses, a null hypothesis, H_O, and an alternative hypothesis, H_A
• calculate a test statistic comparison of the null hypothesis (assuming some characteristic of data).
• The value of the test statistic is to be compared to a critical value for the test, identified for the assumed probability distribution at associated degrees of freedom for the statistical function, and assigned Type I error rate.

We will expand on these statements later in this chapter, so stay with me here. Basically, the null hypothesis is often a statement like the responses of subjects from the treatment and control groups are the same, eg, no treatment effect. Note that the alternative hypothesis, eg, hypertensive patients receiving hydalazine for six weeks have lower systolic blood pressure than patients receiving a placebo (Campbell et al 2011), would be the scientific hypothesis we are most interested in. But in the Frequentist NHST approach we test the null hypothesis, not the alternative hypothesis. This framework over proper hypothesis testing is the basis of the Bayesian vs Frequentist controversy.

Consider the independent sample t-test (see Chapter 8.5 and 8.6), our first example of a parametric test.

After plugging in the sample means and the standard error for the difference between the means, we calculate t, the test statistic of the t-test. The critical value is treated as a cut-off value in the NHST approach. We have to set our Type I error rate before we start the experiment, and we have available the degrees of freedom for the test, which follows from the sample size. With these in hand, the critical value is found by looking in the t-table of probabilities (or better, use R).

For example, what is the critical value of t-test with 10 degrees of freedom and Type I error of 5%?

In Rcmdr, choose Distributions → Continuous distributions → t distribution → t quantiles… (Fig 1).

Figure 1. Screenshot t-quantiles Rcmdr menu.

Note we want Type I equal to 5%. Since there are two tails for our test, we divide 5% by two and enter 0.025 and select the Upper tail.

R output

> qt(c(.025), df=10, lower.tail=FALSE)
[1] 2.228139

which is the same thing we would get if we look up on the t-distribution table (Fig 2).

Figure 2. Screenshot of portion of t-table with highlighted (red) critical value for 10 degrees of freedom.

If the test statistic is greater than the critical value, then the conclusion is that the null hypothesis is to be provisionally rejected. We would like to conclude that the alternative hypothesis should favored as best description of the results. However, we cannot — the p-value simply tells us how likely our results would be obtained and if the null hypothesis was true. Confusingly, however, you cannot interpret the p-value as telling you the probability (how likely) that the null hypothesis is true. If however the test statistic is less than the critical value, then the conclusion is that the null hypothesis is to be provisionally accepted.

The test statistic can be assigned a probability or p-value. This p-value is judged to be large or small relative to an a priori error probability level cut off called the Type I error rate. Thus, NHST as presented in this way may be thought of as a decision path — if the test statistic is greater than the critical value, which will necessarily mean that the p value is less than the Type I error rate, then we make one type of conclusion (reject H_O). In contrast, if the test statistic is less than the critical value, which will mean that the p-value associated with the test statistic will be greater than the Type I error rate, then we conclude something else about the null hypothesis. The various terms used in this description of NHST were defined in Chapter 8.3.

Sounds confusing, but, you say, OK, what exactly is the controversy? The controversy has to do whether the probability or p-value can be interpreted as evidence for a hypothesis. In one sense, the smaller the p-value, the stronger the case to reject the null hypothesis, right? However, just because the p-value is small — the event is rare — how much evidence do we have that the null hypothesis is true? Not necessarily, and so we can only conclude that the p-value is one part of what we may need for evidence for or against a hypothesis (hint: part of the solution is to consider effect size — introduced in Chapter 9.2 — and the statistical power of the test, see Ch 11). What follows was covered by Goodman (1988) and others. Here’s the problem. Consider tossing a fair coin ten times, with the resulting trial yielding nine out of ten heads (eg, a value of one, with tails equal to zero).

R code

set.seed(938291156)
rbinom(10,1,0.5)
[1] 1 1 1 1 1 1 0 1 1 1

While rare (binomial probability 0.0098), do we take this as evidence that the coin is not fair? By itself, the p-value provides no information about the alternative hypothesis. More about p-value follows below in sections What’s wrong with the p-value from NHST? and The real meaning and interpretation of P-values.

Statisticians have been aware of limitations of the NHST approach for years (see editorial by Wasserstein et al 2019), but only now is the message getting attention of researchers in the biosciences and other fields. In fact, the New York Times recently had a nice piece by F.D. Flam (“The Odds, Continually Updated,” 29 Sep 2014) on the controversy and the Bayesian alternative. Like most controversies there are strong voices on either side, and it can be difficult as an outsider to know which position to side with (Fig 3).

Figure 3. “Frequentists vs. Bayesians,” https://xkcd.com/1132/.

The short answer is — as you go forward do realize that there is a limitation to the frequentist approach and to be on the correct side of the controversy, you need to understand what you can conclude from statistical results. NHST is by far the most commonly used approached in biosciences (eg, out of 49 research articles I checked from four randomly selected issues of 2015 PLoS Biology, 43 used NHST, 2 used a likelihood approach, none used Bayesian statistics). The NHST is also the overwhelming manner in which we teach introductory statistics courses (eg, checking out the various MOOC courses at www.coursera.org, all of the courses related to Basic Statistics or Inferential Statistics are taught primarily from the NHST perspective). However, right from the start I want to emphasize the limits of the NHST approach.

If the purpose of science is to increase knowledge, then the NHST approach by itself is an inadequate framework at best, and in the eyes of some, worthless! Now, I think this latter sentiment is way over the top, but there is a need for us to stop before we begin, in effect, to set the ground rules for what can be interpreted from the NHST approach. The critics of NHST have a very important point, and that needs to be emphasized, but we will also defend use and teaching of this approach so that you are not left with the feeling that somehow this is a waste of time or that you are being cheated from learning the latest knowledge on the subject of statistical inference. The controversy hinges on what probability means.

P-values, statistical power, and replicability of research findings

Science, as a way of knowing how the world works, is the only approach that humans have developed that has been empirically demonstrated to work. Note how I narrowed what science is good for — if we are asking questions about the material world, then science should be your toolkit. Some (eg, Platt 1964), may further argue that there are disciplines in science that have been more successful (eg, molecular biology) than others (eg, evolutionary psychology, cf discussion in Ryle 2006) at advancing our knowledge about the material world. However, to the extent research findings are based solely on statistical results there is reason to believe that many studies in fact have not recovered truth (Ioannidis 2005).

In a review of genomics, it was reported that findings of gene expression differences by many microarray studies were not reproducible (Allison et al 2006). The consensus is that confidence in the findings should hold only for the most abundant gene transcripts of many microarray gene expression profiling studies, a conclusion that undercuts the perceived power of the technology to discover new causes of disease and the basis for individual differences for complex phenotypes. Note that when we write about failure of research reproducibility we are not including cases of alleged fraud (Carlson 2012 on Duke University oncogenomics case), we are instead highlighting that these kinds of studies often lack statistical power; hence, when repeated, the experiments yield different results.

Frequentist and Bayesian Probabilities

Turns out there is a lot of philosophical problems around the idea of “probability,” and three schools of thought. In the Fisherian approach to testing, the researcher devises a null hypothesis, H_O, collects the data, then computes a probability (p-value) of the result or outcome of the experiment. If the p-value is small, then this is inferred as little evidence in support of the null hypothesis.

In the Frequentists’ approach, the one we are calling NHST, the researcher devises two hypotheses, the null hypothesis, H_O, and an alternative hypothesis, H_A. The results are collected from the experiment and, prior to testing, a Type I error rate (α, chance) is defined. The Type I error rate is set to some probability and refers to the chance of rejecting the null hypothesis purely due to random chance. The Frequentist then computes a p-value of result of the experiment and applies a decision criterion: If P-value greater than Type I error rate, then provisionally accept null hypothesis. In both the Fisherian and Frequentist approaches, the probability, again defined at the relative frequency of an event over time, is viewed as a physical, objective and well-defined set of values.

Bayesian approach: based on Bayes conditional probability, one identifies the prior (subjective) probability of an hypothesis, then, adjusts the prior probability (down or up) as new results come in. The adjusted probability is known as posterior probability and it is equal to the likelihood function for the problem. The posterior probability is related to the prior probability and this function can be summarized by the Bayes factor as evidence the evidence against the null hypothesis. And that’s what we want, a metric of our evidence for or against the null hypothesis.

We can calibrate the Bayesian probability to the frequentist p-value (Selke et al 2001; Goodman 2008; Held 2010; Greenland and Poole 2012). Methods to achieve this calibration vary, but the Fagan nomogram Held (2010) proposed is a good tool for us as we go forward. We can calculate our NHST p-value, but then convert the p-value to a Bayes factor by looking at the nomogram. I mention this here not as part of your to do list, but rather as a way past the controversy: the NHST p-value can be interpreted as a Bayesian conditional probability, but they do not test the same hypothesis.

Likelihood

Before we move on there is one more concept to introduce, that of likelihood. We describe a model (an equation) we believe can generate the data we observe. By constructing different models with different parameters (hypotheses), you generate a statistic that yields a likelihood value. If the model fits the data, then the likelihood function has a small value. The basic idea then is to compare related, but different models to see which fits the data better. We will use this approach when comparing linear models when we introduce multiple regression models in Chapter 18.

What’s wrong with the p-value from NHST?

Well, really nothing is “wrong” with the p-value.

Where we tend to get into trouble with the p-value concept is when we try and interpret it. See below, Why is this important to me as a beginning student? The p-value is not evidence for a position, it is a statement about error rates. The p-value from NHST can be viewed as the culmination of a process that is intended to minimize the chance that the statistician makes an error.

In Bayesian terms, the p-value from NHST is the probability that we observe the data (eg, the differences between two sample means), assuming the null hypothesis is true. If we want to interpret the p-value in terms of evidence for a proposition, then we want the conditional error probability.

Sellke et al (2001) provided a calibration of p-values and, assuming that the prior probabilities of the null hypothesis and the alternative hypothesis are equal (that is, that each have a prior probability of 0.5), by using a formula provided by them (equation 3), we can correct our NHST p-value into a probability that can be interpreted as evidence in favor of the interpretation that the null hypothesis is true. In Bayesian terms this is called the posterior probability of the null hypothesis. The formula is

where e is Euler’s number, the base of the natural logarithm (ln), and p is the p-value from the NHST. This calibration works as long as   (Sellke et al 2001).

By convention we set the Type I error at 5% (cf Cohen 1994). How strong of evidence is a p-value near 5% against the null hypothesis being true, again, under the assumption that the prior probability of the null hypothesis being true is 50%? Using the above formula I constructed a plot of the calculated conditional error probability values against p-values (Fig 4).

Figure 4. Conditional error probability values plotted against p-values.

As you can see, a p-value of 5% is not strong evidence at just 0.289. Not until p-values are smaller than 0.004 does the conditional error probability value dip below 0.05, suggesting strong evidence against the null hypothesis being true.

R note: For those of you keeping up with the R work, here’s the code for generating this plot. Text after “#” are comments and are not interpreted by R.

At the R prompt type each line

NHSTp = seq(0.00001,0.37,by=0.01)      #create a sequence of numbers between 0.0001 and 0.37 with a step of 0.01
CEP = (1+(-1*exp(1)*NHSTp*log(NHSTp))^-1)^-1    #equation 3 from Sellke et al 2001
plot(NHSTp,CEP,xlab="P-value", ylab="Conditional error probability",type="l",col="blue")

Why is this important to me as a beginning student?

As we go forward I will be making statements about p-values and Type I error rates and null hypotheses and even such things as false positives and false negative. We need to start to grapple with what exactly can be said by p-values in the context of statistical inference, and to recognize that we will sometimes state conclusions that cut some corners when it comes to interpreting p-values. And yet, you (and all consumers of statistics!) are expected to recognize what p-values mean. Always.

The real meaning and interpretation of P-values

This is as good of a time as any to make some clarification about the meaning of p-value and the whole inference concept. Fisher indeed came up with the concept of the p-value, but its use as a decision criterion owes to others and Fisher disagreed strongly with use of the p-value in this way (Fisher 1955; Lehmann 1993).

Here are some common p-value corner-cutting statements to avoid using (after Goodman 2008; Held 2010). P-values are sometimes interpreted, incorrectly, as

Table. Incorrect interpretations of NHST p-values

1. the probability of obtaining the observed data under the assumption of no real effect
2. an observed type-I error rate
3. the false discovery rate, ie, the probability that a significant finding is a “false positive”
4. the (posterior) probability of the null hypothesis.

So, if p-values don’t mean any of these things, what does a p-value mean? It means that we begin by assuming that there is no effect of our treatments — the p-value is then the chance we will get as large of a result (our test statistic) and the null hypothesis is true. Note that this definition does not include a statement about evidence of the null hypothesis being true. To get evidence of “truth” we need additional tools, like the Bayes Factor and the correction of the p-value to the conditional error probability (see above). Why not dump all of the NHST and go directly to a Bayesian perspective, as some advise? The single best explanation was embedded in the assumption we made about the prior probability in order to calculate the conditional error probability. We assumed the prior probability was 50%. For many, many experiments, that is simply a guess. The truth is we generally don’t know what the prior probability is. Thus, if this assumption is incorrect, then the justification for the formula by Sellke et al (2001) is weakened, and we are no closer to establishing evidence than before.

The take home message is that it is unlikely that a single experiment will provide strong evidence for the truth. Thus the message is repeat your experiments — and you already knew that!  And the Bayesians can tell us that with the addition of more and more data reduces the effect of the particular value of the prior probability on our calculation of the conditional error probability. So, that’s the key to this controversy over the p-value.

Reporting p-values

Estimated p-values can never be zero. Students may come to use software that may return p-values like “0” — I’m looking at you Google Sheets re: default results from CHISQ.TEST() — but again, this does not mean the probability of the result is zero. The software simply reports values to two significant figures and failed to round. Some journals may recommend that 0 should be replaced by p < 0.01 or even < 0.05 inequalities, but the former lacks precision and the latter over-emphasizes the 5% Type I error rate threshold, the “statistical significance” of the result. In general, report p-value to three significant figures and four digits. If a p-value is small, use scientific notation and maintain significant digits. Thus, a p-value of 0.004955794 should be reported as 0.00496 and a p-value of 0.0679 should be reported as 0.0679. Use R’s signif() function, for example p-value reported as 6.334e-05, then

signif(6.334e-05,3)
[1] 6.33e-05

Rounding and significant figures were discussed in Chapter 3.5. See Land and Altman (2015) for guidelines on reporting p-values and other statistical results.

Questions

1. Revisit Figure 1 again and consider the following hypothesis — the sun will rise tomorrow.
   • If we take the Frequentist position, what would the null hypothesis be?
   • If we take the Bayesian approach, identify the prior probability.
   • Which approach, Bayesian or Frequentist, is a better approach for testing this hypothesis?
2. Consider the pediatrician who, upon receiving a chest X-ray for a child notes the left lung has a large irregular opaque area in the lower quadrant. Based on the X-ray and other patient symptoms, the doctor diagnoses pneumonia and prescribes a broad-spectrum antibiotic. Is the doctor behaving as a Frequentist or a Bayesian?
3. With the p-value interpretations listed in the table above in hand, select an article from PLoS Biology, or any of your other favorite research journals, and read how the authors report results of significance testing. Compare the precise wording in the results section against the interpretative phrasing in the discussion section. Do the authors fall into any of the p-value corner-cutting traps?

Quiz Chapter 8.2

The controversy over proper hypothesis testing

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Chapter 8 contents

• Introduction
• The null and alternative hypotheses
• The controversy over proper hypothesis testing<.span>
• 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

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

2.5 – Scientific method and where statistics fits

Science isn’t just a recipe.

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The scientific method is what makes science a “powerful way of knowing” (Church and Carpenter 2008), and by now, you should be familiar with the outline: hypothesis, experiment, etc. What distinguishes science from other fields of inquiry is that at its best, science accumulates verifiable evidence about our world. I begin with a disclaimer — our introductory textbooks tend to reify the scientific method (cf. Blachowicz 2009). Outside of the classroom and introductory science text books, I don’t think you’ll find much agreement among practicing scientists precisely what the scientific method entails, or whether strict adherence to a list of steps distinguishes what scientists do compared with other professions. For one difference, typical discussions of the scientific method may emphasize the experiment, which probably brings to mind images of a lab coat and test tubes, but should emphasize the critical thinking “tool kit” (Wivagg and Allchin 2002), eg, model-based inquiry. That said, holding a view that introductory textbooks should present nuances of epistemology seems a big ask.

But we do emphasize experimentation with good reason. In principle it is straightforward to identify what control groups are needed to conduct an experiment in the lab, but what are the control groups for an experiment conducted on the computer? In most cases, one should argue that if an outcome is obtained by random processes, then no additional cause need be addressed. Therefore, the control group for a simulation would be a random process generator.

Disagreements about the scientific method center about how science is really done (eg, from a social perspective), but also because there appear to be differences in approach in sciences that work on historical questions (physical cosmology, evolutionary biology, geology), and those that conform to the classic experimental approach (chemistry, molecular biology, physics). Epistemology is a fascinating area — How do we know what we know, What exactly is science and how is it different from other areas of knowing, etc. But I will leave you only with the tantalizing suggestion to read more and start with you with a list of readings to start (see Readings below). This is the stuff of graduate and professional school; we have work to do.

Despite apparent differences between what scientists say they do and how they actually do sciences, there is broad agreement; science as a way of knowing can be characterized by the following steps (National Academy of Sciences 1999).

1. Begin with facts, which are observations confirmed and treated as true
2. Formulate a hypothesis, with emphasis on hypotheses that are testable statements about relationships observed about the natural world
3. Given the possible outcomes, state predictions derived from the hypothesis.
4. Make observations of perform an experiment designed to test the hypothesis.
5. Analyze the data from the experiment.
6. Evaluate the results of the experiment against the predictions.
7. Repeat.

This deceptively simple list hides much work to be done. Hypotheses are not “educated guesses,” where educated guess implies an idea about some phenomenon is likely the correct explanation because of the skill or knowledge of the person making the guess. Good hypotheses make possible experimental tests whose results can be used to rule out alternative explanations, sensu Platt’s “strong-inference”.

Hypothetico-deductive reasoning.

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I mention this disclaimer about the common (and reassuring) textbook discourse on the scientific method to suggest that, if you have not already reached this point in your career, it is time to move past the cookbook approach to thinking about what it means to do science in practice. Yet another helpful reading is from a paper in Environmental Health by Ozonoff  amd Grandjean (2020).

In other words, what makes “good science”?

There is induction and deduction and probabilistic thinking that must be grappled with, all emphasizing efforts by the individual, and yet science, if it is to make any progress must ultimately be a communal activity (Varmus 2009). In particular, to the extent a researcher consistently applies hypothetico-deductive reasoning, or as Platt (1964) called it, “strong-inference,” then good science can happen (see Fudge 2014 for an update). Strong inference according to Platt implies that researchers should follow three steps (after Fudge 2014):

1. develop alternative hypotheses
2. think of a crucial experiment that can exclude one or more hypotheses
3. perform the experiment and obtain a clean result.

Then, beginning with step 1, repeat the procedure to refine the possibilities that remain.

The list of elements of the scientific method neglected to address the crucial role of scientists engaged in an active community of scientists fits in. However, we can can quickly suppose that every step of the scientific method can involve input from others to help shape, improve, and indeed carry out the activities needed to practice science. What scientists share is critical thinking and the tools of statistics provides a common language.

It’s all about the probability of a particular event.

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Platt (1964) wrote that the observations we make that are puzzling or hard to explain, which implies our understanding is incomplete. We then proceed to ask questions about why the observations are different from our expectations, and we speculate about how the outcome comes about. Thus, we have to consider the probability, or chance, of a particular outcome (event) compare to other possible outcomes. Statistics is about analyzing the probability of outcomes. But there is a twist; there are two distinct, but complementary approaches to statistics. Most of the statistics you have been taught so far comes from the frequentist approach. That is, how often (frequency) will we get the kind of results we observe given the hypothesis? Another approach can be termed Bayesian, where the question is how likely is the hypothesis to be correct given the data? These two approaches view the data obtained from the same experiment differently. A frequentist views the data as random — repeat the experiment and the results will differ — but the hypothesis is fixed (it is either true or it is not). A Bayesian on the other hand views the hypothesis as random with a probability of being true somewhere between zero and 100 percent and the data are fixed.

Which are you, a frequentist, or a Bayesian? Consider the following from xkcd.com no. 1132 (Fig. 1).

Figure 1. “Frequentists vs Bayesians,” xkcd.com no. 1132

The Bayesian approach makes sense when there exists prior knowledge — the Sun came up this morning, as it did the day before, and the day before that, etc. New data are assessed against what we already know. The Frequentist approach, despite philosophical shortcomings, works for analyzing experiments where prior knowledge is lacking about outcomes. Because much of biomedical research is based on the Frequentist approach, most of our efforts, too, will utilize that approach.

A statistical translation of the Scientific Method.

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Given our frequentist perspective, we can outline our Scientific Method as follows.

1. Formulate a Null and an Alternative hypothesis.
2. State predictions from the Null and an Alternative hypotheses.
3. Design an experiment or observation.
4. Analyze the data from the experiment or observation.
5. Interpret the experiment or observation.
6. Evaluate the predictions from the Null and an Alternative hypotheses.
7. Accept (provisionally) or reject (provisionally) the Null hypothesis.
8. Evaluate model fit and robustness
9. Repeat.

In practice, there is more to statistics than these 8(9) steps, but this does provide the outline of what statistical analyses are about. There are nuances to how a Frequentist or a Bayesian views statistical analyses as evidence for or against a conclusion (Goodman 1999a, 1999b). We need to distinguish between when data are acquired in the process; data may come before or after the hypotheses are stated. In epistemology, hypotheses are either a priori or a posterori and we need to add these to our discussion of Scientific Method. The terms are Latin, translated apparently as “from the earlier” and “from the later, respectfully.

Note from the lists described as “the” scientific method how hypothesis comes first. Folks who think and write about how and why we know what we know — the discipline is called epistemology — tell us we are generally on more solid footing when we design experiments with specific intent, specific and testable hypotheses. The branch of statistics concerned with experimental design provides rich context for many practical aspects of how to implement experiments — in other words, how to follow the Scientific Method, even if turns out there isn’t just one universal definition of Scientific Methods out there.

When we say “experiment,” the hypothesis came first.

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More commonly in statistics, the phrases planned, and therefore a priori, and unplanned or a posterori, comparisons are referenced. In practice, biologists design experiments and make observations accordingly to test one or more hypotheses, but they may also address additional hypotheses after the fact. Especially if the experiment generates a lot of data.

You may have also heard of the phrase data mining. Data mining is loosely defined, but mostly refers to sets of protocols and procedures to extract patterns from large data sets stored in databases. Google apparently does lots of data mining as do many other businesses that obtain large amounts of data. Unplanned comparisons includes any data mining protocols, no matter how sophisticated the language sounds (feature selection, classification tree). Data mining is not consistent with classic experiments; it’s different than Step 6 (Step 8 in the second list) listed above because no new data or experiment is carried out. 

Can you get away with coming up with and testing new hypotheses from data gathered from an experiment designed to test a different hypothesis? Yes, and of course, the process can be quite profitable for Google. However, you should proceed with caution and restraint. If you are not careful about how you write it up — you will probably be called on it by a reviewer of your work. Is this a posterori approach still science? Of course, yes! I’d even go so far as to say that when one studies real systems, you can’t limit yourself to only planned hypotheses and testing. At least, one designs experiments and carries out those tests but then also uses current data to generate new ideas. Science is also exploratory — you may not design an experiment at all, but through observations, you probably will develop testable hypotheses! We’ll return to these concepts soon. What you must be aware of in any unplanned comparison or data mining sojourn is the possibility of committing a data dredging (aka p-hacking) sin — searching through data to come up with misleading, but statistically “significant” results (Ioannidis 2007; Stefan and Schönbrodt 2023).

Testing of unplanned a posterori hypotheses is a real concern in science. On the one hand, those who think about how we learn about the world and make sense of it have stated emphatically that the best way we know is to follow the scientific method — and that begins with hypotheses followed by designed experiments to test claims derived from those a priori hypotheses. We (teachers, textbook author) continue to teach science as the act of individuals toiling away in the forest or in the lab, pursuing sets of questions that may involve the collection of measurements on dozens to a few hundred subjects. While this type of science is still in practice, there is no doubt that big science involving many people is more common and, perhaps, better at generating new knowledge (Wuchty et al 2007). One result of “big science” is to generate a lot of data, the very essence now of big data, and there must be room for testing of new hypotheses gathered on data sets. This is the essence of the argument for the ENCODE project (ENCODE Project Consortium 2012) — generate lots of genetic data on the human genome using common techniques and make the data publicly available — new research can be conducted on old data.

Bottom-up, top-down.

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Consider the pronouncements (almost daily it seems!) about the discovery of a new gene for some disease, process, or behavior in humans. Often times, although not always, these “discoveries” are not duplicated by other research groups. Why not? Well, for one, the phrase “gene for” is a dubious shorthand for what is usually a more complex causation. But from our statistical perspective it is problematic because the search for genes is really an a posterori exercise — one begins with phenotypic differences (some have the disease, others do not) and some genetic information (SNPs, DNA sequencing) and then proceeds to see if there are any differences in the genetic material at hand between the two groups. This approach, the Genome Wide Association Study, or GWAS, would be termed “top-down” — begin with the phenotypic differences and search for genetic differences between those that do and those who do not have the condition. Sampling is an issue (are the unaffected subjects a random sample from the entire population?), but the problem also is one of logical design — the hypothesis is made after the fact — a statistical difference between the groups is attributed to a particular genetic difference.

Case-control subjects, where patients with the condition are matched with other individuals who do not have the condition, but match in other ways (eg, age, income, etc.), are enrolled in such studies because there can be no random assignment of subjects to treatment. Case-control subjects are selected such that affected and unaffected individuals are matched by characteristics in as many meaningful ways as possible (eg, age, gender, income, etc.). Because many tests are conducted in GWAS studies, ie, is there a difference between the control and affected group for the first gene, the second gene, and so on up to the number of genes on the microarray chip (10,000 or more genes), the chance that any particular association is a false positive is high, discussed further when we cover Risk Analysis in Chapter 7.

How to interpret a test result.

You are a medical doctor reading the results of a test for three-month average glycated hemoglobin, A1C, levels  for your patient. For example, A1C above 6% is considered strong marker of diabetes. A marker, yes, but not the same as a guarantee that a person has diabetes. A false positive (FP) is the case where a test result is positive, but the subject in fact does not have the condition. False positive is equal to the ratio of

where FP is number of false positive readings and TN refers to the true negative, those who in fact do not have the condition.

In Chapter 7 we will spend some time on risk analysis. To introduce this important subject in biostatistics, we’ll begin with an example. A study of 15,934 subjects without diagnosed diabetes found 3.8% or 605 individuals had elevated A1C levels, or about 7.1 million adults not yet diagnosed with diabetes (Selvin et al 2011). About 90% of these individuals also had fasting glucose levels greater than 100 mg/dL, i., diabetic. This is clearly a good test. However, note that 10% of the 605 individuals had elevated A1C, but fasting glucose levels less than 100 mg/dl, ie, did not meet the diagnostic of type 2 diabetes. Thus, the false positive rate is about 0.4%, or nearly 30,000 adults with elevated A1C without diabetes.

Working through frequencies can be challenging, so applying a natural number approach helps. Having just read through the frequencies and percentages, now look at how they translate to probability tree (Fig. 2). Start by utilizing a per capita rate standard: for proportions in the 10 – 20% range, a standard 1000 persons works well (Fig. 2).

Figure 2. Probability tree diagram with prevalence of type 2 diabetes and sensitivity, specificity of A1C test, data from CDC and Selvin et al 2011. Tree drawn with free diagrams.net app.

How many expected to have A1C above 6%? Twenty-six total positives, only 23% are true positives. An important lesson is that if the prevalence of a condition is low, then any diagnostic test with high sensitivity necessarily will identify many false positives.

False discovery rate: claims must be stringently evaluated.

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The genome scientists involved in GWAS studies generally are aware of false positives, also termed false discovery rate (FDR), conduct statistical corrections to account for false positives (eg, Brzyski et al 2016), and generally are cautious in their interpretation. But not always. Studies of associations between autism and environment come to mind (eg, Waldman et al 2008), and recent developments in the direct-to-consumer genetic testing market also suggest that the limitations of these kinds of studies are not always represented.

There is a broader concern about the reliability of research, and the debate about how to improve reliability comes from a call to understand how to do statistics better and, more importantly to understand how statistics are to be used in making claims from statistical results (Ioannidis 2007; but see Goodman and Greenland 2007). A key element of scientific work is that findings are repeatable: results from one group should apply to other groups. One dictate to improve reproducibility — increase number of subjects in studies — is obvious, but given the cost of GWAS, currently an unreasonable demand.

In some cases, biologists already have a particular gene in mind, whose function is more or less known, and then the exercise follows the Scientific Method listing much more directly. This bottom-up approach leads to a straightforward, testable genetic hypotheses: a specific difference in genomic sequence predicts a difference in phenotype outcome. A good example is the identification of more than 100 different single nucleotide mutations, called single nucleotide polymorphisms or SNPs, in the CFTR gene of patients known to have cystic fibrosis disease (Castellani 2013). As of August 2018, the number of known pathogenic or likely pathogenic SNPs is now listed at 440 in the SNP database; to put this in some context, there are more than 40,000 reported single nucleotide polymorphisms for the CFTR gene (this number includes SNP duplicates).

Projects and the data lifecycle.

A requirement for our biostatistics course, students are asked to design and implement a research project during the semester (Hydorn 2019). The project can either be part of an ongoing student research project or solely confined to the biostatistics course. The topics are diverse, ranging from observational to experimental to systematic reviews from the disciplines of biology, environmental science, and public health. Examples of project ideas implemented in the biostatistics and related courses at Chaminade University include:

• effects of environmental stress on growth rates.
• 2D and 3D shape analysis of mollusk shells.
• Individual variation and fluctuating asymmetry of plant leaves and flowers along an altitudinal gradient
• several systematic reviews (meta analysis) of published works of interest to public health
  • vaccination benefits and risks.
  • pain management of over the counter vs prescription medication.
  • efficacy of “alternative medicine” approaches to health. 
  • comparisons of types of medical insurance on patient outcomes of knee replacement therapy.

The justification for the project learning activity is that it provides hands-on opportunity to apply statistical and data sciences concepts to real-world problems. Students work through the challenges of developing a research question, identifying testable hypotheses, designing a study, collecting data, performing analysis, and interpreting results. Importantly, students must decide for themselves which statistical tests and methods are most appropriate for their unique data, rather than simply following a formula presented in a lecture. This connects abstract statistical ideas to practical application, promoting a deeper, more lasting understanding. We manage the projects by encouraging a research lab community: students discuss with their peers their ideas, their data collection and experimental design challenges, with the instructor serving to guide and mentor the multiple projects and intervene as needed to steer student projects towards completion. 

Throughout the semester we model the project approach. Students are encouraged to create a working folder on their computer, while dividing homework and other assignments into named projects. Students are also asked to address aspects of their project’s data life cycle (Borgman 2019, Ezer and Whitacker 2019, Griffin et al 2018). A quick Google search on the phrase returns millions of hits (see Note 1), but key elements of the concept are that projects have a beginning and an end, and it is helpful — and in the case of our statistics course, required — to address project management.

Projects include data generation and collection, data processing, storage and data management, data exploration and visualization, interpretation and reporting. Thus, students need to develop a data management plan, which includes how data are collected, application of best practices to insure data file and storage of images, and how to share the project results. We ask students to develop 10-min research talks, which are presented and recorded, suitable for ePortfolio inclusion.

Conclusions.

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Epistemology, the theory of how knowledge is acquired, is a complicated business – what I want you to appreciate now is that planned and unplanned comparisons affect interpretation of your statistical results, how the difference is likely also to affect the reproducibility of your work. In the language or clinical trials and experimental design, planned and unplanned accompany prospective and retrospective studies (Chapter 5). A prospective study in the case of GWAS means genetic differences among individuals are known at the start of the study, and phenotypic differences arise naturally during the course of the study. Knowing the strengths and limitations of, for example, a planned retrospective study is at the heart of experimental design (Chapter 5).

In the biostatistics course, students are required to design and carry out an independent research project that integrates statistical methods with real-world problems in biology, environmental science, or public health. The course emphasizes hands-on learning through the full research process—from formulating questions to data collection, analysis, and interpretation—while fostering a collaborative research lab environment. Students also learn project and data management skills and conclude by presenting their findings in recorded research talks suitable for inclusion in an ePortfolio.

Questions.

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help.
2. Above we described how GWAS studies are generally top-down; What would constitute a bottom-up approach to GWAS?
3. In your own words provide pro and con points of view on data mining of large data sets.
4. Create a stylized infographic, like a mind map, to illustrate the data life cycle for a one semester research project. 

Quiz Chapter 2.5

Scientific method and where statistics fits

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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
• References and suggested reading