This differs markedly from Fisher who proposed a general approach for scientific inference conditioned on the null hypothesis only. The second key concept is the control of error rates. This dichotomization allows distinguishing correct results rejecting H0 when there is an effect and not rejecting H0 when there is no effect from errors rejecting H0 when there is no effect, the type I error, and not rejecting H0 when there is an effect, the type II error.
In this context, alpha is the probability of committing a Type I error in the long run. Alternatively, Beta is the probability of committing a Type II error in the long run. The theoretical difference in terms of hypothesis testing between Fisher and Neyman-Pearson is illustrated on Figure 1.
If the p-value is below the level of significance, it is used to reject H0. In the 2 nd case, we set a critical interval based on the a priori effect size and error rates. If an observed statistic value is below and above the critical values the bounds of the confidence region , it is deemed significantly different from H0.
In the NHST framework, the level of significance is in practice assimilated to the alpha level, which appears as a simple decision rule: if the p-value is less or equal to alpha, the null is rejected. It is however a common mistake to assimilate these two concepts.
The figure was prepared with G-power for a one-sided one-sample t-test, with a sample size of 32 subjects, an effect size of 0. Therefore, one can only reject the null hypothesis if the test statistics falls into the critical region s , or fail to reject this hypothesis. In the latter case, all we can say is that no significant effect was observed, but one cannot conclude that the null hypothesis is true. This is another common mistake in using NHST: there is a profound difference between accepting the null hypothesis and simply failing to reject it Killeen, By failing to reject, we simply continue to assume that H0 is true, which implies that one cannot argue against a theory from a non-significant result absence of evidence is not evidence of absence.
CI have been advocated as alternatives to p-values because i they allow judging the statistical significance and ii provide estimates of effect size. Assuming the CI a symmetry and width are correct but see Wilcox, , they also give some indication about the likelihood that a similar value can be observed in future studies. If sample sizes however differ between studies, CI do not however warranty any a priori coverage. The most common mistake is to interpret CI as the probability that a parameter e.
The alpha value has the same interpretation as testing against H0, i. This implies that CI do not allow to make strong statements about the parameter of interest e.
To make a statement about the probability of a parameter of interest e. NHST has always been criticized, and yet is still used every day in scientific reports Nickerson, One question to ask oneself is what is the goal of a scientific experiment at hand?
While a Bayesian analysis is suited to estimate that the probability that a hypothesis is correct, like NHST, it does not prove a theory on itself, but adds its plausibility Lindley, Reporting everything can however hinder the communication of the main result s , and we should aim at giving only the information needed, at least in the core of a manuscript.
Here I propose to adopt optimal reporting in the result section to keep the message clear, but have detailed supplementary material.
For the reader to understand and fully appreciate the results, nothing else is needed. Because science progress is obtained by cumulating evidence Rosenthal, , scientists should also consider the secondary use of the data. It is also essential to report the context in which tests were performed — that is to report all of the tests performed all t, F, p values because of the increase type one error rate due to selective reporting multiple comparisons and p-hacking problems - Ioannidis, I can see from the history of this paper that the author has already been very responsive to reviewer comments, and that the process of revising has now been quite protracted.
That makes me reluctant to suggest much more, but I do see potential here for making the paper more impactful. So my overall view is that, once a few typos are fixed see below , this could be published as is, but I think there is an issue with the potential readership and that further revision could overcome this.
I suspect my take on this is rather different from other reviewers, as I do not regard myself as a statistics expert, though I am on the more quantitative end of the continuum of psychologists and I try to keep up to date.
I think I am quite close to the target readership , insofar as I am someone who was taught about statistics ages ago and uses stats a lot, but never got adequate training in the kinds of topic covered by this paper. The fact that I am aware of controversies around the interpretation of confidence intervals etc is simply because I follow some discussions of this on social media.
I am therefore very interested to have a clear account of these issues. This paper contains helpful information for someone in this position, but it is not always clear, and I felt the relevance of some of the content was uncertain.
So here are some recommendations:. I wondered about changing the focus slightly and modifying the title to reflect this to say something like: Null hypothesis significance testing: a guide to commonly misunderstood concepts and recommendations for good practice. So it might be better to just focus on explaining as clearly as possible the problems people have had in interpreting key concepts.
I think a title that made it clear this was the content would be more appealing than the current one. P 3, col 1, para 3, last sentence. I wondered whether it would be useful here to note that in some disciplines different cutoffs are traditional, e.
Having read the section on the Fisher approach and Neyman-Pearson approach I felt confused. As I understand it, I have been brought up doing null hypothesis testing, so am adopting a Fisher approach. But I also talk about setting alpha to. But the explanation of the difference was hard to follow and I found myself wondering whether it would actually make any difference to what I did in practice.
Maybe it would be possible to explain this better with the tried-and-tested example of tossing a coin. So in Fisher approach you do a number of coin tosses to test whether the coin is unbiased Null hypothesis ; you can then work out p as the probability of the null given a specific set of observations, which is the p—value.
The section on acceptance or rejection of H0 was good, though I found the first sentence a bit opaque and wondered if it could be made clearer. Also I wondered if this rewording would be accurate as it is clearer to me : instead of:. I felt most readers would be interested to read about tests of equivalence and Bayesian approaches, but many would be unfamiliar with these and might like to see an example of how they work in practice — if space permitted.
I understand about difficulties in comparing CI across studies when sample sizes differ, but I did not find the last sentence on p 4 easy to understand. Here too I felt some concrete illustration might be helpful to the reader. I have read this submission. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.
The revisions are OK for me, and I have changed my status to Approved. I believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. On the whole I think that this article is reasonable, my main reservation being that I have my doubts on whether the literature needs yet another tutorial on this subject.
A further reservation I have is that the author, following others, stresses what in my mind is a relatively unimportant distinction between the Fisherian and Neyman-Pearson NP approaches. I see this as being unimportant and not even true. Unless one considers that the person carrying out a hypothesis test original tester is mandated to come to a conclusion on behalf of all scientific posterity, then one must accept that any remote scientist can come to his or her conclusion depending on the personal type I error favoured.
To operate the results of an NP test carried out by the original tester, the remote scientist then needs to know the p-value.
The type I error rate is then compared to this to come to a personal accept or reject decision 1. In fact Lehmann 2 , who was an important developer of and proponent of the NP system, describes exactly this approach as being good practice. See Testing Statistical Hypotheses, 2nd edition P Thus using tail-area probabilities calculated from the observed statistics does not constitute an operational difference between the two systems. A more important distinction between the Fisherian and NP systems is that the former does not use alternative hypotheses 3.
Fisher's opinion was that the null hypothesis was more primitive than the test statistic but that the test statistic was more primitive than the alternative hypothesis. Thus, alternative hypotheses could not be used to justify choice of test statistic.
Only experience could do that. Further distinctions between the NP and Fisherian approach are to do with conditioning and whether a null hypothesis can ever be accepted. I have one minor quibble about terminology. The Null Hypothesis H0 is a broadly accepted phenomenon or value for a given parameter. The H0 is an accepted conjecture but, as the name suggests, while assumed to be true, it is nullifiable which is to say, falsifiable or refutable.
For additional background, the history of the H0 as a refutable claim is connected to the work of the Austrian-English philosopher of science, Karl Popper ; The Ha drives the collection of data, and those data, upon analysis, will ultimately allow the researcher one of two conclusions: either reject the null hypothesis or fail to reject the null hypothesis.
The dual structure of the H0-Ha allows for statistical testing and decision making, the purpose of which is to limit the presence of chance in explaining the difference between the variables in the study. A necessary assumption in this process is that true random and representative samples have been drawn. The H0 and Ha also have a relationship between them such that they are mathematical opposites.
As a result, should data indicate that the H0 is no longer supported, the researcher is able to reject that hypothesis with a level of quantifiable certainty. However, by the same logic, in the event that the data continue to support the established H0, the researcher must then fail to reject it. It should be noted, as well, that any research study neither proves nor disproves the null hypothesis in either rejecting or failing to reject it.
In a criminal court of law, there is a universally held null hypothesis: namely, that a defendant is presumed innocent. The role of the prosecutor, like the role of a researcher, is to present data i. For this reason, there are only two possible outcomes in a criminal proceeding: either the defendant is guilty i. This is much the same in a quantitative research study where a researcher either rejects the null hypothesis or fails to reject it.
And, just as when an individual is found guilty in a court proceeding, that outcome the Ha should not be construed as definitively proved. When a researcher finds statistically significant support to reject the null hypothesis, the alternative hypothesis is simply promoted as the better of the two claims. Like the null hypothesis before it, the alternative hypothesis now comes to occupy a nullifiable position.
Following the analogy of a criminal court proceeding above, the H0 that a defendant is presumed innocent is a statement that there is no statistically significant relationship between the defendant the dependent variable being tested by the prosecutor and the evidence for the crime the independent variable that the prosecutor manipulates to make a case.
Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis.
For example, you could compare the mean exam performance of each group i. This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things.
As such, we can state:. Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points.
Both these issues are dealt with next. The level of statistical significance is often expressed as the so-called p -value. Depending on the statistical test you have chosen, you will calculate a probability i. Another way of phrasing this is to consider the probability that a difference in a mean score or other statistic could have arisen based on the assumption that there really is no difference.
Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population i. So, you might get a p -value such as 0. However, you want to know whether this is "statistically significant".
We reject it because at a significance level of 0. Whilst there is relatively little justification why a significance level of 0. However, if you want to be particularly confident in your results, you can set a more stringent level of 0. When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement.
For example, the alternative hypothesis that was stated earlier is:. The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable s on the dependent variable s? Second, what was the predicted direction of this effect?
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