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In secondary school, I was taught that all laboratory experiments must include an explicit, a priori hypothesis statement with an associated theoretical foundation. For example, one of my lab reports might explicitly state, "I predict that the test tube of xenon will not demonstrate any observable reaction to the heat of the Bunsen burner in the presence of oxygen because xenon is a noble gas and noble gasses do not react except under extreme circumstances." At the end of the paper, we were required to explicitly state whether or not the hypothesis was supported by our findings, e.g. "The results above clearly demonstrate no reaction to applying heat from a Bunsen burner to a test tube of xenon in the presence of oxygen for any of the time intervals in Table A. It has thus been demonstrated that the hypothesis has been shown to be true."

What I've found in the real world is that actual, published studies generally do not include such an explicit hypothesis. The hypothesis may be implicit in the study's formulation or background, but at no point do the researchers actually come out and say, "This is what we think will happen". That is, someone who knows enough about the research field will have a good idea what the hypothesis probably was, but it is not formally stated in the paper. I do see a lot of papers with implicit hypotheses that look something like:

Understanding whether reticulating the splines of wooden spoons produces a GHV value that is positively correlated with the price of rice in China is important to proving the No Free Lunches conjecture [McGillicudy & Jones 2013, Smith 2015, Lopez 2018]. We use the Glover-Branch Method [1995] to apply positive pressure to the negative gradient of our standardized set of wooden spoons to determine the second coefficient....

At no point in this do the authors come out and say, "We think reticulating the splines of wooden spoons produces a GHV value that is positively correlated with the price of rice in China", but rather we perceive it as implicit in the narrative.

When publishing research, when (if ever) is it required to explicitly state an a priori, theoretical hypothesis rather than leaving it implicit?

To be clear, I'm not asking whether it is ok to proceed with research without first identifying a specific hypothesis, but rather whether it is essential to state or disclose the hypothesis to the readers upon publication or whether explicit hypothesis statements are really only a pedagogical tool for students.

In many cases, of course, the hypothesis will be supported by the data and there should be no problem stating so, but I'm thinking about cases where the researchers' hypothesis was found to not be supported. In that case, is it required for the researchers to "admit" that they jumped to a false conclusion in their initial hypothesis or is it sufficient to simply publish the results?

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I don't know whether you are an scientist/experimentalist, but you've hit upon something that is a problem in the modern scientific community, at least in the so-called "softer" sciences, I confess I don't know what physicists do.

The short answer is that, yes, people frequently write their paper to "tell the best story," and often that involves stating a hypothesis that is supported by the paper, or, if you have a little bit of shame, only hinting at the idea that you had an a priori hypothesis. [I might come back and add more about why this is bad, statistically.]

To answer the title question, "When does an explicit hypothesis need to be stated in a paper?," it doesn't, which is why many advocacy groups and journals encourage preregistration, that is the explicit statement of a hypothesis before data collection or analysis.

Preregistration separates hypothesis-generating (exploratory) from hypothesis-testing (confirmatory) research. Both are important. But the same data cannot be used to generate and test a hypothesis, which can happen unintentionally and reduce the credibility of your results. Addressing this problem through planning improves the quality and transparency of your research.

In short, one publishes a hypothesis and analysis plan. Usually it is not peer-reviewed, just time-stamped for posterity. Some journals will peer-review your analysis plan and accept a paper based on this, and promise to publish the results even if your hypothesis is not supported.

In that case, is it required for the researchers to "admit" that they jumped to a false conclusion in their initial hypothesis or is it sufficient to simply publish the results?

No, I don't think this is quite right. A hypothesis isn't a conclusion - I don't think a paper needs to contain some admission of error. A good paper would explain how a hypothesis was generated based on the literature, and explain what the consequences are for the field that your study was not consistent with what an expert would expect from previously published research.

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