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The following is an example of a hypothesis testing from an introductory stats book.

A medical research team has been given the task of evaluating a new laser treatment for certain types of tumors. Consider the following two scenarios:

Scenario 1: The current standard treatment is considered reasonable and safe by the medical community, is not costly, has no major side effects, and has a known success rate of 0.85 (or 85%).

Scenario 2: The current standard treatment sometimes has serious side effects, is costly, and has a known success rate of 0.30 (or 30%).

The solution provided by the books is as follows:

In the first scenario, the research question of interest would probably be “Does the new treatment have a higher success rate than the standard treatment?” Unless there is convincing evidence that the new treatment has a higher success rate, it is unlikely that current medical practice would change. With p representing the true proportion of success for the laser treatment, the following hypotheses would be tested:

H0: p=0.85 versus Ha: p > 0.85

In this case, rejecting the null hypothesis would require convincing evidence that the success rate is higher for the new treatment.

In the second scenario, the current standard treatment does not have much to recommend it. The new laser treatment may be considered preferable because of cost or because it has fewer or less serious side effects, as long as the success rate for the new procedure is no worse than that of the standard treatment. Here, researchers might decide to test

H0: p=0.30 versus Ha: p<0.30

My question is why in the 2nd scenario alternative hypothesis is p<0.30 not p>0.30?

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  • $\begingroup$ p<0.30 means testing the null against the alternative hypothesis that it is worse. You need to power the study so that you could detect if it is significantly worse, i.e you are making a serious attempt to falsify the precise hypothesis which you are testing. $\endgroup$
    – ReneBt
    Mar 25, 2020 at 12:15
  • $\begingroup$ what if I test for p>0.30 instead of p<0.30? This may sound like a dumb question (bear with me, I'm still learning) but it is not crystal clear to me why we're doing it in the reverse direction. The first scenario makes perfect sense to me but when I read the description for the second scenario things are clear as mud for me. $\endgroup$
    – Cody
    Mar 25, 2020 at 12:27
  • $\begingroup$ Because it is less stringent and therefore require less extensive testing, which is a critical consideration from an ethical point of view. The fewer subjects exposed for a clinical trial (which is by definition the point before enough information is known) the better. $\endgroup$
    – ReneBt
    Mar 25, 2020 at 15:38

2 Answers 2

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The proposal 2 is to ensure the new method is not less effective than the existing method. The only current option in this scenario has many undesirable characteristics, so it is simpler to demonstrate that the new method improves these characteristics without compromising quality.

In any investigation resources and risks should be adequately balanced in order to return the most informative results for the least risk. Statistics does not happen in a vacuum, there are always real world demands, limitations and restrictions. When one is planning an investigation it is necessary to fully consider all relevant issues, which may differ from field to field or even between specific applications.

In the medical research scenario presented some (far from all) of the concerns are:

  1. Ethics - the need to minimise exposure of subjects to incompletely quantified risks (although preliminary data for the new method promises improved safety this not as well tested as an established methods with extensive post-market review).

  2. Monetary Cost - biomedical research is not cheap and as the lower hanging fruit is picked, it is getting harder and more expensive to make incremental improvements. Trials can cost tens of thousands or multiples of that per subject, so going for a trial needing 100 vs 1000 people can make a huge difference to the affordability of the research.

  3. time pressure - other teams may be working on competing technologies. If you are third to market you now have to beat the new technology that beat the first one, and it may have improved side effects too. A year long study vs a 3 year one can make a huge difference

  4. patient care - until your product is launched patient will continue to be put at risk by the existing method and so it is ethically imperative that it be improved upon as quickly as possible. How many people would suffer from the side effects from delaying for a larger trial?

So this means there are lots of reasons to test against a less stringent benchmark. Once you have decide that then the alternative hypothesis must be framed in such a way as to provide a rigorous attempt at falsification of that hypothesis. So if you want to test some thing is not worse, then that needs to be the alternative hypothesis ($H_a<0.3$), not one formulated to test a completely different hypothesis such as that it is better ($H_a>0.3$).

Of course a prudent R&D manager will forsee the day competitors will catch up and will plan for continuing to develop the evidence for the new method beyond this trial. This will be easier to do if it is bringing in revenue and is deployed in the healthcare system demonstrating real world safety and efficacy gains.

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H0: p=0.30 versus Ha: p<0.30 is testing if the new treatment is worse than the current treatment, vs it being the same

H0: p=0.30 versus Ha: p>0.30 is testing if the treatment is better than the current treatment.

The choice of alternative hypothesis has to do with the goal of the study. A non inferiority trial would use the former option.

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