I want to know how to proceed in an experimental research, where there is a conflict between different theories with different predictions. I.e. the theory A predicts X > Y, while the theory B predicts X ≠ Y or X < Y.

So, can I, based on the conflicting theories, adopt and test both hypothesis in my research? Or do I really have to pick one?

From a statistical point of view, there is no reason you cannot have the null hypothesis that X = Y and the alternative that X unequal to Y. Then when you get data you can assess which is greater and by how much.

Before you start you should do a 'power computation' to find what the sample sizes need to be in order to have a reasonable probability of detecting a difference of a size that is of practical importance. (Technically, 'power' is the probability of rejecting the null hypothesis, at a particular significance level, if the difference is at least D --- ideally, where D is chosen to be a difference large enough to be worth talking about.)

If you fail to reject the null hypothesis, you will not have a "statictically significant" difference to discuss, but you should be able to cite an updated version of the power computation to say that the true difference, if any, is likely to be smaller then D. If you reject the null hypothesis, you may want to present confidence intervals (say 95% intervals) to give an idea how large the difference is and with what margin of error.

From a political point of view, you may find your adviser or those who are willing to support your work strongly on one side or the other of the debate whether X < Y or X > Y. Then you can write your proposal in terms of testing the null hypothesis (that X = Y) against whichever alternative (research) hypothesis is favored. But if possible, make sure your methods, sample sizes, types of data, and so on, have anticipated the possibility that the truth is in the other direction from the popular view, so that you will still have something to publish, even if the prevailing view is wrong.

Your hypothesis should be clear (following either A or B), however, there should not be a problem to test and compare both theories' predictions in a single research study. Measure/compare X and Y and see which theory explains your results/is supported by your data.

You may have to work on theory A and another researcher on theory B.

It may also be the case that your supervisor expects you to research both theories...

Discuss this with your supervisor, it depends on the amount of research required for each.

I assume you are not talking about mathematics in which things can be definitively proven. In (most) other fields that doesn't happen. Research provides evidence for a theory but doesn't prove or disprove it. If you accept an hypothesis your experiment shows that within certain statistical bounds the hypothesis is likely to be true. Rejecting it doesn't prove it false, it just concludes that there is less evidence that it might be true.

But research with rejected hypotheses is still valid research. Something is learned about the potential truth of the hypothesis. So, if you research something and need to reject the hypothesis as stated, you can, certainly, start to explore alternate, even contradictory, hypotheses.

For a dissertation, writing up a series of experiments (some with accepted and some with rejected hypotheses) would seem to me to be valid, but presenting it all in a single paper would be a bit of overkill and likely confusing to readers.

The worst case is if you accept contradictory hypotheses. Then you have to question the methodology itself. You are doing something wrong. "We have 99% confidence that mumble-matter is black and we have 99% confidence that mumble-matter is white", would be looked at askance.

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