@Barry
As I understand it for PACE, the main (i.e. alternative?) hypothesis is along the lines that patients are locked into a vicious circle of being deconditioned, reinforced and perpetuated by activity avoidance. GET reverses that vicious circle and thereby reconditions patients.
This is the abstract paradigm/'theory' behind the study but you wouldn't put this language in a hypothesis for statistical testing; it's not quantifiable.
Instead you would say something like:
Null Hypothesis: "CFS patients treated with (CBT, GET, or APT) + SMC
will not show any more improvement (i.e. equivalent improvement or less improvement) on (Chalder Fatigue Scale, SF-36 score, etc.) than patients treated with SMC alone.""
Alternative Hypothesis: "CFS patients treated with (CBT, GET, or APT) + SMC
will show greater improvement on (Chalder Fatigue Scale, SF-36 score, etc.) than patients treated with SMC alone."
Then you can run an experiment, and calculate a test statistic and a p-value from your observed results. A p-value is a measure of how unusual your observed result is assuming the null hypothesis is true. I.e. is it reasonable to assume that the deviation in your observed results from the null hypothesis is due to random chance or not. If p=.001, there is a 1/1000 chance that you would get a test statistic as extreme or more than the one you got based on your observations
if the null hypothesis is true - it's reasonable to assume in this case that the null is not true.
(PACE's p-values are in table 3 at
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3065633/)
PACE rejected the null hypothesis for CBT+SMC and GET+SMC. This simply means that it is very unlikely that the results (more improvement on Chalder scale/SF-36 in these groups vs SMC alone) could be explained by random chance. But does this support the abstract theory? Is the improvement on the questionnaires a reliable indicator of improvement in the patients' disease brought about by addressing unhelpful beliefs and reconditioning? No, because of all of the many problems that have been pointed out.
Something to keep in mind is that, even without some of the weird issues specific to PACE, we should expect these results to be quite easily replicable because (a) the unblinded-treatment/subjective-outcome problem - CBT or GET will be tested as the intervention that's supposed to help, so patients receiving it will be biased to say they are better, (b) CBT and GET train patients to say they are better, so it's not an interesting result that patients say they are better after being treated. These results cannot be taken to support the 'deconditioning/unhelpful beliefs' hypothesis.
I hope that's not too muddled. I'm not learned on all of the particulars of PACE methodology so I hope others who are will clean up any mess I made, but I think this will help to conceptualize the basics.
Barry, a book I found helpful for learning the basic concepts of statistics is
Statistics by Freedman, Pisani, and Purves. I think you would find it quite helpful in sorting out your questions about hypothesis testing, and it's quite fun to work through!