This post is under constant updating…

The severity principle proposed by Deborah Mayo is used for accepting a null hypothesis H when:

1. there is no evidence to reject H and

2. H passes the test with high severity.

It seems to me that it is quite similar to my purpose (the s-value), but the measures involved in steps 1. and 2. are different.

An example from the normal distribution with known variance follows. Consider and the hypotheses

vs ,

a) If the sample average is 5, then 5+50*1/10 = 10. That is, we have no evidence against (since the sample average is corroborating ) and strong evidence against (since the sample average is very far away from ). The s-value for is one and the s-value for is almost zero (i.e., passes with high severity).

b) If the sample average is 15, then 15-50*1/10 = 10. That is, we have strong evidence against (since the sample average is very far away from ) and no evidence against (since the sample average is corroborating ). The s-value for is almost zero and the s-value for is one.

c) If the sample average is 9.9 , then 9.9+1*1/10 = 10. That is, we have no evidence against (since the sample average is corroborating ) and also a not strong evidence against (since the sample average is near ). The s-value for is one and the s-value for is approx 0.3 (i.e., does not pass with high severity)

d) If the sample average is 10.1, then 10.1-1*1/10 = 10. That is, we do not have strong evidence against (since the sample average is near ) and also we do not have evidence against (since the sample average is corroborating ). The s-value for is approx 0.3 and the s-value for is one.

My impression is that the Deborah’s approach tries to accept H even when the data do not corroborate with H.

Deborah’s severity:

For my example vs :

Notice that

is the p-value for the following “inverted” hypotheses

vs

Then, SEV is high whenever the p-value for the “inverted” hypotheses is low. This means that the event “” is quite improbable for the observed data. The p-value for the inverted hypotheses vs is:

a) very small for m=10

b) not applied (since we reject the null)

c) not small for m=10

d) not small for m=10.

That is, we have the same conclusions as I showed above with the s-value. If there some think wrong please let me know.

All the best,

Alexandre.

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