3 Made it more explicit which piece of jargon the question refers to, this should make the question more searchable

# The rationale behind using the popular"fail to reject the null" jargon in hypothesis testing?

2 deleted 50 characters in body

In hypothesis testing with a Null and the Alternative hypothesis (assuming that these two cases are mutually inclusive of all the possible "truths"), we usually base our hypothesis selection criterion on a constraint put on type I error, and choose to say "could not reject the null" as opposed to accepting it, but we without any qualms (relatively) say that we "reject the null" in the complimentary case. The reason told to me is this:

Since we usually focus on, and constraint via our selection procedure only the type I error, and do not try to bound the type II error, we do not know how likely is the null if the decision is to go with the Null since we did not know what was the chance of our decision criteria to go with the null if the alternative were true. But if the decision is to go with the alternative, we can reject the null since we set alpha to be very low already.

But this does not make senseseems fallacious. The probability of selecting the alternative given the Null is the truth may be low, but the corresponding probability given the alternative may be even lower! (In which case our rejecting of the Null was misplaced).

While I agree that the general criteria chosen makes it more likely to go with the alternative given the alternative is the truth than given Null is the truth, the absence of quantifiable information which prevents us from selecting the null should also prevent our unequivocally rejecting it. To add, since the conditional probability of selecting the Null given the truth is the Null hypothesis is 1 - probability of selecting the alternative given the null is the truth. Hence, by constraining the type I error, we also bound the former to be greater than a particular value.

If type I error is the only one that is constrained, shouldn't not rejecting it be equivalent to accepting the null within the same level of "confidence" as rejecting it had?

I have been doing the excellent course on Mathematical Biostatistics by Hopkins School of Biostats on Coursera, where I came across the above.

In hypothesis testing with a Null and the Alternative hypothesis (assuming that these two cases are mutually inclusive of all the possible "truths"), we usually base our hypothesis selection criterion on a constraint put on type I error, and choose to say "could not reject the null" as opposed to accepting it, but we without any qualms (relatively) say that we "reject the null" in the complimentary case. The reason told to me is this:

Since we usually focus on, and constraint via our selection procedure only the type I error, and do not try to bound the type II error, we do not know how likely is the null if the decision is to go with the Null since we did not know what was the chance of our decision criteria to go with the null if the alternative were true. But if the decision is to go with the alternative, we can reject the null since we set alpha to be very low already.

But this does not make sense. The probability of selecting the alternative given the Null is the truth may be low, but the corresponding probability given the alternative may be even lower! (In which case our rejecting of the Null was misplaced).

While I agree that the general criteria chosen makes it more likely to go with the alternative given the alternative is the truth than given Null is the truth, the absence of quantifiable information which prevents us from selecting the null should also prevent our unequivocally rejecting it. To add, since the conditional probability of selecting the Null given the truth is the Null hypothesis is 1 - probability of selecting the alternative given the null is the truth. Hence, by constraining the type I error, we also bound the former to be greater than a particular value.

If type I error is the only one that is constrained, shouldn't not rejecting it be equivalent to accepting the null within the same level of "confidence" as rejecting it had?

I have been doing the excellent course on Mathematical Biostatistics by Hopkins School of Biostats on Coursera, where I came across the above.

In hypothesis testing with a Null and the Alternative hypothesis (assuming that these two cases are mutually inclusive of all the possible "truths"), we usually base our hypothesis selection criterion on a constraint put on type I error, and choose to say "could not reject the null" as opposed to accepting it, but we say that we "reject the null" in the complimentary case. The reason told to me is this:

Since we usually focus on, and constraint via our selection procedure only the type I error, and do not try to bound the type II error, we do not know how likely is the null if the decision is to go with the Null since we did not know what was the chance of our decision criteria to go with the null if the alternative were true. But if the decision is to go with the alternative, we can reject the null since we set alpha to be very low already.

But this seems fallacious. The probability of selecting the alternative given the Null is the truth may be low, but the corresponding probability given the alternative may be even lower! (In which case our rejecting of the Null was misplaced).

While I agree that the general criteria chosen makes it more likely to go with the alternative given the alternative is the truth than given Null is the truth, the absence of quantifiable information which prevents us from selecting the null should also prevent our rejecting it. To add, since the conditional probability of selecting the Null given the truth is the Null hypothesis is 1 - probability of selecting the alternative given the null is the truth. Hence, by constraining the type I error, we also bound the former to be greater than a particular value.

If type I error is the only one that is constrained, shouldn't not rejecting it be equivalent to accepting the null within the same level of "confidence" as rejecting it had?

I have been doing the excellent course on Mathematical Biostatistics by Hopkins School of Biostats on Coursera, where I came across the above.

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# The rationale behind using the popular jargon in hypothesis testing?

In hypothesis testing with a Null and the Alternative hypothesis (assuming that these two cases are mutually inclusive of all the possible "truths"), we usually base our hypothesis selection criterion on a constraint put on type I error, and choose to say "could not reject the null" as opposed to accepting it, but we without any qualms (relatively) say that we "reject the null" in the complimentary case. The reason told to me is this:

Since we usually focus on, and constraint via our selection procedure only the type I error, and do not try to bound the type II error, we do not know how likely is the null if the decision is to go with the Null since we did not know what was the chance of our decision criteria to go with the null if the alternative were true. But if the decision is to go with the alternative, we can reject the null since we set alpha to be very low already.

But this does not make sense. The probability of selecting the alternative given the Null is the truth may be low, but the corresponding probability given the alternative may be even lower! (In which case our rejecting of the Null was misplaced).

While I agree that the general criteria chosen makes it more likely to go with the alternative given the alternative is the truth than given Null is the truth, the absence of quantifiable information which prevents us from selecting the null should also prevent our unequivocally rejecting it. To add, since the conditional probability of selecting the Null given the truth is the Null hypothesis is 1 - probability of selecting the alternative given the null is the truth. Hence, by constraining the type I error, we also bound the former to be greater than a particular value.

If type I error is the only one that is constrained, shouldn't not rejecting it be equivalent to accepting the null within the same level of "confidence" as rejecting it had?

I have been doing the excellent course on Mathematical Biostatistics by Hopkins School of Biostats on Coursera, where I came across the above.