Q right way to interpret a test with multiple metrics

[Cosine]

Hi:

We could easily find in the literature that a study used more than one performance metric for the hypothesis test without explicitly and clearly stating what hypothesis this study aims to test. Often the paper only states that it intends to test if a newly developed object (algorithm, drug, device, technique, etc) would perform better than some chosen benchmarks. Then the paper presents some tables summarizing the results of many comparisons. Among the tables, the paper picks those comparisons having better values of some performance metric and showing statistical significance. Finally, the paper claims that the new object is successful since it has some favorable results that are statistically significant.

This looks odd. SHouldn't we clearly define the hypothesis before conducting any tests? For example, shouldn't we define the success of the object to be "having all the chosen metrics have better results"? Otherwise, why would we test so many metrics, instead of only one?

The aforementioned approach looks like this: we do not know what would happen. So let's pick some commonly used metrics to test if we could get some of them to show favorable and significant results.

Anyway, what are the correct or rigorous ways to conduct tests with multiple metrics?

[David Jones]

You might want to search for the terms "multiple testing" and
"Bonferroni correction".

[Rich Ulrich]

On Sat, 18 Mar 2023 01:25:44 -0000 (UTC), "David Jones"
<dajh...@nowherel.com> wrote:

>Cosine wrote:
>
>> Hi:
>>
>> We could easily find in the literature that a study used more than
>> one performance metric for the hypothesis test without explicitly and
>> clearly stating what hypothesis this study aims to test.

That sounds like a journal with reviewers who are not doing their job.
A new method may have better sensitivity or specificity, making it
useful as a second test. If it is cheaper/easier, that virtue might
justify slight inferiority. If it is more expensive, there should be
a gain in accuracy to justify its application (or, it deserves further
development).

> Often the
>> paper only states that it intends to test if a newly developed object
>> (algorithm, drug, device, technique, etc) would perform better than
>> some chosen benchmarks. Then the paper presents some tables
>> summarizing the results of many comparisons. Among the tables, the
>> paper picks those comparisons having better values of some
>> performance metric and showing statistical significance. Finally, the
>> paper claims that the new object is successful since it has some
>> favorable results that are statistically significant.
>>
>> This looks odd. SHouldn't we clearly define the hypothesis before
>> conducting any tests? For example, shouldn't we define the success of
>> the object to be "having all the chosen metrics have better results"?
>> Otherwise, why would we test so many metrics, instead of only one?
>>
>> The aforementioned approach looks like this: we do not know what
>> would happen. So let's pick some commonly used metrics to test if we
>> could get some of them to show favorable and significant results.

I am not comfortable with your use of the word 'metrics' -- I like
to think of improving the metrics of a scale by taking a power
transformation, like, square root for Poisson, etc.

Or, your metric for measuring 'size' might be area, volume, weight....

>>
>> Anyway, what are the correct or rigorous ways to conduct tests
>> with multiple metrics?
>
>You might want to search for the terms "multiple testing" and
>"Bonferroni correction".

That answers the final question -- assuming that you do have
some stated hypothesis or goal.

--
Rich Ulrich

[David Jones]

Not quite. The "Bonferroni correction" is an approximation, and one
needs to think about that, and more deeply than jut the approximation
to 1-(1-p)^n. More deeply, the formula is exact and valid if all the
test-statistics are statistically independent, it is conservative if
there is positive dependence (and so "OK"). But, theoretically, it
might be wildly wrong if there is negative dependence

[David Jones]

My other answer concentrated on the case where you put all attention on
the null hypothesis "no effect of any kind", but one could also think
of finding if any of the alternatives on which the test-statistics are
based are of any importance, and if so, which one(s).

In theory the "Bonferroni correction" approach doesn't deal with this.
One presumably would need to go back to estimates of effect sizes. But,
if the plan was to do further experiments targeted at getting better
estimates of particular effects, how do you choose how many and which
effects to investigate further. The original experiment might suggest
the one with the smallest p-value, but that might just be a chance
event, with some other one being better.

[Rich Ulrich]

On Sun, 19 Mar 2023 10:58:42 -0000 (UTC), "David Jones"

<dajh...@nowherel.com> wrote:

>Rich Ulrich wrote:
>
>> On Sat, 18 Mar 2023 01:25:44 -0000 (UTC), "David Jones"
>> <dajh...@nowherel.com> wrote:
>>
>> > Cosine wrote:
>> >
>> >>
>> >> Anyway, what are the correct or rigorous ways to conduct tests
>> >> with multiple metrics?
>> >
>> > You might want to search for the terms "multiple testing" and
>> > "Bonferroni correction".
>>
>> That answers the final question -- assuming that you do have
>> some stated hypothesis or goal.
>
>Not quite. The "Bonferroni correction" is an approximation, and one

The sufficient answer started with "Search for the terms" -- You
should find much more than "How To" apply Bonferroni correction.

Multiple testing is also a broad topic. The original question was
not very specific, but there should be a GOAL, something about
making some /decision/ or reaching a conclusion.

Here's some open-ended thinking about an open-ended question.

I think I can usually work a decision into some hypothesis; but
"p-level of 0.05" is a convention of social science research. Not
every hypothesis merits that test.

Some areas with tests (new atomic particles) use far more stringent
nominal levels ... I think the official logic incorporates
"bonferroni"-type considerations. But for decisions in general,
in other areas, sometimes we settle for "50%" (or worse).

>needs to think about that, and more deeply than jut the approximation
>to 1-(1-p)^n. More deeply, the formula is exact and valid if all the
>test-statistics are statistically independent, it is conservative if
>there is positive dependence (and so "OK"). But, theoretically, it
>might be wildly wrong if there is negative dependence

--
Rich Ulrich