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JunoExpress

Posted: Mon Mar 10, 2008 3:49 am

Guest

Hi,

I'm starting to read about statistical power analysis for a problem
related to an engineering application. I went back and reviewed a
portion of a *classic* textbook, "Statistical Signal Processing" by
Loius Scharf and came across a statement that floored me.

Scharf is talking about the decision rules for a simple binary
hypothesis test:
H0 : Theta = Theta_0
H1 : Theta = Theta_1

He then discusses a curve he calls the "Receiver Operating
Characteristics", which is a curve of alpha vs beta (or as he puts it,
the Prob of a False Alarm vs the Prob of Detection). The curve Scharf
plots is a continuous convex curve that goes from (0,1) to (1,1).

Here is the statement Scharf makes next," If the prob of false alarm
equals zero, then H0 is always selected, meaning that H1 is never
selected, and that the probability of detection equals zero." (or, the
way I would read it is, alpha=0 implies beta = 0).

I would strongly disagree with his statement that "If the prob of
false alarm equals zero, then H0 is always selected". The prob of a
false alarm is a conditional prob, i.e. it is the prob I choose H1
given H0 is true. Thus to say that the prob of a false alarm equals
zero, means, to me, that H0 is always chosen, given H0 is true. Or to
put it another way, in general, alpha = 0 tells me nothing about what
decision I will make if the conditional hypothesis is not satisfied,
i.e. if H1 is true.

OT1H, the text is a classic and the ROC curves are well-known in
engineering, and it's hard to believe he made a mistake, but OTOH, I
read this, and it just seems plain wrong.

Any suggestions about who is right or if I am misreading something or
misunderstanding Scharf?

Thank-you,

Matt

JunoExpress

Posted: Mon Mar 10, 2008 4:28 am

Guest

OK, I think I may see where Scharf is coming from, but I'm still a bit
puzzled.

I know that:
Prob of choosing H1 = Prob(H1|H1) + Prob(H1|H0) = 1-beta + alpha
which means that
Prob of CORRECTLY choosing H1 = [ 1 - beta ]/(1-beta+alpha)

Now, I agree with a statement along the lines of :
"The Prob of CORRECTLY choosing H1 = 1 iff alpha = 0"
and I can see where he is probably coming from if he what he calls
"probability of detection" is the "The prob of correctly choosing H1".

However, I am still hung up on the fact that it seems to me that the
probability of detection is a conditional probability and should be
just 1-beta.

Any suggestions?

Matt

JunoExpress

Posted: Mon Mar 10, 2008 10:57 am

Guest

Quote:

defines beta as the power: must be a stat signal processing
convention. From this point on, I'll use beta in the "conventional"
sense, and hope that does not cause any confusion.

Quote:

What you have repeated in the final sentence is that
when "alpha = 0" -- a condition that is cannot be met by our
general, continuous statistical tests -- then the power is zero.
That seems entirely reasonably, as an extrapolation on
continuous probabilities. The smaller alpha for a test
has a smaller power, for any fixed difference.

I understand now what you are saying. I mean, how do you get alpha = 0

for a right-tail hypothesis test? Well, set the critical value for the
test statistic at its extreme value. Then, you're guaranteed that H0
is always selected and H1 is never selected and you have a power of
zero (beta=1) regardless of whether H0 is true or not.

I was thinking that the probabilities of the events :
Choose H1 | H0 true
and
Choose H1 | H1 true
are independent, but I can see now that once you set a confidence
level that defines some decision rule, this clearly is not the case.

Thanks,

Matt

Richard Ulrich

Posted: Mon Mar 10, 2008 2:35 pm

Guest

On Mon, 10 Mar 2008 06:49:03 -0700 (PDT), JunoExpress
<MTBrenneman@gmail.com> wrote:

Quote:

You are using beta as power, where I usually think of
beta as (1-power). I don't have a problem with the author,
though I am not as philosophical as some readers.
I think you are not being pragmatic enough.

What you have repeated in the final sentence is that
when "alpha = 0" -- a condition that is cannot be met by our
general, continuous statistical tests -- then the power is zero.
That seems entirely reasonably, as an extrapolation on
continuous probabilities. The smaller alpha for a test
has a smaller power, for any fixed difference.

Quote:

I would strongly disagree with his statement that "If the prob of
false alarm equals zero, then H0 is always selected". The prob of a
false alarm is a conditional prob, i.e. it is the prob I choose H1
given H0 is true. Thus to say that the prob of a false alarm equals
zero, means, to me, that H0 is always chosen, given H0 is true.

Look at the continuous gradient on the ROC chart.
If it is *possible* to choose H1 at the low end, where
alpha is near 0 and power is near 0, then it is possible
to choose H1 erroneously.

This is practically a tautology, something being given
by the definitions -- Where the "false alarm rate" is zero,
the power is zero.

There is something existential or epistemological
here. Possibly, there should be a more complicated
discussion, somewhere, that goes deeper. For now,
Alpha = 0 says, "Under this condition, we are
never going to endorse H1."

Quote:

Or to
put it another way, in general, alpha = 0 tells me nothing about what
decision I will make if the conditional hypothesis is not satisfied,
i.e. if H1 is true.

But we are tagging or identifying conditions, I think,
by some score, in addition to H0 and H1....

Quote:

OT1H, the text is a classic and the ROC curves are well-known in
engineering, and it's hard to believe he made a mistake, but OTOH, I
read this, and it just seems plain wrong.

Any suggestions about who is right or if I am misreading something or
misunderstanding Scharf?

--
Rich Ulrich

http://www.pitt.edu/~wpilib/index.html

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