On Mon, 10 Mar 2008 06:49:03 -0700 (PDT), JunoExpress
<MTBrenneman@[EMAIL PROTECTED]
> wrote:
> Hi,
>
> I'm starting to read about statistical power analysis for a problem
> related to an engineering application. I went back and reviewed a
> ****tion 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).
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.
>
> 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."
> 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....
>
> 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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