I don't think this is correct.

First, as far as the hypotheses are concerned, they are equivalent since 
b = 0 iff bp(1-p) = 0 (because 0 < p < 1). Thus, the only thing that can 
differ between the tests is that one has more power than the other.

What you have are two (asymptotic) t-tests, where t_1 = \hat b/SD(\hat 
b) and t_2 = \hat b \hat p (1 - \hat p) / SD(\hat b \hat p (1 - \hat 
p)). I think, in practice, that the first will tend to be larger than 
the second, though I couldn't determine whether the ratio t_1/t_2 is 
always greater than one. However, I think a Rao-Blackwell argument 
should show that the first is always better (since you're effectively 
just adding some positively correlated noise to the estimate by 
multiplying by \hat p (1 - \hat p). At least, that's my intuition.

Not a bad prelim question ;)

Doug


Larry M. Bartels wrote:
> I think the crucial difference here is p-hat versus p. If you want to
> know whether the effect is "significant" at any specified point on the
> logit curve, that is simply the same thing as asking whether the
> coefficient is "significant" (because, for fixed p, the only thing that
> is random in beta-hat_j*p*(1-p) is beta-hat_j). If you want to know
> whether the effect is "significant" for a particular constellation of
> explanatory variables, then the variance of p-hat enters as well, and
> that depends on the entire covariance matrix of the parameter estimates.
> 
> Larry
> 
> 
> -----Original Message-----
> From: Political Methodology Society [mailto:[log in to unmask]] On
> Behalf Of Franzese, Robert
> Sent: Tuesday, April 17, 2007 10:28 AM
> To: [log in to unmask]
> Subject: [POLMETH] Significance of parameter vs. of effects in
> non-linear-additive models, like logit/probit
> 
> Fellow PolMethers,
> 
> I write with (yet another) simple-but-very-good-and(-perhaps)-deep
> question that the students in my methods class this term raised. They
> noticed, in the course of a problem set, that a coefficient in a logit
> estimation could be statistically distinguishable from zero while
> marginal or first-difference effects of the associated variable are not,
> or vice versa the former may be insignificant but the latter significant
> (in some or all ranges of other variable values). As we all know, in a
> logit model,
> 
> d(p-hat)/dX_j = beta_j*(phat)*(1-phat)
> 
> So, the marginal effect is zero if beta_j is zero. So, for years, I'd
> simply been teaching that one quick way to assess the zero or non-zero
> effect of a variable was to look at our old familiar friend, estimated
> coefficient divided by its standard error, emphasizing however that the
> significance/certainty of estimated effects must be assessed using the
> estimated variance of the above formula. That seemed to satisfy everyone
> before, but, as I said, this is a very deep-thinking cohort.
> 
> The question they raised is what to make of the results when these two
> signals about the significance of some explanatory variable(s) conflict.
> I could assemble my own thoughts on the conundrum, but I thought I'd
> check with this audience to see if there are any extant good discussions
> of the matter and/or what you all think.
> 
> Here are my thoughts so far on the matter:
> My first thought was: Are these close calls? The tests that beta_j=0 and
> that beta_j*phat*(1-phat)=0 might be asymptotically equivalent. In that
> case, the differences amount simply, mostly, to a reminder not to place
> too much emphasis on any knife edge at .05 or anywhere else. On the
> other hand, perhaps they are not even asymptotically equivalent, since
> the covariances of the coefficient estimates enter in the latter but not
> the former. In that case, just as in a time-serial context that we
> discussed some in this same class, where b/(1-rho) might be
> distinguishable from zero when b is not, or vice versa, we might find
> that a particular coefficient is distinguishable from zero whereas its
> effect under various configurations of other coefficient estimates and
> variable values is not or vice versa. What to make of this in the
> logit/probit context seems more complicated than in the time-serial one,
> though. (In the time-serial case, I do not find it particularly
> difficult to comprehend how we might be more certain that the long-run
> impact of some variable is non-zero whereas we are uncertain that that
> the instantaneous/contemporaneous impact is so, or vice versa.)
> Essentially, the difference in this case seems to be more on the order
> of being able to distinguish from zero one parameter in an inseparable
> formula involving multiple parameters vs. being able to distinguish the
> latter whole AND substantively meaningful formula from zero, and I guess
> I would lean toward putting more weight on the latter result.
> 
> Thoughts? Suggested readings? (I seem to recall a recent piece, maybe to
> the PolMeth WPA, on linear interactions in logit/probit that emphasized
> this distinction, e.g.)
> 
> Thanks!
> Rob
> 
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> Robert (Rob) J. Franzese, Jr.                  US Mail:   (ISR Room
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