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Re: st: Standard error of a ratio of two random variables
Austin-
Sergiy defined the denominator population to be:
[Y] population at least 10 years old and less than 11 AND in the 5th
grade
Therefore for Y<., we know whether the child is in school, and, if
so, what the current grade is. This suggests to me that [X] could be
predicted from [Y] and, possibly, from other factors in the
restricted sample. If [Y] is false because the child is in the 5th
grade, but is <10, I would expect P(X =1) to be higher than its
overall value in the restricted sample. If [Y] is false because the
child is >=11 although in the 5th grade, I would expect P(X=1) to be
lower than its overall value in the restricted sample. Lastly, if
[Y] is true, I would expect P(X =1) to be very high. Thus multiple
imputation of X in the restricted sample might improve on your upper
and lower bounds.
"Y" missing is more troublesome. I like the idea of weighting by the
1/P(Y<.), if the data support it.
-Steve
On Jan 15, 2009, at 10:51 AM, Austin Nichols wrote:
Sergiy and Steven--
I think more information is needed, but I would probably avoid
imputation.
Let's say the basic setup is to estimate totals X,Y, and X/Y for:
[X] new entrant (of any age) to 5th grade
[Y] population at least 10 years old and less than 11
Being a new entrant is known if response to the "grade in previous
year" is known, which is subject to recall error, has more missings
than age, etc.
It seems likely that those kids never in school are more likely to
have no response to the "grade in previous
year" i.e. to have missing X. So those who have true X=0 will be more
likely to have observed X=. which means I think you have problems
estimating this no matter how you do it. I think a plausible lower
bound might be constructed by estimating new weights for those cases
with Y<. and then estimating the total of X0=max(0,X) and Y and X0/Y
in that sample. Then a fairly plausible upper bound might be
constructed by estimating new weights for those cases with Y<. and X<.
and then estimating the total of X and Y and X/Y in that much more
restricted sample. The estimation of new weights w1 for the first
case can just inflate each weight by the ratio of the sum of weights
within the restricted sample to the sum of weights for whole survey,
and in the second case inflate each w1 by the ratio of the sum of w1
for each single year of age within the restricted sample to the sum of
w1 for each single year of age. If you have variables that are never
missing in the whole survey, you can use those in a parametric model
to inflate the weights, e.g. estimate a logit of 1(Y<.) on Z to
predict phat and then multiply weights by 1/phat.
Maybe Manski would have better ideas:
http://faculty.wcas.northwestern.edu/~cfm754/
On Thu, Jan 15, 2009 at 10:20 AM, Steven Samuels
<[email protected]> wrote:
Sergiy-
The -ratio- and -svy: ratio- commands in Stata ignore observations
with
missing values in X or Y. So do the Stata programs that you and
Austin
wrote. You risk bias if you estimate numerator and denominator
separately
(ignoring missing values in each); and, if you do that, I don't
know of
simple way of getting standard errors. Of course you also risk
bias if you
use only complete cases. I think that the best approach would be
multiple
imputation of X and Y. Then use the standard commands. I'd guess
that there
are good prospects for predicting your variable "A", at least.
By the way, I often compute SE's for log(X/Y) (use -nlcom- after -
ratio- or
-svy: ratio-) and transform to CI's for X/Y. That way, a CI for X/
Y is
consistent with that for Y/X.
-Steve
On Jan 14, 2009, at 3:36 PM, Sergiy Radyakin wrote:
On Tue, Jan 13, 2009 at 12:01 PM, Austin Nichols
<[email protected]> wrote:
Sergiy Radyakin <[email protected]>:
With different N's you would divide by the product of the sqrt
of N1
and N2 instead. My point was just to point out what you would
need to
multiply rho if you wanted to keep rho for some reason. And
yes, this
is all approximation--if X and Y were normal you might consult
http://www.jstor.org/stable/pdfplus/2334671.pdf
But the example does not seem to bear very well on your actual
application--are the X and Y two variables on the same survey with
different degrees of missingness? The approach given so far
does not
seem the optimal solution in that case...
Thank you Austin, the X and Y are coming from the same survey.
To make things more clear, and since Steven Samuels asked
specifically, here is what I am doing:
I need the mean and the SE for the following indicator "Primary
Completion Rate is: new entrants in the last grade divided by
population being in the last grade and of the proper age (for the
last
grade)". (with a slightly different wording the definition of PCR
and
details are here:
[http://www.un.org/esa/sustdev/natlinfo/indicators/
methodology_sheets/education/intake_education.pdf]
on the first page of the document(see "(b) brief definition")
The definition uses the following characteristics:
[A] being a new entrant (of any age)
[B] being of proper age, e.g. 12y.o., and
Being a new entrant is known if response to the "grade in previous
year" is known, which is subject to recall error, has more missings
than age, etc.
Not all As are Bs and not all Bs are As.
So far I am estimating separately (with svy:total) the numerator and
denominator. Stata returns me the two numbers for each: the mean and
SE.
I then need the mean and SE of the ratio of these two. They are
expected to be well away from zero (if this constitutes a problem),
but note that the ratio is not a proportion, it can be more than
100%.
I then use the formula (see the program in the first post and the
quoted PDF file) to construct the SE for the ratio of the two.
Austin mentioned that there might be a better solution. I hope this
information may help to determine if there is really something
better
to do. I don't care about computing time, but correctness and
conceptual unambiguity are of highest importance.
Thank you, Sergiy Radyakin
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