12:07 p.m.
Well, as you can see from my recent posts I have been using the test
statistic calculator for the last couple of days. Here's another
question about it:
When testing the difference of two means (Utilities -> test statistic
calculator), I cannot reproduce the standard deviation used in the
denominator. For example, using
mu1=1, sigma1=2, n1=25,
mu2=1, sigma2=3, n2=36
gretl's test statistic output is 0/0.687129. What's the formula for that
value? I would use \sqrt(4/25 + 9/36) = 0.6403... in the denominator
instead, and I couldn't get gretl's ouput by some variations.
Thanks for your help,
Sven
11:13 p.m.
On Wed, 7 Jun 2006, Sven Schreiber wrote:
Well, as you can see from my recent posts I have been using the
test statistic calculator for the last couple of days. Here's
another question about it:
When testing the difference of two means (Utilities -> test statistic
calculator), I cannot reproduce the standard deviation used in the
denominator. For example, using
mu1=1, sigma1=2, n1=25,
mu2=1, sigma2=3, n2=36
gretl's test statistic output is 0/0.687129. What's the formula
for that value? I would use \sqrt(4/25 + 9/36) = 0.6403... in
the denominator instead, and I couldn't get gretl's ouput by
some variations.
In this case, since at least one of the sample sizes is less than
30, gretl is calculating the standard error of the difference in
means as follows (in TeX notation):
x = ((n_1 - 1) * s^2_1 + (n_2 - 1) * s^2_2) / (n_1 + n_2 - 2)
se = \sqrt{ x / n_1 + x / n_2 }
I can't recall right now on what authority I have this formula.
I will check with my "bible" on such matters, DeGroot's
Probability and Statistics, 2e.
Allin.
11:37 p.m.
New subject: variance in mean test (again)
On Wed, 7 Jun 2006, Sven Schreiber wrote:
When testing the difference of two means (Utilities -> test
statistic
calculator), I cannot reproduce the standard deviation used in the
denominator. For example, using
mu1=1, sigma1=2, n1=25,
mu2=1, sigma2=3, n2=36
gretl's test statistic output is 0/0.687129. What's the formula for that
value?
Having checked with DeGroot[*], I stand by this value. It's what
you get when you apply equation (5) on page 508. DeGroot writes
the test statistic in this form:
U = \frac{\sqrt{(n_1 + n_2 - 2)} (\bar{X}_1 - \bar{X}_2)}
{\sqrt{(1/n_1 + 1/n_2)} \sqrt{(S^2_1 + S^2_2)}}
where S^2_i is the sum of squared deviations from the mean for
sample i.
[*] Morris H. DeGroot, Probability and Statistics, 2nd edition,
Addison-Wesley, 1986 (Reprinted with corrections, 1989).
Allin
12:12 a.m.
New subject: Behrens-Fisher problem
I wrote:
Having checked with DeGroot, I stand by this value...
As for the case where we're _not_ assuming a constant variance and
both samples are large, I really don't know. In setting up
gretl's hypothesis test calculator I took a formula for the
standard error of the difference of means in this case from some
statistics text (but evidently not DeGroot).
Having read DeGroot and googled on the "Behrens-Fisher problem"
(hypothesis testing on the difference between two means, with the
repespective population means unknown and not assumed equal), the
case seems more or less hopeless. DeGroot says the problem is
"very difficult", and that while "various test procedures have
been proposed", "most of them have been the subject of controversy
in regard to their appropriateness or usefulness".
Allin Cottrell
5:04 p.m.
New subject: Behrens-Fisher problem
Thanks Allin; ok, let's see if I understood your replies correctly:
1. The stand. dev. formula that is used in the 2-means test depends on
whether both samples are large (n>30) or not.
2. If one of the samples is "small", then a formula is used that appears
to be valid only in the case of equal variances.
3. there is no known/accepted way to test an arbitrary difference of
means in small samples with unequal (and unknown) variances
Provided those assessments are correct, I would suggest the following:
ad 1) I would like to see some mention of which assumption/formula is
used in the test output, or make it a user option.
ad 2) This is unfortunate given that the user explicitly choose the
option "do not assume equal variances". The easiest workaround is to use
the asymptotic formula instead and print a warning in the output that
the approximation may be poor due to the sample size.
ad 3) Given that insight, I think gretl should always use asymptotic
formulae for unequal variances and merely warn the user if samples are
small.
Ok, I expect that there are some details where I misunderstood Allin and
then some of these comments would probably be off mark. Please take that
into account.
I just discovered the test statistic calculator and I really like it,
especially for teaching. So I would like to make sure it's transparent
to the user what it does. (and hopefully does the right thing)
Thanks,
Sven
Allin Cottrell schrieb:
I wrote:
> Having checked with DeGroot, I stand by this value...
As for the case where we're _not_ assuming a constant variance and both
samples are large, I really don't know. In setting up gretl's
hypothesis test calculator I took a formula for the standard error of
the difference of means in this case from some statistics text (but
evidently not DeGroot).
Having read DeGroot and googled on the "Behrens-Fisher problem"
(hypothesis testing on the difference between two means, with the
repespective population means unknown and not assumed equal), the
case seems more or less hopeless. DeGroot says the problem is "very
difficult", and that while "various test procedures have been proposed",
"most of them have been the subject of controversy in regard to their
appropriateness or usefulness".
Allin Cottrell
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Gretl-users(a)ricardo.ecn.wfu.edu
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5:41 p.m.
New subject: Behrens-Fisher problem
On Thu, 8 Jun 2006, Sven Schreiber wrote:
Thanks Allin; ok, let's see if I understood your replies
correctly:
1. The stand. dev. formula that is used in the 2-means test depends on
whether both samples are large (n>30) or not.
Yes, that's right.
2. If one of the samples is "small", then a formula is used
that appears
to be valid only in the case of equal variances.
True.
3. there is no known/accepted way to test an arbitrary difference of
means in small samples with unequal (and unknown) variances
So far as I understand it, yes.
Provided those assessments are correct, I would suggest the
following:
ad 1) I would like to see some mention of which assumption/formula is
used in the test output, or make it a user option.
Yes, that should be done.
ad 2) This is unfortunate given that the user explicitly choose the
option "do not assume equal variances". The easiest workaround is to use
the asymptotic formula instead and print a warning in the output that
the approximation may be poor due to the sample size.
OK, we'll do that.
Allin.
6:15 p.m.
New subject: Behrens-Fisher problem
Allin Cottrell schrieb:
> ad 1) I would like to see some mention of which
assumption/formula is
> used in the test output, or make it a user option.
Yes, that should be done.
> ad 2) This is unfortunate given that the user explicitly choose the
> option "do not assume equal variances". The easiest workaround is to use
> the asymptotic formula instead and print a warning in the output that
> the approximation may be poor due to the sample size.
OK, we'll do that.
Thanks a lot! I'm sorry I can't help with those things, but I don't know
C. (to be honest, nor do I want to know it... ;-)
-Sven
10:46 p.m.
New subject: Behrens-Fisher problem
On Thu, June 8, 2006 20:15, Sven Schreiber wrote:
Allin Cottrell schrieb:
>> ad 1) I would like to see some mention of which assumption/formula is
>> used in the test output, or make it a user option.
>
> Yes, that should be done.
>
>> ad 2) This is unfortunate given that the user explicitly choose the
>> option "do not assume equal variances". The easiest workaround is to
use
>> the asymptotic formula instead and print a warning in the output that
>> the approximation may be poor due to the sample size.
>
> OK, we'll do that.
>
Thanks a lot! I'm sorry I can't help with those things, but I don't know
C. (to be honest, nor do I want to know it... ;-)
... and I don't blame you!
But with a little LaTeX, you could summarise this very interesting discussion
in the manual (hint, hint)
Riccardo "Jack" Lucchetti
Dipartimento di Economia
Facoltà di Economia "G. Fuà"
Ancona
2:19 p.m.
New subject: Behrens-Fisher problem
Riccardo Jack Lucchetti schrieb:
But with a little LaTeX, you could summarise this very interesting
discussion
in the manual (hint, hint)
Since I'm a big fan of good documentation I'll see what I can do. But
don't expect it to be ready for the next release... Any specifics
concerning the structure/formatting etc?
-sven
6550
days inactive
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8 comments
3 participants
participants (3)
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Allin Cottrell -
Riccardo Jack Lucchetti -
Sven Schreiber