[SciPy-User] scipy.stats one-sided two-sided less, greater, signed ?

Sun Jun 12 07:30:39 EDT 2011

On Sun, Jun 12, 2011 at 6:20 AM, Ralf Gommers
<ralf.gommers at googlemail.com> wrote:
>
>
> On Wed, Jun 8, 2011 at 12:56 PM, <josef.pktd at gmail.com> wrote:
>>
>> On Tue, Jun 7, 2011 at 10:37 PM, Bruce Southey <bsouthey at gmail.com> wrote:
>> > On Tue, Jun 7, 2011 at 4:40 PM, Ralf Gommers
>> > <ralf.gommers at googlemail.com> wrote:
>> >>
>> >>
>> >> On Mon, Jun 6, 2011 at 9:34 PM, <josef.pktd at gmail.com> wrote:
>> >>>
>> >>> On Mon, Jun 6, 2011 at 2:34 PM, Bruce Southey <bsouthey at gmail.com>
>> >>> wrote:
>> >>> > On 06/05/2011 02:43 PM, josef.pktd at gmail.com wrote:
>> >>> >> What should be the policy on one-sided versus two-sided?
>> >>> > Yes :-)
>> >>> >
>> >>> >> The main reason right now for looking at this is
>> >>> >> http://projects.scipy.org/scipy/ticket/1394 which specifies a
>> >>> >> "one-sided" alternative and provides both lower and upper tail.
>> >>> > That refers to the Fisher's test rather than the more 'traditional'
>> >>> > one-sided tests. Each value of the Fisher's test has special
>> >>> > meanings
>> >>> > about the value or probability of the 'first cell' under the null
>> >>> > hypothesis.  So it is necessary to provide those three values.
>> >>> >
>> >>> >> I would prefer that we follow the alternative patterns similar to R
>> >>> >>
>> >>> >> currently only kstest has    alternative : 'two_sided' (default),
>> >>> >> 'less' or 'greater'
>> >>> >> but this should be added to other tests where it makes sense
>> >>> > I think that these Kolmogorov-Smirnov  tests are not the traditional
>> >>> > meaning either. It is a little mind-boggling to try to think about
>> >>> > cdfs!
>> >>> >
>> >>> >> R fisher.exact
>> >>> >> """alternative        indicates the alternative hypothesis and must
>> >>> >> be
>> >>> >> one
>> >>> >> of "two.sided", "greater" or "less". You can specify just the
>> >>> >> initial
>> >>> >> letter. Only used in the 2 by 2 case."""
>> >>> >>
>> >>> >> mannwhitneyu reports a one-sided test without actually specifying
>> >>> >> which alternative is used  (I thought I remembered other cases like
>> >>> >> this but don't find any right now)
>> >>> >>
>> >>> >> related:
>> >>> >> in many cases in the two-sided tests the test statistic has a sign
>> >>> >> that indicates in which tail the test-statistic falls.
>> >>> >> This is useful in ttests for example, because the one-sided tests
>> >>> >> can
>> >>> >> be backed out from the two-sided tests. (With symmetric
>> >>> >> distributions
>> >>> >> one-sided p-value is just half of the two-sided pvalue)
>> >>> >>
>> >>> >> In the discussion of https://github.com/scipy/scipy/pull/8  I
>> >>> >> argued
>> >>> >> that this might mislead users to interpret a two-sided result as a
>> >>> >> one-sided result. However, I doubt now that this is a strong
>> >>> >> argument
>> >>> >> against not reporting the signed test statistic.
>> >>> > (I do not follow pull requests so is there a relevant ticket?)
>> >>> >
>> >>> >> After going through scipy.stats.stats, it looks like we always
>> >>> >> report
>> >>> >> the signed test statistic.
>> >>> >>
>> >>> >> The test statistic in ks_2samp is in all cases defined as a max
>> >>> >> value
>> >>> >> and doesn't have a sign in R either, so adding a sign there would
>> >>> >> break with the standard definition.
>> >>> >> one-sided option for ks_2samp would just require to find the
>> >>> >> distribution of the test statistics D+, D-
>> >>> >>
>> >>> >> ---
>> >>> >>
>> >>> >> So my proposal for the general pattern (with exceptions for special
>> >>> >> reasons) would be
>> >>> >>
>> >>> >> * add/offer alternative : 'two_sided' (default), 'less' or
>> >>> >> 'greater'
>> >>> >> http://projects.scipy.org/scipy/ticket/1394  for now,
>> >>> >> and adjustments of existing tests in the future (adding the option
>> >>> >> can
>> >>> >> be mostly done in a backwards compatible way and for symmetric
>> >>> >> distributions like ttest it's just a convenience)
>> >>> >> mannwhitneyu seems to be the only "weird" one
>> >>
>> >> This would actually make the fisher_exact implementation more
>> >> consistent,
>> >> since only one p-value is returned in all cases. I just don't like the
>> >> R
>> >> naming much; alternative="greater" does not convey to me that this is a
>> >> one-sided test using the upper tail. How about:
>> >>     test : {"two-tailed", "lower-tail", "upper-tail"}
>> >> with two-tailed the default?
>>
>> I think matlab uses (in general) larger and smaller, the advantage of
>> less/smaller and greater/larger is that it directly refers to the
>> alternative hypothesis, while the meaning in terms of tails is not
>> always clear (in kstest and I guess some others the test statistics is
>> just reversed and uses the same tail in both cases)
>>
>> so greater smaller is mostly "future proof" across tests, while
>> reference to the tail can only be used where this is an unambiguous
>> statement. but see below
>>
> I think I understand your terminology a bit better now, and consistency
> across all tests is important. So I've updated the Fisher's exact patch to
> use alternative={'two-sided', 'less', greater'} and sent a pull request:
> https://github.com/scipy/scipy/pull/32

looks good to me, I added some comments to the pull request.

Josef

>
> Cheers,
> Ralf
>
>>
>>
>> >>
>> >> Ralf
>> >>
>> >>
>> >>>
>> >>> >>
>> >>> >> * report signed test statistic for two-sided alternative (when a
>> >>> >> signed test statistic exists):  which is the status quo in
>> >>> >> stats.stats, but I didn't know that this is actually pretty
>> >>> >> consistent
>> >>> >> across tests.
>> >>> >>
>> >>> >> Opinions ?
>> >>> >>
>> >>> >> Josef
>> >>> >> _______________________________________________
>> >>> >> SciPy-User mailing list
>> >>> >> SciPy-User at scipy.org
>> >>> >> http://mail.scipy.org/mailman/listinfo/scipy-user
>> >>> > I think that there is some valid misunderstanding here (as I was in
>> >>> > the
>> >>> > same situation) regarding what is meant here. My understanding is
>> >>> > that
>> >>> > under a one-sided hypothesis, all the values of the null hypothesis
>> >>> > only
>> >>> > exist in one tail of the test distribution. In contrast the values
>> >>> > of
>> >>> > null distribution exist in both tails with a two-sided hypothesis.
>> >>> > Yet
>> >>> > that interpretation does not have the same meaning as the tails in
>> >>> > the
>> >>> > Fisher or Kolmogorov-Smirnov tests.
>> >>>
>> >>> The tests have a clear Null Hypothesis (equality) and Alternative
>> >>> Hypothesis (not equal or directional, less or greater).
>> >>> So the "alternative" should be clearly specified in the function
>> >>> argument, as in R.
>> >>>
>> >>> Whether this corresponds to left and right tails of the distribution
>> >>> is an "implementation detail" which holds for ttests but not for
>> >>> kstest/ks_2samp.
>> >>>
>> >>> kstest/ks2sample   H0: cdf1 == cdf2  and H1:  cdf1 != cdf2 or H1:
>> >>> cdf1 < cdf2 or H1:  cdf1 > cdf2
>> >>> (looks similar to comparing two survival curves in Kaplan-Meier ?)
>> >>>
>> >>> fisher_exact (2 by 2)  H0: odds-ratio == 1 and H1: odds-ratio != 1 or
>> >>> H1: odds-ratio < 1 or H1: odds-ratio > 1
>> >>>
>> >>> I know the kolmogorov-smirnov tests, but for fisher exact and
>> >>> contingency tables I rely on R
>> >>>
>> >>> from R-help:
>> >>> For 2 by 2 tables, the null of conditional independence is equivalent
>> >>> to the hypothesis that the odds ratio equals one. <...> The
>> >>> alternative for a one-sided test is based on the odds ratio, so
>> >>> alternative = "greater" is a test of the odds ratio being bigger than
>> >>> or.
>> >>> Two-sided tests are based on the probabilities of the tables, and take
>> >>> as ‘more extreme’ all tables with probabilities less than or equal to
>> >>> that of the observed table, the p-value being the sum of such
>> >>> probabilities.
>> >>>
>> >>> Josef
>> >>>
>> >>>
>> >>> >
>> >>> > I never paid much attention to the frequency based tests but it does
>> >>> > not
>> >>> > surprise if there are no one-sided tests. Most are rank-based so it
>> >>> > is
>> >>> > rather hard to do in a simply manner - actually I am not even sure
>> >>> > how
>> >>> > to use a permutation test.
>> >>> >
>> >>> > Bruce
>> >>> >
>> >>> >
>> >>> >
>> >>> > _______________________________________________
>> >>> > SciPy-User mailing list
>> >>> > SciPy-User at scipy.org
>> >>> > http://mail.scipy.org/mailman/listinfo/scipy-user
>> >>> >
>> >>> _______________________________________________
>> >>> SciPy-User mailing list
>> >>> SciPy-User at scipy.org
>> >>> http://mail.scipy.org/mailman/listinfo/scipy-user
>> >>
>> >>
>> >> _______________________________________________
>> >> SciPy-User mailing list
>> >> SciPy-User at scipy.org
>> >> http://mail.scipy.org/mailman/listinfo/scipy-user
>> >>
>> >>
>> >
>> > But that is NOT the correct interpretation  here!
>> > I tried to explain to you that this is the not the usual idea
>> > one-sided vs two-sided tests.
>> > For example:
>> > http://www.msu.edu/~fuw/teaching/Fu_ch10_2_categorical.ppt
>> > "The test holds the marginal totals fixed and computes the
>> > hypergeometric probability that n11 is at least as large as the
>> > observed value"
>>
>> this still sounds like a less/greater test to me
>>
>>
>> > "The output consists of three p-values:
>> > Left: Use this when the alternative to independence is that there is
>> > negative association between the variables.  That is, the observations
>> > tend to lie in lower left and upper right.
>> > Right: Use this when the alternative to independence is that there is
>> > positive association between the variables. That is, the observations
>> > tend to lie in upper left and lower right.
>> > 2-Tail: Use this when there is no prior alternative.
>> > "
>> > There is also the book "Categorical data analysis: using the SAS
>> > system  By Maura E. Stokes, Charles S. Davis, Gary G. Koch" that came
>> > up via Google that also refers to the n11 cell.
>> >
>> > http://www.langsrud.com/fisher.htm
>>
>> I was trying to read the Agresti paper referenced there but it has too
>> much detail to get through in 15 minutes :)
>>
>> > "The output consists of three p-values:
>> >
>> >    Left: Use this when the alternative to independence is that there
>> > is negative association between the variables.
>> >    That is, the observations tend to lie in lower left and upper right.
>> >    Right: Use this when the alternative to independence is that there
>> > is positive association between the variables.
>> >    That is, the observations tend to lie in upper left and lower right.
>> >    2-Tail: Use this when there is no prior alternative.
>> >
>> > NOTE: Decide to use Left, Right or 2-Tail before collecting (or
>> > looking at) the data."
>> >
>> > But you will get a different p-value if you switch rows and columns
>> > because of the dependence on the n11 cell. If you do that then the
>> > p-values switch between left and right sides as these now refer to
>> > different hypotheses regarding that first cell.
>>
>> switching row and columns doesn't change the p-value in R
>> reversing columns changes the definition of less and greater, reverses
>> them
>>
>> The problem with 2 by 2 contingency tables with given marginals, i.e.
>> row and column totals, is that we only have one free entry. Any test
>> on one entry, e.g. element 0,0, pins down all the other ones and
>> (many) tests then become equivalent.
>>
>>
>> http://support.sas.com/documentation/cdl/en/procstat/63104/HTML/default/viewer.htm#procstat_freq_a0000000658.htm
>> some math got lost
>> """
>> For <2 by 2> tables, one-sided -values for Fisher’s exact test are
>> defined in terms of the frequency of the cell in the first row and
>> first column of the table, the (1,1) cell. Denoting the observed (1,1)
>> cell frequency by , the left-sided -value for Fisher’s exact test is
>> the probability that the (1,1) cell frequency is less than or equal to
>> . For the left-sided -value, the set includes those tables with a
>> (1,1) cell frequency less than or equal to . A small left-sided -value
>> supports the alternative hypothesis that the probability of an
>> observation being in the first cell is actually less than expected
>> under the null hypothesis of independent row and column variables.
>>
>> Similarly, for a right-sided alternative hypothesis, is the set of
>> tables where the frequency of the (1,1) cell is greater than or equal
>> to that in the observed table. A small right-sided -value supports the
>> alternative that the probability of the first cell is actually greater
>> than that expected under the null hypothesis.
>>
>> Because the (1,1) cell frequency completely determines the table when
>> the marginal row and column sums are fixed, these one-sided
>> alternatives can be stated equivalently in terms of other cell
>> probabilities or ratios of cell probabilities. The left-sided
>> alternative is equivalent to an odds ratio less than 1, where the odds
>> ratio equals (). Additionally, the left-sided alternative is
>> equivalent to the column 1 risk for row 1 being less than the column 1
>> risk for row 2, . Similarly, the right-sided alternative is equivalent
>> to the column 1 risk for row 1 being greater than the column 1 risk
>> for row 2, . See Agresti (2007) for details.
>> R C Tables
>> """
>>
>> I'm not a user of Fisher's exact test (and I have a hard time keeping
>> the different statements straight), so if left/right or lower/upper
>> makes more sense to users, then I don't complain.
>>
>> To me they are all just independence tests with possible one-sided
>> alternatives that one distribution dominates the other. (with the same
>> pattern as ks_2samp or ttest_2samp)
>>
>> Josef
>>
>> >
>> >
>> > Bruce
>> > _______________________________________________
>> > SciPy-User mailing list
>> > SciPy-User at scipy.org
>> > http://mail.scipy.org/mailman/listinfo/scipy-user
>> >
>> _______________________________________________
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>
>
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