Chi-square as one- or two-tailed test
Michael Lichter
goodness of fit for that matter) was a one-tailed test. In CROSSTABS,
however, SPSS provides a two-tailed probability. I checked it against
Stata and against an on-line probability calculator and found that they
both generated the same probability (for a particular chi-square value).
I just don't understand how the test of independence can be two-tailed
given that the chi-square statistic is unsigned (always positive) and
therefore isn't sensitive to how the relationship between the variables
deviates from the expected values. My image of a two-tailed test is a t
or z test against a Normal distribution with positive and negative
critical areas. In a two-tailed chi-square test, what does the left tail
represent?
On more or less the same topic, SPSS shows two-tailed probabilities for
lambda and phi and other nondirectional measures of association. What
does a two-tailed test mean for these?
Finally, SPSS allows phi to be signed. Phi is supposed to be the square
root of chi-square/N, which by definition can't be negative. It appears
that SPSS is simply reporting r rather than phi. This is incorrect,
isn't it?
Thanks.
Michael
Michael....@colostate.edu
> Hi. I always thought that the chi-square test of independence (or
> goodness of fit for that matter) was a one-tailed test. In CROSSTABS,
> however, SPSS provides a two-tailed probability. I checked it against
> Stata and against an on-line probability calculator and found that they
> both generated the same probability (for a particular chi-square value).
In either of the applications you mention, Chi-Squared is a
one-tailed test, i.e., values in the righthand tail of the
distribution constitute evidence against the null. The
confusion you have, I suspect, is between a directional
test, which these Chi-Squared tests are *not*, and a one-
tailed test, which they *are*. "Directional" and "one-
tailed" go together for t-tests, but not for Chi-Squared.
--
=-=-=-=-=-=-=-=-=-==-=-=-=
Mike Lacy, Ft Collins CO 80523
Clean out the 'junk' to email me.
Bruce Weaver
> Finally, SPSS allows phi to be signed. Phi is supposed to be the square
> root of chi-square/N, which by definition can't be negative. It appears
> that SPSS is simply reporting r rather than phi. This is incorrect,
> isn't it?
Mike Lacy addressed the issue of one-tailed vs directional tests. I'll
just add that the phi-coefficient IS Pearson r computed on a pair of
dichotomous variables. A lot of books do say that phi = the square root
of chi-square/N. What they *should* say is that phi-squared = r-squared
= chi-square/N.
--
Bruce Weaver
bwe...@lakeheadu.ca
www.angelfire.com/wv/bwhomedir
Michael Lichter
how I've always thought of it. SPSS reports two-tailed probabilities.
These correspond to the probabilities you see in a standard chi-square
table in a stat textbook. Either SPSS is incorrect about these
probabilities being two-tailed, or you're/we're incorrect about
chi-square being one-tailed. Which is it?
Michael
Richard Ulrich
This was hashed out in sci.stat.edu a few years ago (2001), in
a discussion with many participants.
Here is the post that google numbers as 41st in the thread,
where Jim Clark discusses the chi-squared test in particular,
in reference to "directionality".
http://groups.google.com/group/sci.stat.edu/msg/39dbc4d4b8b87d66
Google lets you link to the rest of the thread, from there, by
clicking on the name of the thread.
How do we want to use terminology? -- I think of it like this.
There is a two-tailed use of a test, where the two ends are
added together, versus testing a two-tailed "hypothesis" with
one tail, which is what is done by the contingency chi-squared.
Maybe SPSS should omit mentioning anything about tails?
I would not criticize whatever-they-say without seeing
exactly what it is, in context.
--
Rich Ulrich, wpi...@pitt.edu
http://www.pitt.edu/~wpilib/index.html
Michael Lichter
> Mike Lacy addressed the issue of one-tailed vs directional tests. I'll
> just add that the phi-coefficient IS Pearson r computed on a pair of
> dichotomous variables. A lot of books do say that phi = the square root
> of chi-square/N. What they *should* say is that phi-squared = r-squared
> = chi-square/N.
I understand what you're saying, but I'm not certain it's correct.
Almost all of the references I've seen to phi say that it has a range of
0 to +1. This isn't an issue of great significance, but the question
whether I'm going to embarrass myself by treating phi as a signed,
directional measure of association. I suppose that the last (and first)
word on phi is in the article where the statistic was introduced. Thanks.
-ml
Art Kendall
Cohen (2003, p 30-31) points out that the phi coefficient is an easier
hand calculation when both variables are dichotomies.
He is clear that value goes from -1 to +1 . Phi can only reach the
limits when both variables have the same split.
One way to think of it is this. Picture scatterplots.
With any Pearson correlation coefficient a positive correlation can be
read as meaning that high scores on one variable tend to go with high
scores on the other and low scores on the variable tend to go with low
scores on the other.
Like wise a negative correlation can be read as meaning that high
scores on one variable tend to go with low scores on the other and low
scores on the variable tend to go with high scores on the other.
if i have two variables coded zero and one, isn't it possible for ones
on one variable to tend to go with ones on the other, and zeros on one
variable to tend to go with zeros on the other?
isn't it also possible for ones on one variable to tend to go with zeros
on the other, and zeros on one variable to tend to go with ones on the
other?
here is a clip from the Library of Congress card catalog for an
excellent book on regression and correlation.
Main Title: Applied multiple regression/correlation analysis for the
behavioral sciences / Jacob Cohen ... [et al.].
Edition Information: 3rd ed.
Published/Created: Mahwah, N.J. : L. Erlbaum Associates, 2003.
Related Names: Cohen, Jacob, 1923-
Cohen, Jacob, 1923- Applied multiple regression/correlation analysis for
the behavioral sciences.
Description: xxviii, 703 p. : ill. ; 26 cm. + 1 CD-ROM (4 3/4 in.)
ISBN: 0805822232 (hard cover : alk. paper)
Notes: Rev. ed. of: Applied multiple regression/correlation analysis
for the behavioral sciences / Jacob Cohen, Patricia Cohen. 2nd ed. 1983.
The CD-ROM contains the data for almost all examples as well as the
command codes for each of the major statistical packages for the tabular
and other findings in the book.
Includes bibliographical references (p. 655-669) and indexes.
Subjects: Regression analysis.
Correlation (Statistics)
Social sciences--Statistical methods.
LC Classification: HA31.3 .A67 2003
Dewey Class No.: 519.5/36 21
Art
A...@DrKendall.org
Social Research Consultants
Inside the Washington, DC beltway.
(301) 864-5570
Richard Ulrich
This has been discussed in the sci.stat.* groups, I think.
What I seem to recall is that the phi was probably
described 100 years ago as the square root of X^2/n.
But since it is otherwise, exactly, a Pearson product
moment correlation between dichotomies, it seems
silly to preserve the convention that it is positive.
Enough statisticians don't know the old convention that
there would not be much embarrassment in the mistake.
But I don't known how many textbook authors,
in what specialties - or how many pedantic reviewers -
may be on that side.