Thanks to those who responded to my earlier message. I have received a few
ideas for pursuing generation of bivariate beta random variables.
Simon Jackman (jackman@stanford.edu) wrote:
>the multivariate version of the beta distribution is the Dirichlet
>distribution. You can get negatively correlated beta deviates that way
>(i.e., the Dirichlet is vector values, and is such that the correlation
>between its components are always negative).
My desire is to obtain positively correlated variates. Suppose I would take
(x1,x2) as Dirichlet with negative correlation, and use (x1, 1-x2) to get
positively correlated variates. Now need to see about generating Dirichlet
variates.
Tony Lee (ajl@xtra.co.nz) wrote:
>A program (DOS) by Professor Jeff Miller of Otago University has the
>facility to generate the variates you need. See
>http://psy.otago.ac.nz/staff/miller/welcome.htm#Software
The web site contains references to a few DOD-based programs, one of which is
called BIVAR to create bivariate random variables. BIVAR mentions three
procedures to generate such bivariate variables (X1, X2), this first two are
summarized here:
1: Generate bivariate normal (Z1, Z2) with specified correlation p;
transpose to bivariate uniform (U1, U2), where U1=probit(Z1) and U2=probit(Z2);
and finally transpose to desired distribution using the density function
X1=f(U1) and X2=f(U2). Note: correlation of X1 and X2 is of same sign as p
used in generating Z1 and Z2, but may not have the same magnitude. Value of p
must be adjusted to obtain desired correlation between X1 and X2.
2: Using a mixture approach, generate uncorrelated variates Y1 and Y2
having desired marginal distribution; set X1=Y1; with probability p, set
X2=f(F(X1)), where f() and F() are the probability density function for X2 and
cumulative probability function for X1; with probability 1-p, X2=Y2. The value
p determines the degree of correlation between X1 and X2.
The first approach used in BIVAR was also suggested by Matthew A. Kurbatt
<matthew.a.kurbat@ncmi.com>
>two thoughts:
>* Johnson and Kotz (sometimes with a 3rd co-author) have published
>a series of books on distributions. Their multivariate version may have
>what you want (of course, I don't have that one)
>
>* This is just off-the-cuff, but: would it work to generate from a bivariate
>normal then transform to beta? I haven't done the math - just a thought.
>One more thought - because resulting bivariate dist'n is no longer normal,
>"correlation" may not entirely describe the dependence between the
>marginals.
Alan Zaslavsky <zaslavsk@hcp.med.harvard.edu> sent the following reference
>there is an article by Mei-ling Lee which develops such a joint
>distribution. I think this is it:
>AUTHOR = Lee, Mei-Ling Ting
>TITLE = Properties and applications of the Sarmanov family of bivariate
> distributions
>JOURNAL = Communications in Statistics, Part A -- Theory and Methods
>VOLUME = 25
>PAGES = 1207-1222
>YEAR = 1996
>KEYWORDS = Bayes method
>
>I can't remember offhand whether she gives a method for generating from
>it.
One respondent suggested
>Since a beta distribution is one way of representing a binary distribution,
>I wonder whether one solution is for you to use correlated binary
>distributions?
My desire is to generate bivariate multinomial values from which kappa
(inter-rater agreement) would be calculated and use the correlation of the
latent continuous variables to help understand the behavior of kappa. Hence
the desire for correlated beta.
Thanks for the suggestions. I am not sure which to try first.
Jim Pratt
>>> "Jim Pratt" <JimP@sonuspharma.com> 10/13/99 05:28PM >>>
I searched the S-News archive and did not find a suggestion for my situation.
I'd like to generate bivariate beta random variables with specified
correlation. Much has been discussed on S-News recently regarding correlated
binary and normal variates, but have not seen anything on correlated beta
variates. Any suggestions would be greatly appreciated.
What I plan to do with these variates:
The kappa statistic for inter-rater agreement 'seems' to behave poorly when the
objects being rated are not uniformly distributed across the entire scoring
scale. E.g., suppose a 4-point scale (0-3) is used, but 75% of the objects are
scored 3, 24% scored 2, .5% each at 0 and 1. Even though there is high
agreement between the two raters (80% of objects are on diagonal), the observed
kappa was 0.30. I am interested in understanding the behavior of kappa in such
non-uniform situations. The plan is to generate bivariate beta scores as
latent variables with specified correlation, collapse to categorical scores,
and assess how observed kappa relates to correlation of the latent variates.
In a series of papers by Cicchetti and Feinstein, they discuss paradoxes of
kappa statistics in the context of 2-point scoring (yes/no; pos/neg). They
discuss the effect of prevalence on the kappa statistics and how severity of
prevalence is related to the maximum possible kappa value that could be
attained. They propose adjusting the computed kappa by weighting by the
maximum possible. [To me, this indicates that kappa statistics should not be
reviewed in isolation, but with some measure of observed prevalence.] I am
looking for an extension of Cicchetti and Feinstein's work to polytomous
scales, and would appreciate any help on that as well.
Will summarize to the group.
Regards,
James Pratt
Sonus Pharmaceuticals
jimp@sonuspharma.com
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