Estimates the parameters of the beta binomial distribution (D.M. Smith).
Options
PRINT = string tokens |
Controls printed output (estimates, loglikelihood); default esti |
|---|---|
MAXCYCLE = scalar |
Maximum number of iterations; default 50 |
TOLERANCE = scalar |
Convergence criterion; default 10-5 |
Parameters
RBINOMIAL = variates |
Numerator of binomial data |
|---|---|
NBINOMIAL = variates |
Denominator of binomial data or scalars |
MU = scalars |
Mean, expectation of underlying beta distribution |
THETA = scalars |
Shape-determining parameter of underlying beta distribution |
SEMU = scalars |
Standard error of mu |
SETHETA = scalars |
Standard error of theta |
LOGLIKELIHOOD = scalars |
Log likelihood |
NCYCLES = scalars |
Number of iterations |
EXIT = scalars |
Indicator of faults |
Description
BBINOMIAL estimates the two parameters of the beta binomial distribution by maximum likelihood, using the methods of Smith (1983) and Smith & Ridout (1995).
The parameters mu and theta are estimated instead of the standard alpha and beta, as they are more “stable” i.e. reliable to estimate. (See Williams 1975.) The relationship is that
mu = alpha / (alpha + beta)
theta = 1 / (alpha + beta)
or
alpha = mu / theta
beta = (1 – mu) / theta
(Note: in the descriptions of the probability functions, PRBETA etc, alpha and beta are represented as a and b.)
The numbers responding are specified in a variate using the RBINOMIAL parameter, and the corresponding total numbers are specified by the NBINOMIAL parameter in either a variate or a scalar.
Printed output is controlled by the PRINT option, with settings:
estimates |
to print the estimated values of mu and theta, together with the corresponding values, and |
|---|---|
loglikelihood |
to print the log-likelihood. |
The estimates of the two parameters of the distribution can be saved by the parameters MU and THETA, and their standard errors can be saved by parameters SEMU and SETHETA. The LOGLIKELIHOOD parameter can save the value of the log-likelihood.
The NCYCLES parameter can save the number of iterations that were needed. The MAXCYCLE option sets a limit on the total number of iterations (default 50), and the TOLERANCE option sets the convergence criterion (default 10-5). The EXIT parameter can save a scalar to indicate the success or failure of the estimation, as follows.
0 success.
1 a value of NBINOMIAL is less than or equal to 1.
2 all values of RBINOMIAL are zero.
3 all values of RBINOMIAL are equal to NBINOMIAL.
4 a value of RBINOMIAL is greater than NBINOMIAL.
5 some values of either RBINOMIAL or NBINOMIAL are less than zero.
6 if either MU went outside range 0 to 1 or THETA went outside range 0 to infinity, where infinity is the value (106) set inside BBINOMIAL to represent infinity.
7 if the maximum number of iterations (MAXCYCLE) was exceeded.
8 if the damped Newton-Raphson procedure failed.
9 if the minimum value for THETA has been reached and the maximum likelihood estimate of MU found, but moving THETA away from the minimum value slightly increases the log likelihood. The estimate of MU returned is the estimate on the minimum value of THETA. The estimates are not then the overall maximum likelihood estimates.
When EXIT = 1, 2, 3, 4 or 5 BBINOMIAL gives a fault, and MU, THETA, SEMU, SETHETA and LOGLIKELIHOOD are undefined. When EXIT = 6, 7, 8 or 9 BBINOMIAL gives a warning, and MU, THETA and LOGLIKELIHOOD are returned with their current values, while SEMU and SETHETA contain missing values. When EXIT = 6 the out-of-range parameter is set to the appropriate limiting value.
Options: PRINT, MAXCYCLE, TOLERANCE.
Parameters: RBINOMIAL, NBINOMIAL, MU, THETA, SEMU, SETHETA, LOGLIKELIHOOD, NCYCLES, EXIT.
Method
For full details of the methods implemented in these procedures see Smith (1983) and Smith & Ridout (1995). BBINOMIAL has four associated procedures _BBSET, _BBME, _BBL and _BBGDER that are Genstat implementations of various Fortran subroutines of Smith (1983) and Smith & Ridout (1995). They can also be run independently if desired. _BBSET calculates the integer arrays of counts required by _BBL and _BBGDER. _BBME calculates moment estimates of mu and theta. _BBL calculates the log likelihood given mu and theta.
Action with RESTRICT
If either RBINOMIAL or NBINOMIAL are restricted, the analysis will exclude the restricted units.
References
Kupper, L.L. & Haseman, J.K. (1978). The use of a correlated binomial model for the analysis of toxicological experiments. Biometrics, 34, 69-76.
Smith, D.M. (1983). AS 189. Maximum Likelihood Estimation of the Parameters of the Beta Binomial Distribution. Applied Statistics, 32, 196-204.
Smith, D.M. & Ridout, M.S. (1995). AS R93. A remark on AS 189. Maximum Likelihood Estimation of the Parameters of the Beta Binomial Distribution. Applied Statistics, 44, 545-547.
Williams, D.A. (1975). The analysis of binary responses from toxicological experiments involving reproduction and teratogenicity. Biometrics, 31, 949-952.
See also
Directive: DISTRIBUTION.
Procedure: DPROBABILITY.
Commands for: Basic and nonparametric statistics.
Example
CAPTION 'BBINOMIAL example',\
!t('Data from Smith AS189 (1983) Applied Statistics Algorithm:',\
'1) control data from Kupper & Haseman (1978), Biometrics, 34,',\
'69-76.'); STYLE=meta,plain
VARIATE rc,nc,rt,nt; values=\
!(0,2,5(0),1,2,1),!(5,6,2(7),3(8),2(9),10),\
!(0,2,1,0,2,3,0,4,1,6),!(2(5),7,3(8),2(9,10))
BBINOMIAL RBINOMIAL=rc; NBINOMIAL=nc; MU=mu; THETA=theta; SEMU=sem;\
SETHETA=seth; LOGLIKELIHOOD=logl; NCYCLES=iter; EXIT=ifault
CAPTION '2) treatment data from Kupper & Haseman.'
BBINOMIAL RBINOMIAL=rt; NBINOMIAL=nt; MU=mu; THETA=theta; SEMU=sem;\
SETHETA=seth; LOGLIKELIHOOD=logl; NCYCLES=iter; EXIT=ifault