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BBINOMIAL procedure

Estimates the parameters of the beta binomial distribution (D.M. Smith).


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


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


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)
      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.



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.


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

Commands for: Basic and nonparametric statistics.


          !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=\
          SETHETA=seth; LOGLIKELIHOOD=logl; NCYCLES=iter; EXIT=ifault
CAPTION   '2) treatment data from Kupper & Haseman.'
          SETHETA=seth; LOGLIKELIHOOD=logl; NCYCLES=iter; EXIT=ifault
Updated on September 5, 2019

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