dirmul.old: Fitting a Dirichlet-Multinomial Distribution

In VGAM: Vector Generalized Linear and Additive Models

Fitting a Dirichlet-Multinomial Distribution

Description

Fits a Dirichlet-multinomial distribution to a matrix of non-negative integers.

Usage

dirmul.old(link = "loglink", ialpha = 0.01, parallel = FALSE,
           zero = NULL)

Arguments

link	Link function applied to each of the M (positive) shape parameters \alpha_j for j=1,\ldots,M. See Links for more choices. Here, M is the number of columns of the response matrix.
ialpha	Numeric vector. Initial values for the alpha vector. Must be positive. Recycled to length M.
parallel	A logical, or formula specifying which terms have equal/unequal coefficients.
zero	An integer-valued vector specifying which linear/additive predictors are modelled as intercepts only. The values must be from the set {1,2,...,M}. See CommonVGAMffArguments for more information.

Details

The Dirichlet-multinomial distribution, which is somewhat similar to a Dirichlet distribution, has probability function

P(Y_1=y_1,\ldots,Y_M=y_M) = {2y_{*} \choose {y_1,\ldots,y_M}} \frac{\Gamma(\alpha_{+})}{\Gamma(2y_{*}+\alpha_{+})} \prod_{j=1}^M \frac{\Gamma(y_j+\alpha_{j})}{\Gamma(\alpha_{j})}

for \alpha_j > 0, \alpha_+ = \alpha_1 + \cdots + \alpha_M, and 2y_{*} = y_1 + \cdots + y_M. Here, a \choose b means “a choose b” and refers to combinations (see choose). The (posterior) mean is

E(Y_j) = (y_j + \alpha_j) / (2y_{*} + \alpha_{+})

for j=1,\ldots,M, and these are returned as the fitted values as a M-column matrix.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, rrvglm and vgam.

Note

The response should be a matrix of non-negative values. Convergence seems to slow down if there are zero values. Currently, initial values can be improved upon.

This function is almost defunct and may be withdrawn soon. Use dirmultinomial instead.

Author(s)

Thomas W. Yee

References

Lange, K. (2002). Mathematical and Statistical Methods for Genetic Analysis, 2nd ed. New York: Springer-Verlag.

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

Paul, S. R., Balasooriya, U. and Banerjee, T. (2005). Fisher information matrix of the Dirichlet-multinomial distribution. Biometrical Journal, 47, 230–236.

Tvedebrink, T. (2010). Overdispersion in allelic counts and \theta-correction in forensic genetics. Theoretical Population Biology, 78, 200–210.

See Also

dirmultinomial, dirichlet, betabinomialff, multinomial.

Examples

# Data from p.50 of Lange (2002)
alleleCounts <- c(2, 84, 59, 41, 53, 131, 2, 0,
       0, 50, 137, 78, 54, 51, 0, 0,
       0, 80, 128, 26, 55, 95, 0, 0,
       0, 16, 40, 8, 68, 14, 7, 1)
dim(alleleCounts) <- c(8, 4)
alleleCounts <- data.frame(t(alleleCounts))
dimnames(alleleCounts) <- list(c("White","Black","Chicano","Asian"),
                    paste("Allele", 5:12, sep = ""))

set.seed(123)  # @initialize uses random numbers
fit <- vglm(cbind(Allele5,Allele6,Allele7,Allele8,Allele9,
                  Allele10,Allele11,Allele12) ~ 1, dirmul.old,
             trace = TRUE, crit = "c", data = alleleCounts)

(sfit <- summary(fit))
vcov(sfit)
round(eta2theta(coef(fit),
                fit@misc$link,
                fit@misc$earg), digits = 2)  # not preferred
round(Coef(fit), digits = 2)  # preferred
round(t(fitted(fit)), digits = 4)  # 2nd row of Lange (2002, Table 3.5)
coef(fit, matrix = TRUE)


pfit <- vglm(cbind(Allele5,Allele6,Allele7,Allele8,Allele9,
                   Allele10,Allele11,Allele12) ~ 1,
             dirmul.old(parallel = TRUE), trace = TRUE,
             data = alleleCounts)
round(eta2theta(coef(pfit, matrix = TRUE), pfit@misc$link,
                pfit@misc$earg), digits = 2)  # 'Right' answer
round(Coef(pfit), digits = 2)  # 'Wrong' due to parallelism constraint

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.