Fitting a Dirichlet-Multinomial Distribution
Fits a Dirichlet-multinomial distribution to a matrix of non-negative integers.
dirmul.old(link = "loglink", ialpha = 0.01, parallel = FALSE, zero = NULL)
link |
Link function applied to each of the M (positive)
shape parameters alpha_j for j=1,…,M.
See |
ialpha |
Numeric vector. Initial values for the
|
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}. |
The Dirichlet-multinomial distribution, which is somewhat similar to a Dirichlet distribution, has probability function
P(Y_1=y_1,…,Y_M=y_M) = C_{y_1,…,y_M}^{2y_{*}} Gamma(alpha_+) / Gamma( 2y_* + alpha_+) prod_{j=1}^M [ 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, C_b^a 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,…,M, and these are returned as the fitted values as a M-column matrix.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm,
rrvglm
and vgam.
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This function is almost defunct and may be withdrawn soon.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.
Thomas W. Yee
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 θ-correction in forensic genetics. Theoretical Population Biology, 78, 200–210.
# 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 Table 3.5 of Lange (2002)
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' answer due to parallelism constraintPlease choose more modern alternatives, such as Google Chrome or Mozilla Firefox.