seq2binomial: The Two-stage Sequential Binomial Distribution Family Function

Description

Estimation of the probabilities of a two-stage binomial distribution.

Usage

seq2binomial(lprob1 = "logit", lprob2 = "logit",
             iprob1 = NULL,    iprob2 = NULL,
             parallel = FALSE, zero = NULL)

Arguments

lprob1, lprob2

Parameter link functions applied to the two probabilities, called $p$ and $q$ below. See Links for more choices.

iprob1, iprob2

Optional initial value for the first and second probabilities respectively. A NULL means a value is obtained in the initialize slot.

parallel, zero

Details at Links. If parallel = TRUE then the constraint also applies to the intercept.

Value

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

Details

This VGAM family function fits the model described by Crowder and Sweeting (1989) which is described as follows. Each of $m$ spores has a probability $p$ of germinating. Of the $y_1$ spores that germinate, each has a probability $q$ of bending in a particular direction. Let $y_2$ be the number that bend in the specified direction. The probability model for this data is $P(y_1,y_2) =$

(\binom{m}{y_{1}}) p^{y_{1}} (1 - p)^{m - y_{1}} (\binom{y_{1}}{y_{2}}) q^{y_{2}} (1 - q)^{y_{1} - y_{2}}

for $0 < p < 1$, $0 < q < 1$, $y_1=1,\ldots,m$ and $y_2=1,\ldots,y_1$. Here, $p$ is prob1, $q$ is prob2.

Although the Authors refer to this as the bivariate binomial model, I have named it the (two-stage) sequential binomial model. Fisher scoring is used.

References

Crowder, M. and Sweeting, T. (1989). Bayesian inference for a bivariate binomial distribution. Biometrika, 76, 599--603.

Examples

Run this code

sdata <- data.frame(mvector = round(rnorm(nn <- 100, m = 10, sd = 2)),
                    x2 = runif(nn))
sdata <- transform(sdata, prob1 = logit(+2 - x2, inverse = TRUE),
                          prob2 = logit(-2 + x2, inverse = TRUE))
sdata <- transform(sdata, successes1 = rbinom(nn, size = mvector,    prob = prob1))
sdata <- transform(sdata, successes2 = rbinom(nn, size = successes1, prob = prob2))
sdata <- transform(sdata, y1 = successes1 / mvector)
sdata <- transform(sdata, y2 = successes2 / successes1)
fit <- vglm(cbind(y1, y2) ~ x2, seq2binomial, weight = mvector,
            data = sdata, trace = TRUE)
coef(fit)
coef(fit, matrix = TRUE)
head(fitted(fit))
head(depvar(fit))
head(weights(fit, type = "prior"))  # Same as with(sdata, mvector)
# Number of first successes:
head(depvar(fit)[, 1] * c(weights(fit, type = "prior")))
# Number of second successes:
head(depvar(fit)[, 2] * c(weights(fit, type = "prior")) * 
                          depvar(fit)[, 1])

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