Exchangeable Bivariate cloglog Odds-ratio Model From a Zero-inflated Poisson Distribution
Fits an exchangeable bivariate odds-ratio model to two binary responses with a complementary log-log link. The data are assumed to come from a zero-inflated Poisson distribution that has been converted to presence/absence.
zipebcom(lmu12 = "clogloglink", lphi12 = "logitlink", loratio = "loglink",
imu12 = NULL, iphi12 = NULL, ioratio = NULL,
zero = c("phi12", "oratio"), tol = 0.001, addRidge = 0.001)lmu12, imu12 |
Link function, extra argument and optional initial values for
the first (and second) marginal probabilities.
Argument |
lphi12 |
Link function applied to the phi parameter of the
zero-inflated Poisson distribution (see |
loratio |
Link function applied to the odds ratio.
See |
iphi12, ioratio |
Optional initial values for phi and the odds ratio.
See |
zero |
Which linear/additive predictor is modelled as an intercept only?
A |
tol |
Tolerance for testing independence. Should be some small positive numerical value. |
addRidge |
Some small positive numerical value.
The first two diagonal elements of the working weight matrices are
multiplied by |
This VGAM family function fits an exchangeable bivariate odds
ratio model (binom2.or) with a clogloglink link.
The data are assumed to come from a zero-inflated Poisson (ZIP) distribution
that has been converted to presence/absence.
Explicitly, the default model is
cloglog[P(Y_j=1)/(1-phi)] = eta_1,\ \ \ j=1,2
for the (exchangeable) marginals, and
logit[phi] = eta_2,
for the mixing parameter, and
log[P(Y_{00}=1) P(Y_{11}=1) / (P(Y_{01}=1) P(Y_{10}=1))] = eta_3,
specifies the dependency between the two responses. Here, the responses equal 1 for a success and a 0 for a failure, and the odds ratio is often written psi=p00 p11 / (p10 p01). We have p10 = p01 because of the exchangeability.
Suppose a dataset1 comes from a Poisson distribution that has been
converted to presence/absence, and that both marginal probabilities
are the same (exchangeable).
Then binom2.or("clogloglink", exch=TRUE) is appropriate.
Now suppose a dataset2 comes from a zero-inflated Poisson
distribution. The first linear/additive predictor of zipebcom()
applied to dataset2
is the same as that of
binom2.or("clogloglink", exch=TRUE)
applied to dataset1.
That is, the phi has been taken care
of by zipebcom() so that it is just like the simpler
binom2.or.
Note that, for eta_1,
mu12 = prob12 / (1-phi12) where prob12 is the probability
of a 1 under the ZIP model.
Here, mu12 correspond to mu1 and mu2 in the
binom2.or-Poisson model.
If phi=0 then zipebcom() should be equivalent to
binom2.or("clogloglink", exch=TRUE).
Full details are given in Yee and Dirnbock (2009).
The leading 2 x 2 submatrix of the expected
information matrix (EIM) is of rank-1, not 2! This is due to the
fact that the parameters corresponding to the first two
linear/additive predictors are unidentifiable. The quick fix
around this problem is to use the addRidge adjustment.
The model is fitted by maximum likelihood estimation since the full
likelihood is specified. Fisher scoring is implemented.
The default models eta2 and eta3 as
single parameters only, but this
can be circumvented by setting zero=NULL in order to model the
phi and odds ratio as a function of all the explanatory
variables.
An object of class "vglmff" (see vglmff-class).
The object is used by modelling functions such as vglm
and vgam.
When fitted, the fitted.values slot of the object contains the
four joint probabilities, labelled as (Y1,Y2) = (0,0),
(0,1), (1,0), (1,1), respectively.
These estimated probabilities should be extracted with the fitted
generic function.
The fact that the EIM is not of full rank may mean the model is
naturally ill-conditioned.
Not sure whether there are any negative consequences wrt theory.
For now
it is certainly safer to fit binom2.or to bivariate binary
responses.
The "12" in the argument names reinforce the user about the
exchangeability assumption.
The name of this VGAM family function stands for
zero-inflated Poisson exchangeable bivariate complementary
log-log odds-ratio model or ZIP-EBCOM.
See binom2.or for details that are pertinent to this
VGAM family function too.
Even better initial values are usually needed here.
The xij (see vglm.control) argument enables
environmental variables with different values at the two time points
to be entered into an exchangeable binom2.or model.
See the author's webpage for sample code.
Yee, T. W. and Dirnbock, T. (2009). Models for analysing species' presence/absence data at two time points. Journal of Theoretical Biology, 259(4), 684–694.
zdata <- data.frame(x2 = seq(0, 1, len = (nsites <- 2000)))
zdata <- transform(zdata, eta1 = -3 + 5 * x2,
phi1 = logitlink(-1, inverse = TRUE),
oratio = exp(2))
zdata <- transform(zdata, mu12 = clogloglink(eta1, inverse = TRUE) * (1-phi1))
tmat <- with(zdata, rbinom2.or(nsites, mu1 = mu12, oratio = oratio, exch = TRUE))
zdata <- transform(zdata, ybin1 = tmat[, 1], ybin2 = tmat[, 2])
with(zdata, table(ybin1, ybin2)) / nsites # For interest only
## Not run:
# Various plots of the data, for interest only
par(mfrow = c(2, 2))
plot(jitter(ybin1) ~ x2, data = zdata, col = "blue")
plot(jitter(ybin2) ~ jitter(ybin1), data = zdata, col = "blue")
plot(mu12 ~ x2, data = zdata, col = "blue", type = "l", ylim = 0:1,
ylab = "Probability", main = "Marginal probability and phi")
with(zdata, abline(h = phi1[1], col = "red", lty = "dashed"))
tmat2 <- with(zdata, dbinom2.or(mu1 = mu12, oratio = oratio, exch = TRUE))
with(zdata, matplot(x2, tmat2, col = 1:4, type = "l", ylim = 0:1,
ylab = "Probability", main = "Joint probabilities"))
## End(Not run)
# Now fit the model to the data.
fit <- vglm(cbind(ybin1, ybin2) ~ x2, zipebcom, data = zdata, trace = TRUE)
coef(fit, matrix = TRUE)
summary(fit)
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