zipebcom {VGAM} | R Documentation |
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="cloglog", lphi12="logit", loratio="loge", emu12=list(), ephi12=list(), eoratio=list(), imu12=NULL, iphi12=NULL, ioratio = NULL, zero=2:3, tol=0.001, addRidge=0.001)
lmu12, emu12, imu12 |
Link function, extra argument and optional initial values for
the first (and second) marginal probabilities.
Arguments lmu12 and emu12 should be left alone.
Argument imu12 may be of length 2 (one element for each response).
|
lphi12 |
Link function applied to the phi parameter of the
zero-inflated Poisson distribution (see zipoisson ).
See Links for more choices.
|
loratio |
Link function applied to the odds ratio.
See Links for more choices.
|
iphi12, ioratio |
Optional initial values for phi and the odds ratio.
See CommonVGAMffArguments for more details.
In general, good initial values (especially for iphi12 )
are often required, therefore use these
arguments if convergence failure occurs.
If inputted, the value of iphi12 cannot be more than the sample
proportions of zeros in either response.
|
ephi12, eoratio |
List. Extra argument for each of the links.
See earg in Links for general information.
|
zero |
Which linear/additive predictor is modelled as an intercept only?
A NULL means none.
The default has both phi and the odds ratio as
not being modelled as a function of the explanatory variables (apart
from an intercept).
|
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 1+addRidge to make it diagonally dominant,
therefore positive-definite.
|
This VGAM family function fits an exchangeable bivariate odds
ratio model (binom2.or
) with a cloglog
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.
The second linear/additive predictor models the phi
parameter (see zipoisson
).
The third linear/additive predictor is the same as binom2.or
,
viz., the log odds ratio.
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("cloglog", 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("cloglog", 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("cloglog", exch=TRUE)
.
Full details are given in Yee and Dirnbock (2008).
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 "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.
Thomas W. Yee
Yee, T. W. and Dirnbock, T. (2008) A model for species presence/absence data at two time points based on an odds ratio and zero-inflated Poisson distribution. In preparation.
binom2.or
,
zipoisson
,
cloglog
,
CommonVGAMffArguments
.
mydat = data.frame(x = seq(0, 1, len=(nsites <- 2000))) mydat = transform(mydat, eta1 = -3 + 5 * x, phi1 = logit(-1, inverse=TRUE), oratio = exp(2)) mydat = transform(mydat, mu12 = cloglog(eta1, inverse=TRUE) * (1-phi1)) tmat = with(mydat, rbinom2.or(nsites, mu1=mu12, oratio=oratio, exch=TRUE)) mydat = transform(mydat, ybin1 = tmat[,1], ybin2 = tmat[,2]) with(mydat, table(ybin1,ybin2)) / nsites # For interest only ## Not run: # Various plots of the data, for interest only par(mfrow=c(2,2)) with(mydat, plot(jitter(ybin1) ~ x, col="blue")) with(mydat, plot(jitter(ybin2) ~ jitter(ybin1), col="blue")) with(mydat, plot(x, mu12, col="blue", type="l", ylim=0:1, ylab="Probability", main="Marginal probability and phi")) with(mydat, abline(h=phi1[1], col="red", lty="dashed")) tmat2 = with(mydat, dbinom2.or(mu1=mu12, oratio=oratio, exch=TRUE)) with(mydat, matplot(x, 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) ~ x, fam=zipebcom, dat=mydat, trace=TRUE) coef(fit, matrix=TRUE) summary(fit) vcov(fit)