Hi,
two more questions, sorry.
6.
could it be that where you wrote
normal.evd<-function(x, mu, v){
exp(-exp(x))*dnorm(x, mu, sqrt(v))
}
you're not actually passing the variance but the standard deviation? so the
functions should be
normal.evd<-function(x, mu, sd){
exp(-exp(x))*dnorm(x, mu, sd)
}
normal.zt<-function(x, mu, sd){
exp(x)/(1-exp(-exp(x)))*dnorm(x, mu, sd)
}
Because at least in the example, you were taking the square root here
already: sd_1<-mean(sqrt(m1$VCV[,1]))
Maybe this was a copy-paste from the source where you do it differently? Or
I misunderstood...
An addition to question 5:
5. When I use the correct inverse link function in my old code, but don't
do double integration, I do get a "reasonable" amount of uncertainty around
the reference factor. The predicted means are too low, which I put down to
this approach ignoring the VCV. I was just wondering how this could happen
(actually seeing the "reasonable" amount of uncertainty was what made me
hopeful that my approach might not be entirely wrong).
Best,
Ruben
On Tue, Mar 17, 2015 at 7:53 PM Ruben Arslan <rubenarslan at gmail.com> wrote:
Hi Jarrod,
thanks for the extensive reply! This helps a lot, though it sounds like I
was hubristic to attempt this myself.
I tried using the approach you mapped out in the function gist
<https://gist.github.com/rubenarslan/aeacdd306b3d061819a6> I posted. I
simply put the pred function in a loop, so that I wouldn't make any
mistakes while vectorising and since I don't care about performance at this
point.
Of course, I have some follow up questions though.. I'm sorry if I'm being
a pain, I really appreciate the free advice, but understand of course if
other things take precedence.
1. You're not "down with" developing publicly are you? Because I sure
would like to test-drive the newdata prediction and simulation functions..
2. Could you make sure that I got this right: "When predictions are to be
taken after marginalising the random effects (including the `residual'
over-dispersion) it is not possible to obtain closed form expressions."
That is basically my scenario, right? In the example I included, I also
had a group-level random effect (family). Ore are you talking about the
"trait" as the random effect (as in your example) and my scenario is
different and I cannot apply the numerical double integration procedure you
posted?
To be clear about my prediction goal without using language that I might
be using incorrectly: I want to show what the average effect in the
response unit, number of children, is in my population(s). I have data on
whole populations and am using all of it (except individuals that don't
have completed fertility yet, because I have yet to find a way to model
both zero-inflation and right censoring).
3. "Numerical integration could be extended to double integration in
which case covariance between the Poisson part and the binary part could
be handled." That is what you posted an example of and it applies to my
scenario, because I specified a prior R=list(V=diag(2), nu=1.002, fix=2)
and rcov=~idh(trait):units, random=~idh(trait):idParents?
But this double integration approach is something you just wrote
off-the-cuff and I probably shouldn't use it in a publication? Or is this
in the forthcoming MCMCglmm release and I might actually be able to refer
to it once I get to submitting?
4. Could I change my model specification to forbid covariance between the
two parts and not shoot myself in the foot? Would this allow for a more
valid/tested approach to prediction?
5. When I use your method on my real data, I get less variation around the
prediction "for the reference level" than for all other factor levels.
My reference level actually has fewer cases than the others, so this
isn't "right" in a way.
Is this because I'm not doing newdata prediction? I get the "right"
looking uncertainty if I bootstrap newdata predictions in lme4,
Sorry if this is children's logic :-)
Here's an image of the prediction
<http://rpubs.com/rubenarslan/mcmcglmm_pred> and the raw data
<http://rpubs.com/rubenarslan/raw>.
Many thanks for any answers that you feel inclined to give.
Best wishes,
Ruben
On Tue, Mar 17, 2015 at 5:31 PM Jarrod Hadfield <j.hadfield at ed.ac.uk>
wrote:
Hi,
Sorry - I should have replied to this post earlier. I've been working
on predict/simulate methods for all MCMcglmm models (including
zero-inflated/altered/hurdle/truncated models) and these are now
largely complete (together with newdata options).
When predictions are to be taken after marginalising the random
effects (including the `residual' over-dispersion) it is not possible
to obtain closed form expressions. The options that will be available
in MCMCglmm are:
1) algebraic approximation
2) numerical integration
3) simulation
1) and 2) are currently only accurate when the random/residual effect
structure implies no covariance between the Poisson part and the
binary part.
1) is reasonably accurate for zero-inflated distributions, but can be
pretty poor for the remainder because they all involve zero-truncated
Poisson log-normal distributions and my taylor approximation for the
mean is less than ideal (any suggestions would be helpful).
2) could be extended to double integration in which case covariance
between the Poisson part and the binary part could be handled.
In your code, part of the problem is that you have fitted a zapoisson,
but the prediction is based on a zipoisson (with complementary log-log
link rather than logt link).
In all zero-inflated/altered/hurdle/truncated models
E[y] = E[(1-prob)*meanc]
where prob is the probabilty of a zero in the binary part and meanc is
the mean of a Poisson distribution (zipoisson) or a zero-truncated
poisson (zapoisson and hupoisson). If we have eta_1 as the linear
predictor for the poisson part and eta_2 as the linear predictor for
the binary part:
In zipoisson: prob = plogis(eta_2) and meanc = exp(eta_1)
In zapoisson: prob = exp(-exp(eta_2)) and meanc =
exp(eta_1)/(1-exp(-exp(eta_1)))
In hupoisson: prob = plogis(eta_2) and meanc =
exp(eta_1)/(1-exp(-exp(eta_1)))
In ztpoisson: prob = 0 and meanc =
exp(eta_1)/(1-exp(-exp(eta_1)))
In each case the linear predictor has a `fixed' part and a `random'
part which I'll denote as `a' and `u' respectively. Ideally we would
want
E[(1-prob)*meanc] taken over u_1 & u_2
if prob and meanc are independent this is a bit easier as
E[y] = E[1-prob]E[meanc]
and the two expectations ony need to taken with repsect to their
respective random effects. If we have sd_1 and sd_2 as the standard
deviations of the two sets of random effects then for the zapoisson:
normal.evd<-function(x, mu, v){
exp(-exp(x))*dnorm(x, mu, sqrt(v))
}
normal.zt<-function(x, mu, v){
exp(x)/(1-exp(-exp(x)))*dnorm(x, mu, sqrt(v))
}
pred<-function(a_1, a_2, sd_1, sd_2){
prob<-1-integrate(normal.evd, qnorm(0.0001, a_2,sd_2),
qnorm(0.9999, a_2,sd_2), a_2,sd_2)[[1]]
meanc<-integrate(normal.zt, qnorm(0.0001, a_1,sd_1),
qnorm(0.9999, a_1,sd_1), a_1,sd_1)[[1]]
prob*meanc
}
# gives the expected value with reasonable accuracy. As an example:
x<-rnorm(300)
l1<-rnorm(300, 1/2+x, sqrt(1))
l2<-rnorm(300, 1-x, sqrt(1))
y<-rbinom(300, 1, 1-exp(-exp(l2)))
y[which(y==1)]<-qpois(runif(sum(y==1), dpois(0,
exp(l1[which(y==1)])), 1), exp(l1[which(y==1)]))
# cunning sampler from Peter Dalgaard (R-sig-mixed)
data=data.frame(y=y, x=x)
prior=list(R=list(V=diag(2), nu=0.002, fix=2))
m1<-MCMCglmm(y~trait+trait:x-1, rcov=~idh(trait):units, data=data,
family="zapoisson", prior=prior)
b_1<-colMeans(m1$Sol)[c(1,3)]
b_2<-colMeans(m1$Sol)[c(2,4)]
sd_1<-mean(sqrt(m1$VCV[,1]))
sd_2<-mean(sqrt(m1$VCV[,2]))
# note it is more accurate to take the posterior mean prediction
rather than the prediction from the posterior means as I've done here,
but for illustration:
x.pred<-seq(-3,3,length=100)
p<-1:100
for(i in 1:100){
p[i]<-pred(a_1 = b_1[1]+x.pred[i]*b_1[2], a_2 =
b_2[1]+x.pred[i]*b_2[2], sd_1=sd_1, sd_2=sd_2)
}
plot(y~x)
lines(p~x.pred)
Cheers,
Jarrod
Quoting Ruben Arslan <rubenarslan at gmail.com> on Tue, 17 Mar 2015
13:33:25 +0000:
Dear list,
I've made a reproducible example of the zero-altered prediction,
in the hope that someone will have a look and reassure me that I'm going
about this the right way.
I was a bit confused by the point about p_i and n_i being correlated
and if you don't want to run the simulation fit yourself, I've put an
Dear Dr Bolker,
I'd thought about something like this, one point of asking was to
see whether
a) it's implemented already, because I'll probably make dumb mistakes
trying b) it's not implemented because it's a bad idea.
Your response and the MCMCglmm course notes make me hope that it's c)
implemented because nobody did yet or d) it's so simple that everybody
it on-the-fly.
So I tried my hand and would appreciate corrections. I am sure there is
some screw-up or an inelegant approach in there.
I included code for dealing with mcmc.lists because that's what I have
I'm not entirely sure how I deal with them is correct either.
I started with a zero-altered model, because those fit fastest and
according to the course notes have the least complex likelihood.
Because I know not what I do, I'm not dealing with my random effects at
all.
I pasted a model summary below to show what I've applied the below
function to. The function gives the following quantiles when applied
chains of that model.
5% 50% 95%
5.431684178 5.561211207 5.690655200
5% 50% 95%
5.003974382 5.178192327 5.348246558
Warm regards,
Ruben Arslan
HPDpredict_za = function(object, predictor) {
if(class(object) != "MCMCglmm") {
if(length( object[[1]]$Residual$nrt )>1) {
object = lapply(object,FUN=function(x) { x$Residual$nrt<-2;x })
}
Sol = mcmc.list(lapply(object,FUN=function(x) { x$Sol}))
vars = colnames(Sol[[1]])
} else {
Sol = as.data.frame(object$Sol)
vars = names(Sol)
}
za_predictor = vars[ vars %ends_with% predictor & vars %begins_with%
"traitza_"]
za_intercept_name = vars[ ! vars %contains% ":" & vars %begins_with%
"traitza_"]
intercept = Sol[,"(Intercept)"]
za_intercept = Sol[, za_intercept_name]
l1 = Sol[, predictor ]
l2 = Sol[, za_predictor ]
if(is.list(object)) {
intercept = unlist(intercept)
za_intercept = unlist(za_intercept)
l1 = unlist(l1)
l2 = unlist(l2)
}
py_0 = dpois(0, exp(intercept + za_intercept))
y_ygt0 = exp(intercept)
at_intercept = (1-py_0) * y_ygt0
py_0 = dpois(0, exp(intercept + za_intercept + l2))
y_ygt0 = exp(intercept + l1)
at_predictor_1 = (1-py_0) * y_ygt0
print(qplot(at_intercept))
print(qplot(at_predictor_1))
df = data.frame("intercept" = at_intercept)
df[, predictor] = at_predictor_1
print(qplot(x=variable,
y=value,data=suppressMessages(melt(df)),fill=variable,alpha=I(0.40),
'violin'))
print(quantile(at_intercept, probs = c(0.05,0.5,0.95)))
print(quantile(at_predictor_1, probs = c(0.05,0.5,0.95)))
invisible(df)
}
Iterations = 100001:299901
Thinning interval = 100
Sample size = 2000
DIC: 349094
G-structure: ~idh(trait):idParents
post.mean l-95% CI u-95% CI eff.samp
children.idParents 0.0189 0.0164 0.0214 1729
za_children.idParents 0.2392 0.2171 0.2622 1647
R-structure: ~idh(trait):units
post.mean l-95% CI u-95% CI eff.samp
children.units 0.144 0.139 0.148 1715
za_children.units 1.000 1.000 1.000 0
Location effects: children ~ trait * (maternalage.factor +
maternalloss + center(nr.siblings) + birth.cohort + urban + male +
paternalage.mean + paternalage.diff)
post.mean l-95% CI u-95%
eff.samp pMCMC
(Intercept) 2.088717 2.073009
2000 <0.0005 ***
traitza_children -1.933491 -1.981945
2000 <0.0005 ***
maternalage.factor(14,20] 0.007709 -0.014238
1500 0.460
maternalage.factor(35,50] 0.006350 -0.009634
2000 0.462
paternallossTRUE 0.000797 -0.022716
2000 0.925
maternallossTRUE -0.015542 -0.040240
2000 0.226
center(nr.siblings) 0.005869 0.004302
2000 <0.0005 ***
birth.cohort(1703,1722] -0.045487 -0.062240
2000 <0.0005 ***
birth.cohort(1722,1734] -0.055872 -0.072856
2000 <0.0005 ***
birth.cohort(1734,1743] -0.039770 -0.056580
2000 <0.0005 ***
birth.cohort(1743,1750] -0.030713 -0.048301
2000 0.002 **
urban -0.076748 -0.093240
2567 <0.0005 ***
male 0.106074 0.095705
2000 <0.0005 ***
paternalage.mean -0.024119 -0.033133
2000 <0.0005 ***
paternalage.diff -0.018367 -0.032083
2000 0.007 **
traitza_children:maternalage.factor(14,20] -0.116510 -0.182432
1876 0.001 ***
traitza_children:maternalage.factor(35,50] -0.045196 -0.094485
2000 0.075 .
traitza_children:paternallossTRUE -0.171957 -0.238218
2000 <0.0005 ***
traitza_children:maternallossTRUE -0.499539 -0.566825
2000 <0.0005 ***
traitza_children:center(nr.siblings) -0.023723 -0.028676
1848 <0.0005 ***
traitza_children:birth.cohort(1703,1722] -0.026012 -0.074250
2000 0.319
traitza_children:birth.cohort(1722,1734] -0.279418 -0.329462
2000 <0.0005 ***
traitza_children:birth.cohort(1734,1743] -0.260165 -0.312659
2130 <0.0005 ***
traitza_children:birth.cohort(1743,1750] -0.481457 -0.534568
2000 <0.0005 ***
traitza_children:urban -0.604108 -0.645169
1702 <0.0005 ***
traitza_children:male -0.414988 -0.444589
2000 <0.0005 ***
traitza_children:paternalage.mean 0.006545 -0.018570
2000 0.651
traitza_children:paternalage.diff -0.097982 -0.136302
2000 <0.0005 ***
On Mon, Mar 9, 2015 at 10:12 PM Ben Bolker <bbolker at gmail.com> wrote:
Ruben Arslan <rubenarslan at ...> writes:
Dear list,
I wanted to ask: Is there any (maybe just back of the envelope) way
obtain a response prediction for zero-inflated or hurdle type
I've fit such models in MCMCglmm, but I don't work in ecology and my
previous experience with explaining such models to "my audience"
bode well. When it comes to humans, the researchers I presented to
used to offspring count being zero-inflated (or acquainted with that
concept), but in my historical data with high infant mortality, it
modern data it's actually slightly underdispersed).
Currently I'm using lme4 and simply splitting my models into two
(finding a mate and having offspring).
That's okay too, but in one population the effect of interest is not
clearly visible in either stage, only when both are taken together
then the outcome is zero-inflated).
I expect to be given a hard time for this and hence thought I'd use
binomial model with the outcome offspring>0 as my main model, but
turns out to be hard to explain too and doesn't
really do the data justice.
Basically I don't want to be forced to discuss my smallest
non-replication of the effect because I was insufficiently able to
the statistics behind my reasoning that the effect shows.
I think the back-of-the envelope answer would be that for a
model with a prediction of p_i for the probability of having a
response (or in the case of zero-inflated models, the probability of
_not_ having a structural zero) and a prediction of n_i for the
conditional
part of the model, the mean predicted value is p_i*n_i and the
variance is _approximately_ (p_i*n_i)^2*(var(p_i)/p_i^2 +
(this is assuming
that you haven't built in any correlation between p_i and n_i, which
would be hard in lme4 but _might_ be possible under certain
via a multitype model in MCMCglmm).
Does that help?