Predictions from zero-inflated or hurdle models
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
(they
are in my case), but I think this was a red herring for me since I'm not deriving the variance analytically. The script is here: https://gist.github.com/rubenarslan/
aeacdd306b3d061819a6
and if you don't want to run the simulation fit yourself, I've put an
RDS
file of the fit here: https://dl.dropboxusercontent.com/u/1078620/m1.rds Best regards, Ruben Arslan On Tue, Mar 10, 2015 at 1:22 PM Ruben Arslan <rubenarslan at gmail.com>
wrote:
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
while
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)
not
implemented because nobody did yet or d) it's so simple that everybody
does
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
and
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
to 19
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),
geom =
'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) }
summary(object[[1]])
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 +
paternalloss +
maternalloss + center(nr.siblings) + birth.cohort + urban + male +
paternalage.mean + paternalage.diff)
post.mean l-95% CI u-95%
CI
eff.samp pMCMC (Intercept) 2.088717 2.073009
2.103357
2000 <0.0005 *** traitza_children -1.933491 -1.981945
-1.887863
2000 <0.0005 *** maternalage.factor(14,20] 0.007709 -0.014238
0.027883
1500 0.460 maternalage.factor(35,50] 0.006350 -0.009634
0.024107
2000 0.462 paternallossTRUE 0.000797 -0.022716
0.025015
2000 0.925 maternallossTRUE -0.015542 -0.040240
0.009549
2000 0.226 center(nr.siblings) 0.005869 0.004302
0.007510
2000 <0.0005 *** birth.cohort(1703,1722] -0.045487 -0.062240
-0.028965
2000 <0.0005 *** birth.cohort(1722,1734] -0.055872 -0.072856
-0.036452
2000 <0.0005 *** birth.cohort(1734,1743] -0.039770 -0.056580
-0.020907
2000 <0.0005 *** birth.cohort(1743,1750] -0.030713 -0.048301
-0.012214
2000 0.002 ** urban -0.076748 -0.093240
-0.063002
2567 <0.0005 *** male 0.106074 0.095705
0.115742
2000 <0.0005 *** paternalage.mean -0.024119 -0.033133
-0.014444
2000 <0.0005 *** paternalage.diff -0.018367 -0.032083
-0.005721
2000 0.007 ** traitza_children:maternalage.factor(14,20] -0.116510 -0.182432
-0.051978
1876 0.001 *** traitza_children:maternalage.factor(35,50] -0.045196 -0.094485
0.002640
2000 0.075 . traitza_children:paternallossTRUE -0.171957 -0.238218
-0.104820
2000 <0.0005 *** traitza_children:maternallossTRUE -0.499539 -0.566825
-0.430637
2000 <0.0005 *** traitza_children:center(nr.siblings) -0.023723 -0.028676
-0.018746
1848 <0.0005 *** traitza_children:birth.cohort(1703,1722] -0.026012 -0.074250
0.026024
2000 0.319 traitza_children:birth.cohort(1722,1734] -0.279418 -0.329462
-0.227187
2000 <0.0005 *** traitza_children:birth.cohort(1734,1743] -0.260165 -0.312659
-0.204462
2130 <0.0005 *** traitza_children:birth.cohort(1743,1750] -0.481457 -0.534568
-0.426648
2000 <0.0005 *** traitza_children:urban -0.604108 -0.645169
-0.562554
1702 <0.0005 *** traitza_children:male -0.414988 -0.444589
-0.387005
2000 <0.0005 *** traitza_children:paternalage.mean 0.006545 -0.018570
0.036227
2000 0.651 traitza_children:paternalage.diff -0.097982 -0.136302
-0.060677
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
to
obtain a response prediction for zero-inflated or hurdle type
models?
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"
did not
bode well. When it comes to humans, the researchers I presented to
are
not
used to offspring count being zero-inflated (or acquainted with that concept), but in my historical data with high infant mortality, it
is
(in
modern data it's actually slightly underdispersed). Currently I'm using lme4 and simply splitting my models into two
stages
(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
(but
then the outcome is zero-inflated). I expect to be given a hard time for this and hence thought I'd use
a
binomial model with the outcome offspring>0 as my main model, but
that
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
population
as a
non-replication of the effect because I was insufficiently able to
explain
the statistics behind my reasoning that the effect shows.
I think the back-of-the envelope answer would be that for a
two-stage
model with a prediction of p_i for the probability of having a
non-zero
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 +
var(n_i)/n_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
circumstances
via a multitype model in MCMCglmm). Does that help?
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