# [ADMB Users] Random effect as finite mixture

dave fournier otter at otter-rsch.com
Sun Feb 22 16:32:26 PST 2009

```Shige,

I think the "ADMB team" consists mostly of me. So I'll try and answer
very little idea what growth models were, but in writing the simulator I
hope I am getting the idea.

Actually finite mixtures are handled directly so it is not necessary to
use the random effects package. This is actually an advantage since it
greatly speeds up the calculations.
To illustrate the ideas I built a simulator to create a data set
something like you describe in

This could be done in R but I used ADMB since it illustrates some of the
techniques which are useful for building the analyzer. since you didn't
identify the growth curves you used I use the von Bertalanfy curve

L_t = Linf * ( 1 - exp(-k*age))

where L_0=0 is assumed. Of course it is a simple matter to switch to a
different curve There are two curves one with Linf=100 and the other
with Linf=120. For both cases k=0.11 (time measured in years).

I generated 2000 samples. First the size measured in 2 month intervals
and then at ages 8,11,15,19

The individuals were generated with a probability of 0.4 of belonging to
group1 (linf=100) and 0.6 of belonging to group 2. After the first
period the individual were changed at random to groups 1 or 2 with a
transition matrix

.8  prob stay in group 1
.2  prob to move from group 2 to group 1
.7  prob to say in group 2
.3  prob to move from group 2 to group 1

The the size data was generator using the appropriate growth curve.
the resulting sizes were then perturbed using  multiplicative lognormal
random variables (std dev of .08) which were highly positively
correlated (0.9)

The resulting output was analyzed with a finite mixture model where
the 12 initial measuremeasurements are assumed to come from a mixture of
two multivariate log-normal distributions with a correlation matrix of
the form

1     rho rho^2  .... rho^11
rho    1
rho^2   .......

etc.

The model was fit and the Hessian evaluated in under a minute.

these are the parameter estimates for a typical run.

# Number of parameters = 10  Objective function value = -84436.9
# p1coff:
0.379774 0.620226
# p2coff:
0.482844 0.517158
# Linf:
99.2992 119.646
# k:
0.109928 0.110294
# rho:
0.882750993879
# lsigma:
-2.54165198426

For each individual the conditional probabilities of belonging in either
of the two groups was calculated after observing the first sizes and
again after observing the second group of sizes. this was used to
estimate the transition probabilities. (Of course this could be done in
the model which would be more efficient.)
~
~ 0.793901 0.206099
0.292103 0.707897
~
which is almost perfect.

the C++ code for the simulator is

~
~

dvector growth_function(const dvector & age,const dvector& theta)
{
dvector tmp=theta(1)*(1.0-exp(-theta(2)*age)); // use you own
return tmp;                           // growth function here
}

int main(int argc,char * argv[])
{
int n=2000;
int nm=16;
int iseed=289;
random_number_generator rng(iseed);
dvector p1(1,2);  // priori probability that a child is in group 1 or 2
p1(1)=.4; p1(2)=.6;
dvector p2(1,2);
dmatrix M(1,2,1,2);   // transition probabilities
M(1,1)=.8;  // 1 remains 1
M(2,1)=.3;  // 2 moves to 1
M(1,2)=.2;  // 1 moves to 2
M(2,2)=.7;  // 2 remains 2
dvector choose(1,n);
dvector choose1(1,n);
double sigma=0.08;
double rho=0.9;
dmatrix size(1,n,1,nm);
imatrix group(1,n,1,2);
dvector age(1,nm);
dvector eta(1,nm);
dvector eps(1,nm);
age(13)=8;
age(14)=11;
age(15)=15;
age(16)=19;
dmatrix theta(1,2,1,2);  //parameters for growth curves
theta(1,1)=100;
theta(2,1)=120;
theta(1,2)=.11;
theta(2,2)=.11;

choose.fill_randu(rng);  // use to pick group that each child belongs to
choose1.fill_randu(rng);  // use to pick group that each child belongs to
int i;
for (i=1;i<=n;i++)   // loop over children
{
if (choose(i)<p1(1))
{
group(i,1)=1;  //child in group1
}
else
{
group(i,1)=2; // child in group2
}
if (group(i,1)==1)  // use probability that group 1 remains in group1
if (choose1(i)<M(1,1))
group(i,2)=1;    // remained in group1
else
group(i,2)=2; // moved to group2
else
if (choose1(i)<M(2,1)) // use probability that group 2 changes to
group1
group(i,2)=1;    // changed to group1
else
group(i,2)=2; // remained in group2
}

for (i=1;i<=n;i++)   // loop over children and get size data
{
size(i)(1,12)=growth_function(age(1,12),theta(group(i,1)));
size(i)(13,16)=growth_function(age(13,16),theta(group(i,2)));
}
double rho2=rho*rho;
for (i=1;i<=n;i++)   // loop over children and  add noise
{
eta.fill_randn(rng);
eps(1)=eta(1);
int j;
for (j=2;j<=4;j++)   // loop over children and  add noise
{
eps(j)=rho*eps(j-1)+sqrt(1.0-rho2)*eta(j);
}
eps(5)=eta(1);
for (j=6;j<=16;j++)   // loop over children and  add noise
{
eps(j)=rho*eps(j-1)+sqrt(1.0-rho2)*eta(j);
}
size(i)=elem_prod(size(i),exp(sigma*eps));  // lognormal error
}

ofstream ofs("growth.dat");
ofs << "# nobs" << endl << n << endl ;
ofs << "# nmeasure" << endl << nm << endl ;
ofs << "# ages" << endl << setprecision(3) << setw(7) << age << endl ;
ofs << "# observations" << endl << setprecision(3) << setw(7) << size
<< endl ;
ivector icount1(1,2);
icount1.initialize();
ivector icount2(1,2);
icount2.initialize();
for (i=1;i<=n;i++)   // loop over children
{
icount1(group(i,1))++;
icount2(group(i,2))++;
}
cout << "proportions in groups " << icount1/sum(icount1) << endl;
cout << "proportions in groups " << icount2/sum(icount2) << endl;
}

while the TPL code for the analyzer is

DATA_SECTION
init_int n
init_int nm
init_vector ages(1,nm)
init_matrix sizes(1,n,1,nm)
matrix q0(1,n,1,2)
matrix q1(1,n,1,2)
number lmin
number lmax
LOC_CALCS
ages(13)=8;
ages(14)=11;
ages(15)=15;
ages(16)=19;
lmin=min(column(sizes,nm));
lmax=max(column(sizes,nm));
PARAMETER_SECTION
init_bounded_vector p1coff(1,2,.001,1.1)
init_bounded_vector p2coff(1,2,.001,1.1)
vector p1(1,2)
vector p2(1,2)
number psum1
number psum2
init_bounded_vector Linf(1,2,0.9*lmin,1.1*lmax,2)
init_bounded_vector k(1,2,.01,.3)
init_bounded_number rho(-.1,.95,3)
init_bounded_number lsigma(-5.0,5.0)
number sigma
matrix cor1(1,12,1,12)
matrix cor2(1,4,1,4)
matrix is1(1,12,1,12)
matrix is2(1,4,1,4)
matrix ic1(1,12,1,12)
matrix ic2(1,4,1,4)
matrix pred_size(1,2,1,nm)
number lds1
number lds2
objective_function_value f
PROCEDURE_SECTION
get_correlation();
get_proportions();
get_predicted_sizes();
get_likelihood();

FUNCTION get_correlation
f+=square(rho);

sigma=exp(lsigma);
dvariable vinv=1.0/(sigma*sigma);
int i,j;
for (i=1;i<=12;i++)
for (j=1;j<=12;j++)
cor1(i,j)=pow(rho,fabs(i-j));

for (i=1;i<=4;i++)
for (j=1;j<=4;j++)
cor2(i,j)=pow(rho,fabs(i-j));

ic1=inv(cor1);
ic2=inv(cor2);

is1=ic1*vinv;
is2=ic2*vinv;

lds1=ln_det(is1);
lds2=ln_det(is2);

FUNCTION get_proportions
psum1=sum(p1coff);
p1=p1coff/psum1;
f+=100.*square(log(psum1));

psum2=sum(p2coff);
p2=p2coff/psum2;
f+=100.*square(log(psum2));

FUNCTION get_predicted_sizes
lmax=max(column(sizes,nm));
pred_size(1)=log(Linf(1)*(1.0-exp(-k(1)*ages)));
pred_size(2)=log(Linf(2)*(1.0-exp(-k(2)*ages)));

FUNCTION get_likelihood
int i;
dvar_matrix res(1,2,1,nm);
for (i=1;i<=n;i++)
{
res(1)=log(sizes(i))-pred_size(1);
res(2)=log(sizes(i))-pred_size(2);
dvar_vector res1a=res(1)(1,12);
dvar_vector res1b=(res(1)(13,16)).shift(1);
dvar_vector res2a=res(2)(1,12);
dvar_vector res2b=(res(2)(13,16)).shift(1);
dvariable l1a=exp(-0.5*res1a*(is1*res1a));
dvariable l1b=exp(-0.5*res1b*(is2*res1b));
dvariable l2a=exp(-0.5*res2a*(is1*res2a));
dvariable l2b=exp(-0.5*res2b*(is2*res2b));
f-=log(1.e-10+(p1(1)*l1a+p1(2)*l2a));
f-=log(1.e-10+(p2(1)*l1b+p2(2)*l2b));
f-=0.5*(lds1+lds2);

q0(i,1)=value(p1(1))*value(l1a);
q0(i,2)=value(p1(2))*value(l2a);
q0(i)/=sum(q0(i));
q1(i,1)=value(p2(1))*value(l1b);
q1(i,2)=value(p2(2))*value(l2b);
q1(i)/=sum(q1(i));
}
REPORT_SECTION
report << setprecision(4) << setw(8) << inv(is1) << endl;
report << setprecision(4) << setw(8) << inv(is2) << endl;

report << setprecision(4) << setw(8) << cor1 << endl;
report << setprecision(4) << setw(8) << cor2 << endl;

report << colsum(q0)/n << endl;
report << colsum(q1)/n << endl;

report << n << endl;
report << q0 << endl;
report << q1 << endl;

I hope this illustrates how ADMB might be applied to growth models.

Dave

--
David A. Fournier
P.O. Box 2040,
Sidney, B.C. V8l 3S3