LogNormal
Parametrisation
The LogNormal has density1 √
12f (y) = √τ exp − τ (log y − µ) ,2y 2πy>0whereτ > 0 is the precision parameter,µ is the mean parameter.

Link-function
The parameter µ is linked to the linear predictor as:
η=µHyperparameters
The τ parameter is represented asθ = log τand the prior is defined on θ.

Specification• family="lognormal" for regression models and family="lognormalsurv" for survival models.
• Required arguments: y. Given in a format by using inla.surv() function for "lognormal.surv"
Hyperparameter spesification and default values
lognormal
doc The log-Normal likelihood
hyper
theta
hyperid 77101
name log precision
short.name prec
output.name Precision for the lognormal observations
output.name.intern Log precision for the lognormal observations
initial 0
fixed FALSE
prior loggamma
param 1 5e-05
to.theta function(x) log(x)from.theta function(x) exp(x)survival FALSE
discrete FALSE
link default identity
pdf lognormal1lognormalsurv
doc The log-Normal likelihood (survival)
hyper
theta1hyperid 78001
name log precision
short.name prec
output.name Precision for the lognormalsurv observations
output.name.intern Log precision for the lognormalsurv observations
initial 0
fixed FALSE
prior loggamma
param 1 5e-05
to.theta function(x) log(x)from.theta function(x) exp(x)theta2
hyperid 78002
name beta1
short.name beta1
output.name beta1 for logNormal-Cure
output.name.intern beta1 for logNormal-Cureinitial -7
fixed FALSE
prior normal
param -4 100
to.theta function(x) x
from.theta function(x) xtheta3
hyperid 78003
name beta2
short.name beta2
output.name beta2 for logNormal-Cureoutput.name.intern beta2 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta4
hyperid 78004
name beta3
short.name beta3
output.name beta3 for logNormal-Cure2

output.name.intern beta3 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta5
hyperid 78005
name beta4
short.name beta4
output.name beta4 for logNormal-Cure
output.name.intern beta4 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta6
hyperid 78006
name beta5
short.name beta5
output.name beta5 for logNormal-Cure
output.name.intern beta5 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta7
hyperid 78007
name beta6
short.name beta6
output.name beta6 for logNormal-Cure
output.name.intern beta6 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta8hyperid 78008
3name beta7
short.name beta7
output.name beta7 for logNormal-Cureoutput.name.intern beta7 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta9hyperid 78009
name beta8
short.name beta8
output.name beta8 for logNormal-Cureoutput.name.intern beta8 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta10hyperid 78010
name beta9
short.name beta9
output.name beta9 for logNormal-Cure
output.name.intern beta9 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x
from.theta function(x) xtheta11
hyperid 78011
name beta10
short.name beta10
output.name beta10 for logNormal-Cure
output.name.intern beta10 for logNormal-Cureinitial 0
fixed FALSE
prior normal
param 0 100
to.theta function(x) x4

from.theta function(x) x
survival TRUE
discrete FALSE
link default identity
pdf lognormal

ExampleIn the following example we estimate the parameters in a simulated casen = 300x = c(scale(runif(n)))
eta = 1+2.2*xy = exp(rnorm(n, mean = eta, sd = 1))
data = list(y=y, event=rep(1, n), x=x)
formula = inla.surv(y, event) ~ 1 + x
r=inla(formula, family ="lognormalsurv", data=data)summary(r)data = data.frame(y, x)
formula = y ~ 1 + x
r=inla(formula, family ="lognormal", data=data)summary(r)Notes• lognormalsurv can be used for right censored, left censored, interval censored data. A generalframework to represent time is given by inla.surv. If the observed times y are large/huge,then this can cause numerical overflow, and if you encounter this problem, try to scale theobservatios, like time = time / max(time).

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