likelihood: Specify first two levels of hierarchical model.
In StatisticsNZ/demest: Bayesian Demographic Estimation and Forecasting
likelihood R Documentation
Specify first two levels of hierarchical model.
Description
Specify the likelihood and part of the prior for a
Poisson, binomial, or normal hierarchical model.
Usage
Poisson(formula, useExpose = TRUE, structuralZeros = NULL, boxcox = 0)
CMP(
formula,
dispersion = Dispersion(),
useExpose = TRUE,
structuralZeros = NULL,
boxcox = 0
)
Binomial(formula, structuralZeros = NULL)
Normal(formula, sd = NULL, priorSD = HalfT())
Arguments
formula
A formula with response
mean and names of dimensions of demographic
series or dataset being modelled.
useExpose
Whether the model includes an
exposure term. Defaults to TRUE.
structuralZeros
Location of any structural zeros
in the data. An object of class Values,
or, for the typical case, the word "diag".
boxcox
Parameter determining transformation of rates
or counts used in Poisson or CMP models. Defaults to 0,
implying that a log transform is used.
dispersion
The dispersion parameter for a CMP model.
An object of class Dispersion,
typically created by a call to function Dispersion.
sd
Standard deviation in the likelihood for the
normal model. If a value is not supplied,
it is estimated from the data.
priorSD
An object of class HalfT specifying
a non-default prior for the standard deviation in the
likelihood for the normal model.
Details
Specify a likelihood and prior of the form
y_i \sim Poisson(\gamma_i n_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
y_i \sim Poisson(\gamma_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
y_i \sim CMP(\gamma_i n_i, \nu_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
log(\nu_i) \sim N(m, s^2)
y_i \sim CMP(\gamma_i, \nu_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
log(\nu_i) \sim N(m, s^2)
y_i \sim binomial(n_i, \gamma_i)
logit(\gamma_i) \sim N(x_i \beta, \sigma_i),
or
y_i \sim normal(\gamma_i, \phi^2 / w_i)
\gamma_i \sim N(x_i \beta, \sigma_i)
.
Subscript i denotes a cell within a multiway array,
such as an array with dimensions age, sex, and time. In
Poisson and binomial models, y_i is a count.
The n_i term is an exposure in the case of Poisson
and CMP (COMPoisson) models and a
sample size in the case of binomial models.
It is not supplied in calls to function Poisson
or Binomial. In normal models y_i
is a cell-specific value such as a mean,
and w_i is a cell-specific weight. Weights
are not supplied in calls to function Normal.
Vector \beta contains main effects
and interactions, such as age effects, time effects,
and age-region interactions. Vector x_i is the
ith row from the design matrix X.
The main effects and interactions are specified
via the formula argument. For instance,
mean ~ age * sex + time
specifies a model with age, sex, and time main effects,
and an age-sex interaction.
The main effects and interactions in a hierarchical model are only weakly
identified: see the documentation for function fetch for
details.
If a model has two or more levels, the second level
typically contains more than just main effects
and interactions. For instance, the second level of
Poisson, binomial, and normal hierarchical models
contains a variance term. The remaining parts of
the second level, such as the variances, as well
as any higher levels, are specified
in calls to function Model, or to
functions estimateModel,
estimateCounts, or estimateAccount.
Poisson and CMP models allow structural zeros in the data,
that is, cells whose value must be zero by definition.
Examples include the number of pregnant males or the
number of people transitioning straight from "single"
to "divorced". The most general way to specify structural zeros
is to supply an object of class Values,
with zeros in the places where structural zeros are expected,
and non-zero values elsewhere. The Values object only has to
contain enough dimensions to specify the positions where the structural
zeros appear: other dimensions will be added as needed. See below
for an example.
The most common situation where structural zeros
are required is when the data has origin and destination dimensions,
and values on the diagonal are always zero. Setting
structuralZeros to "diag" is equivalent to
supplying a Values object with zeros on the diagonal.
Value
An object of class SpecLikelihood.
See Also
Functions Poisson, Binomial, and Normal are
used as part of a call to function Model.
Examples
## age effect, sex effect, age-sex interaction,
## and time effect
Poisson(mean ~ age * sex + time)
## same model, but without exposure term
Poisson(mean ~ age * sex + time, useExpose = FALSE)
## use formula notation to specify second-order interactions
Binomial(mean ~ (age + sex + region)^2)
Normal(mean ~ age + education + income)
## specify the exact value of the standard deviation
Normal(mean ~ age + education + income,
sd = 0.3)
## specify a non-default prior for the standard deviation
Normal(mean ~ age + education + income,
priorSD = HalfT(scale = 100))
## Specify structural zero on the diagonal
struc.zeros <- Values(array(c(0, 1, 1,
1, 0, 1,
1, 1, 0),
dim = c(3, 3),
dimnames = list(region_orig = c("A", "B", "C"),
region_dest = c("A", "B", "C"))))
## Note that the model contains age and sex dimensions. These
## The pattern of zeros specified by 'struc.zeros' will be
## replicated for each combination of these dimensions.
Poisson(mean ~ region_orig * region_dest + age * sex,
structuralZeros = struc.zeros)
## The same pattern of structural zeros, with zeros on the diagonal
## formed by the origina and destination dimensions, can be specified
## by using the word "diag"
Poisson(mean ~ region_orig * region_dest + age * sex,
structuralZeros = "diag")
StatisticsNZ/demest documentation built on Nov. 2, 2023, 7:56 p.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
| likelihood | R Documentation |
Specify first two levels of hierarchical model.
Description
Specify the likelihood and part of the prior for a Poisson, binomial, or normal hierarchical model.
Usage
Poisson(formula, useExpose = TRUE, structuralZeros = NULL, boxcox = 0)
CMP(
formula,
dispersion = Dispersion(),
useExpose = TRUE,
structuralZeros = NULL,
boxcox = 0
)
Binomial(formula, structuralZeros = NULL)
Normal(formula, sd = NULL, priorSD = HalfT())
Arguments
formula |
A |
useExpose |
Whether the model includes an
exposure term. Defaults to |
structuralZeros |
Location of any structural zeros
in the data. An object of class |
boxcox |
Parameter determining transformation of rates or counts used in Poisson or CMP models. Defaults to 0, implying that a log transform is used. |
dispersion |
The dispersion parameter for a CMP model.
An object of class |
sd |
Standard deviation in the likelihood for the normal model. If a value is not supplied, it is estimated from the data. |
priorSD |
An object of class |
Details
Specify a likelihood and prior of the form
y_i \sim Poisson(\gamma_i n_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
y_i \sim Poisson(\gamma_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
y_i \sim CMP(\gamma_i n_i, \nu_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
log(\nu_i) \sim N(m, s^2)
y_i \sim CMP(\gamma_i, \nu_i)
log(\gamma_i) \sim N(x_i \beta, \sigma_i),
log(\nu_i) \sim N(m, s^2)
y_i \sim binomial(n_i, \gamma_i)
logit(\gamma_i) \sim N(x_i \beta, \sigma_i),
or
y_i \sim normal(\gamma_i, \phi^2 / w_i)
\gamma_i \sim N(x_i \beta, \sigma_i)
.
Subscript i denotes a cell within a multiway array,
such as an array with dimensions age, sex, and time. In
Poisson and binomial models, y_i is a count.
The n_i term is an exposure in the case of Poisson
and CMP (COMPoisson) models and a
sample size in the case of binomial models.
It is not supplied in calls to function Poisson
or Binomial. In normal models y_i
is a cell-specific value such as a mean,
and w_i is a cell-specific weight. Weights
are not supplied in calls to function Normal.
Vector \beta contains main effects
and interactions, such as age effects, time effects,
and age-region interactions. Vector x_i is the
ith row from the design matrix X.
The main effects and interactions are specified
via the formula argument. For instance,
mean ~ age * sex + time
specifies a model with age, sex, and time main effects, and an age-sex interaction.
The main effects and interactions in a hierarchical model are only weakly
identified: see the documentation for function fetch for
details.
If a model has two or more levels, the second level
typically contains more than just main effects
and interactions. For instance, the second level of
Poisson, binomial, and normal hierarchical models
contains a variance term. The remaining parts of
the second level, such as the variances, as well
as any higher levels, are specified
in calls to function Model, or to
functions estimateModel,
estimateCounts, or estimateAccount.
Poisson and CMP models allow structural zeros in the data,
that is, cells whose value must be zero by definition.
Examples include the number of pregnant males or the
number of people transitioning straight from "single"
to "divorced". The most general way to specify structural zeros
is to supply an object of class Values,
with zeros in the places where structural zeros are expected,
and non-zero values elsewhere. The Values object only has to
contain enough dimensions to specify the positions where the structural
zeros appear: other dimensions will be added as needed. See below
for an example.
The most common situation where structural zeros
are required is when the data has origin and destination dimensions,
and values on the diagonal are always zero. Setting
structuralZeros to "diag" is equivalent to
supplying a Values object with zeros on the diagonal.
Value
An object of class SpecLikelihood.
See Also
Functions Poisson, Binomial, and Normal are
used as part of a call to function Model.
Examples
## age effect, sex effect, age-sex interaction,
## and time effect
Poisson(mean ~ age * sex + time)
## same model, but without exposure term
Poisson(mean ~ age * sex + time, useExpose = FALSE)
## use formula notation to specify second-order interactions
Binomial(mean ~ (age + sex + region)^2)
Normal(mean ~ age + education + income)
## specify the exact value of the standard deviation
Normal(mean ~ age + education + income,
sd = 0.3)
## specify a non-default prior for the standard deviation
Normal(mean ~ age + education + income,
priorSD = HalfT(scale = 100))
## Specify structural zero on the diagonal
struc.zeros <- Values(array(c(0, 1, 1,
1, 0, 1,
1, 1, 0),
dim = c(3, 3),
dimnames = list(region_orig = c("A", "B", "C"),
region_dest = c("A", "B", "C"))))
## Note that the model contains age and sex dimensions. These
## The pattern of zeros specified by 'struc.zeros' will be
## replicated for each combination of these dimensions.
Poisson(mean ~ region_orig * region_dest + age * sex,
structuralZeros = struc.zeros)
## The same pattern of structural zeros, with zeros on the diagonal
## formed by the origina and destination dimensions, can be specified
## by using the word "diag"
Poisson(mean ~ region_orig * region_dest + age * sex,
structuralZeros = "diag")
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
Embedding an R snippet on your website
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.