- formula
A model formula containing the group-level binomial
response on the left-hand side, and general group-level covariates on the
right-hand side. For example,
cbind(n.cases, population) ~ mean.income + deprivation.index
If formula is not specified, then there is assumed to be only
individual-level data, and iformula should be supplied.
- binary
An optional model formula with an empty left-hand
side. The right-hand side should contain the names of any group-level
proportions, which are to modelled as individual-level binary
predictors of the response given in formula. For example,
~ p.smokers + p.nonwhite + p.unemployed
- categorical
An optional list of matrices or data frames.
Each element corresponds to a categorical covariate. Each element
has the same number of rows as the aggregate data, and number of columns
corresponding to the number of levels of the categorical covariate.
The cells give the number or proportion of individuals in the area in each
category. These will be modelled as individual-level predictors of
the response given in formula.
- normal
An optional model formula with an empty left-hand
side. The right-hand side should list variables containing the
group-level means of normally-distributed covariates. These will be
modelled as individual-level predictors of the response given in
formula. For example
~ pollution + income.
- iformula
A model for the corresponding individual-level data. The
individual-level binary response should be on the right-hand side,
and the individual-level covariates should be on the left-hand side.
They should represent the same covariates, in the same order, as given in
formula and binary respectively. However they need not
have the same names. For example
outcome ~ mean.income + deprivation.index + smoking + nonwhite + unemployed.
If iformula is not specified, then there is assumed to be
only aggregate data, and formula should be supplied.
- data
Data frame containing the group-level variables given in
formula and binary.
- idata
Data frame containing the individual-level variables
given in iformula.
- groups
A group-level variable containing the group identifiers
to be matched with the groups given in igroups. Defaults to
the row numbers of the aggregate data. Only necessary if the model
includes random group effects.
- igroups
An individual-level variable containing the group
identifiers of the individual-level data to be matched with the
groups given in groups. Only necessary if the model
includes random group effects.
- strata
A matrix with the same number of rows as the aggregate
data. Rows representing groups, and columns
representing strata occupancy probabilities, often estimated as
observed occupancy proportions. The relative risks for the strata
will be included as fixed offsets in the underlying logistic
regression, using the probabilites supplied in pstrata. This
is to save the computational burden of estimating the "nuisance"
strata-specific risks from the data.
- istrata
A variable containing the individual-level variable
indicating the stratum an individual occupies. This should be a
factor with the levels corresponding to the columns of the matrix
strata.
- pstrata
A vector with one element for each stratum, giving
the assumed baseline outcome probabilities for the strata.
- cross
A matrix giving the joint within-area distribution of
all the covariates supplied in binary and categorical
and any strata. This should have the same number of rows as the
aggregate data, and number of columns equal to the product of the
numbers of levels of the covariates and strata, for example
\(2^n\) if there are \(n\) binary covariates. Each
cell gives the proportion of individuals in the area occupying a
category defined by a unique combination of the covariates. The
categories are given in the order
column 1: covariate 1 absent, covariate 2 absent, ..., covariate n-1 absent, covariate n absent
column 2: covariate 1 present, covariate 2 absent, ..., covariate n-1 absent, covariate n absent
column 3: covariate 1 absent, covariate 2 present, ..., covariate n-1 absent, covariate n absent
column 4: covariate 1 present, covariate 2 present, ..., covariate n-1 absent, covariate n absent
etc.
(assuming \(n\) binary covariates, with the obvious generalisation
for categorical covariates) If strata are used, these are taken as
covariate n+1.
- norm.var
A data frame, matrix or list, supplying the within-area covariances of the continuous covariates.
If norm.var is a data frame or matrix, then the continuous
covariates are assumed to be independent within areas. It should
have rows corresponding to areas, columns corresponding to continuous
covariates, each cell giving the within-area standard deviation of
the covariate.
If norm.var is a list, then it should have the same number of
elements as the number of areas, and each element should be the
within-area covariance matrix of the continuous covariates.
norm.var can also be the name of a variable in data
which contains the standard deviation of a single continuous
covariate.
- random
If TRUE then a normally-distributed random
group-level intercept, with zero mean, is also included in the model.
- pars
Vector of initial values of the model parameters, given
in the following order:
logit-scale intercept,
coefficients for group-level covariates,
coefficients for individual-level covariates,
random effects standard deviation.
If not supplied, the initial values are 0 for all covariate effects,
1 for the random effects standard deviation. The intercept is
initialised to the logit mean outcome proportion over groups from
the aggregate data.
- fixed
If TRUE then eco just calculates the
likelihood with all parameters are fixed at their
initial values.
- model
If "marginal" then the ecological group-level risk is based on
integrating over binary individual-level covariates. This is suitable if the
aggregate exposures are estimated using a survey of individuals in the area.
If "conditional" then the binary individual-level covariates are
conditioned on, and the group-level risk is the normal approximation
model described by Wakefield (2004). This is suitable if the
aggregate exposures are estimated using a full population census.
- outcome
Distribution of the aggregate outcome, by default
"binomial". outcome="poisson" can be specified for rare
outcomes.
- gh.points
Number of points for Gauss-Hermite numerical
integration in the random effects model.
- iter.adapt
Number of adaptive iterations to estimate the mode and scale
for Gauss-Hermite numerical integration in the random-effects
model.
- ...
Arguments passed to optim.