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\title{Ecological inference with R: the \Rpackage{ecoreg} package \vskip 0.2in
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\author{Christopher Jackson\\MRC Biostatistics Unit, Cambridge\\ \texttt{chris.jackson@mrc-bsu.cam.ac.uk} \footnote{Written while a member of the BIAS project at the Department of Epidemiology and Public Health, Imperial College London} }
\date{March 2006}
\bibliographystyle{unsrt}

\begin{document}

\maketitle

\begin{abstract}
  In typical small-area studies of health and environment we wish to
  make inference on the relationship between individual-level
  quantities using aggregate, or ecological, data.  Such ecological
  inference is often subject to bias and imprecision, due to the lack
  of individual-level information in the data.  Simple regressions of
  area-level mean outcomes on area-level mean exposures are usually
  biased.  To alleviate bias, the within-area distribution of
  exposures should be accounted for.  The \Rpackage{ecoreg} package
  can be used to fit this class of models for ecological inference
  from aggregate data.  In addition, full outcome and covariate
  information from a survey of individuals within the areas can be
  used to improve bias and precision.  \Rpackage{ecoreg} can be used
  in this way to analyse ecological and individual data simultaneously,
  using \emph{hierarchical related regression}.
\end{abstract}


\section{Ecological inference} 
\label{sec:ecological}

Ecological studies analyse data defined at a group level, but aim to
make inferences about the individuals within the groups. To make
reliable individual-level inferences from these studies, a number of
problems must be overcome.  One crucial difficulty is that the
group-level exposure-response relationship may not reflect the
individual-level relationship, a problem known as {\em ecological
  bias}, or the {\em ecological fallacy}.  See, for example,
\cite{richardson87,GreenlandM,GreenlandR,richardson:monfort} for discussion of these issues.

Denote the outcome count in area $i$, with population $N_i$, by
$y_{i}$.  To model $y_{i}$ in terms of exposures measured as
aggregate-level summaries, the usual model is a simple binomial or
Poisson regression on the area-level covariate means $\overline
z_{i}$.  With binary covariates, $\overline z_{i}$ is the proportion
exposed over the area.  However, this only models the relationship
between the \emph{aggregate} exposures and outcomes.  It is only
justified as a method of estimating individual-level relationships if
all individuals in the area have the same covariate value, or there is
the same exposure-response relationship at the individual and
aggregate levels, which is generally only true for linear models.
With non-linear models, such as Binomial or Poisson models, using the
same model form at both levels will lead to ecological bias
\cite{richardson87}.

Despite these problems, aggregate data \emph{can} provide information
about individual-level relationships.  As the ratio of the
between-area to the within-area variability of the exposure increases,
the aggregate data summarise the true distribution of the exposure
more accurately, and contain more information about the true
individual-level exposure-outcome relationship.  With sufficient
exposure information, and with \emph{correctly-specified} models for
the mean outcome
\cite{richardson87,wakefield:salway,PrenticeS},
ecological bias can be reduced to negligible levels.

In general, successful ecological inference requires samples of
individual-level data within areas.  Individual exposure data are
usually required to reduce ecological bias by accounting for the
within-area variability of exposures.  But further improvements can
also be made by using samples of exposures and \emph{outcomes} for
selected individuals, as discussed by Wakefield~\cite{wakefield2by2}
for 2 $\times$ 2 tables, and more generally by Jackson \etal
\cite{improving:eco}.


\section{Models for aggregate and individual data}
\label{sec:models}

The models described here are also described in the papers by
Jackson \etal \cite{improving:eco} \cite{my:hrr}.

\subsection{Individual data}
\label{sec:indiv}

We begin by specifying the form of the relationship between the
individual-level risk of the binary outcome and the covariates.  If
individual-level exposure and outcome data are available, this is used
to model them. It will also be used as the basis for an equivalent model
for the aggregate data, as described in the next section.  The risk $p_{ij}$ of the individual-level
outcome $y_{ij}$ for the $j$th individual in area $i$ is assumed to be
a logit-linear function of the covariates.  The most general model we
consider is \begin{equation}
  \label{eq:indiv}
  \logit(p_{ij}) = \mu_i + \sum_r \alpha_r x_{ir} + \sum_r \beta_r z_{ijr} + \gamma_{s_{ij}}
\end{equation}
where $x_{ir}$ are group-level covariates, and $z_{ijr}$ are
individual-level covariates.  The group-level covariates may include
descriptions of the socio-economic status of the area, or the health
service provision in the area.  Individual-level covariates might
comprise individual behaviours such as smoking, demographics such as
ethnicity, or individual indicators of wealth and social class.
Individuals may be influenced by the overall average exposure in the
area, in addition to their own, so that the group-level variables may
include the means of certain individual-level variables.  $\gamma_s$
represents an additional contribution to the baseline risk for an
individual occupying one of several strata $s$, usually defined by age
and sex.

The baseline risk $\mu_i$ may be fixed at $\mu$ or considered as a
random effect with some distribution across areas.  This can account
for any remaining overdispersion and heterogeneity between areas,
after adjusting for observed area-level variables.  A random effect
also allows the borrowing of information across areas and can
stabilise estimation from areas with small populations
\cite{richardson:best}.





\subsection{Aggregate data}
\label{sec:agg}

\subsubsection{Marginal model}
Suppose the area-level exposures have been estimated from a survey.
For example, in the UK census, aggregate data on social class and
education are calculated using a 10\% sample to maintain
confidentiality. The proportion of smokers in the area might also be
estimated from sales figures instead of a census.  Then, individual
outcomes can be assumed to be independent and identically distributed,
with risk equal to some marginal ``group-level risk'' $p_{i}$.  We assume
\begin{equation}
  \label{eq:bin}
  y_{i} \sim Bin(N_{i}, p_{i}),
\end{equation}
where $p_{i}$ is determined by integrating the individual-level model over
the joint within-area distribution of covariates
\cite{richardson87,wakefield:salway}.  Thus, $p_i$ is the average risk
for an individual in group $i$. 
\begin{equation}
  \label{eq:integral}
  p_i = \int p_{ij}(\mathbf x) f_i(\mathbf x) d \mathbf x = E_{\mathbf x} (p_{ij}(\mathbf x) | i)    
\end{equation}
For a \emph{single binary covariate}, observing the proportion exposed
gives us enough information to estimate a binomial within-area
distribution.  For \emph{continuous covariates} the mean is not
sufficient to estimate the within-area distribution, and we generally
need samples of individual covariate data to be able to estimate the
within-area variability.  For \emph{multiple binary covariates} the
joint distribution is estimated by the cross-classification of
individuals between covariates. Typical census data do not usually
have this cross-classification, and we need individual data to
estimate it.

\subsubsection{Conditional model} An alternative to the marginal model
was proposed by Wakefield \cite{wakefield2by2}.  The binomial
model~(\ref{eq:bin}) is based on the assumption that each individual
in group $i$ has an identical marginal probability $p_i$ of outcome,
integrating over their unknown exposures.  If the binary exposure is
known for \emph{all} the individuals in the area from a full
population census, but not necessarily coupled with exposures for the
same individuals, then these should be conditioned on, leading to a
likelihood based on a convolution of binomial distributions.  The
exact likelihood is related to the extended (or non-central)
hypergeometric distribution, however, Wakefield describes a more
computationally convenient normal approximation.


\subsubsection{Binary covariates} 

For clarity we demonstrate the marginal model for 3 binary covariates,
however, the framework extends immediately to any number of binary
covariates.  The integral to obtain the group-level risk is equivalent
to a sum.  Each individual falls into one of $S \times 2^3$
categories, defined by the distinct combinations of the 3 covariates
and the $S$ age-sex strata, and indexed by $k$.  Let $\phi_{ik}$ be
the probability that an individual occupies category $k$.  Let
$q_{ik}$ be the probability of the individual outcome conditionally on
occupying category $k$.  Rewriting the index $k$ as $\{k_1, k_2, k_3,
s\}$, where $k_r$ (0 or 1) indicates the presence or absence of
covariate $r = 1, \ldots, 3$.  \begin{equation}
  \label{eq:bin3}
  p_{i} = \sum_{k} \phi_{ik} q_{ik}  = \sum_{k_1, k_2, k_3, s} \phi_{\{i, k_1, k_2, k_3, s\}} q_{\{i, k_1, k_2, k_3, s\}}.
\end{equation}
The outcome model conditionally on the unobserved category is
\begin{equation}
  \label{eq:q}
  \logit ( q_{\{i, k_1, k_2, k_3, s\}} ) = \mu_i  + \logit(e_s) + \sum_r \alpha_r x_{ir} +  \sum_r k_r \beta_{r}
\end{equation}
The effect $\beta_r$ of the binary individual-level covariate $r$ only
enters into this equation when the covariate $r$ is present.  This is
a generalisation of the model presented for two covariates by Lasserre
\etal \cite{lasserre:binary}, except that the outcome is assumed to be
non-rare and binomial instead of Poisson.  $\logit(e_s)$ is a fixed
offset, where $e_s$ is the risk of the outcome in stratum $s$,
estimated from national population data. This is similar in spirit to
``indirect standardisation'', see, for example, \cite{clayton:hills}.
If population and outcome totals are available for strata within
areas, then we could generalise this to a separate binomial model for
each stratum within area, with coefficients for the other covariates
shared between the models, assuming no stratum-covariate interactions.
It is also important to check whether \emph{exposures} are correlated
with strata. If not accounted for, this can lead to ``mutual
standardisation bias'' \cite{rosenbaum:rubin:stand}.  

We normally replace $\phi_{ik}$ in (\ref{eq:bin3}) by an estimate
$\hat \phi_{ik}$.  Ideally this \emph{probability} would be estimated
by the \emph{proportion} of individuals in area $i$ occupying category
$k$.  But typical census data are not sufficient to know the complete
cross-classification of individuals between all of these covariate and
strata categories.  Usually, our only information is the marginal
proportions of single covariates, for example, the proportion
$\phi_i^{(1)}$ of individuals who are economically inactive.  If we
assume that the covariates are independent, then $\phi_{ik}$ can be
estimated by the product of the proportions of individuals occupying
each marginal category defining $k$.  But generally, socioeconomic
indicators, such as unemployment and social class, are highly
correlated.  \cite{lasserre:binary} demonstrated that, in a typical
case, bias is negligible when the joint covariate distribution is
estimated by the product of the two marginal distributions, even when
the covariates are correlated.  However, we may wish to study more
than two covariates.

To estimate $\phi_{ik}$ we can often use a combination of marginal
proportions $\overline z_{ir}$ and individual covariate data
$z_{ijr}$.  In the context of the UK census, for example, these
correspond to district-level aggregate data and the Samples of
Anonymised Records.  Let $C_{ik}$ be the number of individuals in area
$i$ in this individual-level dataset occupying category $k$, computed
from the $z_{ijr}$.  We use the following two principles.  Firstly,
the estimated $\hat\phi_{ik}$ corresponding to categories $k$ in which
binary covariate $r$ is 1 must sum to $z_{ir}$.  Secondly, the ratio
of estimates $\hat\phi_{i_k} / \hat\phi_{i_l}$ must be the same as the
ratio $C_{ik} / C_{il}$.  This gives, for example, where $R$ is the
set of categories in which covariate $r$ is 1, \begin{equation}
  \label{eq:phiik}
  \hat \phi_{ik} = C_{ik} \overline z_{ir} / \sum_{l \in R} {C_{il}}
\end{equation}



\subsubsection{Continuous covariates} 
Suppose now the individual-level model~(\ref{eq:indiv}) depends only
on an intercept and one continuous individual-level covariate
$x_{ij}$: $\logit(p_{ij}) = \mu_i + \beta x_{ij}$.  The ecological
data consist of the within-area mean $m_i$ of $x_{ij}$.  In some
cases, as well as the within-area mean, we may also have an estimate
$s_i^2$ of the within-area variance of $x_{ij}$, for example, from
geographical modelling of an environmental exposure surface (e.g.
Best \etal \cite{best:leuk}).  Then we suppose that these exposures
are normally distributed, with $x_{ij} \sim N(m_i, s_i^2)$.  If an
exposure is not naturally normally distributed, it can often be
transformed to normality. We can then calculate the area-specific
risks~(\ref{eq:integral}) by integrating over $f_i$, here the density
function of the normal distribution.
\begin{eqnarray}
  \label{eq:qint}
  p_i & = & \int \expit(\mu_i + \beta x) f_i(x) dx
\end{eqnarray}
Our assumed underlying model for $p_{ij}(x)$ is logit-linear. In this
case, the integral is not available in closed form.  However, if we
approximate the logit by a probit link function, then (\ref{eq:qint})
evaluates to
\begin{eqnarray}
  \label{eq:normal:binomial}
  p_i & = & \expit  \left \{ (1 + c^2 \beta^2 s_i^2) ^ {-1/2} (\mu_i + \beta m_i) \right \}    
\end{eqnarray}
where $c = 16\sqrt{3} / (15 \pi)$ (Salway and Wakefield
\cite{nonrare}).

If, instead, we were using a Poisson model for a rare outcome, $y_i
\sim \mbox{Pois}(N_i p_i)$, and a log-linear individual-level model
$\log(p_{ij}) = \mu_{i} + \beta x_{ij}$, then the integrals can be
evaluated explicitly without an approximation. Instead
of~(\ref{eq:normal:binomial}), we would have (Richardson \etal
\cite{richardson87})
\begin{eqnarray}
\label{eq:normal:poisson}
p_i = \exp \left \{ \mu_i + \beta m_i + \frac{\beta^2 s_i^2}{2} \right \} \qquad
\end{eqnarray}
These methods generalise to multiple jointly normally-distributed
covariates.

\subsubsection{Semi-parametric approach}

We have described a fully parametric model for ecological inference.
Prentice and Sheppard \cite{PrenticeS} described an alternative
semi-parametric approach based on estimating functions. This is often
simply called the \emph{aggregate data} method.  Suppose a sample of
covariates (binary or continuous) are available from a subset of $n_i$
individuals, but not the corresponding outcomes on the same
individuals.  Broadly, the mean and variance of the total disease
count $y_i$ are calculated in terms of an \emph{aggregate} risk
$\frac{1}{n_i}\sum_j p_{ij}$.  $p_{ij}$ is the risk for individual $j$
in area $i$, conditionally on their covariate values.  This method is
not implemented in the \Rpackage{ecoreg} R package.  This approach
does not require a within-area distribution to be specified for the
covariates.  It requires samples of covariate data as an explicit part
of the model.  On the other hand, the parametric approaches described
above, and implemented in \Rpackage{ecoreg}, require individual
covariate data implicitly to estimate an appropriate within-area
distribution.



\subsection{Combining aggregate and individual data}
\label{sec:aggindiv}

To summarise the model for the ecological data, we have a binomial
model~(\ref{eq:bin}) for the area-level outcome $y_i$.  The
corresponding area-level risk $p_i$ is calculated explicitly in terms
of the transformed group baseline risk $\mu_i$, the individual-level
covariate effects, and the within-area distributions of the covariates.

It is easy to extend this model to include information from a sample
of individuals in each area whose outcomes and exposures are known.
We simply model the risk of the individual level outcome with the
logistic regression~(\ref{eq:indiv}).  Then the covariate effects
$\alpha$ and $\beta$ and the intercept $\mu_i$ are shared by the
models for both the aggregate and individual-level data.  Thus, we can
fit a joint model which combines the information from the two
sources of data.  This is termed \emph{hierarchical related
  regression}.

Note that it is not necessary to have individual data within all of
the areas $i$. In practice, sample survey data will be available from
varying numbers of individuals between areas.




\section{Using \Rpackage{ecoreg}}
\label{sec:ecoreg}

The \Rpackage{ecoreg} package implements a fairly general case of the
models described in Section~\ref{sec:models}.  We assume you have
already downloaded and installed the \Rpackage{ecoreg} package.  To
apply these methods, you should have one or both of

\begin{itemize}
\item An aggregate dataset with one record for each aggregate group,
  for example a geographical area, or a stratum within area, for
  example from a population census.  This contains aggregate outcomes
  and exposures, and optionally some indication of the within-area
  distribution of the exposures.

\item An individual-level dataset, for example from a sample survey
  study.  There need not be the same number of individuals per area,
  and there may be some areas in the aggregate dataset with no
  individuals.  This contains full individual-level covariate and
  outcome data.
\end{itemize}

First we load the \Rpackage{ecoreg} package into the working R
session.
<<>>=
library(ecoreg)
@ 
The main R function in \Rpackage{ecoreg} is called \Rfunction{eco}.
This is used to fit a model to one of these datasets, or a combination
of the two.  The R help page for \emph{eco} fully describes each of
the function's arguments.  


\subsection{Example}

The use of \Rfunction{eco} is illustrated here using a simulated
dataset.  We simulate aggregate data consisting of 50 groups of 100
individuals.  Two contextual covariates (labelled deprivation and mean
income) are generated as standard normal variables.  Two covariates
which are binary at the individual level, and available at the
aggregate level as the proportion of non-white individuals and smokers
in each area, are generated from uniform distributions.  The data
frame \Robject{sim.df} contains the ecological covariate data.

<<>>=
ng <- 50
N <- rep(100, ng)
set.seed(31412) 
ctx <- cbind(deprivation = rnorm(ng), mean.income = rnorm(ng))
phi <- cbind(nonwhite = runif(ng), smoke = runif(ng))
sim.df <- as.data.frame(cbind(ctx, phi))
sim.df[1:5,]
@  
A disease outcome with approximate 5\% baseline prevalence, and odds
ratios of 1.01, 1.02, 1.5 and 2 respectively for the four covariates,
is now simulated.  The function \Rfunction{sim.eco} is provided to
simulate ecological outcome data and individual sample data, in terms
of known covariates, baseline risks and odds ratios.  
<<>>=
mu <- qlogis(0.05)
alpha.c <- log(c(1.01, 1.02))
alpha <- log(c(1.5, 2))
sim1 <- sim.eco(N, ctx=~deprivation+mean.income, binary=~nonwhite+smoke, 
                data = sim.df,  mu=mu, alpha.c=alpha.c, alpha=alpha, 
                isam=10)
sim1$y[1:5]
aggdata <- as.data.frame(cbind(y=sim1$y, N=N, sim.df))
@ 
\paragraph{Format of aggregate dataset} 
We have now got an aggregate dataset \Robject{aggdata} suitable for
use with the \Rfunction{eco} function.  Generally, the aggregate
dataset should be a data frame with a row corresponding to an area or
group.  It should have at mininum an outcome variable. This is
available either as a proportion of individuals with the outcome, or
the number of events and the population at risk in the area. In
addition, any number of aggregate covariates can be specified.  Often
these will be binary covariates, expressed as proportions over the
area.  For covariates that are continuous at the individual level,
these should be specified as within-area means, and if possible,
within-area variances, after transformation to an approximate normal
distribution.  For example, we print the first ten rows of the
\Robject{aggdata} data.
<<>>=
aggdata[1:5,]
@ 
The number of individuals with the disease, the population of the
area, the deprivation index, the mean income, the proportion of
non-white individuals, the proportion of smokers, and the mean and
standard deviation of the pollution exposure are labelled \Robject{y},
\Robject{N}, \Robject{deprivation},
\Robject{mean.income}, \Robject{nonwhite} and \Robject{smoke}, respectively.

The return value of \Rfunction{sim.eco} has a component \Robject{y}
containing the ecological outcome data (the number of individuals in
each area with the outcome), and a component \Robject{idata}
containing the individual sample data.  Here we have specified
\Rfunarg{isam=10} in the call to \Rfunction{sim.eco}, producing an
individual sample dataset with 10 individuals for each of the 50 areas.
<<>>=
indivdata <- sim1$idata
@ 

\paragraph{Format of individual dataset} 
We have now got an individual dataset \Robject{indivdata} suitable for
use with the \Rpackage{ecoreg} package.  The individual dataset should
be a data frame with each row corresponding to an individual.
Variables may include a binary outcome and any number of covariates.
For example, the first 15 rows of \Robject{indivdata} are illustrated.
<<>>=
indivdata[1:15,]
@  

The area indicator, the disease status of the individual, the
deprivation index and the mean income of the area in which the
individual lives, indicators for non-white ethnicity and whether the
individual smoked, the pollution exposure and the area indicator are
labelled \Robject{group}, \Robject{y}, \Robject{deprivation},
\Robject{mean.income}, \Robject{nonwhite} and \Robject{smoke}
respectively.  Binary indicators are 0 or 1 corresponding to no and
yes respectively.  The area indicator is only necessary when using
models with random area effects.




\subsection{Calling \Rfunction{eco}}

Now we give examples of calling the \emph{eco} function to fit models
to the aggregate and individual datasets.  

\subsubsection{Aggregate data alone} 

Firstly, we fit the correct model to the simulated data, with two
contextual covariates and two individual binary covariates, using the
aggregate data alone.
<<>>=
agg.eco <- eco(cbind(y, N) ~ deprivation + mean.income,
               binary = ~ nonwhite + smoke,  data = aggdata)
@ 
\begin{itemize}
\item The first argument of \Rfunction{eco} is a formula, as used in
most statistical modelling functions in R such as \Rfunction{lm} and
\Rfunction{glm}. It specifies the \emph{aggregate} component of the
model, that is, the names of any covariates included in $x_{ir}$
(equation~\ref{eq:indiv}).  

\item The argument \Rfunarg{data} specifies a data frame which should
contain all aggregate variables specified in the call to
\Rfunction{eco}.

\item The \Rfunarg{binary} argument to \Rfunction{eco} is a formula whose
right hand side should contain the names of any aggregate covariates
considered as \emph{individual-level} rather than contextual effects,
here, non-white ethnicity and smoking.  These should be binary at
the individual level.  \Rfunction{eco} will fit the marginal model
(\ref{eq:bin}--\ref{eq:integral}) by default for binary covariates.
To fit the convolution normal-approximation model of
Wakefield~\cite{wakefield2by2} specify \Rfunarg{model=conditional} in the
call to \Rfunction{eco}.
\end{itemize}

The \Rfunction{eco} function returns objects of class
\Rfunction{ecoreg}.  Printing an object of this class displays the
estimated odds ratios $\exp(\alpha)$ associated with aggregate-level
covariates, and odds ratios $\exp(\beta)$ associated with individual
covariates (equation~\ref{eq:indiv}), along with their 95\%
confidence intervals, and $-2 \times$ the maximised log-likelihood.
In this example, the estimates are close to the true values used 
for simulating the data. 
<<>>=
agg.eco
@ 

\subsubsection{Combining aggregate and individual data} 

Next we combine the aggregate data with the information from samples
of individuals, as described in Section~\ref{sec:aggindiv}.  Firstly,
we form a reduced aggregate dataset, removing the individual outcomes
and population totals which appear in the individual data.  We assume
that the sampled individuals did not contribute to the estimation of
the aggregate covariates.
<<>>=
aggdata.sub <- aggdata
aggdata.sub$y <- aggdata$y - tapply(indivdata$y, indivdata$group, sum)
aggdata.sub$N <- aggdata.sub$N - 10
@ 
Again, we fit the correct model to the simulated data, with two
contextual covariates and two individual binary covariates.  The
individual-level regression model is given in the \Rfunarg{iformula}
argument.  The name of the individual-level dataset, in which the
variables in the individual-level model should appear, is given in the
\Rfunarg{idata} argument.
<<>>=
agg.indiv.eco <- eco(cbind(y, N) ~ deprivation + mean.income,
               binary = ~ nonwhite + smoke,
               iformula = y ~ deprivation + mean.income + nonwhite + smoke, 
               data = aggdata.sub, idata=sim1$idata)
agg.indiv.eco
@ 
In this example, combining with the individual sample data does not
noticeably improve the precision of the estimates.

\subsubsection{Importance of the between-area exposure contrasts}

Suppose now that we simulate data with a much lower between-area
exposure variance, such as a uniform(0, 0.2) distribution for
proportion of non-white ethnicity and uniform(0.1, 0.3) for proportion
of smokers.  The aggregate data now contain less information about the
individual-level effects.  The amount of individual-level information
in ecological data decreases as the between-area to the within-area
variability of the exposure decreases.  When there are low exposure
contrasts between areas, inference may be improved by combining the
ecological data with individual-level data~\cite{improving:eco}.

When fitting the true individual model to the aggregate data alone, we
obtain highly imprecise estimates for the individual-level effects.  
<<>>=
phi <- cbind(nonwhite = runif(ng, 0, 0.2), smoke = runif(ng, 0.1, 0.3))
sim.df <- as.data.frame(cbind(ctx, phi))
sim1 <- sim.eco(N, ctx=~deprivation+mean.income, binary=~nonwhite+smoke, 
                data = sim.df,  mu=mu, alpha.c=alpha.c, alpha=alpha, 
                isam=10)
aggdata <- as.data.frame(cbind(y=sim1$y, N=N, sim.df))
indivdata <- sim1$idata
agg.eco <- eco(cbind(y, N) ~ deprivation + mean.income,
               binary = ~ nonwhite + smoke,  data = aggdata)
agg.eco
@ 
To be able to estimate the individual-level effects more accurately,
we combine the aggregate data with the individual-level sample data.
<<>>=
aggdata.sub <- aggdata
aggdata.sub$y <- aggdata$y - tapply(indivdata$y, indivdata$group, sum)
aggdata.sub$N <- aggdata.sub$N - 10
agg.indiv.eco <- eco(cbind(y, N) ~ deprivation + mean.income,
               binary = ~ nonwhite + smoke,
               iformula = y ~ deprivation + mean.income + nonwhite + smoke, 
               data = aggdata.sub, idata=indivdata)
agg.indiv.eco
@ 
This raises the question of whether the aggregate data do contribute
any information in this case, and whether we can do just as well by
analysing the individual data alone.  But we find that precision is
lower when only the individual data are included. 

<<>>=
indiv.eco <- eco(iformula = y ~ deprivation + mean.income + nonwhite + smoke, 
               idata=indivdata)
indiv.eco
@ 


\subsubsection{Normally-distributed covariates}

We now add a continuous covariate to the simulated data, representing
estimated exposure to air pollution.  This will have a constant
within-area standard deviation of 2, and within-area means varying
around 1.24 with between-area standard deviation 0.1.  A disease
outcome is simulated with an odds ratio of 1.2 for one unit of
pollution exposured, and the same odds ratios as before for the other
covariates.

<<>>=
phi <- cbind(nonwhite = runif(ng), smoke = runif(ng))
sim.df <- as.data.frame(cbind(ctx, phi))
sim.df$poll <- rnorm(ng, 1.24, 0.1)
sim.df$poll.sd <- rep(0.2, ng)
sim1 <- sim.eco(N, ctx=~deprivation+mean.income, binary=~nonwhite+smoke,
                m=sim.df["poll"], S=sim.df["poll.sd"], beta=log(2), 
                data = sim.df,  mu=mu, alpha.c=alpha.c, alpha=alpha, 
                isam=10)
aggdata <- as.data.frame(cbind(y=sim1$y, N=N, sim.df))
aggdata[1:5,]
@ 
The area mean of the covariate is called \Robject{poll} in the
aggregate dataset, and the area standard deviation is called
\Robject{poll.sd}.  We now model these data as before, including
pollution as another individual-level covariate in the model.

<<>>=
agg.eco <- eco(cbind(y, N) ~ deprivation + mean.income, 
               normal= ~ poll, norm.var=poll.sd, 
               binary = ~ nonwhite + smoke,  data = aggdata)
agg.eco
@ 

The \Rfunarg{normal} argument to \Rfunction{eco} is a formula whose
right-hand side should contain variables denoting the group-level
means of the normally-distributed covariates.  These covariates will
then be fitted as individual-level effects, using a model of the form
of equation~(\ref{eq:normal:binomial}) by default,
or~(\ref{eq:normal:poisson}) if the outcome is rare and
\Rfunarg{model = ``poisson''} is specified.  The \Rfunarg{norm.var} is
used to supply the corresponding group-level variances.

The true odds ratio of 2 for pollution is fairly well estimated.  Now
suppose the pollution data had a lower ratio of between-area to
within-area standard deviation.

<<>>=
sim.df$poll <- rnorm(ng, 1.24, 0.1)
sim.df$poll.sd <- rep(0.2, ng)
sim1 <- sim.eco(N, ctx=~deprivation+mean.income, binary=~nonwhite+smoke,
                m=sim.df["poll"], S=sim.df["poll.sd"], beta=log(2), 
                data = sim.df,  mu=mu, alpha.c=alpha.c, alpha=alpha, isam=10)
aggdata <- as.data.frame(cbind(y=sim1$y, N=N, sim.df))
agg.eco <- eco(cbind(y, N) ~ deprivation + mean.income, 
               normal= ~ poll, norm.var=poll.sd, 
               binary = ~ nonwhite + smoke,  data = aggdata)
agg.eco
@ 
Now the confidence interval for the pollution odds ratio estimate is
much wider, as there is less information in the aggregate data.
However, the estimate is not biased.  Again, to improve precision we
can incorporate the individual-level data.  Remember to include
pollution in the individual level model \Rfunarg{iformula}. 

<<>>=
indivdata <- sim1$idata
indivdata[1:5,]
aggdata.sub <- aggdata
aggdata.sub$y <- aggdata$y - tapply(indivdata$y, indivdata$group, sum)
aggdata.sub$N <- aggdata.sub$N - 10
agg.indiv.eco <- eco(cbind(y, N) ~ deprivation + mean.income, 
                     normal= ~ poll, norm.var=poll.sd, 
                     binary = ~ nonwhite + smoke,  data = aggdata.sub,
                     iformula = y ~ deprivation + mean.income + 
                            nonwhite + smoke + poll, 
                     idata=indivdata
                     )
agg.indiv.eco
@ 
The precision of the estimate for pollution is improved. 


\subsubsection{Within-area distribution of binary covariates} 

To build the model~(\ref{eq:integral}) with more than one binary
covariate, by default \Rfunction{eco} assumes that the covariates are
independent within areas. Often this assumption is not appropriate,
especially when considering, for example, socio-economically related
factors.

To account for the joint within-area distribution of a set of binary
or categorical covariates, use the \Rfunarg{cross} argument to
\Rfunction{eco}.  This should be a matrix containing the same number
of rows as the aggregate data, and number of columns equal to the
distinct number of covariate categories into which an individual can
belong.  For full details of how to specify \Rfunarg{cross}, refer to
the help page for the \Rfunction{eco} function.  The \Rfunarg{cross}
needs to be calculated by the user before calling \Rfunction{eco}.
Individual data may be required to estimate \Rfunarg{cross}, as
typical census data do not give detailed cross-classification tables.

This is now illustrated with a hypothetical example.  We introduce
another covariate called \Robject{soclass} into the \Robject{aggdata}
data representing the proportion of individuals in a lower social
class.  This is likely to be correlated with smoking at both aggregate
and individual level.  Suppose that we have an information from an
individual-level survey, suggesting that an individual is twice as
likely to smoke if in a lower social class. We wish to use this
information to construct a matrix with 100 rows and four columns,
representing estimates of the proportion of individuals who are in
each of four categories:

\begin{enumerate}
\item neither smoke nor are in the lower social class ($p_{00}$)
\item smoke but are not in the lower social class  ($p_{10}$)
\item do not smoke but are in the lower social class ($p_{01}$)
\item smoke and are in the lower social class.  ($p_{11}$)
\end{enumerate}
Let $p_A$ be the proportion of smokers, and $p_B$ be the proportion of
individuals in the lower social class, in an area.  We know that
$p_{10} + p_{11} = p_A$, $p_{01} + p_{11} = p_B$ and, from the survey,
$(p_{11}/p_B) / (p_{10}/(1 - p_B)) = 2$.  This leads, for example, to
\[
p_{11} = 2 p_A p_B / (1 + p_B)
\]
This is enough information to construct all the estimated cross-classification probabilities, 
as illustrated by the following R code. 
<<>>=
aggdata$soclass <- plogis(qlogis(aggdata$smoke) + runif(ng, -1, 1))
pa <- aggdata$smoke
pb <- aggdata$soclass
p11 <- pa*pb* 2 /(1 + pb)
p10 <- pa - p11
p01 <- pb - p11
p00 <- 1 - (p01 + p10 + p11)
cross <- cbind(p00, p10, p01, p11)
cross[1:5,]
@
This is the cross-classification matrix that we can supply to
\Rfunction{eco} if we wanted to construct an ecological model
including both smoking and social class, for example
<<results=hide>>=
eco(cbind(y, N) ~ 1, binary = ~ smoke + soclass, cross=cross, data=aggdata)
@
To account for the joint within-area distribution of a set of
continuous covariates, use the \Rfunarg{norm.var} argument to
\Rfunction{eco} to specify the joint covariance matrix of the
covariates, assumed normally distributed, for each area.  For further
details of how to specify \Rfunarg{norm.var} in this way, refer to the
help page for the \Rfunction{eco} function.

\Rpackage{ecoreg} does not support specifying the within-area
covariance between binary and continuous covariates.  These are
assumed to be independent in the model.


\subsubsection{Stratification} 

Suppose that outcome data and covariate data are available by age and 
To account for differing disease risks in strata defined by age and
sex, using fixed offsets determined from the whole population (as in
equations \ref{eq:indiv} and \ref{eq:q}), use the \Rfunarg{strata},
\Rfunarg{pstrata} and \Rfunarg{istrata} arguments to \Rfunction{eco}.

\begin{itemize}
\item \Rfunarg{pstrata} should be a vector with one element for each
  stratum, giving the assumed baseline outcome probabilities for the
  strata.

\item \Rfunarg{strata} should be a matrix with the same number of rows
  as the aggregate data.  Rows represent areas, and columns represent
  the strata occupancy \emph{proportions} for those areas (which are
  used as estimates of the observed occupancy \emph{probabilities}).
  Alternatively, to account for within-area correlation between strata
  membership and binary covariate status, the cross-classification
  between strata and covariates can be specified in the
  \Rfunarg{cross} argument.  See the help page to \Rfunction{eco}.
  
\item If individual data are modelled, \Rfunarg{istrata} should be a
  variable containing the individual-level variable indicating the
  stratum an individual occupies. This should be a factor, whose
  levels correspond to the columns of the matrix \Rfunarg{strata}.
\end{itemize}


\subsubsection{Categorical covariates} 

As well as binary covariates, categorical covariates can also be
fitted as individual-level predictors.  The aggregate data for
categorical covariates must be supplied separately from the main
aggregate dataset, in the \Rfunarg{categorical} argument to
\Rfunction{eco}.  See the help page for \Rfunction{eco}.  In practice,
there is not likely to be enough information in ecological data for
successful ecological inference on categorical variables with large
numbers of categories.





\subsubsection{Random effects models}

By default, \Rfunction{eco} assumes the baseline risk $\mu_i$
(equation~\ref{eq:indiv}) is constant $\mu$ between areas $i$.
Optionally, \Rfunction{eco} can also fit $\mu_i$ as a
normally-distributed random effect.  If \Rfunarg{random=TRUE} is
specified in the call to \Rfunction{eco}, an area-level random
intercept is included in the model.  In this case the data should
indicate which area each row of the data corresponds to.  In the
individual data, \Rfunarg{igroups} should give the name of a variable
containing the group identifiers of the individual-level data.  In the
aggregate data, by default, the groups are the row numbers of the
dataset.  Alternatively, \Rfunarg{groups} specifies a group-level
variable containing the group identifiers to be matched with the
groups given in \Rfunarg{igroups}.

\Rfunction{eco} uses adaptive Gauss-Hermite integration
\cite{liu:pierce} to fit random effects models.  The Gauss-Hermite
integration can be controlled by the arguments \Rfunarg{gh.points} and
\Rfunarg{iter.adapt} to \Rfunction{eco}.  \Rfunarg{gh.points} gives
the number of points to use for quadrature, while \Rfunarg{iter.adapt}
gives the number of iterations to use for the adaptive phase of the
algorithm.

Random effects model fitting is relatively slow, and it may be useful
to view the progress of the model fitting by specifying a
\Rfunction{control} argument, such as \Rfunction{control=list(trace=1,
  REPORT=1)}. This is passed from \Rfunction{eco} to
\Rfunction{optim}, the R function which performs optimisation of the
likelihood. See \Rfunction{help(optim)} for further options to control
optimisation.


\section{Warnings and limitations} 

\begin{itemize}
  
\item It is easy to over-fit models, especially with several
  covariates.  Often there is not enough information available in
  aggregate data.
  
\item When fitting many covariates, it is essential to account for
  their within-area distribution.
  
\item Continuous covariates must be normally-distributed or able to be
  transformed to normality. 
  
\item Only limited error-checking is performed at the moment.
  \Rfunction{eco} may fail with an incomprehensible error message if
  the model or data are specified wrongly or inconsistently.

\end{itemize}



\section{\Rpackage{eco} reference guide}

The R help page for \Rfunction{eco} gives details of all the allowed
arguments and options to the \Rfunction{eco} function.  To view this
online in R, type:

<<eval=FALSE>>=
help(eco)
@ 
Similarly, all other functions in the package have help pages,
which should always be consulted in case of doubt about how to call
them.  The web-browser based help interface may be convenient - type

<<eval=FALSE>>=
help.start()
@ 
and navigate to \textsf{Packages} $\ldots$ \textsf{ecoreg}, which brings
up a list of all the functions in the package with links to their
documentation, and a link to this manual in PDF format.



\section{Similar software}

\begin{itemize}
  
\item WinBUGS (\texttt{http://www.mrc-bsu.cam.ac.uk/bugs}) can be used
  to fit these models from a Bayesian perspective using Markov Chain
  Monte Carlo simulation.  While computationally slower, this approach
  is amenable to extension to account for complexities such as random
  intercepts, random coefficients, spatial correlation and measurement
  error.  WinBUGS files containing worked examples of a range of these
  models, using methods described by Jackson
  \etal~\cite{improving:eco}~\cite{my:hrr} are provided at
  \texttt{http://www.bias-project.org.uk/software}.

\item The R package \Rpackage{eiPack} implements ecological inference
  for R $\times$ C contingency tables.
  
\item The R package \Rpackage{MCMCpack}, available from CRAN,
  implements ecological inference for $2 \times 2$ tables using a
  Bayesian hierarchical model described by Wakefield
  \cite{wakefield2by2}.
  
\item The R package \Rpackage{eco}, available from CRAN, implements
  ecological inference for $2 \times 2$ tables, using methods
  described by Imai and Lu~\cite{imai:lu}.
  
\item \emph{EI} and \emph{EzI} \cite{king:ei} by Kenneth Benoit and
  Gary King, implementing methods from King~\cite{king:solution}.

\end{itemize}




\bibliography{abbrev,ecoreg}

\end{document}
