1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
|
\name{ei.reg}
\alias{ei.reg}
\title{Ecological regression}
\description{ Estimate an ecological regression using least squares. }
\usage{
ei.reg(formula, data, ...)
}
\arguments{
\item{formula}{An R formula object of the form \code{cbind(c1, c2, ...)
~ cbind(r1, r2, ...)}}
\item{data}{data frame containing the variables specified in \code{formula}}
\item{\dots}{Additional arguments passed to \code{\link[stats]{lm}}. }
}
\value{A list containing
\item{call}{the call to \code{ei.reg}}
\item{coefficients}{an \eqn{R \times C}{R x C} matrix of estimated
population cell fractions}
\item{se}{an \eqn{R \times C}{R x C} matrix of standard errors
for \code{coefficients}.}
\item{cov.matrices}{A list of the \eqn{C}{C} scaled variance-covariance
matrices for each of the ecological regressions}
}
\details{
For \eqn{i \in 1,\ldots,C}{i in 1,...,C}, C regressions of the form
\code{c_i ~ cbind(r1, r2, ...)} are performed.
These regressions make use of the accounting identities
and the constancy assumption, that \eqn{\beta_{rci} =
\beta_{rc}}{beta_rci = beta_rc} for all \eqn{i}{i}.
The accounting identities include
\itemize{
\item{--}{defining the population cell fractions
\eqn{\beta_{rc}}{beta_rc} such that \eqn{\sum_{c=1}^{C} \beta_{rc}
= 1}{sum_{c=1}^{C} beta_rc = 1} for every \eqn{r}{r}}
\item{--}{\eqn{\sum_{c=1}^{C} \beta_{rci} = 1}{sum_{c=1}^{C} beta_rci =
1} for \eqn{r = 1, \ldots, R}{r = 1,...,R} and \eqn{i = 1, \ldots,
n}{i = 1,...,n}}
\item{--}{\eqn{T_{ci} = \sum_{r=1}^R \beta_{rci}X_{ri}}{T_ci = sum_{r=1}^R
beta_rci X_ri} for \eqn{c = 1,\ldots,C}{c = 1,...,C} and \eqn{i =
1\ldots,n}{i = 1,...,n}}
}
Then regressing \deqn{T_{ci} = \beta_{rc} X_{ri} + \epsilon_{ci}}{T_ci
= beta_rc X_ri + epsilon_ci} for \eqn{c = 1,\dots,C}{c = 1,...C}
recovers the population parameters \eqn{\beta_{rc}}{beta_rc} when the
standard linear regression assumptions apply, including
\eqn{E[\epsilon_{ci}] = 0}{E[epsilon_ci] = 0} and
\eqn{Var[\epsilon_{ci}] = \sigma_c^2}{Var[epsilon_ci] = sigma_c^2} for
all \eqn{i}{i}.
}
\references{
Leo Goodman. 1953. ``Ecological Regressions and the Behavior of Individuals.''
\emph{American Sociological Review} 18:663--664.
}
\author{
Olivia Lau <\email{olivia.lau@post.harvard.edu}> and Ryan T. Moore <\email{rtm@american.edu}>
}
\keyword{models}
|
