ei.reg: Ecological regression

Description

Estimate an ecological regression using least squares.

Usage

ei.reg(formula, data, ...)

Arguments

formula

An R formula object of the form

cbind(c1, c2, ...)
    ~ cbind(r1, r2, ...)

data

data frame containing the variables specified in formula

...

Additional arguments passed to lm.

Value

call: the call to ei.reg
coefficients: an $R x C$ matrix of estimated population cell fractions
se: an $R x C$ matrix of standard errors for coefficients.
cov.matrices: A list of the $C$ scaled variance-covariance matrices for each of the ecological regressions

Details

For $i in 1,...,C$, C regressions of the form c_i ~ cbind(r1, r2, ...) are performed.

These regressions make use of the accounting identities and the constancy assumption, that $beta_rci = beta_rc$ for all $i$.

The accounting identities include

--defining the population cell fractions $beta_rc$ such that $sum_{c=1}^{C} beta_rc = 1$ for every $r$
--$sum_{c=1}^{C} beta_rci = 1$ for $r = 1,...,R$ and $i = 1,...,n$
--$T_ci = sum_{r=1}^R beta_rci X_ri$ for $c = 1,...,C$ and $i = 1,...,n$

Then regressing $$T_{ci} = \beta_{rc} X_{ri} + \epsilon_{ci}$$ for $c = 1,...C$ recovers the population parameters $beta_rc$ when the standard linear regression assumptions apply, including $E[epsilon_ci] = 0$ and $Var[epsilon_ci] = sigma_c^2$ for all $i$.

References

Leo Goodman. 1953. ``Ecological Regressions and the Behavior of Individuals.'' American Sociological Review 18:663--664.