ecoBD: Calculating the Bounds for Ecological Inference in RxC Tables

Description

ecoBD is used to calculate the bounds for missing internal cells of $R \times C$ ecological table. The data can be entered either in the form of counts or proportions.

Usage

ecoBD(formula, data = parent.frame(), N = NULL)

Value

An object of class ecoBD containing the following elements (When three dimensional arrays are used, the first dimension indexes the observations, the second dimension indexes the row numbers, and the third dimension indexes the column numbers):

call: The matched call.
X: A matrix of the observed row margin, $X$ .
Y: A matrix of the observed column margin, $Y$ .
N: A vector of the size of ecological tables, $N$ .
aggWmin: A three dimensional array of aggregate lower bounds for proportions.
aggWmax: A three dimensional array of aggregate upper bounds for proportions.
Wmin: A three dimensional array of lower bounds for proportions.
Wmax: A three dimensional array of upper bounds for proportions.
Nmin: A three dimensional array of lower bounds for counts.
Nmax: A three dimensional array of upper bounds for counts.

The object can be printed through print.ecoBD.

Arguments

formula: A symbolic description of ecological table to be used, specifying the column and row margins of $R \times C$ ecological tables. Details and specific examples are given below.
data: An optional data frame in which to interpret the variables in formula. The default is the environment in which ecoBD is called.
N: An optional variable representing the size of the unit; e.g., the total number of voters. If formula is entered as counts and the last row and/or column is omitted, this input is necessary.

Details

The data may be entered either in the form of counts or proportions. If proportions are used, formula may omit the last row and/or column of tables, which can be calculated from the remaining margins. For example, Y ~ X specifies Y as the first column margin and X as the first row margin in $2 \times 2$ tables. If counts are used, formula may omit the last row and/or column margin of the table only if N is supplied. In this example, the columns will be labeled as X and not X, and the rows will be labeled as Y and not Y.

For larger tables, one can use cbind() and +. For example, cbind(Y1, Y2, Y3) ~ X1 + X2 + X3 + X4) specifies $3 \times 4$ tables.

An $R \times C$ ecological table in the form of counts:

$n_{i 11}$	$n_{i 12}$	...	$n_{i 1 C}$	$n_{i 1.}$
$n_{i 21}$	$n_{i 22}$	...	$n_{i 2 C}$	$n_{i 2.}$
...	...	...	...	...
$n_{i R 1}$	$n_{i R 2}$	...	$n_{i R C}$	$n_{i R .}$
$n_{i .1}$	$n_{i .2}$	...	$n_{i . C}$	$N_{i}$

where $n_{n r .}$ and $n_{i . c}$ represent the observed margins, $N_{i}$ represents the size of the table, and $n_{i r c}$ are unknown variables. Note that for each $i$ , the following deterministic relationships hold; $n_{i r .} = \sum_{c = 1}^{C} n_{i r c}$ for $r = 1, \dots, R$ , and $n_{i . c} = \sum_{r = 1}^{R} n_{i r c}$ for $c = 1, \dots, C$ . Then, each of the unknown inner cells can be bounded in the following manner, $max (0, n_{i r .} + n_{i . c} - N_{i}) \leq n_{i r c} \leq min (n_{i r .}, n_{i . c}) .$ If the size of tables, N, is provided,

An $R \times C$ ecological table in the form of proportions:

$W_{i 11}$	$W_{i 12}$	...	$W_{i 1 C}$	$Y_{i 1}$
$W_{i 21}$	$W_{i 22}$	...	$W_{i 2 C}$	$Y_{i 2}$
...	...	...	...	...
$W_{i R 1}$	$W_{i R 2}$	...	$W_{i R C}$	$Y_{i R}$
$X_{i 1}$	$X_{i 2}$	...	$X_{i C}$

where $Y_{i r}$ and $X_{i c}$ represent the observed margins, and $W_{i r c}$ are unknown variables. Note that for each $i$ , the following deterministic relationships hold; $Y_{i r} = \sum_{c = 1}^{C} X_{i c} W_{i r c}$ for $r = 1, \dots, R$ , and $\sum_{r = 1}^{R} W_{i r c} = 1$ for $c = 1, \dots, C$ . Then, each of the inner cells of the table can be bounded in the following manner, $max (0, (X_{i c} + Y_{i r} - 1) / X_{i c}) \leq W_{i r c} \leq min (1, Y_{i r} / X_{i r}) .$

References

Imai, Kosuke, Ying Lu and Aaron Strauss. (2011) “eco: R Package for Ecological Inference in 2x2 Tables” Journal of Statistical Software, Vol. 42, No. 5, pp. 1-23.

Imai, Kosuke, Ying Lu and Aaron Strauss. (2008) “Bayesian and Likelihood Inference for 2 x 2 Ecological Tables: An Incomplete Data Approach” Political Analysis, Vol. 16, No. 1, (Winter), pp. 41-69.

Examples

Run this code



## load the registration data
data(reg)

## calculate the bounds
res <- ecoBD(Y ~ X, N = N, data = reg)
## print the results
print(res)

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