Description Usage Arguments Details Value Author(s) References See Also Examples
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.
1  ecoBD(formula, data = parent.frame(), N = NULL)

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

N 
An optional variable representing the size of the unit; e.g., the
total number of voters. If 
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_{i11}  n_{i12}  ...  n_{i1C}  n_{i1.} 
n_{i21}  n_{i22}  ...  n_{i2C}  n_{i2.} 
...  ...  ...  ...  ... 
n_{iR1}  n_{iR2}  ...  n_{iRC}  n_{iR.} 
n_{i.1}  n_{i.2}  ...  n_{i.C}  N_i 
where n_{nr.} and n_{i.c} represent the observed margins, N_i represents the size of the table, and n_{irc} are unknown variables. Note that for each i, the following deterministic relationships hold; n_{ir.} = ∑_{c=1}^C n_{irc} for r=1,…,R, and n_{i.c}=∑_{r=1}^R n_{irc} for c=1,…,C. Then, each of the unknown inner cells can be bounded in the following manner,
\max(0, n_{ir.}+n_{i.c}N_i) ≤ n_{irc} ≤ \min(n_{ir.}, n_{i.c}).
If the size of tables, N
, is
provided,
An R \times C ecological table in the form of proportions:
W_{i11}  W_{i12}  ...  W_{i1C}  Y_{i1} 
W_{i21}  W_{i22}  ...  W_{i2C}  Y_{i2} 
...  ...  ...  ...  ... 
W_{iR1}  W_{iR2}  ...  W_{iRC}  Y_{iR} 
X_{i1}  X_{i2}  ...  X_{iC} 
where Y_{ir} and X_{ic} represent the observed margins, and W_{irc} are unknown variables. Note that for each i, the following deterministic relationships hold; Y_{ir} = ∑_{c=1}^C X_{ic} W_{irc} for r=1,…,R, and ∑_{r=1}^R W_{irc}=1 for c=1,…,C. Then, each of the inner cells of the table can be bounded in the following manner,
\max(0, (X_{ic} + Y_{ir}1)/X_{ic}) ≤ W_{irc} ≤ \min(1, Y_{ir}/X_{ir}).
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
.
Kosuke Imai, Department of Politics, Princeton University [email protected], http://imai.princeton.edu/; Ying Lu, Center for Promoting Research Involving Innovative Statistical Methodology (PRIISM), New York University [email protected]
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. 123. available at http://imai.princeton.edu/software/eco.html
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. 4169. available at http://imai.princeton.edu/research/eiall.html
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