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Last data update: 2014.03.03

R: Frechet bounds of cells in a contingency table

Frechet.bounds.cat

R Documentation

Frechet bounds of cells in a contingency table

Description

This function permits to derive the bounds for cell probabilities of the table Y vs. Z starting from the marginal tables (X vs. Y), (X vs. Z) and the joint distribution of the X variables.

Usage

Frechet.bounds.cat(tab.x, tab.xy, tab.xz, print.f="tables", tol= 0.001)

Arguments

`tab.x`	A R table crossing the X variables. This table must be obtained by using the function `xtabs` or `table`, e.g. `tab.x <- xtabs(~x1+x2+x3, data=data.all)`. When `tab.x = NULL` then only `tab.xy` and `tab.xz` must be supplied.
`tab.xy`	A R table of X vs. Y variable. This table must be obtained by using the function `xtabs` or `table`, e.g. `table.xy <- xtabs(~x1+x2+x3+y, data=data.A)`. A single categorical Y variable is allowed. One or more categorical variables can be considered as X variables (common variables). Obviously, the same X variables in `tab.x` must be available in `tab.xy`. Moreover, it is assumed that the joint distribution of the X variables computed from `tab.xy` is equal to `tab.x`; a warning appears if this is not true (see argument `tol`). When `tab.x = NULL` then `tab.xy` should be a one–dimensional table providing the marginal distribution of the Y variable.
`tab.xz`	A R table of X vs. Z variable. This table must be obtained by using the function `xtabs` or `table`, e.g. `tab.xz <- xtabs(~x1+x2+x3+z, data=data.B)`. A single categorical Z variable is allowed. One or more categorical variables can be considered as X variables (common variables). The same X variables in `tab.x` must be available in `tab.xz`. Moreover, it is assumed that the joint distribution of the X variables computed from `tab.xz` is equal to `tab.x`; a warning appears if this is not true (see argument `tol`). When `tab.x = NULL` then `tab.xz` should be a one–dimensional table providing the marginal distribution of the Z variable.
`print.f`	A string: when `print.f="tables"` (default) all the cells' estimates will be saved as tables in a list. On the contrary, if `print.f="data.frame"`, they will be saved as columns of a data.frame.
`tol`	Tolerance used in comparing joint distributions as far as X variables are considered (default `tol= 0.001`); the joint distribution of the X variables computed from `tab.xy` and `tab.xz` should be equal to that in `tab.x`.

Details

This function permits to compute the Frechet bounds for the relative frequencies in the contingency table of Y vs. Z, starting from the distributions P(Y|X), P(Z|X) and P(X). The bounds for the relative frequencies p(y=j,z=k) in the table Y vs. Z are:

p(Y=j,Z=k) >= sum_i(p(X=i) * max(0; p(Y=j|X=i) + p(Z=k|X=i) - 1) )

p(Y=j,Z=k) <= sum_i(p(X=i) * min(p(Y=j|X=i),p(Z=k|X=i)))

The relative frequencies p(X=i)=n_i/n are computed from the frequencies in tab.x;
the relative frequencies p(Y=j|X=i)=n_ij/n_i. are computed from the tab.xy,
finally, p(Z=k|X=i)=n_ik/n_i. are derived from tab.xy.

It is assumed that the marginal distribution of the X variables is the same in all the input tables: tab.x, tab.xy and tab.xz. If this is not true a warning message will appear.

Note that the cells bounds for the relative frequencies in the contingency table of Y vs. Z are computed also without considering the X variables:

max(0;p(Y=j)+p(Z=k)-1) <= p(Y=j,Z=k) <= min(p(Y=j);p(Z=k))

These bounds represent the unique output when tab.x = NULL.

Finally, the contingency table of Y vs. Z estimated under the Conditional Independence Assumption (CIA) is obtained by considering:

p(Y=i,Z=k) = p(Y=j|X=i)*p(Z=k|X=i)*p(X=i)

When tab.x = NULL then it is also provided the expected table under the assumption of independence between Y and Z:

p(Y=i,Z=k) = p(Y=j)*p(Z=k)*

Note that in the presence of too many cells with 0s in the input contingency tables is an indication of sparseness; this is an unappealing situation when estimating the cells' relative frequencies needed to derive the bounds; in such cases the corresponding results may be unreliable. A possible alternative way of working consists in estimating the required parameters by considering a pseudo-Bayes estimator (see pBayes); in practice the input tab.x, tab.xy and tab.xz should be the ones provided by the pBayes function.

Value

When print.f="tables" (default) a list with the following components:

`low.u`	The estimated lower bounds for the relative frequencies in the table Y vs. Z without conditioning on the X variables.
`up.u`	The estimated upper bounds for the relative frequencies in the table Y vs. Z without conditioning on the X variables.
`CIA`	The estimated relative frequencies in the table Y vs. Z under the Conditional Independence Assumption (CIA).
`low.cx`	The estimated lower bounds for the relative frequencies in the table Y vs. Z when conditioning on the X variables.
`up.cx`	The estimated upper bounds for the relative frequencies in the table Y vs. Z when conditioning on the X variables.
`uncertainty`	The uncertainty associated to input data, summarized in terms of average width of uncertainty bounds with and without conditioning on the X variables

When print.f="data.frame" the output list contains just two components:

`bounds`	A data.frame whose columns reports the estimated uncertainty bounds.
`uncertainty`	The uncertainty associated to input data, summarized in terms of average width of uncertainty bounds with and without conditioning on the X variables

Author(s)

Marcello D'Orazio madorazi@istat.it

References

Ballin, M., D'Orazio, M., Di Zio, M., Scanu, M. and Torelli, N. (2009) “Statistical Matching of Two Surveys with a Common Subset”. Working Paper, 124. Dip. Scienze Economiche e Statistiche, Univ. di Trieste, Trieste.

D'Orazio, M., Di Zio, M. and Scanu, M. (2006). Statistical Matching: Theory and Practice. Wiley, Chichester.

Examples


data(quine, package="MASS") #loads quine from MASS
str(quine)

# split quine in two subsets
set.seed(7654)
lab.A <- sample(nrow(quine), 70, replace=TRUE)
quine.A <- quine[lab.A, 1:3]
quine.B <- quine[-lab.A, 2:4]

# compute the tables required by Frechet.bounds.cat()
freq.x <- xtabs(~Sex+Age, data=quine.A)
freq.xy <- xtabs(~Sex+Age+Eth, data=quine.A)
freq.xz <- xtabs(~Sex+Age+Lrn, data=quine.B)

# apply Frechet.bounds.cat()
bounds.yz <- Frechet.bounds.cat(tab.x=freq.x, tab.xy=freq.xy,
        tab.xz=freq.xz, print.f="data.frame")
bounds.yz

#compare marg. distribution of Xs in A and B
comp.prop(p1=margin.table(freq.xy,c(1,2)), p2=margin.table(freq.xz,c(1,2)), 
          n1=nrow(quine.A), n2=nrow(quine.B))

# harmonize distr. of Sex vs. Age before applying
# Frechet.bounds.cat()

N <- nrow(quine)
quine.A$pop <- N
quine.A$f <- N/70 # reciprocal sampling fraction
quine.B$pop <- N
quine.B$f <- N/(N-70)

# derive the table of Sex vs. Age related to the whole data set
tot.sex.age <- colSums(model.matrix(~Sex*Age-1, data=quine))
tot.sex.age

# use hamonize.x() to harmonize the Sex vs. Age between
# quine.A and quine.B

# create svydesign objects
require(survey)
svy.qA <- svydesign(~1, weights=~f, fpc=~pop, data=quine.A)
svy.qB <- svydesign(~1, weights=~f, fpc=~pop, data=quine.B)

# apply harmonize.x 
out.hz <- harmonize.x(svy.A=svy.qA, svy.B=svy.qB, form.x=~Sex*Age-1, x.tot=tot.sex.age)

# compute the new tables required by Frechet.bounds.cat()
freq.x <- xtabs(out.hz$weights.A~Sex+Age, data=quine.A)
freq.xy <- xtabs(out.hz$weights.A~Sex+Age+Eth, data=quine.A)
freq.xz <- xtabs(out.hz$weights.B~Sex+Age+Lrn, data=quine.B)

#compare marg. distribution of Xs in A and B
comp.prop(p1=margin.table(freq.xy,c(1,2)), p2=margin.table(freq.xz,c(1,2)), 
          n1=nrow(quine.A), n2=nrow(quine.B))

# apply Frechet.bounds.cat()
bounds.yz <- Frechet.bounds.cat(tab.x=freq.x, tab.xy=freq.xy,
        tab.xz=freq.xz, print.f="data.frame")
bounds.yz