StatMatch (version 1.2.3)

Frechet.bounds.cat: 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= 0.0001)

Arguments

tab.x

A Rtable 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

tab.xy

A Rtable 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=

tab.xz

A Rtable 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=da

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= 0.0001); the joint distribution of the X variables computed from tab.xy and tab.xz should be equal to t

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

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 $p_{j,k}$ in the table Y vs. Z are:

$$p^{(low)}_{Y=j,Z=k} = \sum_{i} p_{X=i}\max (0; p_{Y=j|X=i} + p_{Z=k|X=i} - 1 )$$

$$p^{(up)}_{Y=j,Z=k} = \sum_{i} p_{X=i} \min ( p_{Y=j|X=i}; p_{Z=k|X=i})$$

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, $p_{Z=k|X=i}=n_{ik}/n_{k+}$ 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:

$$\max{0; p_{Y=j} + p_{Z=k} - 1} \leq p_{Y=j,Z=k} \leq \min { p_{Y=j}; p_{Z=k}}$$

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:

$$p_{Y=j,Z=k} = p_{Y=j|X=i} \times p_{Z=k|X=i} \times p_{X=i}.$$

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

$$p_{Y=j,Z=k} = p_{Y=j} \times p_{Z=k}.$$

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.

See Also

Fbwidths.by.x, harmonize.x

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