# comb.samples: Statistical Matching of data from complex sample surveys In StatMatch: Statistical Matching or Data Fusion

 comb.samples R Documentation

## Statistical Matching of data from complex sample surveys

### Description

This function permits to cross-tabulate two categorical variables, Y and Z, observed separately in two independent surveys (Y is collected in survey A and Z is collected in survey B) carried out on the same target population. The two surveys share a number of common variables X. When it is available a third survey C, carried on the same population, in which both Y and Z are collected, these data are used as a source of auxiliary information.

The statistical matching is performed by carrying out calibration of the survey weights, as suggested in Renssen (1998).

It is possible also to use the function to derive the estimates that a unit falls in one of the categories of the target variable (estimation are based on Liner Probability Models and are obtained as a by-product of the Renssen's method).

### Usage

```comb.samples(svy.A, svy.B, svy.C=NULL, y.lab, z.lab, form.x,
estimation=NULL, micro=FALSE, ...)
```

### Arguments

 `svy.A` A `svydesign` R object that stores the data collected in the survey A and all the information concerning the corresponding sampling design. This object can be created by using the function `svydesign` in the package survey. All the variables specified in `form.x` and by `y.lab` must be available in `svy.A`. `svy.B` A `svydesign` R object that stores the data collected in the survey B and all the information concerning the corresponding sampling design. This object can be created by using the function `svydesign` in the package survey. All the variables specified in `form.x` and by `z.lab` must be available in `svy.B`. `svy.C` A `svydesign` R object that stores the data collected in the the survey C and all the information concerning the corresponding sampling design. This object can be created by using the function `svydesign` in the package survey. When `svy.C=NULL` (default), i.e. no auxiliary information is available, the function returns an estimate of the contingency table of Y vs. Z under the Conditional Independence assumption (CIA) (see Details for major information). When `svy.C` is available, if `estimation="incomplete"` then it must contain at least `y.lab` and `z.lab` variables. On the contrary, when `estimation="synthetic"` all the variables specified in `form.x`, `y.lab` and `z.lab` must be available in `svy.C`. `y.lab` A string providing the name of the Y variable, available in survey A and in survey C (if available). The Y variable can be a categorical variable (`factor` in R) or a continuous one (in this latter case `z.lab` should be categorical). `z.lab` A string providing the name of the Z variable available in survey B and in survey C (if available). The Z variable can be a categorical variable (`factor` in R) or a continuous one (in this latter case `y.lab` should be categorical). `form.x` A R formula specifying the matching variables (subset of the X variables) collected in all the surveys, and how have to be considered in combining samples. For instance, `form.x=~x1+x2` means that the variables x1 and x2 have to be considered marginally without taking into account their cross-tabulation; just their marginal distribution is considered. In order to skip the intercept the formula has to be written as `form.x=~x1+x2-1`. When dealing with categorical variables, `form.x=~x1:x2-1` means that the the joint distribution of the two variables (table of x1 vs. x2) has to be considered. To better understand the usage of `form.x` see `model.matrix` (see also `formula`). Due to weights calibration features, it is preferable to work with categorical X variables. In some cases, procedure may be successful when a single continuous variable is considered jointly with one or more categorical variables but, often, it may be necessary to categorize the continuous variable (see Details). `estimation` A character string that identifies the method to be used to estimate the table of Y vs. Z when data from survey C are available. As suggested in Renssen (1998), two alternative methods are available: (i) Incomplete Two-Way Stratification (`estimation="incomplete"`, or `estimation="ITWS"`, the default one) and (ii) Synthetic Two-Way Stratification (`estimation="synthetic"` or `estimation="STWS"`). In the first case (`estimation="incomplete"`) only Y and Z variables must be available in `svy.C`. On the contrary, when `estimation="synthetic"` the survey C must contain all the X variables specified via `form.x`, as well as the Y and Z variables. See Details for further information. `micro` Logical, when `TRUE` predictions of Z in A and of Y in B are provided. In particular when Y and Z are both categorical variables it is provided the estimated probability that a unit falls in each of the categories of the given variable. These probabilities are estimated as a by-product of the whole procedure by considering Linear Probability Models, as suggested in Renssen (1998) (see Details) `...` Further arguments that may be necessary for calibration. In particular, the argument `calfun` allows to specify the calibration function: (i) `calfun="linear"` for linear calibration (default); (ii) `calfun="raking"` to rake the survey weights; and (iii) `calfun="logit"` for logit calibration. See `calibrate` for major details. Note that when `calfun="linear"` calibration may return negative weights. Generally speaking, in sample surveys weights are expected to be greater than or equal to 1, i.e. `bounds=c(1, Inf)`. The number of iterations used in calibration can be modified by using the argument `maxit` (by default `maxit=50`). See `calibrate` for further details.

### Details

This function estimates the contingency table of Y vs. Z by performing a series of calibrations of the survey weights. In practice the estimation is carried out on data in survey C by exploiting all the information from surveys A and B. When survey C is not available the table of Y vs. Z is estimated under the assumption of Conditional Independence (CIA), i.e. p(Y,X)=p(Y|X)*p(Z|X)*p(X).

When data from survey C are available (Renssen, 1998), the table of Y vs. Z can be estimated by: Incomplete Two-Way Stratification (ITWS) or Synthetic Two-Way Stratification (STWS). In the first case (ITWS) the weights of the units in survey C are calibrated so that the new weights allow to reproduce the marginal distributions of Y estimated on survey A, and that of Z estimated on survey B. Note that the distribution of the X variables in survey A and in survey B, must be harmonized before performing ITWS (see `harmonize.x`).

The Synthetic Two-Way Stratification allows to estimate the table of Y vs. Z by considering also the X variables observed in C. This method consists in correcting the table of Y vs. Z estimated under the CIA according to the relationship between Y and Z observed in survey C (for major details see Renssen, 1998).

When the argument `micro` is set to `TRUE` the function provides also `Z.A` and `Y.B`. The first data.frame has the same rows as `svy.A` and the number of columns equals the number of categories of the Z variable specified via `z.lab`. Each row provides the estimated probabilities of assuming a value in the various categories. The same happens for `Y.B` which presents the estimated probabilities of assuming a category of `y.lab` for each unit in B. The estimated probabilities are obtained by applying the linear probability models (for major details see Renssen, 1998). Unfortunately, such models may provide estimated probabilities less than 0 or greater than 1. Much caution should be used in using such predictions for practical purposes.

### Value

A R list with the results of the calibration procedure according to the input arguments.

 `yz.CIA` The table of Y (`y.lab`) vs. Z (`z.lab`) estimated under the Conditional Independence Assumption (CIA). `cal.C` The survey object `svy.C` after the calibration. Only when `svy.C` is provided. `yz.est` The table of Y (`y.lab`) vs. Z (`z.lab`) estimated under the method specified via `estimation` argument. Only when `svy.C` is provided. `Z.A` Only when `micro=TRUE`. It is a data frame with the same rows as in `svy.A` and the number of columns is equal to the number of categories of the variable `z.lab`. Each row provides the estimated probabilities for a unit being in the various categories of `z.lab`. `Y.B` Only when `micro=TRUE`. It is a data frame with the same rows as in `svy.B` and the number of columns is equal to the number of categories of `y.lab`. Each row provides the estimated probabilities for a unit being in the various categories of `y.lab`. `call` Stores the call to this function with all the values specified for the various arguments (`call=match.call()`).

### Author(s)

Marcello D'Orazio mdo.statmatch@gmail.com

### References

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

Renssen, R.H. (1998) “Use of Statistical Matching Techniques in Calibration Estimation”. Survey Methodology, 24, pp. 171–183.

`calibrate`, `svydesign`, `harmonize.x`

### Examples

```
data(quine, package="MASS") #loads quine from MASS
str(quine)
quine\$c.Days <- cut(quine\$Days, c(-1, seq(0,20,10),100))
table(quine\$c.Days)

# split quine in two subsets

suppressWarnings(RNGversion("3.5.0"))
set.seed(124)
lab.A <- sample(nrow(quine), 70, replace=TRUE)
quine.A <- quine[lab.A, c("Eth","Sex","Age","Lrn")]
quine.B <- quine[-lab.A, c("Eth","Sex","Age","c.Days")]

# create svydesign objects
require(survey)
quine.A\$f <- 70/nrow(quine) # sampling fraction
quine.B\$f <- (nrow(quine)-70)/nrow(quine)
svy.qA <- svydesign(~1, fpc=~f, data=quine.A)
svy.qB <- svydesign(~1, fpc=~f, data=quine.B)

# Harmonizazion wrt the joint distribution
# of ('Sex' x 'Age' x 'Eth')

# vector of population total known
# estimated from the full data set
# note the formula!
tot.m <- colSums(model.matrix(~Eth:Sex:Age-1, data=quine))
tot.m

out.hz <- harmonize.x(svy.A=svy.qA, svy.B=svy.qB, x.tot=tot.m,
form.x=~Eth:Sex:Age-1, cal.method="linear")

# estimation of 'Lrn' vs. 'c.Days' under the CIA

svy.qA.h <- out.hz\$cal.A
svy.qB.h <- out.hz\$cal.B

out.1 <- comb.samples(svy.A=svy.qA.h, svy.B=svy.qB.h,
svy.C=NULL, y.lab="Lrn", z.lab="c.Days",
form.x=~Eth:Sex:Age-1)

out.1\$yz.CIA

#
# incomplete two-way stratification

# select a sample C from quine
# and define a survey object

suppressWarnings(RNGversion("3.5.0"))
set.seed(4321)
lab.C <- sample(nrow(quine), 50, replace=TRUE)
quine.C <- quine[lab.C, c("Lrn","c.Days")]
quine.C\$f <- 50/nrow(quine) # sampling fraction
svy.qC <- svydesign(~1, fpc=~f, data=quine.C)

# call comb.samples
out.2 <- comb.samples(svy.A=svy.qA.h, svy.B=svy.qB.h,
svy.C=svy.qC, y.lab="Lrn", z.lab="c.Days",
form.x=~Eth:Sex:Age-1, estimation="incomplete",
calfun="linear", maxit=100)

summary(weights(out.2\$cal.C))
out.2\$yz.est # estimated table of 'Lrn' vs. 'c.Days'
# difference wrt the table 'Lrn' vs. 'c.Days' under CIA

# synthetic two-way stratification
# only macro estimation

quine.C <- quine[lab.C, ]
quine.C\$f <- 50/nrow(quine) # sampling fraction
svy.qC <- svydesign(~1, fpc=~f, data=quine.C)

out.3 <- comb.samples(svy.A=svy.qA.h, svy.B=svy.qB.h,
svy.C=svy.qC, y.lab="Lrn", z.lab="c.Days",
form.x=~Eth:Sex:Age-1, estimation="synthetic",
calfun="linear",bounds=c(.5,Inf), maxit=100)

summary(weights(out.3\$cal.C))

out.3\$yz.est # estimated table of 'Lrn' vs. 'c.Days'
# difference wrt the table of 'Lrn' vs. 'c.Days' under CIA
# diff wrt the table of 'Lrn' vs. 'c.Days' under incomplete 2ws

# synthetic two-way stratification
# with micro predictions

out.4 <- comb.samples(svy.A=svy.qA.h, svy.B=svy.qB.h,
svy.C=svy.qC, y.lab="Lrn", z.lab="c.Days",
form.x=~Eth:Sex:Age-1, estimation="synthetic",
micro=TRUE, calfun="linear",bounds=c(.5,Inf),
maxit=100)