survey (version 4.0)

# calibrate: Calibration (GREG) estimators

## Description

Calibration, generalized raking, or GREG estimators generalise post-stratification and raking by calibrating a sample to the marginal totals of variables in a linear regression model. This function reweights the survey design and adds additional information that is used by `svyrecvar` to reduce the estimated standard errors.

## Usage

```calibrate(design,...)
# S3 method for survey.design2
calibrate(design, formula, population,
aggregate.stage=NULL, stage=0, variance=NULL,
bounds=c(-Inf,Inf), calfun=c("linear","raking","logit"),
maxit=50,epsilon=1e-7,verbose=FALSE,force=FALSE,trim=NULL,
bounds.const=FALSE, sparse=FALSE,...)
# S3 method for svyrep.design
calibrate(design, formula, population,compress=NA,
aggregate.index=NULL, variance=NULL, bounds=c(-Inf,Inf),
calfun=c("linear","raking","logit"),
maxit=50, epsilon=1e-7, verbose=FALSE,force=FALSE,trim=NULL,
bounds.const=FALSE, sparse=FALSE,...)
# S3 method for twophase
calibrate(design, phase=2,formula, population,
calfun=c("linear","raking","logit","rrz"),...)
grake(mm,ww,calfun,eta=rep(0,NCOL(mm)),bounds,population,epsilon,
verbose,maxit,variance=NULL)
cal_names(formula,design,...)```

## Arguments

design

Survey design object

formula

Model formula for calibration model, or list of formulas for each margin

population

Vectors of population column totals for the model matrix in the calibration model, or list of such vectors for each cluster, or list of tables for each margin. Required except for two-phase designs

compress

compress the resulting replicate weights if `TRUE` or if `NA` and weights were previously compressed

stage

See Details below

variance

Coefficients for variance in calibration model (heteroskedasticity parameters) (see Details below)

aggregate.stage

An integer. If not `NULL`, make calibration weights constant within sampling units at this stage.

aggregate.index

A vector or one-sided formula. If not `NULL`, make calibration weights constant within levels of this variable

bounds

Bounds for the calibration weights, optional except for `calfun="logit"`

bounds.const

Should be `TRUE` if `bounds` have been spcified as constant values rather than multiplicative values

trim

Weights outside this range will be trimmed to these bounds.

...

Options for other methods

calfun

Calibration function: see below

maxit

Number of iterations

epsilon

Tolerance in matching population total. Either a single number or a vector of the same length as `population`

verbose

Print lots of uninteresting information

force

Return an answer even if the specified accuracy was not achieved

phase

Phase of a two-phase design to calibrate (only `phase=2` currently implemented.)

mm

Model matrix

ww

Vector of weights

eta

Starting values for iteration

sparse

Use sparse matrices for faster computation

## Value

A survey design object.

## Details

The `formula` argument specifies a model matrix, and the `population` argument is the population column sums of this matrix. The function `cal_names` shows what the column names of this model matrix will be.

For the important special case where the calibration totals are (possibly overlapping) marginal tables of factor variables, as in classical raking, the `formula` and `population` arguments may be lists in the same format as the input to `rake`.

If the `population` argument has a names attribute it will be checked against the names produced by `model.matrix(formula)` and reordered if necessary. This protects against situations where the (locale-dependent) ordering of factor levels is not what you expected.

Numerical instabilities may result if the sampling weights in the `design` object are wrong by multiple orders of magnitude. The code now attempts to rescale the weights first, but it is better for the user to ensure that the scale is reasonable.

The `calibrate` function implements linear, bounded linear, raking, bounded raking, and logit calibration functions. All except unbounded linear calibration use the Newton-Raphson algorithm described by Deville et al (1993). This algorithm is exposed for other uses in the `grake` function. Unbounded linear calibration uses an algorithm that is less sensitive to collinearity. The calibration function may be specified as a string naming one of the three built-in functions or as an object of class `calfun`, allowing user-defined functions. See `make.calfun` for details.

The `bounds` argument can be specified as global upper and lower bounds e.g `bounds=c(0.5, 2)` or as a list with lower and upper vectors e.g. `bounds=list(lower=lower, upper=upper)`. This allows for individual boundary constraints for each unit. The lower and upper vectors must be the same length as the input data. The bounds can be specified as multiplicative values or constant values. If constant, `bounds.const` must be set to `TRUE`.

Calibration with bounds, or on highly collinear data, may fail. If `force=TRUE` the approximately calibrated design object will still be returned (useful for examining why it failed). A failure in calibrating a set of replicate weights when the sampling weights were successfully calibrated will give only a warning, not an error.

When calibration to the desired set of bounds is not possible, another option is to trim weights. To do this set `bounds` to a looser set of bounds for which calibration is achievable and set `trim` to the tighter bounds. Weights outside the bounds will be trimmed to the bounds, and the excess weight distributed over other observations in proportion to their sampling weight (and so this may put some other observations slightly over the trimming bounds). The projection matrix used in computing standard errors is based on the feasible bounds specified by the `bounds` argument. See also `trimWeights`, which trims the final weights in a design object rather than the calibration adjustments.

For two-phase designs `calfun="rrz"` estimates the sampling probabilities using logistic regression as described by Robins et al (1994). `estWeights` will do the same thing.

Calibration may result in observations within the last-stage sampling units having unequal weight even though they necessarily are sampled together. Specifying `aggegrate.stage` ensures that the calibration weight adjustments are constant within sampling units at the specified stage; if the original sampling weights were equal the final weights will also be equal. The algorithm is as described by Vanderhoeft (2001, section III.D). Specifying `aggregate.index` does the same thing for replicate weight designs; a warning will be given if the original weights are not constant within levels of `aggregate.index`.

In a model with two-stage sampling, population totals may be available for the PSUs actually sampled, but not for the whole population. In this situation, calibrating within each PSU reduces with second-stage contribution to variance. This generalizes to multistage sampling. The `stage` argument specifies which stage of sampling the totals refer to. Stage 0 is full population totals, stage 1 is totals for PSUs, and so on. The default, `stage=NULL` is interpreted as stage 0 when a single population vector is supplied and stage 1 when a list is supplied. Calibrating to PSU totals will fail (with a message about an exactly singular matrix) for PSUs that have fewer observations than the number of calibration variables.

The variance in the calibration model may depend on covariates. If `variance=NULL` the calibration model has constant variance. If `variance` is not `NULL` it specifies a linear combination of the columns of the model matrix and the calibration variance is proportional to that linear combination. Alternatively `variance` can be specified as a vector of values the same length as the input data specifying a heteroskedasticity parameter for each unit.

The design matrix specified by formula (after any aggregation) must be of full rank, with one exception. If the population total for a column is zero and all the observations are zero the column will be ignored. This allows the use of factors where the population happens to have no observations at some level.

In a two-phase design, `population` may be omitted when `phase=2`, to specify calibration to the phase-one sample. If the two-phase design object was constructed using the more memory-efficient `method="approx"` argument to `twophase`, calibration of the first phase of sampling to the population is not supported.

## References

Deville J-C, Sarndal C-E, Sautory O (1993) Generalized Raking Procedures in Survey Sampling. JASA 88:1013-1020

Kalton G, Flores-Cervantes I (2003) "Weighting methods" J Official Stat 19(2) 81-97

Lumley T, Shaw PA, Dai JY (2011) "Connections between survey calibration estimators and semiparametric models for incomplete data" International Statistical Review. 79:200-220. (with discussion 79:221-232)

Sarndal C-E, Swensson B, Wretman J. "Model Assisted Survey Sampling". Springer. 1991.

Rao JNK, Yung W, Hidiroglou MA (2002) Estimating equations for the analysis of survey data using poststratification information. Sankhya 64 Series A Part 2, 364-378.

Robins JM, Rotnitzky A, Zhao LP. (1994) Estimation of regression coefficients when some regressors are not always observed. Journal of the American Statistical Association, 89, 846-866.

Vanderhoeft C (2001) Generalized Calibration at Statistics Belgium. Statistics Belgium Working Paper No 3.

`postStratify`, `rake` for other ways to use auxiliary information

`twophase` and `vignette("epi")` for an example of calibration in two-phase designs

`survey/tests/kalton.R` for examples replicating those in Kalton & Flores-Cervantes (2003)

`make.calfun` for user-defined calibration distances.

`trimWeights` to trim final weights rather than calibration adjustments.

## Examples

Run this code
```# NOT RUN {
data(api)
dclus1<-svydesign(id=~dnum, weights=~pw, data=apiclus1, fpc=~fpc)

cal_names(~stype, dclus1)

pop.totals<-c(`(Intercept)`=6194, stypeH=755, stypeM=1018)

## For a single factor variable this is equivalent to
## postStratify

(dclus1g<-calibrate(dclus1, ~stype, pop.totals))

svymean(~api00, dclus1g)
svytotal(~enroll, dclus1g)
svytotal(~stype, dclus1g)

## Make weights constant within school district
(dclus1agg<-calibrate(dclus1, ~stype, pop.totals, aggregate=1))
svymean(~api00, dclus1agg)
svytotal(~enroll, dclus1agg)
svytotal(~stype, dclus1agg)

cal_names(~stype+sch.wide, dclus1)
(dclus1g2 <- calibrate(dclus1, ~stype+sch.wide, c(pop.totals, sch.wideYes=5122)))

svymean(~api00, dclus1g2)
svytotal(~enroll, dclus1g2)
svytotal(~stype, dclus1g2)

## Finally, calibrate on 1999 API and school type

cal_names(~stype+api99, dclus1)
(dclus1g3 <- calibrate(dclus1, ~stype+api99, c(pop.totals, api99=3914069)))

svymean(~api00, dclus1g3)
svytotal(~enroll, dclus1g3)
svytotal(~stype, dclus1g3)

## Same syntax with replicate weights
rclus1<-as.svrepdesign(dclus1)

(rclus1g3 <- calibrate(rclus1, ~stype+api99, c(pop.totals, api99=3914069)))

svymean(~api00, rclus1g3)
svytotal(~enroll, rclus1g3)
svytotal(~stype, rclus1g3)

(rclus1agg3 <- calibrate(rclus1, ~stype+api99, c(pop.totals,api99=3914069), aggregate.index=~dnum))

svymean(~api00, rclus1agg3)
svytotal(~enroll, rclus1agg3)
svytotal(~stype, rclus1agg3)

###
## Bounded weights
range(weights(dclus1g3)/weights(dclus1))
dclus1g3b <- calibrate(dclus1, ~stype+api99, c(pop.totals, api99=3914069),bounds=c(0.6,1.6))
range(weights(dclus1g3b)/weights(dclus1))

svymean(~api00, dclus1g3b)
svytotal(~enroll, dclus1g3b)
svytotal(~stype, dclus1g3b)

## Individual boundary constraints as constant values
# the first weight will be bounded at 40, the rest free to move
bnds <- list(
lower = rep(-Inf, nrow(apiclus1)),
upper = c(40, rep(Inf, nrow(apiclus1)-1)))
dclus1g3b1 <- calibrate(dclus1, ~stype+api99, c(pop.totals, api99=3914069),
bounds=bnds, bounds.const=TRUE)
svytotal(~api.stu, dclus1g3b1)

## trimming
dclus1tr <- calibrate(dclus1, ~stype+api99, c(pop.totals, api99=3914069),
bounds=c(0.5,2), trim=c(2/3,3/2))
svymean(~api00+api99+enroll, dclus1tr)
svytotal(~stype,dclus1tr)
range(weights(dclus1tr)/weights(dclus1))

rclus1tr <- calibrate(rclus1, ~stype+api99, c(pop.totals, api99=3914069),
bounds=c(0.5,2), trim=c(2/3,3/2))
svymean(~api00+api99+enroll, rclus1tr)
svytotal(~stype,rclus1tr)

## Input in the same format as rake() for classical raking
pop.table <- xtabs(~stype+sch.wide,apipop)
pop.table2 <- xtabs(~stype+comp.imp,apipop)
dclus1r<-rake(dclus1, list(~stype+sch.wide, ~stype+comp.imp),
list(pop.table, pop.table2))
gclus1r<-calibrate(dclus1, formula=list(~stype+sch.wide, ~stype+comp.imp),
population=list(pop.table, pop.table2),calfun="raking")
svymean(~api00+stype, dclus1r)
svymean(~api00+stype, gclus1r)

## generalised raking
dclus1g3c <- calibrate(dclus1, ~stype+api99, c(pop.totals,
api99=3914069), calfun="raking")
range(weights(dclus1g3c)/weights(dclus1))

(dclus1g3d <- calibrate(dclus1, ~stype+api99, c(pop.totals,
api99=3914069), calfun=cal.logit, bounds=c(0.5,2.5)))
range(weights(dclus1g3d)/weights(dclus1))

## Ratio estimators are calibration estimators
dstrat<-svydesign(id=~1,strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)
svytotal(~api.stu,dstrat)

common<-svyratio(~api.stu, ~enroll, dstrat, separate=FALSE)
predict(common, total=3811472)

pop<-3811472
## equivalent to (common) ratio estimator
dstratg1<-calibrate(dstrat,~enroll-1, pop, variance=1)
svytotal(~api.stu, dstratg1)

# Alternatively specifying the heteroskedasticity parameters directly
dstratgh <- calibrate(dstrat,~enroll-1, pop, variance=apistrat\$enroll)
svytotal(~api.stu, dstratgh)

# }
```

Run the code above in your browser using DataCamp Workspace