Compute the variance of a total under multistage sampling, using a recursive descent algorithm.

svyrecvar(x, clusters, stratas,fpcs, postStrata = NULL,
lonely.psu = getOption("survey.lonely.psu"),
one.stage=getOption("survey.ultimate.cluster"))

## Arguments

x

Matrix of data or estimating functions

clusters

Data frame or matrix with cluster ids for each stage

stratas

Strata for each stage

fpcs

Information on population and sample size for each stage, created by as.fpc

postStrata

post-stratification information as created by postStratify or calibrate

lonely.psu

How to handle strata with a single PSU

one.stage

If TRUE, compute a one-stage (ultimate-cluster) estimator

## Details

The main use of this function is to compute the variance of the sum of a set of estimating functions under multistage sampling. The sampling is assumed to be simple or stratified random sampling within clusters at each stage except perhaps the last stage. The variance of a statistic is computed from the variance of estimating functions as described by Binder (1983).

Use one.stage=FALSE for compatibility with other software that does not perform multi-stage calculations, and set options(survey.ultimate.cluster=TRUE) to make this the default.

The idea of a recursive algorithm is due to Bellhouse (1985). Texts such as Cochran (1977) and Sarndal et al (1991) describe the decomposition of the variance into a single-stage between-cluster estimator and a within-cluster estimator, and this is applied recursively.

If one.stage is a positive integer it specifies the number of stages of sampling to use in the recursive estimator.

If pps="brewer", standard errors are estimated using Brewer's approximation for PPS without replacement, option 2 of those described by Berger (2004). The fpc argument must then be specified in terms of sampling fractions, not population sizes (or omitted, but then the pps argument would have no effect and the with-replacement standard errors would be correct).

## Value

A covariance matrix

## References

Bellhouse DR (1985) Computing Methods for Variance Estimation in Complex Surveys. Journal of Official Statistics. Vol.1, No.3, 1985

Berger, Y.G. (2004), A Simple Variance Estimator for Unequal Probability Sampling Without Replacement. Journal of Applied Statistics, 31, 305-315.

Binder, David A. (1983). On the variances of asymptotically normal estimators from complex surveys. International Statistical Review, 51, 279-292.

Brewer KRW (2002) Combined Survey Sampling Inference (Weighing Basu's Elephants) [Chapter 9]

Cochran, W. (1977) Sampling Techniques. 3rd edition. Wiley.

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

## Note

A simple set of finite population corrections will only be exactly correct when each successive stage uses simple or stratified random sampling without replacement. A correction under general unequal probability sampling (eg PPS) would require joint inclusion probabilities (or, at least, sampling probabilities for units not included in the sample), information not generally available.

The quality of Brewer's approximation is excellent in Berger's simulations, but the accuracy may vary depending on the sampling algorithm used.

svrVar for replicate weight designs

svyCprod for a description of how variances are estimated at each stage

## Examples

data(mu284)
dmu284<-svydesign(id=~id1+id2,fpc=~n1+n2, data=mu284)
svytotal(~y1, dmu284)
#>    total     SE
#> y1 15080 2274.3

data(api)
# two-stage cluster sample
dclus2<-svydesign(id=~dnum+snum, fpc=~fpc1+fpc2, data=apiclus2)
summary(dclus2)
#> 2 - level Cluster Sampling design
#> With (40, 126) clusters.
#> svydesign(id = ~dnum + snum, fpc = ~fpc1 + fpc2, data = apiclus2)
#> Probabilities:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#> 0.003669 0.037743 0.052840 0.042390 0.052840 0.052840
#> Population size (PSUs): 757
#> Data variables:
#>  [1] "cds"      "stype"    "name"     "sname"    "snum"     "dname"
#>  [7] "dnum"     "cname"    "cnum"     "flag"     "pcttest"  "api00"
#> [13] "api99"    "target"   "growth"   "sch.wide" "comp.imp" "both"
#> [19] "awards"   "meals"    "ell"      "yr.rnd"   "mobility" "acs.k3"
#> [25] "acs.46"   "acs.core" "pct.resp" "not.hsg"  "hsg"      "some.col"
#> [31] "col.grad" "grad.sch" "avg.ed"   "full"     "emer"     "enroll"
#> [37] "api.stu"  "pw"       "fpc1"     "fpc2"
svymean(~api00, dclus2)
#>         mean     SE
#> api00 670.81 30.099
svytotal(~enroll, dclus2,na.rm=TRUE)
#>          total     SE
#> enroll 2639273 799638

# bootstrap for multistage sample
mrbclus2<-as.svrepdesign(dclus2, type="mrb", replicates=100)
svytotal(~enroll, mrbclus2, na.rm=TRUE)
#>          total     SE
#> enroll 2639273 857469

# two-stage with replacement'
dclus2wr<-svydesign(id=~dnum+snum, weights=~pw, data=apiclus2)
summary(dclus2wr)
#> 2 - level Cluster Sampling design (with replacement)
#> With (40, 126) clusters.
#> svydesign(id = ~dnum + snum, weights = ~pw, data = apiclus2)
#> Probabilities:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max.
#> 0.003669 0.037743 0.052840 0.042390 0.052840 0.052840
#> Data variables:
#>  [1] "cds"      "stype"    "name"     "sname"    "snum"     "dname"
#>  [7] "dnum"     "cname"    "cnum"     "flag"     "pcttest"  "api00"
#> [13] "api99"    "target"   "growth"   "sch.wide" "comp.imp" "both"
#> [19] "awards"   "meals"    "ell"      "yr.rnd"   "mobility" "acs.k3"
#> [25] "acs.46"   "acs.core" "pct.resp" "not.hsg"  "hsg"      "some.col"
#> [31] "col.grad" "grad.sch" "avg.ed"   "full"     "emer"     "enroll"
#> [37] "api.stu"  "pw"       "fpc1"     "fpc2"
svymean(~api00, dclus2wr)
#>         mean     SE
#> api00 670.81 30.712
svytotal(~enroll, dclus2wr,na.rm=TRUE)
#>          total     SE
#> enroll 2639273 820261

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