# svyquantile

##### Quantiles for sample surveys

Compute quantiles for data from complex surveys.

##### Usage

```
# S3 method for survey.design
svyquantile(x, design, quantiles, alpha=0.05,
ci=FALSE, method = "linear", f = 1,
interval.type=c("Wald","score","betaWald"), na.rm=FALSE,se=ci,
ties=c("discrete","rounded"), df=NULL,...)
# S3 method for svyrep.design
svyquantile(x, design, quantiles,
method ="linear", interval.type=c("probability","quantile"), f = 1,
return.replicates=FALSE, ties=c("discrete","rounded"),na.rm=FALSE,
alpha=0.05,df=NULL,...)
# S3 method for svyquantile
SE(object,...)
```

##### Arguments

- x
A formula, vector or matrix

- design
`survey.design`

or`svyrep.design`

object- quantiles
Quantiles to estimate

- method
see

`approxfun`

- f
see

`approxfun`

- ci
Compute a confidence interval? (relatively slow; needed for

`svyby`

)- se
Compute standard errors from the confidence interval length?

- alpha
Level for confidence interval

- interval.type
See Details below

- ties
See Details below

- df
Degrees of freedom for a t-distribution.

`Inf`

requests a Normal distribution,`NULL`

uses`degf`

. Not relevant for`type="betaWald"`

- return.replicates
Return the replicate means?

- na.rm
Remove

`NA`

s?- ...
arguments for future expansion

- object
Object returned by

`svyquantile.survey.design`

##### Details

The definition of the CDF and thus of the quantiles is ambiguous in
the presence of ties. With `ties="discrete"`

the data are
treated as genuinely discrete, so the CDF has vertical steps at tied
observations. With `ties="rounded"`

all the weights for tied
observations are summed and the CDF interpolates linearly between
distinct observed values, and so is a continuous function. Combining
`interval.type="betaWald"`

and `ties="discrete"`

is (close
to) the proposal of Shah and Vaish(2006) used in some versions of SUDAAN.

Interval estimation for quantiles is complicated, because the
influence function is not continuous. Linearisation cannot be used
directly, and computing the variance of replicates is valid only for
some designs (eg BRR, but not jackknife). The `interval.type`

option controls how the intervals are computed.

For `survey.design`

objects the default is
`interval.type="Wald"`

. A 95% Wald confidence interval is
constructed for the proportion below the estimated quantile. The
inverse of the estimated CDF is used to map this to a confidence
interval for the quantile. This is the method of Woodruff
(1952). For `"betaWald"`

the same procedure is used, but the
confidence interval for the proportion is computed using the exact
binomial cdf with an effective sample size proposed by Korn &
Graubard (1998).

If `interval.type="score"`

we use a method described by Binder
(1991) and due originally to Francisco and Fuller (1986), which
corresponds to inverting a robust score test. At the upper and lower
limits of the confidence interval, a test of the null hypothesis that
the cumulative distribution function is equal to the target quantile
just rejects. This was the default before version 2.9. It is much
slower than `"Wald"`

, and Dorfman & Valliant (1993) suggest it is
not any more accurate.

Standard errors are computed from these confidence intervals by
dividing the confidence interval length by `2*qnorm(alpha/2)`

.

For replicate-weight designs, ordinary replication-based standard errors
are valid for BRR and Fay's method, and for some bootstrap-based
designs, but not for jackknife-based designs.
`interval.type="quantile"`

gives these replication-based
standard errors. The default, `interval.type="probability"`

computes confidence on the probability scale and then transforms
back to quantiles, the equivalent of `interval.type="Wald"`

for
`survey.design`

objects (with `alpha=0.05`

).

There is a `confint`

method for `svyquantile`

objects; it
simply extracts the pre-computed confidence interval.

##### Value

returns a list whose first component is the quantiles and second
component is the confidence intervals. For replicate weight designs,
returns an object of class `svyrepstat`

.

##### References

Binder DA (1991) Use of estimating functions for interval estimation
from complex surveys. *Proceedings of the ASA Survey Research
Methods Section* 1991: 34-42

Dorfman A, Valliant R (1993) Quantile variance estimators in complex surveys. Proceedings of the ASA Survey Research Methods Section. 1993: 866-871

Korn EL, Graubard BI. (1998) Confidence Intervals For Proportions With Small Expected Number of Positive Counts Estimated From Survey Data. Survey Methodology 23:193-201.

Francisco CA, Fuller WA (1986) Estimation of the distribution function with a complex survey. Technical Report, Iowa State University.

Shao J, Tu D (1995) *The Jackknife and Bootstrap*. Springer.

Shah BV, Vaish AK (2006) Confidence Intervals for Quantile Estimation from Complex Survey Data. Proceedings of the Section on Survey Research Methods.

Woodruff RS (1952) Confidence intervals for medians and other position measures. JASA 57, 622-627.

##### See Also

`svykm`

for quantiles of survival curves

`svyciprop`

for confidence intervals on proportions.

##### Examples

```
# NOT RUN {
data(api)
## population
quantile(apipop$api00,c(.25,.5,.75))
## one-stage cluster sample
dclus1<-svydesign(id=~dnum, weights=~pw, data=apiclus1, fpc=~fpc)
svyquantile(~api00, dclus1, c(.25,.5,.75),ci=TRUE)
svyquantile(~api00, dclus1, c(.25,.5,.75),ci=TRUE,interval.type="betaWald")
svyquantile(~api00, dclus1, c(.25,.5,.75),ci=TRUE,df=NULL)
dclus1<-svydesign(id=~dnum, weights=~pw, data=apiclus1, fpc=~fpc)
(qapi<-svyquantile(~api00, dclus1, c(.25,.5,.75),ci=TRUE, interval.type="score"))
SE(qapi)
#stratified sample
dstrat<-svydesign(id=~1, strata=~stype, weights=~pw, data=apistrat, fpc=~fpc)
svyquantile(~api00, dstrat, c(.25,.5,.75),ci=TRUE)
#stratified sample, replicate weights
# interval="probability" is necessary for jackknife weights
rstrat<-as.svrepdesign(dstrat)
svyquantile(~api00, rstrat, c(.25,.5,.75), interval.type="probability")
# BRR method
data(scd)
repweights<-2*cbind(c(1,0,1,0,1,0), c(1,0,0,1,0,1), c(0,1,1,0,0,1),
c(0,1,0,1,1,0))
scdrep<-svrepdesign(data=scd, type="BRR", repweights=repweights)
svyquantile(~arrests+alive, design=scdrep, quantile=0.5, interval.type="quantile")
svyquantile(~arrests+alive, design=scdrep, quantile=0.5, interval.type="quantile",df=NULL)
# }
```

*Documentation reproduced from package survey, version 4.0, License: GPL-2 | GPL-3*