# deffS: Spencer design effect for _pps_ sampling In PracTools: Tools for Designing and Weighting Survey Samples

## Description

Compute the Spencer design effect for single-stage samples selected with probability proportional to a measure of size.

## Usage

 `1` ```deffS(p, w, y) ```

## Arguments

 `p` vector of 1-draw selection probabilities, i.e., the probability that each unit would be selected in a sample of size 1. `w` vector of inverses of selection probabilities for a sample `y` vector of the sample values of an analysis variable

## Details

The Spencer design effect is the ratio of the variance of the pwr-estimator of the total of y, assuming that a single-stage sample is selected with replacement, to the variance of the total estimated in srswr. Varying selection probabilities can be used.

## Value

numeric design effect

## Author(s)

Richard Valliant, Jill A. Dever, Frauke Kreuter

## References

Park, I., and Lee, H. (2004). Design Effects for the Weighted Mean and Total Estimators under Complex Survey Sampling. Survey Methodology, 30, 183-193.

Spencer, B. D. (2000). An Approximate Design Effect for Unequal Weighting When Measurements May Correlate With Selection Probabilities. Survey Methodology, 26, 137-138.

Valliant, R., Dever, J., Kreuter, F. (2013, chap. 14). Practical Tools for Designing and Weighting Survey Samples. New York: Springer.

`deff`, `deffCR`, `deffH`, `deffK`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14``` ```set.seed(-500398777) # generate population using HMT function pop.dat <- as.data.frame(HMT()) mos <- pop.dat\$x pop.dat\$prbs.1d <- mos / sum(mos) # select pps sample require(sampling) n <- 80 pk <- n * pop.dat\$prbs.1d sam <- UPrandomsystematic(pk) sam <- sam==1 sam.dat <- pop.dat[sam, ] dsgn.wts <- 1/pk[sam] deffS(p=sam.dat\$prbs.1d, w=dsgn.wts, y=sam.dat\$y) ```