# Spencer design effect for pps sampling

### 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.

### See Also

`deffS`

, `deffH`

### Examples

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)
``` |