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

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

`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 |

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.

numeric design effect

Richard Valliant, Jill A. Dever, Frauke Kreuter

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.

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)
``` |

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