This help page describes variance estimators
which are commonly used for survey samples. These variance estimators
can be used as the basis of the generalized replication methods, implemented
with the functions `as_fays_gen_rep_design()`

,
`as_gen_boot_design()`

,
`make_fays_gen_rep_factors()`

,
or `make_gen_boot_factors()`

## Shared Notation

Let \(s\) denote the selected sample of size \(n\), with elements \(i=1,\dots,n\).
Element \(i\) in the sample had probability \(\pi_i\) of being included in the sample.
The *pair* of elements \(ij\) was sampled with probability \(\pi_{ij}\).

The population total for a variable is denoted \(Y = \sum_{i \in U}y_i\), and the Horvitz-Thompson estimator for \(\hat{Y}\) is denoted \(\hat{Y} = \sum_{i \in s} y_i/\pi_i\). For convenience, we denote \(\breve{y}_i = y_i/\pi_i\).

The true sampling variance of \(\hat{Y}\) is denoted \(V(\hat{Y})\), while an estimator of this sampling variance is denoted \(v(\hat{Y})\).

## Horvitz-Thompson

The **Horvitz-Thompson** variance estimator:
$$
v(\hat{Y}) = \sum_{i \in s}\sum_{j \in s} (1 - \frac{\pi_i \pi_j}{\pi_{ij}}) \frac{y_i}{\pi_i} \frac{y_j}{\pi_j}
$$

## Yates-Grundy

The **Yates-Grundy** variance estimator:
$$
v(\hat{Y}) = -\frac{1}{2}\sum_{i \in s}\sum_{j \in s} (1 - \frac{\pi_i \pi_j}{\pi_{ij}}) (\frac{y_i}{\pi_i} - \frac{y_j}{\pi_j})^2
$$

## Poisson Horvitz-Thompson

The **Poisson Horvitz-Thompson** variance estimator
is simply the Horvitz-Thompson variance estimator, but
where \(\pi_{ij}=\pi_i \times \pi_j\), which is the case for Poisson sampling.

## Stratified Multistage SRS

The **Stratified Multistage SRS** variance estimator is the recursive variance estimator
proposed by Bellhouse (1985) and used in the 'survey' package's function svyrecvar.
In the case of simple random sampling without replacement (with one or more stages),
this estimator exactly matches the Horvitz-Thompson estimator.

The estimator can be used for any number of sampling stages. For illustration, we describe its use
for two sampling stages.
$$
v(\hat{Y}) = \hat{V}_1 + \hat{V}_2
$$
where
$$
\hat{V}_1 = \sum_{h=1}^{H} (1 - \frac{n_h}{N_h})\frac{n_h}{n_h - 1} \sum_{i=1}^{n_h} (y_{hi.} - \bar{y}_{hi.})^2
$$
and
$$
\hat{V}_2 = \sum_{h=1}^{H} \frac{n_h}{N_h} \sum_{i=1}^{n_h}v_{hi}(y_{hi.})
$$
where \(n_h\) is the number of sampled clusters in stratum \(h\),
\(N_h\) is the number of population clusters in stratum \(h\),
\(y_{hi.}\) is the weighted cluster total in cluster \(i\) of stratum \(h\),
\(\bar{y}_{hi.}\) is the mean weighted cluster total of stratum \(h\),
(\(\bar{y}_{hi.} = \frac{1}{n_h}\sum_{i=1}^{n_h}y_{hi.}\)), and
\(v_{hi}(y_{hi.})\) is the estimated sampling variance of \(y_{hi.}\).

## Ultimate Cluster

The **Ultimate Cluster** variance estimator is simply the stratified multistage SRS
variance estimator, but ignoring variances from later stages of sampling.
$$
v(\hat{Y}) = \hat{V}_1
$$
This is the variance estimator used in the 'survey' package when the user specifies
`option(survey.ultimate.cluster = TRUE)`

or uses `svyrecvar(..., one.stage = TRUE)`

.
When the first-stage sampling fractions are small, analysts often omit the finite population corrections \((1-\frac{n_h}{N_h})\)
when using the ultimate cluster estimator.

## SD1 and SD2 (Successive Difference Estimators)

The **SD1** and **SD2** variance estimators are "successive difference"
estimators sometimes used for systematic sampling designs.
Ash (2014) describes each estimator as follows:
$$
\hat{v}_{S D 1}(\hat{Y}) = \left(1-\frac{n}{N}\right) \frac{n}{2(n-1)} \sum_{k=2}^n\left(\breve{y}_k-\breve{y}_{k-1}\right)^2
$$
$$
\hat{v}_{S D 2}(\hat{Y}) = \left(1-\frac{n}{N}\right) \frac{1}{2}\left[\sum_{k=2}^n\left(\breve{y}_k-\breve{y}_{k-1}\right)^2+\left(\breve{y}_n-\breve{y}_1\right)^2\right]
$$
where \(\breve{y}_k = y_k/\pi_k\) is the weighted value of unit \(k\)
with selection probability \(\pi_k\). The SD1 estimator is recommended by Wolter (1984).
The SD2 estimator is the basis of the successive difference replication estimator commonly
used for systematic sampling designs and is more conservative. See Ash (2014) for details.

For multistage samples, SD1 and SD2 are applied to the clusters at each stage, separately by stratum.
For later stages of sampling, the variance estimate from a stratum is multiplied by the product
of sampling fractions from earlier stages of sampling. For example, at a third stage of sampling,
the variance estimate from a third-stage stratum is multiplied by \(\frac{n_1}{N_1}\frac{n_2}{N_2}\),
which is the product of sampling fractions from the first-stage stratum and second-stage stratum.

## Beaumont-Emond

The **"Beaumont-Emond"** variance estimator was proposed by Beaumont and Emond (2022),
intended for designs that use fixed-size, unequal-probability random sampling without replacement.
The variance estimator is simply the Horvitz-Thompson
variance estimator with the following approximation for the joint inclusion
probabilities.
$$
\pi_{kl} \approx \pi_k \pi_l \frac{n - 1}{(n-1) + \sqrt{(1-\pi_k)(1-\pi_l)}}
$$
In the case of cluster sampling, this approximation is
applied to the clusters rather than the units within clusters,
with \(n\) denoting the number of sampled clusters. and the probabilities \(\pi\)
referring to the cluster's sampling probability. For stratified samples,
the joint probability for units \(k\) and \(l\) in different strata
is simply the product of \(\pi_k\) and \(\pi_l\).

For multistage samples, this approximation is applied to the clusters at each stage, separately by stratum. For later stages of sampling, the variance estimate from a stratum is multiplied by the product of sampling probabilities from earlier stages of sampling. For example, at a third stage of sampling, the variance estimate from a third-stage stratum is multiplied by \(\pi_1 \times \pi_{(2 | 1)}\), where \(\pi_1\) is the sampling probability of the first-stage unit and \(\pi_{(2|1)}\) is the sampling probability of the second-stage unit within the first-stage unit.

## Deville 1 and Deville 2

The **"Deville-1"** and **"Deville-2"** variance estimators
are clearly described in Matei and Tillé (2005),
and are intended for designs that use
fixed-size, unequal-probability random sampling without replacement.
These variance estimators have been shown to be effective
for designs that use a fixed sample size with a high-entropy sampling method.
This includes most PPSWOR sampling methods,
but unequal-probability systematic sampling is an important exception.

These variance estimators take the following form: $$ \hat{v}(\hat{Y}) = \sum_{i=1}^{n} c_i (\breve{y}_i - \frac{1}{\sum_{i=k}^{n}c_k}\sum_{k=1}^{n}c_k \breve{y}_k)^2 $$ where \(\breve{y}_i = y_i/\pi_i\) is the weighted value of the the variable of interest, and \(c_i\) depend on the method used:

**"Deville-1"**: $$c_i=\left(1-\pi_i\right) \frac{n}{n-1}$$**"Deville-2"**: $$c_i = (1-\pi_i) \left[1 - \sum_{k=1}^{n} \left(\frac{1-\pi_k}{\sum_{k=1}^{n}(1-\pi_k)}\right)^2 \right]^{-1}$$

In the case of simple random sampling without replacement (SRSWOR), these estimators are both identical to the usual stratified multistage SRS estimator (which is itself a special case of the Horvitz-Thompson estimator).

For multistage samples, "Deville-1" and "Deville-2" are applied to the clusters at each stage, separately by stratum. For later stages of sampling, the variance estimate from a stratum is multiplied by the product of sampling probabilities from earlier stages of sampling. For example, at a third stage of sampling, the variance estimate from a third-stage stratum is multiplied by \(\pi_1 \times \pi_{(2 | 1)}\), where \(\pi_1\) is the sampling probability of the first-stage unit and \(\pi_{(2|1)}\) is the sampling probability of the second-stage unit within the first-stage unit.

## Deville-Tillé

See Section 6.8 of Tillé (2020) for more detail on this estimator, including an explanation of its quadratic form. See Deville and Tillé (2005) for the results of a simulation study comparing this and other alternative estimators for balanced sampling.

The estimator can be written as follows: $$ v(\hat{Y})=\sum_{k \in S} \frac{c_k}{\pi_k^2}\left(y_k-\hat{y}_k^*\right)^2, $$ where $$ \hat{y}_k^*=\mathbf{z}_k^{\top}\left(\sum_{\ell \in S} c_{\ell} \frac{\mathbf{z}_{\ell} \mathbf{z}_{\ell}^{\prime}}{\pi_{\ell}^2}\right)^{-1} \sum_{\ell \in S} c_{\ell} \frac{\mathbf{z}_{\ell} y_{\ell}}{\pi_{\ell}^2} $$ and \(\mathbf{z}_k\) denotes the vector of auxiliary variables for observation \(k\) included in sample \(S\), with inclusion probability \(\pi_k\). The value \(c_k\) is set to \(\frac{n}{n-q}(1-\pi_k)\), where \(n\) is the number of observations and \(q\) is the number of auxiliary variables.

## References

Ash, S. (2014). "*Using successive difference replication for estimating variances*."
**Survey Methodology**, Statistics Canada, 40(1), 47–59.

Beaumont, J.-F.; Émond, N. (2022). "*A Bootstrap Variance Estimation Method for Multistage Sampling and Two-Phase Sampling When Poisson Sampling Is Used at the Second Phase*."
**Stats**, *5*: 339–357.
https://doi.org/10.3390/stats5020019

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

Deville, J.‐C., and Tillé, Y. (2005). "*Variance approximation under balanced sampling.*"
**Journal of Statistical Planning and Inference**, 128, 569–591.

Tillé, Y. (2020). "*Sampling and estimation from finite populations*." (I. Hekimi, Trans.). Wiley.

Matei, Alina, and Yves Tillé. (2005).
“*Evaluation of Variance Approximations and Estimators
in Maximum Entropy Sampling with Unequal Probability and Fixed Sample Size.*”
**Journal of Official Statistics**, 21(4):543–70.