# Form replication factors using Fay's generalized replication method

Source:`R/fays_generalized_replication.R`

`make_fays_gen_rep_factors.Rd`

Generate a matrix of replication factors using Fay's generalized replication method. This method yields a fully efficient variance estimator if a sufficient number of replicates is used.

## Arguments

- Sigma
A quadratic form matrix corresponding to a target variance estimator. Must be positive semidefinite.

- max_replicates
The maximum number of replicates to allow. The function will attempt to create the minimum number of replicates needed to produce a fully-efficient variance estimator. If more replicates are needed than

`max_replicates`

, then the full number of replicates needed will be created, but only a random subsample will be retained.- balanced
If

`balanced=TRUE`

, the replicates will all contribute equally to variance estimates, but the number of replicates needed may slightly increase.

## Value

A matrix of replicate factors,
with the number of rows matching the number of rows of `Sigma`

and the number of columns less than or equal to `max_replicates`

.
To calculate variance estimates using these factors,
use the overall scale factor given by calling
`attr(x, "scale")`

on the result.

## Statistical Details

See Fay (1989) for a full explanation of Fay's generalized replication method. This documentation provides a brief overview.

Let \(\boldsymbol{\Sigma}\) be the quadratic form matrix for a target variance estimator,
which is assumed to be positive semidefinite.
Suppose the rank of \(\boldsymbol{\Sigma}\) is \(k\),
and so \(\boldsymbol{\Sigma}\) can be represented by the spectral decomposition
of \(k\) eigenvectors and eigenvalues, where the \(r\)-th eigenvector and eigenvalue
are denoted \(\mathbf{v}_{(r)}\) and \(\lambda_r\), respectively.
$$
\boldsymbol{\Sigma} = \sum_{r=1}^k \lambda_r \mathbf{v}_{(r)} \mathbf{v^{\prime}}_{(r)}
$$
If `balanced = FALSE`

, then we let \(\mathbf{H}\) denote an identity matrix
with \(k' = k\) rows/columns. If `balanced = TRUE`

, then we let \(\mathbf{H}\) be a Hadamard matrix (with all entries equal to \(1\) or \(-1\)),
of order \(k^{\prime} \geq k\). Let \(\mathbf{H}_{mr}\) denote the entry in row
\(m\) and column \(r\) of \(\mathbf{H}\).

Then \(k^{\prime}\) replicates are formed as follows. Let \(r\) denote a given replicate, with \(r = 1, ..., k^{\prime}\), and let \(c\) denote some positive constant (yet to be specified).

The \(r\)-th replicate adjustment factor \(\mathbf{f}_{r}\) is formed as: $$ \mathbf{f}_{r} = 1 + c \sum_{m=1}^k H_{m r} \lambda_{(m)}^{\frac{1}{2}} \mathbf{v}_{(m)} $$

If `balanced = FALSE`

, then \(c = 1\). If `balanced = TRUE`

,
then \(c = \frac{1}{\sqrt{k^{\prime}}}\).

If any of the replicates
are negative, you can use `rescale_reps`

,
which recalculates the replicate factors with a smaller value of \(c\).

If all \(k^{\prime}\) replicates are used, then variance estimates are calculated as:
$$
v_{rep}\left(\hat{T}_y\right) = \sum_{r=1}^{k^{\prime}}\left(\hat{T}_y^{*(r)}-\hat{T}_y\right)^2
$$
For population totals, this replication variance estimator
will *exactly* match the target variance estimator
if the number of replicates \(k^{\prime}\) matches the rank of \(\Sigma\).

## The Number of Replicates

If `balanced=TRUE`

, the number of replicates created
may need to increase slightly.
This is due to the fact that a Hadamard matrix
of order \(k^{\prime} \geq k\) is used to balance the replicates,
and it may be necessary to use order \(k^{\prime} > k\).

If the number of replicates \(k^{\prime}\) is too large for practical purposes,
then one can simply retain only a random subset of \(R\) of the \(k^{\prime}\) replicates.
In this case, variances are calculated as follows:
$$
v_{rep}\left(\hat{T}_y\right) = \frac{k^{\prime}}{R} \sum_{r=1}^{R}\left(\hat{T}_y^{*(r)}-\hat{T}_y\right)^2
$$
This is what happens if `max_replicates`

is less than the
matrix rank of `Sigma`

: only a random subset
of the created replicates will be retained.

Subsampling replicates is only recommended when
using `balanced=TRUE`

, since in this case every replicate
contributes equally to variance estimates. If `balanced=FALSE`

,
then randomly subsampling replicates is valid but may
produce large variation in variance estimates since replicates
in that case may vary greatly in their contribution to variance
estimates.

## Reproducibility

If `balanced=TRUE`

, a Hadamard matrix
is used as described above. The Hadamard matrix is
deterministically created using the function
`hadamard()`

from the 'survey' package.
However, the order of rows/columns is randomly permuted
before forming replicates.

In general, column-ordering of the replicate weights is random.
To ensure exact reproducibility, it is recommended to call
`set.seed()`

before using this function.

## References

Fay, Robert. 1989. "Theory And Application Of Replicate Weighting For Variance Calculations." In, 495–500. Alexandria, VA: American Statistical Association. http://www.asasrms.org/Proceedings/papers/1989_033.pdf

## See also

Use `rescale_reps`

to eliminate negative adjustment factors.

## Examples

```
if (FALSE) { # \dontrun{
library(survey)
# Load an example dataset that uses unequal probability sampling ----
data('election', package = 'survey')
# Create matrix to represent the Horvitz-Thompson estimator as a quadratic form ----
n <- nrow(election_pps)
pi <- election_jointprob
horvitz_thompson_matrix <- matrix(nrow = n, ncol = n)
for (i in seq_len(n)) {
for (j in seq_len(n)) {
horvitz_thompson_matrix[i,j] <- 1 - (pi[i,i] * pi[j,j])/pi[i,j]
}
}
## Equivalently:
horvitz_thompson_matrix <- make_quad_form_matrix(
variance_estimator = "Horvitz-Thompson",
joint_probs = election_jointprob
)
# Make generalized replication adjustment factors ----
adjustment_factors <- make_fays_gen_rep_factors(
Sigma = horvitz_thompson_matrix,
max_replicates = 50
)
attr(adjustment_factors, 'scale')
# Compute the Horvitz-Thompson estimate and the replication estimate
ht_estimate <- svydesign(data = election_pps, ids = ~ 1,
prob = diag(election_jointprob),
pps = ppsmat(election_jointprob)) |>
svytotal(x = ~ Kerry)
rep_estimate <- svrepdesign(
data = election_pps,
weights = ~ wt,
repweights = adjustment_factors,
combined.weights = FALSE,
scale = attr(adjustment_factors, 'scale'),
rscales = rep(1, times = ncol(adjustment_factors)),
type = "other",
mse = TRUE
) |>
svytotal(x = ~ Kerry)
SE(rep_estimate)
SE(ht_estimate)
SE(rep_estimate) / SE(ht_estimate)
} # }
```