Rescale replicate factors — rescale

Rescale replicate factors. The main application of this rescaling is to ensure that all replicate weights are strictly positive.

Note that this rescaling has no impact on variance estimates for totals (or other linear statistics), but variance estimates for nonlinear statistics will be affected by the rescaling.

Usage

rescale_reps(x, tau = NULL, min_wgt = 0.01, digits = 2)

Arguments

x: Either a replicate survey design object, or a numeric matrix of replicate weights.
tau: Either a single positive number, or NULL. This is the rescaling constant $\tau$ used in the transformation $\frac{w + \tau - 1}{\tau}$, where $w$ is the original weight.
If tau=NULL or is left unspecified, then the argument min_wgt should be used instead, in which case, $\tau$ is automatically set to the smallest value needed to rescale the replicate weights such that they are all at least min_wgt.
min_wgt: Should only be used if tau=NULL or tau is left unspecified. Specifies the minimum acceptable value for the rescaled weights, which will be used to automatically determine the value $\tau$ used in the transformation $\frac{w + \tau - 1}{\tau}$, where $w$ is the original weight. Must be at least zero and must be less than one.
digits: Only used if the argument min_wgt is used. Specifies the number of decimal places to use for choosing tau. Using a smaller number of digits is useful simply for producing easier-to-read documentation.

Value

If the input is a numeric matrix, returns the rescaled matrix. If the input is a replicate survey design object, returns an updated replicate survey design object.

For a replicate survey design object, results depend on whether the object has a matrix of replicate factors rather than a matrix of replicate weights (which are the product of replicate factors and sampling weights). If the design object has combined.weights=FALSE, then the replication factors are adjusted. If the design object has combined.weights=TRUE, then the replicate weights are adjusted. It is strongly recommended to only use the rescaling method for replication factors rather than the weights.

For a replicate survey design object, the scale element of the design object will be updated appropriately, and an element tau will also be added. If the input is a matrix instead of a survey design object, the result matrix will have an attribute named tau which can be retrieved using attr(x, 'tau').

Details

Let $\mathbf{A} = \left[ \mathbf{a}^{(1)} \cdots \mathbf{a}^{(b)} \cdots \mathbf{a}^{(B)} \right]$ denote the $(n \times B)$ matrix of replicate adjustment factors. To eliminate negative adjustment factors, Beaumont and Patak (2012) propose forming a rescaled matrix of nonnegative replicate factors $\mathbf{A}^S$ by rescaling each adjustment factor $a_k^{(b)}$ as follows: $$ a_k^{S,(b)} = \frac{a_k^{(b)} + \tau - 1}{\tau} $$ where $\tau \geq 1 - a_k^{(b)} \geq 1$ for all $k$ in $\left\{ 1,\ldots,n \right\}$ and all $b$ in $\left\{1, \ldots, B\right\}$.

The value of $\tau$ can be set based on the realized adjustment factor matrix $\mathbf{A}$ or by choosing $\tau$ prior to generating the adjustment factor matrix $\mathbf{A}$ so that $\tau$ is likely to be large enough to prevent negative adjustment factors.

If the adjustment factors are rescaled in this manner, it is important to adjust the scale factor used in estimating the variance with the bootstrap replicates. For example, for bootstrap replicates, the adjustment factor becomes $\frac{\tau^2}{B}$ instead of $\frac{1}{B}$. $$ \textbf{Prior to rescaling: } v_B\left(\hat{T}_y\right) = \frac{1}{B}\sum_{b=1}^B\left(\hat{T}_y^{*(b)}-\hat{T}_y\right)^2 $$ $$ \textbf{After rescaling: } v_B\left(\hat{T}_y\right) = \frac{\tau^2}{B}\sum_{b=1}^B\left(\hat{T}_y^{S*(b)}-\hat{T}_y\right)^2 $$

References

This method was suggested by Fay (1989) for the specific application of creating replicate factors using his generalized replication method. Beaumont and Patak (2012) provided an extended discussion on this rescaling method in the context of rescaling generalized bootstrap replication factors to avoid negative replicate weights.

The notation used in this documentation is taken from Beaumont and Patak (2012).

- Beaumont, Jean-François, and Zdenek Patak. 2012. "On the Generalized Bootstrap for Sample Surveys with Special Attention to Poisson Sampling: Generalized Bootstrap for Sample Surveys." International Statistical Review 80 (1): 127–48. https://doi.org/10.1111/j.1751-5823.2011.00166.x.

- 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

Examples

# Example 1: Rescaling a matrix of replicate weights to avoid negative weights

 rep_wgts <- matrix(
   c(1.69742746694909, -0.230761178913411, 1.53333377634192,
     0.0495043413294782, 1.81820367441039, 1.13229198793703,
     1.62482013925955, 1.0866133494029, 0.28856654131668,
     0.581930729719006, 0.91827012312825, 1.49979905894482,
     1.26281337410693, 1.99327362761477, -0.25608700039304),
   nrow = 3, ncol = 5
 )

 rescaled_wgts <- rescale_reps(rep_wgts, min_wgt = 0.01)

 print(rep_wgts)
#>            [,1]       [,2]      [,3]      [,4]      [,5]
#> [1,]  1.6974275 0.04950434 1.6248201 0.5819307  1.262813
#> [2,] -0.2307612 1.81820367 1.0866133 0.9182701  1.993274
#> [3,]  1.5333338 1.13229199 0.2885665 1.4997991 -0.256087
 print(rescaled_wgts)
#>            [,1]      [,2]      [,3]      [,4]       [,5]
#> [1,] 1.54915549 0.2515782 1.4919844 0.6708116 1.20693966
#> [2,] 0.03089671 1.6442549 1.0681995 0.9356458 1.78210522
#> [3,] 1.41994786 1.1041669 0.4398162 1.3935426 0.01095512
#> attr(,"tau")
#> [1] 1.27
 
 # Example 2: Rescaling replicate weights with a specified value of 'tau'
 
 rescaled_wgts <- rescale_reps(rep_wgts, tau = 2)
 print(rescaled_wgts)
#>           [,1]      [,2]      [,3]      [,4]      [,5]
#> [1,] 1.3487137 0.5247522 1.3124101 0.7909654 1.1314067
#> [2,] 0.3846194 1.4091018 1.0433067 0.9591351 1.4966368
#> [3,] 1.2666669 1.0661460 0.6442833 1.2498995 0.3719565
#> attr(,"tau")
#> [1] 2

 # Example 3: Rescaling replicate weights of a survey design object
 set.seed(2023)
 library(survey)
 data('mu284', package = 'survey')

 ## First create a bootstrap design object
 svy_design_object <- svydesign(
   data = mu284,
   ids = ~ id1 + id2,
   fpc = ~ n1 + n2
 )

 boot_design <- as_gen_boot_design(
   design = svy_design_object,
   variance_estimator = "Stratified Multistage SRS",
   replicates = 5, tau = 1
 )

 ## Rescale the weights
 rescaled_boot_design <- boot_design |>
   rescale_reps(min_wgt = 0.01)

 boot_wgts <- weights(boot_design, "analysis")
 rescaled_boot_wgts <- weights(rescaled_boot_design, 'analysis')

 print(boot_wgts)
#>           REP_1      REP_2     REP_3       REP_4     REP_5
#>  [1,] 34.071074  -3.352195  7.031013  35.4547243 18.681422
#>  [2,] -3.271131  12.579037 57.474328   9.3992012 25.014379
#>  [3,] 12.204302  16.611770 14.029208   6.9869037 -8.727739
#>  [4,] 40.124053  62.587721 29.834150  31.6263954 10.057763
#>  [5,]  6.857688  48.936835  5.029175  42.1974204 67.126670
#>  [6,] 38.866284  -7.883877  6.363613  35.3323661 14.104502
#>  [7,] -2.705981   5.310800 51.191780 -18.8838184 34.232137
#>  [8,] 23.948409  19.740921 21.950039   0.8683187 -2.397135
#>  [9,] 38.102201  56.396305 39.516036  39.6713935 31.130900
#> [10,]  7.987330  41.986885  8.545987  47.8769538 66.314653
#> [11,] 35.747939 -13.746937  9.901870  41.9315736  8.610798
#> [12,]  1.384506   2.579634 50.469377 -26.8411850 19.800463
#> [13,] 22.153736  11.250766 19.117806   0.9281633 -1.226728
#> [14,] 48.183146  68.452257 28.322524  31.3003309 12.972211
#> [15,]  7.066647  63.713091 11.462660  41.8092990 64.604278
#> attr(,"tau")
#> [1] 1
#> attr(,"scale")
#> [1] 0.2
#> attr(,"rscales")
#> [1] 1 1 1 1 1
 print(rescaled_boot_wgts)
#>          REP_1     REP_2    REP_3     REP_4     REP_5
#>  [1,] 25.24027  6.805158 11.92004 25.921867 17.659157
#>  [2,] 11.91898 19.726947 41.84285 18.160526 25.852732
#>  [3,] 14.46846 16.639624 15.36743 11.898311  4.157107
#>  [4,] 34.98722 46.053065 29.91830 30.801180 20.176238
#>  [5,] 15.21725 35.945896 14.31651 32.625987 44.906406
#>  [6,] 27.60244  4.572803 11.59127 25.861593 15.404516
#>  [7,] 12.19738 16.146535 38.74800  4.228004 30.393500
#>  [8,] 20.25373 18.181078 19.26931  8.884229  7.275631
#>  [9,] 33.99123 43.003106 34.68770 34.764233 30.557093
#> [10,] 15.77373 32.522275 16.04893 35.423787 44.506397
#> [11,] 26.06631  1.684596 13.33425 29.112434 12.698258
#> [12,] 14.21240 14.801133 38.39214  0.308119 23.284300
#> [13,] 19.36966 13.998735 17.87412  8.913709  7.852187
#> [14,] 38.95721 48.941999 29.17366 30.640557 21.611926
#> [15,] 15.32019 43.224840 17.48571 32.434794 43.663848
#> attr(,"tau")
#> [1] 2.03
#> attr(,"scale")
#> [1] 0.82418
#> attr(,"rscales")
#> [1] 1 1 1 1 1