Rescale replicate factors

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_replicates(x, new_scale = NULL, min_wgt = 0.01, digits = 2)

Arguments

x

Either a replicate survey design object, or a numeric matrix of replicate weights. If x is a matrix, then it must have an attribute scale specifying the scale factor for variance estimation.

new_scale

Either a single positive number, or NULL. If supplied, new_scale will be the new scale factor used for estimating variances. For example, with B=10 bootstrap replicates, specifying new_scale=0.2 will change the scale factor for variance estimation from \(B^{-1}=0.1\) to \(0.2\).
If new_scale=NULL or is left unspecified, then the argument min_wgt should be used instead.

min_wgt

Should only be used if new_scale=NULL or new_scale is left unspecified. Specifies the minimum acceptable value for the rescaled weights, which will be used to automatically determine the new scale. 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 the scale adjustment factor, so that the ratio new_scale/orig_scale will be a number with at most digits decimal places. For example, let orig_scale denote the original scale and new_scale denote the new scale. Then if the user specifies digits=2, the ratio new_scale/orig_scale will be a number such as 0.25 or 0.10. This results in documentation that is easier to read.

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.

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. The overall scale factor \(C\) is used for estimating variances with the formula $$v(\hat{y}) = C \sum_{b=1}^{B} (\hat{y}_b - \hat{y})^2$$

The scale factor is changed from \(C\) to \(C^{\prime}\) by rescaling the replicate factor matrix \(\mathbf{A}\) using the transformation $$ 1 + \sqrt{\frac{C}{C^{\prime}}}(\mathbf{A}-1) $$

References

Rescaling was suggested by Fay (1989) for the specific application of creating replicate factors using his generalized replication method. This kind of rescaling is commony used in balanced repeated replication to implement Fay's method of balanced repeated replication. Beaumont and Patak (2012) provided an extended discussion of rescaling methods in the context of rescaling generalized bootstrap replication factors to avoid negative replicate weights.

- 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
 )
 attr(rep_wgts, 'scale') <- 1/5

 rescaled_wgts <- rescale_replicates(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
#> attr(,"scale")
#> [1] 0.2
 print(rescaled_wgts)
#>            [,1]      [,2]      [,3]      [,4]       [,5]
#> [1,] 1.54964984 0.2509045 1.4924273 0.6705153 1.20712596
#> [2,] 0.03002431 1.6448348 1.0682609 0.9355878 1.78280928
#> [3,] 1.42032590 1.1042607 0.4393119 1.3938968 0.01006476
#> attr(,"scale")
#> [1] 0.322
 
 # Example 2: Rescaling replicate weights with a specified value of 'tau'
 
 rescaled_wgts <- rescale_replicates(rep_wgts, new_scale = 1/10)
 print(rescaled_wgts)
#>            [,1]       [,2]         [,3]      [,4]       [,5]
#> [1,]  1.9863114 -0.3442039  1.883629115 0.4087608  1.3716742
#> [2,] -0.7405592  2.1571147  1.122489773 0.8844165  2.4047010
#> [3,]  1.7542479  1.1870891 -0.006118846 1.7068226 -0.7763753
#> attr(,"scale")
#> [1] 0.1

 # 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
 )

 ## Rescale the weights
 rescaled_boot_design <- boot_design |>
   rescale_replicates(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,] 25.25163  6.792090 11.91375 25.9341321 17.660472
#>  [2,] 11.89944 19.717751 41.86297 18.1492541 25.851653
#>  [3,] 14.46554 16.639588 15.36570 11.8919917  4.140530
#>  [4,] 34.99383 46.074337 29.91819 30.8022417 20.163220
#>  [5,] 15.20650 35.962609 14.30456 32.6383013 44.934993
#>  [6,] 27.61693  4.556777 11.58454 25.8737772 15.402844
#>  [7,] 12.17821 16.132594 38.76401  4.1982693 30.398438
#>  [8,] 20.25849 18.183084 19.27276  8.8739163  7.263187
#>  [9,] 33.99652 43.020337 34.69391 34.7705466 30.557832
#> [10,] 15.76371 32.534452 16.03927 35.4398085 44.534455
#> [11,] 26.07876  1.664742 13.32984 29.1289262 12.692999
#> [12,] 14.19590 14.785410 38.40768  0.2731898 23.279818
#> [13,] 19.37324 13.995200 17.87572  8.9034355  7.840506
#> [14,] 38.96908 48.967100 29.17256 30.6414059 21.600811
#> [15,] 15.30957 43.251199 17.47796 32.4468551 43.690789
#> attr(,"scale")
#> [1] 0.822
#> attr(,"rscales")
#> [1] 1 1 1 1 1