rescale_reps {svrep}

Rescale replicate factors

Description

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.

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

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').

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)
 print(rescaled_wgts)
 
 # Example 2: Rescaling replicate weights with a specified value of 'tau'
 
 rescaled_wgts <- rescale_reps(rep_wgts, tau = 2)
 print(rescaled_wgts)

 # 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)
 print(rescaled_boot_wgts)