assign_hadamard_rows function

Create a row-assignment matrix for Successive Difference Replication-Method

Creates a row assignment matrix: for each row of a dataset of size $n$, assigns two rows of a Hadamard matrix.

assign_hadamard_rows(
  n,
  hadamard_order,
  number_of_cycles = ceiling(n/hadamard_order),
  use_first_row = TRUE,
  circular = TRUE
)

Arguments

• n: The sample size of the data.
• hadamard_order: The order of the Hadamard matrix (i.e., the number of rows/columns)
• number_of_cycles: The number of cycles to use in the row assignment. Must be at least as large as n/hadamard_order. Only applies when n exceeds the number of available rows in the Hadamard matrix. The number of available rows is hadamard_order when use_first_row = TRUE, and hadamard_order - 1 when use_first_row = FALSE.
• use_first_row: Whether to use the first row of the Hadamard matrix. The first row of a Hadamard matrix is often all 1's, and so using the first row to create replicate factors leads to the creation of a replicate whose weights exactly match the full-sample weights. Thus, using the first row of the Hadamard matrix may be undesirable for practical purposes, even if it is valid for the purpose of variance estimation.
• circular: TRUE or FALSE. Only applies when the number of available rows in the Hadamard matrix is at least as large as n. Whether to make a circular row assignment, so that the resulting successive-difference replication variance estimator is equivalent to the SD2 variance estimator rather than the SD1 variance estimator (see Ash 2014). The number of available rows is hadamard_order when use_first_row = TRUE, and hadamard_order - 1 when use_first_row = FALSE.

Returns

A matrix with n rows and two columns. Each row gives the assignment of two rows of a Hadamard matrix to the row of data.

Details

Implements row-assignment methods described in Ash (2014) and in Fay and Train (1995). The row-assignment method depends on the number of available rows of the Hadamard matrix used. The number of available rows is hadamard_order when use_first_row = TRUE, and hadamard_order - 1 when use_first_row = FALSE.

When the number of available Hadamard rows is at least as large as n, then the row assignment is as follows. Let $i=1,\dots,n$ be the index for the data that will receive row assignments, and let $a_j$ denote row $j$ of a Hadamard matrix. For $i < n$, the assignments are entries $a_i$ and $a_{i+1}$ when use_first_row = TRUE, and when use_first_row = FALSE, the assignments are entries $a_{i+1}$ and $a_{i+2}$. The assignment for $i=n$ depends on whether circular = TRUE. If $circular=TRUE$, then the assignment for $i=n$ is entries $a_i$ and $a_1$ if use_first_row = TRUE, and entries $a_{i+1}$ and $a_2$ if use_first_row = FALSE.

When the number of available Hadamard rows is less than n, then the row assignment method is the method denoted as RA1 in Ash (2014). This method uses the argument number_of_cycles and does not use the argument circular.

References

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