make_sdr_replicate_factors function

Create matrix of replicate factors to use for Successive Difference Replication Method

make_sdr_replicate_factors(
  n,
  target_number_of_replicates,
  use_normal_hadamard = FALSE
)

Arguments

• n: The number of sampling units
• target_number_of_replicates: The target number of replicates to create. This will determine the order of the Hadamard matrix to use when creating replicate factors. The actual number of replicates will be a multiple of 4. If use_normal_hadamard = FALSE, then the actual number of replicates will be $4 \times 2^k$ for some integer $k$, which means that the actual number of replicates might be much larger than the target.
• use_normal_hadamard: Whether to use a normal Hadamard matrix: that is, a matrix whose first row and first column only have entries equal to 1.

Returns

A matrix of replicate factors, with n rows and the number of columns corresponding to the order of the Hadamard matrix used (which is greater than or equal to target_number_of_replicates).

Examples

# Note that one of the replicates has every factor equal to 1
# Also note that this matches Table 1 in Ash (2014)
make_sdr_replicate_factors(
  n = 4,
  target_number_of_replicates = 4,
  use_normal_hadamard = TRUE
)

# Note the difference when using a non-normal Hadamard matrix
rep_factors <- make_sdr_replicate_factors(
  n = 4,
  target_number_of_replicates = 4,
  use_normal_hadamard = FALSE
)
print(rep_factors)

# These replicate factors are equivalent
# to the SD2 variance estimator
tcrossprod(rep_factors - 1)

# Compare to the quadratic form of the SD2 estimator
sd2_quad_form <- make_quad_form_matrix(
  variance_estimator = "SD2",
  cluster_ids        = matrix(1:4, ncol = 1),
  sort_order         = matrix(1:4, ncol = 1)
)
print(sd2_quad_form)