Decompose DER into Components

Decomposes each parameter's DER into its constituent factors. For fixed effects, the protection factor $R_k$ is computed structurally from the working-weighted within/between split of covariate $k$'s identifying variation: $$R_k = \frac{\sum_j B_j\, a_j \bar x_{jk}^2} {\sum_j (W_{jkk} + a_j \bar x_{jk}^2)},$$ where $a_j$ is the working information of group $j$, $\bar x_{jk}$ the working-weighted group mean of covariate $k$, and $W_{jkk}$ its within-group working sum of squares. A pure within-group covariate has $R_k = 0$; a pure between-group covariate has $R_k$ equal to the information-weighted average shrinkage factor. The prediction is $DER_k \approx DEFF \cdot (1 - R_k)$ with the global Kish DEFF.

Usage

der_decompose(x)

Arguments

x: A svyder object from svyder >= 0.2.0 (the stored $data slots are required).

Value

A data.frame with one row per parameter and columns:

param: Parameter name.
param_type: "fe_within", "fe_between", or "re".
der: Observed Design Effect Ratio.
deff_used: Design effect entering the prediction: the global Kish DEFF for fixed effects, the per-group $\mathrm{DEFF}_j$ for random effects.
B_used: Shrinkage factor $B_j$ (random-effect rows only; NA for fixed effects).
R_k: Structural between-group share of covariate $k$'s identifying variation (fixed-effect rows only; NA for random effects).
kappa: Finite-group correction $\kappa_j$ (random-effect rows only; NA for fixed effects).
der_predicted: Theorem-based prediction: $\mathrm{DEFF}\, (1 - R_k)$ for fixed effects (Theorem 1), $B_j\,\mathrm{DEFF}_j\,\kappa_j$ for random effects (Theorem 2).

Details

For random effects the per-group prediction $DER_j \approx B_j \cdot DEFF_j \cdot \kappa_j$ is used (matching der_theorem_check()), not a pooled mean.

Because der_predicted is structural, comparing it with the observed der is a genuine check of the decomposition theorems' mechanism, within their stated model class.

The prediction is row-specific: der_predicted uses Theorem 1 ($\mathrm{DEFF}\cdot(1 - R_k)$) for fixed-effect rows and Theorem 2 ($B_j\cdot\mathrm{DEFF}_j\cdot\kappa_j$) for random-effect rows. Consequently the component columns are populated per row type: B_used and kappa are NA for fixed effects (whose prediction uses only DEFF and $R_k$), and R_k is NA for random effects (whose prediction uses only $B_j$, $\mathrm{DEFF}_j$ and $\kappa_j$).

References

Lee, J., Williams, M. R., & Savitsky, T. D. (2026). Design Effect Ratios for Bayesian Survey Models: A Diagnostic Framework for Identifying Survey-Sensitive Parameters. Journal of Survey Statistics and Methodology. Submitted.

Kish, L. (1965). Survey Sampling. John Wiley & Sons.

Fay, R. E., & Herriot, R. A. (1979). Estimates of income for small places: An application of James–Stein procedures to census data. Journal of the American Statistical Association, 74(366), 269–277. doi:10.1080/01621459.1979.10482505

Examples

data(nsece_demo)
result <- der_diagnose(
  nsece_demo$draws,
  y = nsece_demo$y, X = nsece_demo$X,
  group = nsece_demo$group, weights = nsece_demo$weights,
  cluster = nsece_demo$psu, family = "binomial",
  sigma_theta = nsece_demo$sigma_theta,
  param_types = nsece_demo$param_types
)
decomp <- der_decompose(result)
head(decomp)
#>      param param_type       der deff_used    B_used        R_k     kappa
#> 1  beta[1] fe_between 0.2626885  2.727828        NA 0.93273657        NA
#> 2  beta[2]  fe_within 2.6894416  2.727828        NA 0.01430246        NA
#> 3  beta[3] fe_between 0.3430527  2.727828        NA 0.92282567        NA
#> 4 theta[1]         re 3.3858856  3.479991 0.6441524         NA 0.9467869
#> 5 theta[2]         re 0.6759045  2.417686 0.6072076         NA 0.9515495
#> 6 theta[3]         re 1.1182861  1.878577 0.5847477         NA 0.9540496
#>   der_predicted
#> 1     0.1834831
#> 2     2.6888138
#> 3     0.2105183
#> 4     2.1223599
#> 5     1.3969105
#> 6     1.0480173