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Generalized Variance Function (GVF) Smoothing for Sampling Variances

Source: R/gvf.R
gvf_smooth.Rd

Fits Generalized Variance Function (GVF) models to smooth unstable direct sampling variance estimates (\(D_i\)) in Small Area Estimation. Supports log-linear regression (Wolter, 2007), Gamma generalized linear models (GLM with log link), and CV power models.

Usage

gvf_smooth(
  vardir,
  y = NULL,
  n = NULL,
  method = c("log_linear", "gamma_glm", "cv_power"),
  data = NULL,
  formula = NULL,
  bias_correction = TRUE
)

Arguments

vardir

vector, column name, or one-sided formula referencing the direct sampling variances in data.

y

optional vector, column name, or one-sided formula referencing the direct survey estimates.

n

optional vector, column name, or one-sided formula referencing domain sample sizes.

method

character string specifying the GVF smoothing model:

"log_linear"

Log-linear model: \(\log(D_i) = \alpha_0 + \alpha_1 \log(y_i) + \alpha_2 \log(n_i) + \varepsilon_i\). Includes second-order bias correction upon exponentiation.

"gamma_glm"

Gamma GLM with log link: \(\mathrm{E}[D_i] = \mu_i, \log(\mu_i) = \alpha_0 + \dots\). Naturally suited for strictly positive variances without back-transformation bias.

"cv_power"

Power model on direct coefficients of variation: \(\log(CV_i^2) = \alpha_0 + \dots\).

Default is "log_linear".

data

optional data frame containing the variables.

formula

optional user-supplied formula overriding the default model specification.

bias_correction

logical. For "log_linear", whether to apply log-normal bias correction \(\tilde{D}_i = \exp(\hat{\mu}_i + \hat{\sigma}^2 / 2)\). Default is TRUE.

Value

An object of class "gvf_smooth" and "fastsae_gvf", containing:

smooth_vardir

numeric vector of smoothed sampling variances \(\tilde{D}_i\).

vardir

original raw sampling variances.

fitted_model

the fitted lm or glm model object.

method

method used.

r_squared

coefficient of determination \(R^2\) (or pseudo-\(R^2\) for GLM).

df

data frame with raw vardir, smooth_vardir, and covariates.

call

the matched function call.

References

  1. Wolter, K. M. (2007). Introduction to Variance Estimation (2nd ed.). Springer.

  2. Rao, J. N. K., & Molina, I. (2015). Small Area Estimation (2nd ed.). John Wiley & Sons.

  3. Rivest, L. P., & Belmonte, E. (2000). A conditional mean squared error of small area estimators. Survey Methodology, 26(1), 67-78.

Examples

library(fastsae)

# Smooth direct variances using direct estimates y and sample size n
gvf_res <- gvf_smooth(
  vardir = "vardir",
  y = "y",
  n = "n",
  data = mys,
  method = "log_linear"
)

# Inspect summary
summary(gvf_res)
#> -- Generalized Variance Function (GVF) Smoothing --
#> Method: "log_linear"
#> Model formula: log_vardir ~ log_y + log_n
#> R-squared / Pseudo R-squared: 0.2321
#> Number of domains: 42
#> Raw vardir range: [0, 10.12571]
#> Smoothed vardir range: [0, 3.24426]
#> 
#> -- Model Coefficients --
#>               Estimate Std. Error   t value   Pr(>|t|)
#> (Intercept)  2.2968261  1.1981633  1.916956 0.06514378
#> log_y        0.4782909  0.2327915  2.054589 0.04902744
#> log_n       -0.3356704  0.1418147 -2.366966 0.02482499

# Use smoothed variances directly in Fay-Herriot model
fit_fh <- eblup_fh(
  y ~ x1 + x2 + x3,
  data = mys,
  vardir = gvf_res$smooth_vardir
)
#> 
#> ── Fast Small Area Estimation (fastsae) ────────────────────────────────────────
#> Call:
#> eblup_fh(formula = y ~ x1 + x2 + x3, vardir = gvf_res$smooth_vardir, data =
#> mys)
#> 
#> ✔ Convergence: Yes (in 5 iterations)
#> Model: Fay-Herriot (Area-level)
#> Method: REML
#> Random effect variance (sigma2_u): 2.042318 
#> 
#> Fixed Effects Coefficients:
#>                   beta  std.error     zvalue pvalue
#> (Intercept)  2.9422920  0.7376432  3.9887742 0.0001
#> x1          -0.0047696  0.0100273 -0.4756575 0.6343
#> x2           0.0782809  0.0577486  1.3555478 0.1752
#> x3           0.0172653  0.0527029  0.3275958 0.7432
#> 
#> EBLUP Estimates (First 6 domains):
#>   domain        y    eblup   vardir random_effect       mse      rse
#> 1      1 8.359527 6.507267 1.858058    2.03594397 1.1309699 16.34282
#> 2      2 7.599650 5.676565 2.463466    1.59431901 1.2938894 20.03840
#> 3      3 5.514137 4.817897 2.478087    0.57380608 1.3523650 24.13734
#> 4      4 3.869326 4.655423 1.717030   -0.93502085 1.0972074 22.50014
#> 5      5 6.305063 6.331141 1.322897   -0.04025957 0.9197609 15.14801
#> 6      6 3.926807 4.421276 1.996022   -0.50593814 1.2099341 24.87902
#> ... and 36 more rows.
#>