Generate Gaussian latent site effects

Draw J standardized site effects from the standard normal distribution and apply the shared Layer 1 location-scale wrapper to produce response-scale effects \(\tau_j = \tau + X_j\boldsymbol{\beta} + \sigma_\tau\,z_j\). The Gaussian shape is the canonical baseline of the eight-distribution catalog — pick it when you have no specific reason to prefer a heavier-tailed or asymmetric shape, or when matching the JEBS paper's reference design.

Usage

gen_effects_gaussian(
  J,
  tau = 0,
  sigma_tau = 0.2,
  variance = 1,
  formula = NULL,
  beta = NULL,
  data = NULL
)

Arguments

J

Integer. Number of sites.

tau

Numeric. Grand mean on the response scale. Default 0.

sigma_tau

Numeric (\(\ge 0\)). Between-site standard deviation on the response scale. Default 0.20.

variance

Numeric. Legacy Gaussian variance argument. Default 1. The unit-variance convention requires variance = 1; passing any other value aborts. Control heterogeneity through sigma_tau instead.

formula

One-sided formula for site-level covariates (e.g., ~ x1 + x2), or NULL.

beta

Numeric coefficient vector matching formula, or NULL.

data

A data.frame with the predictors named in formula, or NULL.

Value

A tibble with one row per site and columns site_index (integer 1:J), z_j (standard-normal residual), tau_j (response-scale effect), plus any covariate columns from data.

Details

The standardized residuals \(z_j\) are drawn from \(\mathcal{N}(0, 1)\) directly via rnorm; no rescaling is needed because the standard normal already satisfies the unit-variance convention shared by all Layer 1 generators (see gen_effects).

For the broader catalog of \(G\) distributions and a decision rubric on when to choose a heavier-tailed or asymmetric shape, see the G-distribution catalog and standardization vignette.

References

Lee, J., Che, J., Rabe-Hesketh, S., Feller, A., & Miratrix, L. (2025). Improving the estimation of site-specific effects and their distribution in multisite trials. Journal of Educational and Behavioral Statistics, 50(5), 731–764. doi:10.3102/10769986241254286 .

See also

gen_effects for the dispatcher and the full eight-shape catalog; gen_effects_studentt and gen_effects_skewn for heavier-tailed and asymmetric variants; the M2 G-distribution catalog vignette.

Other family-effects: gen_effects(), gen_effects_ald(), gen_effects_dpm(), gen_effects_mixture(), gen_effects_pmslab(), gen_effects_skewn(), gen_effects_studentt(), gen_effects_user()

Examples

# Minimal: ten standardized Gaussian effects.
gen_effects_gaussian(J = 10L)
#> # A tibble: 10 × 3
#>    site_index     z_j    tau_j
#>         <int>   <dbl>    <dbl>
#>  1          1 -0.0242 -0.00484
#>  2          2  0.825   0.165  
#>  3          3 -0.750  -0.150  
#>  4          4  0.536   0.107  
#>  5          5 -1.57   -0.315  
#>  6          6 -0.986  -0.197  
#>  7          7  1.98    0.397  
#>  8          8 -1.85   -0.370  
#>  9          9 -0.910  -0.182  
#> 10         10 -1.95   -0.390  

# Larger draw with a non-zero grand mean and modest heterogeneity.
gen_effects_gaussian(J = 50L, tau = 0.2, sigma_tau = 0.15)
#> # A tibble: 50 × 3
#>    site_index     z_j    tau_j
#>         <int>   <dbl>    <dbl>
#>  1          1 -0.800   0.0800 
#>  2          2 -1.87   -0.0804 
#>  3          3 -0.751   0.0874 
#>  4          4 -0.591   0.111  
#>  5          5 -0.742   0.0887 
#>  6          6  0.694   0.304  
#>  7          7 -0.0595  0.191  
#>  8          8 -1.86   -0.0796 
#>  9          9 -1.27    0.00882
#> 10         10 -1.78   -0.0673 
#> # ℹ 40 more rows