Kappa Utilities

Functions for computing and exploring the kappa diagnostic, which measures theoretical imputation bias in the conditional independence test.

compute_kappa(r2_x_z, beta_yx, gamma_x)

Theoretical imputation bias for CI testing.

Parameters:

Name	Type	Description	Default
r2_x_z	float	R-squared of X on observed covariates Z (i.e. 1 - Var(V)/Var(X)). In the single-proxy case this equals rho^2.	required
beta_yx	float	Coefficient of X in the Y equation.	required
gamma_x	float	Loading of X in the missingness equation.	required

Returns:

Type	Description
float	kappa = gamma_x * beta_yx * (1 - R2) / (1 + beta_yx^2 * (1 - R2))

kappa_calibration_table(r2_grid=None, beta_grid=None, gamma_grid=None)

Produce a calibration table of kappa over realistic parameter ranges.

Default grids span typical survey / administrative data settings: R2 from poor to strong imputation, and partial effects in the 0.1-0.5 standardised range.

Returns:

Type	Description
pd.DataFrame with columns r2_x_z, beta_yx, gamma_x, kappa, abs_kappa.

print_calibration_pivot(df=None, beta_yx=0.3)

Return a readable pivot: rows = R2, cols = gamma_x, for fixed beta_yx.

Useful for the appendix table. Call once per beta_yx value you want to display.