Rescaling of logit coefficients across nested models — morie_logit

In a latent-index model with logistic error (variance \(\pi^2/3\)) and an omitted covariate independent of the others carrying latent variance omitted_var, the coefficient a logit identifies shrinks by the factor \(\sqrt{(\pi^2/3)/(\pi^2/3 + v)}\) when that covariate is dropped, with no confounding at all (Research.P5.reduced_coefficient, rescale_lt_one, rescale_eq_one_iff); the apparent odds-ratio change is \(\exp((c-1)\beta)\) (ratio_is_rescaling). Coefficients or odds ratios of a risk score therefore cannot be compared across models with different covariate sets without a variance normalisation (Karlson, Holm & Breen 2012). The identity is exact for a probit index (error_var = 1: normal plus normal is normal); for the logit the logistic-plus-normal mixture is not logistic and the factor is the standard approximation, accurate to a few percent in simulation.

Usage

morie_logit_rescale(beta, omitted_var, error_var = pi^2/3)

Arguments

beta: Identified coefficient(s) in the full model.
omitted_var: Latent variance carried by the omitted independent covariate(s) (\(\gamma^2\tau^2\)), non-negative.
error_var: Error variance of the latent index; \(\pi^2/3\) for the logit, 1 for the probit.

Value

A list with rescale, beta_reduced (what the reduced model identifies), odds_ratio_full, odds_ratio_reduced, apparent_change (their ratio, pure rescaling) and theorems.

Examples

morie_logit_rescale(beta = 0.8, omitted_var = 1)
#> $rescale
#> [1] 0.875724
#> 
#> $beta_reduced
#> [1] 0.7005792
#> 
#> $odds_ratio_full
#> [1] 2.225541
#> 
#> $odds_ratio_reduced
#> [1] 2.014919
#> 
#> $apparent_change
#> [1] 0.9053617
#> 
#> $theorems
#> [1] "Research.P5.rescale_lt_one"      "Research.P5.rescale_eq_one_iff" 
#> [3] "Research.P5.reduced_coefficient" "Research.P5.ratio_is_rescaling" 
#>