add_confregions: Compute the confidence regions of covariate effects

In plot3logit: Ternary Plots for Trinomial Regression Models

Compute the confidence regions of covariate effects

Description

Given the confidence level, it computes the confidence regions of the effects for each arrow of the field3logit or multifield3logit object given in input. If the field3logit or multifield3logit object already contains the confidence regions, they will be updated if the value of conf is different.

Usage

add_confregions(x, conf = 0.95, npoints = 100)

Arguments

x	an object of class field3logit or multifield3logit.
conf	confidence level of the regions.
npoints	number of points of the borders of the regions.

Details

Given a reference probability distribution π_0 over the simplex S=\{(π^{(1)}, π^{(2)}, π^{(3)})\in[0,1]^3\colon π^{(1)}+π^{(2)}+π^{(3)}=1\}, and a change Δ\in\mathbf{R}^k of covariate values, the confidence region of the probability distribution resulting from the covariate change Δ is computed by means of the Wald statistics \insertCiteseverini2000plot3logit, which should satisfy the following condition \insertCitewooldridge2010plot3logit:

(δ-\hatδ)^\top [(I_2\otimesΔ)^\top\,\hatΞ\,(I_2\otimesΔ)]^{-1} (δ-\hatδ) ≤qχ^2_2(1-α)

where \hatδ=\hat{B}^\topΔ\in\mathbf{R}^2 is the point estimate of change of natural parameters associated to Δ, \hat{B}=[β^{(2)}, β^{(3)}]\in\mathbf{R}^{k\times 2} is the matrix of point estimates of regression coefficients, I_2 is the identity matrix of order two, \otimes is the Kronecker product, \hatΞ\in\mathbf{R}^{2k\times2k} is the covariance matrix of vec(\hat{B}), and finally, χ^2_2(1-α) is the (1-α) quantile of χ^2_2.

The set of points which satisfy the previous inequality with equal sign delimits the border of the confidence region for δ.

If we denote with \mathcal{R}_δ the set of points δ which satisfy the previous inequality, it is possible to obtain the confidence region of the effect of the covariate change Δ over the simplex S as follows:

\mathcal{R}=\{g^≤ftarrow(g(π_0)+δ)\colon δ\in\mathcal{R}_δ\} \subseteq S

where g\colon S\to\mathbf{R}^2 and g^≤ftarrow\colon\mathbf{R}^2\to S are respectively the link function of the trinomial logit model and its inverse. They are defined as follows:

g(π)= g([π^{(1)},π^{(2)},π^{(3)}]^\top) =≤ft[\ln\frac{π^{(2)}}{π^{(1)}}\,,\quad\ln\frac{π^{(3)}}{π^{(1)}}\right]^\top

g^≤ftarrow(η)=g^≤ftarrow([η_2,η_3]^\top) =≤ft[ \frac{1}{1+e^{η_2}+e^{η_3}}\,,\quad \frac{e^{η_2}}{1+e^{η_2}+e^{η_3}}\,,\quad \frac{e^{η_3}}{1+e^{η_2}+e^{η_3}} \right]^\top\,.

For further details and notation see \insertCitesanti2022;textualplot3logit and \insertCitesanti2019;textualplot3logit.

Value

Object of class field3logit or multifield3logit with updated confidence regions.

References

\insertAllCited

Examples

data(cross_1year)

mod0 <- nnet::multinom(employment_sit ~ gender + finalgrade,
  data = cross_1year)
field0 <- field3logit(mod0, 'genderFemale')
field0
add_confregions(field0)

plot3logit documentation built on July 19, 2022, 9:07 a.m.