add_confregions: 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)

Value

Object of class field3logit or multifield3logit with updated confidence regions.

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 $\pi_0$ over the simplex $S=\{(\pi^{(1)}, \pi^{(2)}, \pi^{(3)})\in[0,1]^3\colon \pi^{(1)}+\pi^{(2)}+\pi^{(3)}=1\}$, and a change $\Delta\in\mathbf{R}^k$ of covariate values, the confidence region of the probability distribution resulting from the covariate change $\Delta$ is computed by means of the Wald statistics severini2000plot3logit, which should satisfy the following condition wooldridge2010plot3logit: $$ (\delta-\hat\delta)^\top [(I_2\otimes\Delta)^\top\,\hat\Xi\,(I_2\otimes\Delta)]^{-1} (\delta-\hat\delta) \leq\chi^2_2(1-\alpha) $$ where $\hat\delta=\hat{B}^\top\Delta\in\mathbf{R}^2$ is the point estimate of change of natural parameters associated to $\Delta$, $\hat{B}=[\beta^{(2)}, \beta^{(3)}]\in\mathbf{R}^{k\times 2}$ is the matrix of point estimates of regression coefficients, $I_2$ is the identity matrix of order two, $\otimes$ is the Kronecker product, $\hat\Xi\in\mathbf{R}^{2k\times2k}$ is the covariance matrix of $vec(\hat{B})$, and finally, $\chi^2_2(1-\alpha)$ is the $(1-\alpha)$ quantile of $\chi^2_2$.

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

If we denote with $\mathcal{R}_\delta$ the set of points $\delta$ which satisfy the previous inequality, it is possible to obtain the confidence region of the effect of the covariate change $\Delta$ over the simplex $S$ as follows: $$ \mathcal{R}=\{g^\leftarrow(g(\pi_0)+\delta)\colon \delta\in\mathcal{R}_\delta\} \subseteq S $$ where $g\colon S\to\mathbf{R}^2$ and $g^\leftarrow\colon\mathbf{R}^2\to S$ are respectively the link function of the trinomial logit model and its inverse. They are defined as follows: $$ g(\pi)= g([\pi^{(1)},\pi^{(2)},\pi^{(3)}]^\top) =\left[\ln\frac{\pi^{(2)}}{\pi^{(1)}}\,,\quad\ln\frac{\pi^{(3)}}{\pi^{(1)}}\right]^\top $$ $$ g^\leftarrow(\eta)=g^\leftarrow([\eta_2,\eta_3]^\top) =\left[ \frac{1}{1+e^{\eta_2}+e^{\eta_3}}\,,\quad \frac{e^{\eta_2}}{1+e^{\eta_2}+e^{\eta_3}}\,,\quad \frac{e^{\eta_3}}{1+e^{\eta_2}+e^{\eta_3}} \right]^\top\,. $$ For further details and notation see santi2022;textualplot3logit and santi2019;textualplot3logit.

References

Examples

Run this code

data(cross_1year)

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

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