Generalized Information Criteria for the Best Logit Model

Abstract

In this paper the $\gamma$–order Generalized Fisher’s entropy type Information measure ($\gamma$–GFI) is adopted as a criterion for the selection of the best Logit model. Thus the appropriate Relative Risk model can be evaluated through an algorithm. The case of the entropy power is also discussed as such a criterion. Analysis of a real breast cancer data set is conducted to demonstrate the proposed algorithm, while algorithm’s realizations, through MATLAB scripts, are cited in Appendix.

Appendices

Appendix 1

A study on the min–max behavior of the generalized entropy power $\mathrm N_\delta (X_\gamma )$ applied on a $\gamma$-order normally distributed random variable is given below.

Fig. 3

Graphs of $\mathrm N_\delta (X_\gamma )$ along $\delta$ for various $\gamma$ values, where $X_\gamma \sim {\fancyscript{N}}_\gamma (0,\sigma ^2)$ with $\sigma = 0.8,1,1.5$

Let $X_\gamma \in {\fancyscript{N}}^p_\gamma (\mu ,{\Sigma })$. For $\delta > 1$, numerically can be verified (see Fig. 1) that,

$$ \underset{\gamma > 1}{\max }\mathrm N_\delta (X_\gamma ) = \underset{\gamma < 0}{\min }\mathrm N_\delta (X_\gamma ) = \mathrm N_\delta (X_{+\infty }), $$

$$ \underset{\delta \ge 1}{\min \;}\underset{\gamma > 1}{\max }\mathrm N_\delta (X_\gamma ) = \mathrm N_{+\infty }(X_{+\infty }) = 1,\;\;\text {when}\;\; |\det {\Sigma }| = 1, $$

$$ \underset{\delta \ge 1}{\max \;}\underset{\gamma < 0}{\min }\mathrm N_\delta (X_\gamma ) = \mathrm N_1(X_{+\infty }) = e|\det {\Sigma }|^\frac{1}{2p}. $$

The dual case hold for $\delta < 0$ ($\max /\min$ reversion). Moreover, for $|\det {\Sigma }| \le 1$, see also Fig. 3, the following holds:

$$ \underset{\delta \ge 1}{\max }\mathrm N_\delta (X_\gamma ) = \mathrm N_1(X_\gamma ),\quad \gamma > 1, $$

$$ \underset{\gamma \ge 1}{\min \;}\underset{\delta \ge 1}{\max }\mathrm N_\delta (X_\gamma ) = \mathrm N_1(X_1) = |\det {\Sigma }|^\frac{1}{2p}, $$

For $|\det {\Sigma }| \ge 1$, Fig. 3, we have

$$ \underset{\gamma > 1}{\max }\left\{ \underset{\delta \ge 0}{\min }\mathrm N_\delta (X_\gamma ) \ne 0\right\} = \underset{\delta > 1}{\min }\mathrm N_\delta (X_{+\infty }). $$

Appendix 2

A study on the min–max behavior of $\mathrm N_\delta (X_{j;\gamma })$ and $\mathrm J_\delta (X_{j;\gamma })$ applied on the $\gamma$-order normally distributed risk variables $X_{j,\gamma }$, $j = 1,2,\dots ,5$ is given below.

Fig. 4

Graphs of $\mathrm N_\delta (X_{2;\gamma })$ along $\gamma$ for various $\delta$ values, where $X_{2;\gamma }\sim {\fancyscript{N}}_\gamma (\mu _2,\sigma ^2_{2;\gamma })$

Figure 4, verifies that

$$ \mathop {\max }_{\gamma > 1}\left\{ \mathrm N_\delta (X_{j;\gamma })\right\} = \mathrm N_\delta (X_{j;2}) = \mathrm N_\delta (X_j),\quad \delta > 1, $$

i.e. the $\max _{\gamma > 1}\mathrm N_\delta (X_{j;\gamma })$ corresponds, for $\delta > 1$, to the usual Normal distribution, and therefore,

$$ \mathop {\min }_{\delta > 1}\mathop {\max }_{\gamma > 0}\left\{ \mathrm N_\delta (X_{j;\gamma })\right\} = m_j, $$

with $m_j$ as in Table 3. Moreover, the $\max _{\gamma > 1}\mathrm N_\delta (X_{j;\gamma })$ for $\delta < 0$ corresponds to the usual Laplace distribution, i.e.

$$ \mathop {\max }_{\gamma > 1}\left\{ \mathrm N_\delta (X_{j;\gamma })\right\} = \mathrm N_\delta (X_{j;-\infty }),\quad \delta < 0. $$

while for $\delta > 1$, through (23) with $\frac{\gamma -1}{\gamma }\rightarrow 1$, is given by

$$ \mathop {\max }_{\gamma <0}\left\{ \mathrm N_\delta (X_{j;\gamma })\right\} =\mathrm N_\delta (X_{j;-\infty }) = e(\tfrac{\delta -1}{\delta })^{\delta -1} {\Gamma }^{-\delta }\left( \tfrac{\delta -1}{\delta }+1\right) \sigma _j/\sqrt{2},\quad \delta > 1,\;\;\text {i.e.} $$

$$ m^*_j := \mathop {\min }_{\delta > 1}\mathop {\max }_{\gamma < 0}\left\{ \mathrm N_\delta (X_{j;\gamma })\right\} = \mathrm N_1(X_{j;-\infty }) = \tfrac{1}{2}e\sqrt{2}\sigma _j, $$

with $m^*_j$ as in Table 6 (compared with Table 3).

Table 6 Evaluations of $m^*_j$ and $m^{**}_j$, $j = 2,3,4,5$

For the $\delta$–GFI, we have that $\mathrm J_\delta (X_\gamma )$ is a monotone function of $\delta \ge \gamma (1.4628-1)+1$ for all $X_\gamma \sim {\fancyscript{N}}_{\gamma \ge 1}(\mu ,1)$, see proof of Proposition 3.1 in [20]. Thus $\mathrm J_\delta (X_\gamma )$ is an increasing function of $\delta \ge 2$. Therefore,

$$\mathop {\min }_{\delta \ge 2}\mathrm J_\delta (X_\gamma ) = \mathrm J_2(X_\gamma ) = \mathrm J(X_\gamma ),\quad \gamma \ge 1. $$

The above relation holds for $X_\gamma \sim {\fancyscript{N}}_{\gamma \ge 1}(\mu ,\sigma ^2)$, due to (16), assuming $\sigma \ge 1$, and hence

$$ \mathop {\min }_{\delta \ge 2}\mathrm J_\delta (X_{j;\gamma }) = (\tfrac{\gamma -1}{\gamma })^{2/\gamma }\frac{{\Gamma }(\frac{3\gamma -1}{\gamma })}{{\Gamma }(\frac{\gamma -1}{\gamma })}\sigma ^{-2}_{j;\gamma }, \quad \gamma \ge 1. $$

Through (23) we have

$$ \mathop {\min }_{\delta \ge 2}\mathrm J_\delta (X_{j;\gamma }) = (\tfrac{\gamma -1}{\gamma })^{2\frac{2-\gamma }{\gamma }}\frac{{\Gamma }(3\frac{\gamma -1}{\gamma }){\Gamma }(\frac{3\gamma -1}{\gamma })}{{\Gamma }^2(\frac{\gamma -1}{\gamma })}\sigma ^{-2}_j,\quad \gamma \ge 1. $$

Numerically we can derive that (see Table 6)

$$ m_j^{**} := \mathop {\max }_{\gamma \ge 1}\mathop {\min }_{\delta \ge 2}\mathrm J(X_{j;\gamma }) = \mathrm J_\delta (X_{j;+\infty }). $$

Appendix 3

Appendix 3.1. MATLAB script for the evaluations of Table 3. The 5th column of the Data matrix correspond to the data for Age, COMT, CYP17, Menarche and Menopause risk variables, while the b0 and b1 arrays contain the coefficients from Table 2.

Appendix 3.2. MATLAB script for the evaluations of Table 4. Array s2 and variable k are defined in the previous script.

Appendix 3.3. MATLAB script for the evaluations of Table 5. Array V and MATLAB function Var are defined in the previous script.