Bartlett corrections for zero-adjusted generalized linear models

Abstract

Zero-adjusted generalized linear models (ZAGLMs) are used in many areas to fit variables that are discrete at zero and continuous on the positive real numbers. As in other classes of regression models, hypothesis testing inference in the class of ZAGLMs is usually performed using the likelihood ratio statistic. However, the LR test is substantially size distorted when the sample size is small. In this work, we derive an analytical Bartlett correction of the LR statistic. We also consider two different adjustments for the LR statistic based on bootstrap. Monte Carlo simulation studies show that the improved LR tests have null rejection rates close to the nominal levels in small sample sizes and similar power. An application illustrates the usefulness of the improved statistics.

Appendix

The remaining quantities to define the Bartlett correction factor, see equation (8), are:

$$\begin{aligned} \varepsilon _{\beta _{p_{1}}}&= \frac{1}{4} \text{ tr } \left\{ {{\varvec{\Delta }}} {{\varvec{\Phi }}} {{\varvec{H}}}_{1} {\varvec{Z}}_{\beta d}^2 \right\} - \frac{1}{3} {{\varvec{1}}}^{\top } {{\varvec{\Delta }}} {{\varvec{\Phi }}} {{\varvec{G}}} {{\varvec{Z}}}_{\beta }^{(3)} ({{\varvec{F}}} + {{\varvec{G}}}) {{\varvec{\Phi }}} {{\varvec{\Delta }}} {{\varvec{1}}} \\&+ \frac{1}{12} {{\varvec{1}}}^{\top } {{\varvec{\Delta }}} {{\varvec{\Phi }}} {{\varvec{F}}} \left( 2 {{\varvec{Z}}}_{\beta }^{(3)} + 3 {{\varvec{Z}}}_{\beta d} {{\varvec{Z}}}_{\beta } {{\varvec{Z}}}_{\beta d} \right) {{\varvec{F}}} {{\varvec{\Phi }}} {{\varvec{\Delta }}} {{\varvec{1}}},\\ \varepsilon _{\delta _{p_{2}}}&= \frac{1}{4} \text{ tr } \left\{ {{\varvec{\Delta }}} {{\varvec{H}}}_{2} {{\varvec{Z}}}_{\delta d}^2 \right\} + \frac{1}{12} {{\varvec{1}}}^{\top } {{\varvec{\Delta }}} {{\varvec{D}}}_{3} {{\varvec{\Phi }}}_1^3 \left( 2 {{\varvec{Z}}}_{\delta }^{(3)} + 3 {{\varvec{Z}}}_{\delta d} {{\varvec{Z}}}_{\delta } {{\varvec{Z}}}_{\delta d} \right) {{\varvec{\Phi }}}_1^3 {{\varvec{D}}}_{3} {{\varvec{\Delta }}} {{\varvec{1}}} \\&+ \frac{1}{4} {{\varvec{1}}}^{\top } {{\varvec{\Delta }}} {{\varvec{D}}}_{2} {{\varvec{\Phi }}}_1 {{\varvec{\Phi }}}_2 \left( {{\varvec{Z}}}_{\delta d} {{\varvec{Z}}}_{\delta } {{\varvec{Z}}}_{\delta d} - 2 {{\varvec{Z}}}_{\delta }^{(3)} \right) {{\varvec{\Phi }}}_2 {{\varvec{\Phi }}}_1 {{\varvec{D}}}_{2} {{\varvec{\Delta }}} {{\varvec{1}}} \\&+ \frac{1}{2} {{\varvec{1}}}^{\top } {{\varvec{\Delta }}} {{\varvec{D}}}_{3} {{\varvec{\Phi }}}_1^3 {{\varvec{Z}}}_{\delta d} {{\varvec{Z}}}_{\delta } {{\varvec{Z}}}_{\delta d} {{\varvec{\Phi }}}_2 {{\varvec{\Phi }}}_1 {{\varvec{D}}}_{2} {{\varvec{\Delta }}} {{\varvec{1}}},\\ \varepsilon _{\gamma _{p_{3}}}&= \frac{1}{4} \text{ tr } \left\{ {{\varvec{H}}}_{3} {{\varvec{Z}}}_{\gamma d}^2 \right\} - \frac{1}{3} {{\varvec{1}}}^{\top } {{\varvec{G}}}_{\pi } {{\varvec{Z}}}_{\gamma }^{(3)} ({{\varvec{F}}}_{\pi } + {{\varvec{G}}}_{\pi }) {{\varvec{1}}} \\&+ \frac{1}{12} {{\varvec{1}}}^{\top } {{\varvec{F}}}_{\pi } \left( 2 {{\varvec{Z}}}_{\gamma }^{(3)} + 3 {{\varvec{Z}}}_{\gamma d} {{\varvec{Z}}}_{\gamma } {{\varvec{Z}}}_{\gamma d} \right) {{\varvec{F}}}_{\pi } {{\varvec{1}}}, \end{aligned}$$

${{\varvec{F}}} = \text{ diag }\{f_{11}, \ldots , f_{nn}\}$, ${{\varvec{G}}} = \text{ diag }\{g_{11}, \ldots , g_{nn}\}$, ${{\varvec{F}}}_{\pi } = \text{ diag }\{f_{\pi 11}, \ldots , f_{\pi nn}\}$, ${{\varvec{G}}}_{\pi } = \text{ diag }\{g_{\pi 11}, \ldots , g_{\pi nn}\}$, ${{\varvec{H}}}_{i} = \text{ diag }\{h_{i,11}, \ldots , h_{i,nn}\}$, $i = 1, 2, 3$, where

$$\begin{aligned} f_{\ell \ell }&= \frac{1}{V_{\ell }} \frac{d\mu _{\ell }}{d\eta _{1\ell }} \frac{d^2\mu _{\ell }}{d\eta _{1\ell }^2}, \ \ g_{\ell \ell } = \frac{1}{V_{\ell }} \frac{d\mu _{\ell }}{d\eta _{1\ell }} \frac{d^2\mu _{\ell }}{d\eta _{1\ell }^2} - \frac{1}{V_{\ell }^2} \frac{d V_{\ell }}{d\mu _{\ell }} \left( \frac{d\mu _{\ell }}{d\eta _{1\ell }}\right) ^3, \\ f_{\pi \ell \ell }&= \frac{1}{V_{\pi \ell }} \frac{d\pi _{\ell }}{d\eta _{3\ell }} \frac{d^2\pi _{\ell }}{d\eta _{3\ell }^2}, \ \ g_{\pi \ell \ell } = \frac{1}{V_{\pi \ell }} \frac{d\pi _{\ell }}{d\eta _{3\ell }} \frac{d^2\pi _{\ell }}{d\eta _{3\ell }^2} - \frac{1}{V_{\pi \ell }^2} \frac{d V_{\pi \ell }}{d\pi _{\ell }} \left( \frac{d\pi _{\ell }}{d\eta _{3\ell }}\right) ^3, \\ h_{1,\ell \ell }&= w_{\ell }^2 \left[ \frac{2}{V_{\ell }} \left( \frac{d V_{\ell }}{d \mu _{\ell }}\right) ^{2} - \frac{d^2 V_{\ell }}{d \mu _{\ell }^2} \right] + \frac{1}{V_{\ell }} \frac{d^2\mu _{\ell }}{d\eta _{1\ell }^2} \left[ \frac{d^2\mu _{\ell }}{d\eta _{1\ell }^2} - 4 w_{\ell } \frac{d V_{\ell }}{d \mu _{\ell }} \right] , \\ h_{2,\ell \ell }&= \frac{d^4 d_2(\phi _{\ell })}{d\phi _{\ell }^4} \left( \frac{d\phi _{\ell }}{d\eta _{2\ell }}\right) ^4 + 2 \frac{d^3 d_2(\phi _{\ell })}{d\phi _{\ell }^3} \left( \frac{d\phi _{\ell }}{d\eta _{2\ell }}\right) ^2 \frac{d^2\phi _{\ell }}{d\eta _{2\ell }^2} - \frac{d^2 d_2(\phi _{\ell })}{d\phi _{\ell }^2} \left( \frac{d^2\phi _{\ell }}{d\eta _{2\ell }^2}\right) ^2, \\ h_{3,\ell \ell }&= 2 \left[ \frac{1}{\pi _{\ell }^{2}(1 - \pi _{\ell })^{2}} + \frac{(1 - 2\pi _{\ell })^2}{\pi _{\ell }^{3}(1 - \pi _{\ell })^{3}} \right] \left( \frac{d\pi _{\ell }}{d\eta _{3\ell }}\right) ^4 \\&- \frac{4(1 - 2\pi _{\ell })}{\pi _{\ell }^{2}(1 - \pi _{\ell })^{2}} \left( \frac{d\pi _{\ell }}{d\eta _{3\ell }}\right) ^2 \frac{d^2\pi _{\ell }}{d\eta _{3\ell }^2} + \frac{1}{\pi _{\ell }(1 - \pi _{\ell })} \left( \frac{d^2\pi _{\ell }}{d^2\eta _{3\ell }}\right) ^2. \end{aligned}$$

Although deriving the expression for $\varepsilon _p$ entails a great deal of algebra, this expression only involves simple operations of diagonal matrices, i.e. they are simple expressions to implement.