Inference for a Poisson-Inverse Gaussian Model with an Application to Multiple Sclerosis Clinical Trials

Abstract

Magnetic resonance imaging (MRI) based new brain lesion counts are widely used to monitor disease progression in relapsing remitting multiple sclerosis (RRMS) clinical trials. These data generally tend to be overdispersed with respect to a Poisson distribution. It has been shown that the Poisson-Inverse Gaussian (P-IG) distribution fits better than the negative binomial to MRI data in RRMS patients that have been selected for lesion activity during the baseline scan. In this paper we use the P-IG distribution to model MRI lesion count data from RRMS parallel group trials. We propose asymptotic and simulation based exact parametric tests for the treatment effect such as the likelihood ratio (LR), score and Wald tests. The exact tests maintain precise Type I error levels whereas the asymptotic tests fail to do so for small samples. The LR test remains empirically unbiased and results in 30–50 % reduction in sample sizes required when compared to the Wilcoxon rank sum (WRS) test. The Wald test has the highest power to detect a reduction in the number of lesion counts and provides a 40–57 % reduction in sample sizes when compared to the WRS test.

Appendices

Appendix 1

The following lemma presents some useful results needed to obtain the score vector and the matrix of the second order derivatives.

Lemma 1

$$\begin{aligned} \frac{\partial \tau _1}{\partial \gamma }&= \ 0; \quad \ \quad \frac{\partial \tau _1}{\partial \mu } = \frac{\tau _1^3}{\mu ^3}; \ \ \quad \frac{\partial \tau _1}{\partial \lambda } = \frac{\tau _1^3}{\lambda ^2}; \\ \frac{\partial \tau _2}{\partial \gamma }&= \frac{\tau _2^3}{\gamma ^3\mu ^2}; \quad \frac{\partial \tau _2}{\partial \mu } = \frac{\tau _2^3}{\gamma ^2\mu ^3}; \quad \frac{\partial \tau _2}{\partial \lambda } = \frac{\tau _2^3}{\lambda ^2}. \end{aligned}$$

Using the above results we obtain

$$\begin{aligned} \frac{\partial \omega _1}{\partial \gamma }&= \ 0; \quad \ \quad \quad \frac{\partial \omega _1}{\partial \mu } = - \frac{\lambda \tau _1}{\mu ^3}; \quad \frac{\partial \omega _1}{\partial \lambda } = \frac{\lambda -\tau _1^2}{\lambda \tau _1}; \\ \frac{\partial \omega _2}{\partial \gamma }&= -\frac{\lambda \tau _2}{\gamma ^3\mu ^2}; \quad \frac{\partial \omega _2}{\partial \mu } = - \frac{\lambda \tau _2}{\gamma ^2\mu ^3}; \quad \frac{\partial \omega _2}{\partial \lambda } = \frac{\lambda -\tau _2^2}{\lambda \tau _2}. \end{aligned}$$

The following lemma provides results for modified Bessel functions that simplify the derivation of the score vector and the second order derivatives.

Lemma 2

(Modified Bessel function of the third kind (See Sect. 9.6, [2])).

The following relations hold for the modified Bessel function of the third kind $K_{\nu }(z)$:

$$\begin{aligned} K_{-\nu }(z)&= K_{\nu }(z) \nonumber \\ K_{-\frac{1}{2}}(z)&= K_{\frac{1}{2}}(z) = \sqrt{\frac{\pi }{2z}}e^{-z}\nonumber \\ K_{\nu +1}(z)&= K_{\nu -1}(z) + \frac{2\nu }{z}K_{\nu }(z) \\ \frac{\partial }{\partial z} K_{\nu }(z)&= K^\prime _{\nu }(z) = - K_{\nu +1}(z)+ \frac{\nu }{z}K_{\nu }(z).\nonumber \end{aligned}$$

(15)

The ratio of modified Bessel functions $R_{\nu }(z) = \frac{K_{\nu +1}(z)}{K_{\nu }(z)}$, satisfies the following relations:

$$\begin{aligned} R_{-\frac{1}{2}}(z)&= 1; \nonumber \\ R_{\nu }(z)&= \frac{2\nu }{z} + \frac{1}{R_{\nu -1}(z)}, \quad \nu = \frac{1}{2}, \frac{3}{2},\frac{5}{2}, \ldots ; \\ \frac{\partial }{\partial z} R_{\nu }(z)&= R^\prime _{\nu }(z) = R^2_{\nu }(z) - \frac{2(\nu +1/2)}{z}R_{\nu }(z) - 1.\nonumber \end{aligned}$$

(16)

Appendix 2

1.1 First and Second Order Derivatives

The score vector components for the log-likelihood function given in (9) are

$$\begin{aligned} \frac{\partial \ell (\gamma ,\mu ,\lambda )}{\partial \gamma }&= \frac{\lambda \tau _2}{\gamma ^3\mu ^2}\sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2) - \frac{n_2\lambda }{\gamma ^2\mu }, \end{aligned}$$

(17)

$$\begin{aligned} \frac{\partial \ell (\gamma ,\mu ,\lambda )}{\partial \mu }&= \frac{\lambda \tau _1}{\mu ^3} \sum _{i=1}^{n_1} R_{y_{1i}-\frac{1}{2}}(\omega _1) + \frac{\lambda \tau _2}{\gamma ^2\mu ^3} \sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2) - \frac{\lambda (n_1\gamma +n_2)}{\gamma \mu ^2}, \end{aligned}$$

(18)

$$\begin{aligned} \frac{\partial \ell (\gamma ,\mu ,\lambda )}{\partial \lambda }&= \frac{n_1\bar{y}_1+n_2\bar{y}_2}{\lambda } + \frac{n_1\gamma +n_2}{\gamma \mu } - \left( \frac{\lambda -\tau _1^2}{\lambda \tau _1}\right) \sum _{i=1}^{n_1} R_{y_{1i}-\frac{1}{2}}(\omega _1) \nonumber \\&\quad -\, \left( \frac{\lambda -\tau _2^2}{\lambda \tau _2}\right) \sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2). \end{aligned}$$

(19)

The second order derivatives of the log-likelihood function in (9) are

$$\begin{aligned} \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \gamma ^2}&= -\frac{\lambda ^2\tau _2^2}{\gamma ^6\mu ^4}\sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\lambda \tau _2}{\gamma ^6\mu ^4}(\tau ^2_2-3\gamma ^2\mu ^2) \sum _{j=1}^{n_2}R_{y_{2j}-\frac{1}{2}}(\omega _2)+\frac{2n_2\lambda }{\gamma ^3\mu } , \nonumber \\ \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \gamma \partial \mu }&= - \frac{\lambda ^2\tau _2^2}{\gamma ^5\mu ^5} \sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\lambda \tau _2^3}{\gamma ^5\mu ^5}(\tau _2^2-2\gamma ^2\mu ^2)\sum _{j=1}^{n_2}R_{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{n_2\lambda }{\gamma ^2\mu ^2}, \nonumber \\ \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \gamma \partial \lambda }&= \left( \frac{\lambda -\tau _2^2}{\gamma ^3\mu ^2} \right) \sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\tau _2}{\gamma ^3\mu ^2\lambda } (\tau _2^2+\lambda )\sum _{j=1}^{n_2}R_{y_{2j}-\frac{1}{2}}(\omega _2) - \frac{n_2}{\gamma ^2\mu }, \\ \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \mu ^2}&= -\frac{\lambda ^2\tau _1^2}{\mu ^6}\sum _{i=1}^{n_1} R^\prime _{y_{1i}-\frac{1}{2}}(\omega _1) + \frac{\lambda \tau _1}{\mu ^6}(\tau _1^2-3\mu ^2)\sum _{i=1}^{n_1} R_{y_{1i}-\frac{1}{2}}(\omega _1) \nonumber \\&\quad -\,\frac{\lambda ^2\tau _2^2}{\gamma ^4\mu ^6}\sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\lambda \tau _2}{\gamma ^4\mu ^6}(\tau _2^2-3\gamma ^2\mu ^2)\sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2) \nonumber \\&\quad +\,\frac{2\lambda (n_1\gamma +n_2)}{\gamma \mu ^3} ,\nonumber \\ \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \mu \partial \lambda }&= \left( \frac{\lambda - \tau _1^2}{\mu ^3} \right) \sum _{i=1}^{n_1} R^\prime _{y_{1i}-\frac{1}{2}}(\omega _1) + \frac{\tau _1}{\lambda \mu ^3}(\tau _1^2+\lambda ) \sum _{i=1}^{n_1} R_{y_{1i}-\frac{1}{2}}(\omega _1) \nonumber \\&\quad +\, \left( \frac{\lambda - \tau _2^2}{\gamma ^2\mu ^3} \right) \sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\tau _2}{\lambda \gamma ^2\mu ^3}(\tau _2^2+\lambda ) \sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2) \nonumber \\&\quad -\, \frac{n_1\gamma +n_2}{\gamma \mu ^2}, \nonumber \\ \frac{\partial ^2 \ell (\gamma ,\mu ,\lambda )}{\partial \lambda ^2}&= -\,\frac{n_1\bar{y}_1+n_2\bar{y}_2}{\lambda ^2} - \left( \frac{\lambda -\tau _1^2}{\lambda \tau _1} \right) ^2 \sum _{i=1}^{n_1} R^\prime _{y_{1i}-\frac{1}{2}}(\omega _1) + \frac{\tau _1^3}{\lambda ^3} \sum _{i=1}^{n_1} R_{y_{1i}-\frac{1}{2}}(\omega _1) \nonumber \\&\quad - \,\left( \frac{\lambda -\tau _2^2}{\lambda \tau _2} \right) ^2 \sum _{j=1}^{n_2} R^\prime _{y_{2j}-\frac{1}{2}}(\omega _2) + \frac{\tau _2^3}{\lambda ^3} \sum _{j=1}^{n_2} R_{y_{2j}-\frac{1}{2}}(\omega _2). \nonumber \end{aligned}$$

(20)

1.2 Fisher Information Matrix

Since $E(\bar{Y}_1) = \mu$ and $E(\bar{Y}_2)=\gamma \mu$, and $Y_{1i},\ i=1,\ldots ,n_1$ and $Y_{2j},\ j=1,\ldots ,n_2$ are respectively identically distributed, the elements $\big (I(\varvec{\theta })\big )_{i,j} = -E\left\{ \frac{\partial ^2\ell (\varvec{\theta })}{\partial \varvec{\theta }_i\partial \varvec{\theta }_j}\right\}$ of the FIM $\mathbf {I}(\varvec{\theta })$ with the parameter vector $\varvec{\theta }= (\gamma ,\mu ,\lambda )$ can be expressed as follows:

$$\begin{aligned} I_{11}(\varvec{\theta })&= \frac{n_2\lambda ^2\tau _2^2}{\gamma ^6\mu ^4} E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2\lambda \tau _2}{\gamma ^6\mu ^4}(\tau ^2_{2}-3\gamma ^2\mu ^2) E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2) \right\} -\frac{2n_2\lambda }{\gamma ^3\mu } , \\ I_{12}(\varvec{\theta })&= \frac{ n_2\lambda ^2\tau _2^2}{\gamma ^5\mu ^5} E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2) \right\} - \frac{n_2\lambda \tau _2^3}{\gamma ^5\mu ^5}(\tau _2^2-2\gamma ^2\mu ^2)E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2\lambda }{\gamma ^2\mu ^2}, \\ I_{13}(\varvec{\theta })&= -\,n_2 \left( \frac{\lambda -\tau _2^2}{\gamma ^3\mu ^2} \right) E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2\tau _2}{\gamma ^3\mu ^2\lambda } (\tau _2^2+\lambda ) E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2)\right\} + \frac{n_2}{\gamma ^2\mu },\\ I_{22}(\varvec{\theta })&= \frac{n_1\lambda ^2\tau _1^2}{\mu ^6} E\left\{ R^\prime _{Y_1-\frac{1}{2}}(\theta _1)\right\} - \frac{n_1\lambda \tau _1}{\mu ^6}(\tau _1^2-3\mu ^2)E\left\{ R_{Y_1-\frac{1}{2}}(\theta _1)\right\} - \frac{2\lambda (n_1\gamma +n_2)}{\gamma \mu ^3} \\&\quad + \frac{n_2\lambda ^2\tau _2^2}{\gamma ^4\mu ^6} E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2\lambda \tau _2}{\gamma ^4\mu ^6}(\tau _2^2-3\gamma ^2\mu ^2)E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2)\right\} ,\\ I_{23}(\varvec{\theta })&= -\, n_1 \left( \frac{\lambda - \tau _1^2}{\mu ^3} \right) E\left\{ R^\prime _{Y_1-\frac{1}{2}}(\theta _1)\right\} - \frac{n_1\tau _1}{\lambda \mu ^3}(\tau _1^2+\lambda ) E\left\{ R_{Y_1-\frac{1}{2}}(\theta _1)\right\} + \frac{n_1\gamma +n_2}{\gamma \mu ^2} \\&\quad - n_2 \left( \frac{\lambda - \tau _2^2}{\gamma ^2\mu ^3} \right) E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2\tau _2}{\lambda \gamma ^2\mu ^3}(\tau _2^2+\lambda )E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2)\right\} , \\ I_{33}(\varvec{\theta })&= \frac{n_1\mu +n_2\gamma \mu }{\lambda ^2} + n_1 \left( \frac{\lambda -\tau _1^2}{\lambda \tau _1} \right) ^2 E\left\{ R^\prime _{Y_1-\frac{1}{2}}(\theta _1)\right\} - \frac{n_1 \tau _1^3}{\lambda ^3} E\left\{ R_{Y_1-\frac{1}{2}}(\theta _1)\right\} \\&\quad +\, n_2 \left( \frac{\lambda -\tau _2^2}{\lambda \tau _2} \right) ^2 E\left\{ R^\prime _{Y_2-\frac{1}{2}}(\theta _2)\right\} - \frac{n_2 \tau _2^3}{\lambda ^3} E\left\{ R_{Y_2-\frac{1}{2}}(\theta _2) \right\} . \end{aligned}$$

Further, $I_{ij}=I_{ji}$ for $i \ne j = 1, 2, 3$. The above expressions involve evaluating the expectation of R(.), which is a ratio of two modified Bessel functions of the third kind, which cannot be computed in closed form. Instead, the observed information evaluated at the MLEs are used.