Modified Chi-Squared Goodness-of-Fit Tests for Continuous Right-Skewed Response Generalized Linear Models

Abstract

Generalized linear models are applied for data analysis in various areas. One of the most important steps in fitting the model is to check the goodness-of-fit; however, there is a lack of such tests. Modified chi-squared goodness-of-fit tests for generalized linear models were constructed. Models with continuous right-skewed, possibly censored responses were considered. Explicit formulas of test statistics are provided in the case of gamma and inverse Gaussian models. The test power was investigated by simulation. The article presents real data examples to illustrate the application of tests.

1. Introduction

Generalized linear models (GLMs) [1] are among the most commonly used regression models in practice. The most frequently applied continuous GLM models are normal (Gaussian), gamma, and inverse Gaussian. The gamma and inverse Gaussian regression models are used to model right-skewed response variables, for example, modeling the lifetime distribution in reliability theory [2,3], claims prediction and premium computations in insurance [4,5,6], healthcare costs analysis [7,8], and estimation of outcomes in psychology [9].

The Gaussian GLM coincides with the most applied normal regression model, and the theory of normal regression can be found in an enormous number of articles and books on theoretical or applied statistics.

The fitting of a regression model consists of a set of steps, and one of the key steps is to check the goodness-of-fit (GOF). However, there is a lack of such tests, especially for continuous GLM such as gamma and inverse Gaussian regression. Just a few articles consider formal tests.

In many textbooks, the chi-squared approximation of Pearson and deviance statistics is recommended to test the gamma and inverse Gaussian regression models’ fit. It can lead to erroneous conclusions because this approximation is true if the shape parameter is large. It is clearly demonstrated in [10,11], for example. In [11], the authors propose approximations of Pearson and deviance statistics quantiles for the gamma regression model. Unfortunately, these approximations are given only in the case of a known shape parameter $\nu$. The case of unknown $\nu$ is not investigated, so these results can not be used for goodness-of-fit. In [10], GOF tests for gamma and inverse Gaussian models are proposed by applying modifications of Cramer–von Mises and Anderson–Darling statistics. These statistics are computed using transformations of the responses via parametric estimates of their cumulative distribution functions (c.d.f.) and the inverse of the c.d.f. of the standard normal distribution. The theory is not developed rigorously: the asymptotic distributions of the test statistics are not found, and approximations of the distributions of the test statistics for finite sample sizes are not given. These tests can not be applied if the data are censored.

The score test for inverse Gaussian regression against inverse Gaussian mixture was constructed in [12]. The authors considered cases with complete and censored data, and critical values were obtained using the bootstrap. The disadvantage is that this test is not the omnibus test. It is recommended only in the case when only one of the possible alternatives (mixture) is suspected.

The current paper is a natural continuation of our paper [13]. In [13], modified chi-square tests were constructed for parametric accelerated failure time (AFT) models (see also [14]). To obtain these tests, at first, some asymptotic results for general parametric regression models (the AFT models being particular cases of these models) were rigorously obtained. In particular, asymptotic properties of the random vector of differences between the numbers of observed and “expected” failures in the intervals of a data partition (the partition is received using a uniquely defined rule) are derived. Application of general results for the following AFT models was considered: exponential, shape-scale (Weibull, log-normal, and log-logistics).

In the current article, we apply the general theorems of our paper [13] to obtain modified chi-squared goodness-of-fit tests for continuous right-skewed possibly censored GLM models.

The inverse Gaussian regression is GLM but not the AFT; thus, new tests are needed. The gamma regression is GLM, and it is also the AFT model; however, the article [13] on GOF for AFT models did not consider this model. Thus, we are currently investigating it.

Tests for gamma and inverse Gaussian regression models were investigated in detail. The Gaussian GLM by exponential transformation is transferred to log-normal AFT, which is considered in [13]. We did not write the formulas for this model because GOF tests for the normal regression are well-known and investigated in many papers.

Some authors consider diagnostic plots based on residuals for the gamma and inverse Gaussian models, but they are not formal GOF tests, so they can not be compared with the proposed tests because their significance and power can not be investigated. However, diagnostic graphs are useful at the initial stage of analysis, and in conjunction with formal GOFs, provide a broader view of data. The authors [15] proposed two new methods for the detection of influential observations in the case of the inverse Gaussian regression, and also presented a review of existing methods. In the article [16], adjusted deviance residuals for the gamma regression model were proposed and used for influence diagnostics. The construction of partial residuals for the inverse Gaussian regression was carried out in [17] for graphical model diagnostics.

The structure of the article is as follows: firstly (see Section 2), continuous GLMs are discussed; furthermore, in Section 3, the methodology of the modified chi-squared test is provided, the approach of choosing grouping intervals is explained, and the limit distribution of the test statistic is obtained. The results of the simulation study and the application for real data are presented in Section 4 and Section 5, respectively.

2. Gamma and Inverse Gaussian Regression Models

Let us consider the parametrization of the gamma distribution, denoted by $\mathrm{\Gamma }(\nu ,\mu )$, $\nu >0$, and $\mu >0$, with the following probability density function (p.d.f.):

$$f(t,\nu ,\mu )={\displaystyle \frac{{\nu }^{\nu }}{{\mu }^{\nu }\mathrm{\Gamma }\left(\nu \right)}}{t}^{\nu -1}exp\{-(\nu /\mu )t\},t>0,$$

(1)

where $\nu$ is the shape parameter.

If T is a random variable with distribution $\mathrm{\Gamma }(\nu ,\mu )$, then the mean and the variance are

$$\mathbf{E}\left(T\right)=\mu ,\mathbf{Var}\left(T\right)={\mu }^2/\nu ,$$

and the cumulative distribution function (c.d.f.) is

$$F(t,\nu ,\mu )=F_{{\chi }_{2\nu }^2}(2\nu t/\mu )={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\gamma \left(\nu ,{\displaystyle \frac{\nu t}{\mu }}\right),$$

where $F_{{\chi }_{2\nu }^2}$ is the c.d.f. of the chi-squared distribution with $2\nu$ degrees of freedom; $\gamma (s,x)={\int }_0^x{u}^{s-1}{e}^{-u}du,$ i.e., the lower incomplete gamma function.

Let us consider the parametrization of the inverse Gaussian distribution (also known as Wald distribution), denoted by $IG(\nu ,\mu )$, $\nu >0$, $\mu >0$, with the following p.d.f.:

$$f(t,\nu ,\mu )=\sqrt{{\displaystyle \frac{\nu }{2\pi t^3}}}exp\left\{-{\displaystyle \frac{\nu {(t-\mu )}^2}{2{\mu }^{2}t}}\right\},t>0,$$

(2)

where $\nu$ is the shape parameter. If T is a random variable with distribution $IG(\nu ,\mu )$, then the mean and the variance are

$$\mathbf{E}\left(T\right)=\mu ,\mathbf{Var}\left(T\right)={\mu }^3/\nu ,$$

and the c.d.f. is

$$F(t,\nu ,\mu )=\mathrm{\Phi }\left(\sqrt{{\displaystyle \frac{\nu }{t}}}\left({\displaystyle \frac{t}{\mu }}-1\right)\right)+e^{{\displaystyle \frac{2\nu }{\mu }}}\mathrm{\Phi }\left(-\sqrt{{\displaystyle \frac{\nu }{t}}}\left({\displaystyle \frac{t}{\mu }}+1\right)\right),t>0,$$

where $\mathrm{\Phi }$ is the c.d.f. of the standard normal distribution.

The gamma and the IG distributions belong to the exponential family with a p.d.f. of the following form:

$$f(t,\theta ,\varphi )=exp\left\{{\displaystyle \frac{\theta t-b\left(\theta \right)}{a\left(\varphi \right)}}+c(t,\varphi )\right\},t>0.$$

(3)

For the gamma distribution,

$$\theta =-{\displaystyle \frac{1}{\mu }},\varphi =1/\nu ,b\left(\theta \right)=-ln(-\theta ),a\left(\varphi \right)=\varphi ,$$

$$c(t,\varphi )={\varphi }^{-1}ln(t/\varphi )-lnt-ln\mathrm{\Gamma }\left({\varphi }^{-1}\right),$$

and for the IG distribution,

$$\theta =-{\displaystyle \frac{1}{2{\mu }^2}},\varphi =1/\nu ,b\left(\theta \right)=-\sqrt{-2\theta },a\left(\varphi \right)=\varphi ,c(t,\varphi )={\displaystyle \frac{1}{2}}ln{\displaystyle \frac{{\varphi }^{-1}}{2\pi t^3}}-{\displaystyle \frac{{\varphi }^{-1}}{2t}}.$$

Gamma regression model: The distribution of response T given the shape parameter $\nu$ and a vector of covariates $\mathit{z}={(1,z_1,\dots ,z_m)}^T$ is $\mathcal{G}(\nu ,\mu (\mathit{z}\left)\right)$ and the link function is logarithmic:

$$log\left(\mu \left(\mathit{z}\right)\right)={\mathit{\beta }}^T\mathit{z}={\beta }_0+{\beta }_1{z}_1+\dots +{\beta }_m{z}_m⟺\mu \left(z\right)=e^{{\mathit{\beta }}^T\mathit{z}}.$$

Thus, the p.d.f., c.d.f., mean, and variance given the vector of covariates are as follows:

$$f(t,\nu ,\mathit{\beta })={\displaystyle \frac{{\nu }^{\nu }}{e^{\nu {\mathit{\beta }}^T\mathit{z}}\mathrm{\Gamma }\left(\nu \right)}}{t}^{\nu -1}exp\{-(\nu /e^{{\mathit{\beta }}^T\mathit{z}})t\},t>0,$$

(4)

$$F(t,\nu ,\mathit{\beta })=F_{{\chi }_{2\nu }^2}(2\nu t/e^{{\mathit{\beta }}^T\mathit{z}})={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\gamma (\nu ,\nu X_i/e^{{\mathit{\beta }}^T\mathit{z}}),$$

where $\gamma$ is the lower incomplete gamma function and

$$\mathbf{E}\left(T\right|\mathit{z})=\mu \left(\mathit{z}\right)=e^{{\mathit{\beta }}^T\mathit{z}},\mathbf{Var}\left(T\right|\mathit{z})={\mu }^2\left(\mathit{z}\right)/\nu =e^{2{\mathit{\beta }}^T\mathit{z}}/\nu .$$

Sometimes the canonical (inverse) link function is used:

$${\displaystyle \frac{1}{\mu \left(\mathit{z}\right)}}={\mathit{\beta }}^T\mathit{z}={\beta }_0+{\beta }_1{z}_1+\dots +{\beta }_m{z}_m.$$

Inverse Gaussian regression model: The distribution of the response T given the shape parameter $\nu$ and a vector of covariates $\mathit{z}={(1,z_1,\dots ,z_m)}^T$ is $IG(\nu ,\mu (\mathit{z}\left)\right)$ and the link function is logarithmic:

$$log\left(\mu \left(\mathit{z}\right)\right)={\mathit{\beta }}^T\mathit{z}={\beta }_0+{\beta }_1{z}_1+\dots +{\beta }_m{z}_m.$$

Thus, the p.d.f, c.d.f, mean, and variance given the vector of covariates are as follows:

$$f(t,\nu ,\mathit{\beta })=\sqrt{{\displaystyle \frac{\nu }{2\pi t^3}}}exp\left\{-{\displaystyle \frac{\nu {(t-e^{{\mathit{\beta }}^T\mathit{z}})}^2}{2t{e}^{2{\mathit{\beta }}^T\mathit{z}}}}\right\},t>0,$$

(5)

$$F(t,\nu ,\mathit{\beta })=\mathrm{\Phi }\left(\sqrt{{\displaystyle \frac{\nu }{t}}}\left({\displaystyle \frac{t}{e^{{\mathit{\beta }}^T\mathit{z}}}}-1\right)\right)+exp\left\{{\displaystyle \frac{2\nu }{e^{{\mathit{\beta }}^T\mathit{z}}}}\right\}\mathrm{\Phi }\left(-\sqrt{{\displaystyle \frac{\nu }{t}}}\left({\displaystyle \frac{t}{e^{{\mathit{\beta }}^T\mathit{z}}}}+1\right)\right),t>0,$$

$$\mathbf{E}\left(T\right|\mathit{z})=\mu \left(z\mathit{\right)}=e^{{\mathit{\beta }}^T\mathit{z}},\mathbf{Var}\left(T\right|\mathit{z})={\mu }^3\left(\mathit{z}\right)/\nu =e^{3{\mathit{\beta }}^T\mathit{z}}/\nu .$$

Sometimes, the canonical (inverse squared) link function is used:

$${\displaystyle \frac{1}{{\mu }^2\left(\mathit{z}\right)}}={\mathit{\beta }}^T\mathit{z}={\beta }_0+{\beta }_1{z}_1+\dots +{\beta }_m{z}_m.$$

The gamma regression model is also an AFT model, and the IG model is not an AFT model.

3. Chi-Squared GOF Tests for Gamma and Inverse Gaussian Regression

3.1. Parameter Estimation

Let us consider the possibility of right-censored regression data:

$$(X_1,{\delta }_1,{\mathit{z}}_1),\dots ,(X_n,{\delta }_n,{\mathit{z}}_n),$$

where $X_i=Y_i\wedge C_i,$ ${\delta }_i={\mathbf{1}}_{\{Y_i\le C_i\}},$ $Y_i$ are responses and $C_i$ are censorings.

Denote

$$\lambda (x,\mathit{\theta })=f(x,\mathit{\theta })/(1-F(x,\mathit{\theta }\left)\right)=f(x,\mathit{\theta })/S(x,\mathit{\theta }),\mathrm{\Lambda }(x,\mathit{\theta })=-lnS(x,\mathit{\theta })$$

as the hazard and the cumulative hazard functions, respectively, depending on a finite-dimensional parameter $\mathit{\theta }$. In the case of gamma and inverse Gaussian regression models, $\mathit{\theta }={(\mu ,\nu )}^T$.

The parametric log-likelihood function is the following:

$$\mathit{\ell }\left(\mathit{\theta }\right)=\sum _{i=1}^n({\delta }_{i}ln{\lambda }_i(X_i,\mathit{\theta })-{\mathrm{\Lambda }}_i(X_i,\mathit{\theta })),$$

where $ln{\lambda }_i$ and ${\displaystyle \frac{\partial }{\partial \mathit{\theta }}}ln{\lambda }_i(t,\mathit{\theta })$ are presented in Section 3.2 and Section 3.3.

3.2. Gamma Regression

In the case of gamma regression, the following results were obtained:

$$ln{\lambda }_i(X_i,\mathit{\theta })=\nu ln\nu +(\nu -1)ln{X}_i-{\displaystyle \frac{\nu }{{\mu }_i}}{X}_i-\nu ln{\mu }_i-ln\mathrm{\Gamma }\left(\nu \right)-ln\left(1-F_i(X_i,\nu ,{\mu }_i)\right),$$

$${\mu }_i={\mu }_i(z_i,\mathit{\beta })=\left\{\begin{array}{cc}{e}^{{\mathit{\beta }}^T{\mathit{z}}_i}\hfill & \mathrm{i}\mathrm{f} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c},\hfill \\ 1/{\mathit{\beta }}^T{\mathit{z}}_i\hfill & \mathrm{i}\mathrm{f} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e},\hfill \end{array}\right.$$

(6)

$${\displaystyle \frac{\partial }{\partial \nu }}ln{\lambda }_i(X_i,\mathit{\theta })=1+ln\nu +ln{X}_i-X_i/{\mu }_i-ln{\mu }_i-\psi \left(\nu \right)+{\displaystyle \frac{\partial F_i(X_i,\nu ,{\mu }_i)/\partial \nu }{1-F_i(X_i,\nu ,{\mu }_i)}},$$

(7)

where $F_i$ is the c.d.f. of the ith response, and $\psi \left(\nu \right)$ is the digamma function.

Note that the derivative of the probability density function of the ith response with respect to $\nu$ is

$${\displaystyle \frac{\partial f_i(t,\nu ,\mathit{\beta })}{\partial \nu }}={\displaystyle \frac{{\nu }^{\nu }}{{\mu }^{\nu }\mathrm{\Gamma }\left(\nu \right)}}{t}^{\nu -1}exp\{-\nu t/{\mu }_i\}(1+ln\nu -ln{\mu }_i-\psi \left(\nu \right)+lnt-t/{\mu }_i),t>0.$$

(8)

This implies that the derivative of the c.d.f. $F_i$ with respect to $\nu$ is the integral of the derivative of the p.d.f.:

$${\displaystyle \frac{\partial F_i(X_i,\nu ,\mu )}{\partial \nu }}={\displaystyle \frac{{\nu }^{\nu }}{{\mu }_i^{\nu }\mathrm{\Gamma }\left(\nu \right)}}\left\{(1+ln\nu -ln{\mu }_i-\psi \left(\nu \right)){\int }_0^{X_i}{u}^{\nu -1}exp\{-\nu u/{\mu }_i\}du+\right.$$

$$\left.{\int }_0^{X_i}(lnu-u/{\mu }_i)u^{\nu -1}exp\{-\nu u/{\mu }_i\}du\right\}={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\gamma (\nu ,\nu X_i/{\mu }_i)(1+ln\nu -ln{\mu }_i-\psi \left(\nu \right))+$$

$$+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\left({\gamma }_1^{\prime }(\nu ,{\nu }_i{X}_i/{\mu }_i)+ln({\mu }_i/\nu )\gamma (\nu ,\nu X_i/{\mu }_i)\right)-{\displaystyle \frac{1}{\nu \mathrm{\Gamma }\left(\nu \right)}}\gamma (\nu +1,\nu X_i/{\mu }_i),$$

where ${\gamma }_1^{\prime }$ is the derivative of the lower incomplete gamma function with respect to the first argument (could be obtained using the function pgamma.deriv.unscaled of the R software version 4.4.3 package VGAM); the function gammainc of the R software version 4.4.3 package pracma computes the lower incomplete gamma functions, and functions gamma and digamma of the R version 4.4.3 package base return values of gamma and diggama functions, respectively.

$${\displaystyle \frac{\partial }{\partial {\beta }_i}}ln{\lambda }_i(X_i,\mathit{\theta })={\displaystyle \frac{\nu }{{\mu }_i}}\left({\displaystyle \frac{X_i}{{\mu }_i}}-1\right){\displaystyle \frac{\partial {\mu }_i}{\partial {\beta }_i}}+{\displaystyle \frac{\partial F_i(X_i,\nu ,{\mu }_i)/\partial {\beta }_i}{1-F_i(X_i,\nu ,{\mu }_i)}},$$

(9)

$${\displaystyle \frac{\partial F_i(X_i,\nu ,{\mu }_i)}{\partial {\beta }_i}}=\left({\displaystyle \frac{1}{{\mu }_i\mathrm{\Gamma }\left(\nu \right)}}\gamma (\nu +1,\nu X_i/{\mu }_i)-{\displaystyle \frac{\nu }{{\mu }_i}}{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\nu \right)}}\gamma (\nu ,\nu X_i/{\mu }_i)\right){\displaystyle \frac{\partial {\mu }_i}{\partial {\beta }_i}},$$

where ${\mu }_i$ is defined by (6).

3.3. Inverse Gaussian Regression

In the case of inverse Gaussian regression, the following was obtained:

$$ln{\lambda }_i(X_i,\mathit{\theta })=0.5\left(ln\nu -ln\left(2\pi X_i^3\right)\right)-{\displaystyle \frac{\nu {(X_i-{\mu }_i)}^2}{2{\mu }_i^2{X}_i}}-$$

$$-ln\left\{1-\mathrm{\Phi }\left(\sqrt{{\displaystyle \frac{\nu }{X_i}}}\left({\displaystyle \frac{X_i}{{\mu }_i}}-1\right)\right)-e^{2\nu /{\mu }_i}\mathrm{\Phi }\left(-\sqrt{{\displaystyle \frac{\nu }{X_i}}}\left({\displaystyle \frac{X_i}{{\mu }_i}}+1\right)\right)\right\},$$

$${\mu }_i={\mu }_i(\mathit{z},\mathit{\beta })=\left\{\begin{array}{cc}{e}^{{\mathit{\beta }}^T{\mathit{z}}_i}\hfill & \mathrm{i}\mathrm{f} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{m}\mathrm{i}\mathrm{c},\hfill \\ 1/\sqrt{{\mathit{\beta }}^T{\mathit{z}}_i}\hfill & \mathrm{i}\mathrm{f} \mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k} \mathrm{i}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{e} \mathrm{s}\mathrm{q}\mathrm{u}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{d}.\hfill \end{array}\right.$$

(10)

$${\displaystyle \frac{\partial }{\partial \nu }}ln{\lambda }_i(X_i,\mathit{\theta })={\displaystyle \frac{1}{2\nu }}-{\displaystyle \frac{{(X_i-{\mu }_i)}^2}{2{\mu }_i^2{X}_i}}+{\left(1-\mathrm{\Phi }\left(a_i\right)-e^{2\nu /{\mu }_i}\mathrm{\Phi }\left(b_i\right)\right)}^{-1}\times$$

$$\left\{{\displaystyle \frac{a\phi \left(a_i\right)}{2\nu }}+e^{2\nu /{\mu }_i}\left({\displaystyle \frac{2\mathrm{\Phi }\left(b_i\right)}{{\mu }_i}}+{\displaystyle \frac{b\phi \left(b_i\right)}{2\nu }}\right)\right\};$$

(11)

$${\displaystyle \frac{\partial }{\partial {\beta }_i}}ln{\lambda }_i(X_i,\mathit{\theta })={\displaystyle \frac{1}{{\mu }_i^2}}\left({\displaystyle \frac{\nu (t-{\mu }_i)}{{\mu }_i}}-{\left(1-\mathrm{\Phi }\left(a_i\right)-e^{2\nu /{\mu }_i}\mathrm{\Phi }\left(b_i\right)\right)}^{-1}\right.\times$$

$$\left.\left\{\phi \left(a_i\right)\sqrt{\nu t}+e^{2\nu /{\mu }_i}\left(2\nu \mathrm{\Phi }\left(b_i\right)-\phi \left(b_i\right)\sqrt{\nu t}\right)\right\}\right){\displaystyle \frac{\partial {\mu }_i}{\partial {\beta }_i}},$$

(12)

where ${\mu }_i$ is defined by (10) and

$$a_i=\sqrt{{\displaystyle \frac{\nu }{X_i}}}\left({\displaystyle \frac{X_i}{{\mu }_i}}-1\right),b_i=-\sqrt{{\displaystyle \frac{\nu }{X_i}}}\left({\displaystyle \frac{X_i}{{\mu }_i}}+1\right).$$

3.4. Grouping Intervals

Let $\widehat{\mathit{\theta }}$ be the ML estimator of $\mathit{\theta }.$ Set

$$N_i\left(t\right)={{\displaystyle \mathbf{1}}}_{\{X_i\le t,{\delta }_i=1\}},N\left(t\right)=\sum _{i=1}^n{N}_i\left(t\right),$$

where $N\left(t\right)$ is the number of responses in the interval $[0,t]$. The mean of $N\left(t\right)$ is

$$\mathbf{E}N\left(t\right)=\mathbf{E}\sum _{i=1}^n{\mathrm{\Lambda }}_i(t\wedge X_i,\mathit{\theta }).$$

So, ${\sum }_{i=1}^n{\mathrm{\Lambda }}_i(t\wedge X_i,\widehat{\mathit{\theta }})$ may be interpreted as the expected number of responses in the interval $[0,t]$ when the parametric model is true.

If the parametric model is true, then the difference $N\left(t\right)-{\sum }_{i=1}^n{\mathrm{\Lambda }}_i(t\wedge X_i,\widehat{\mathit{\theta }})$ should take smaller values than in the case when the model is false.

Denote $X_{\left(1\right)}\le \dots \le X_{\left(n\right)}$ as the ordered $X_1,\dots ,X_n$. Set ${\mathrm{\Lambda }}_{\left(i\right)}(t,\mathit{\theta })=\mathrm{\Lambda }(t,{\mathit{z}}^{\left(\right(i\left)\right)},\mathit{\theta })$; here, ${\mathit{z}}^{\left(\right(i\left)\right)}$ is the vector of covariates corresponding to $X_{\left(i\right)}$ in the sample. Define

$$E_k=\sum _{i=1}^n{\mathrm{\Lambda }}_i(X_i,\widehat{\mathit{\theta }})=\sum _{i=1}^n{\mathrm{\Lambda }}_{\left(i\right)}(X_{\left(i\right)},\widehat{\mathit{\theta }}),E_j={\displaystyle \frac{j}{k}}{E}_k,j=1,\dots ,k,$$

(13)

where $E_k$ is the estimator (under the true model) of the expected number of responses in the interval $[0,X_{\left(n\right)}]$. If the model is true, then this value should not be far from n.

Divide the interval $[0,X_{\left(n\right)}]$ into k smaller intervals $I_j=(a_{j-1},a_j]$. $a_0=0$ and $a_k=X_{\left(n\right)}$ (do not identify $a_j$ with $a_j$ from Section 3.3) have the same expected number of responses in each interval. More precisely, the point $a_j$ is defined in the following way:

$$g\left(a_j\right)=E_j,g\left(a\right)=\sum _{l=1}^n{\mathrm{\Lambda }}_{\left(l\right)}(a\wedge X_{\left(l\right)}).$$

The function $g\left(a\right)$ is strictly increasing: $g\left(0\right)=0$ and $g\left(X_{\left(n\right)}\right)=E_k$.

Set $X_{\left(0\right)}=0$. Let us use notation ${\sum }_{l=1}^0{c}_l=0$. Define

$$b_i=g\left(X_{\left(i\right)}\right)=\sum _{l=1}^n{\mathrm{\Lambda }}_{\left(l\right)}(X_{\left(i\right)}\wedge X_{\left(l\right)},\widehat{\mathit{\theta }})=\sum _{l=i+1}^n{\mathrm{\Lambda }}_{\left(l\right)}(X_{\left(i\right)},\widehat{\mathit{\theta }})+\sum _{l=1}^i{\mathrm{\Lambda }}_{\left(l\right)}(X_{\left(l\right)},\widehat{\mathit{\theta }}).$$

Note that $b_0=0$, $b_n=E_k$, and $E_j\in (0,E_k]$, $j=1,...,k$. Hence, there exists $i_j$ such that $E_j\in (b_{i_j-1},b_{i_j}]$, which implies that $a_j\in (X_{(i_j-1)},X_{\left(i_j\right)}]$. So, at first, $i_j$ is found. Then, $a_j$ is obtained, which is the unique root of the function $h_j\left(a\right)=g\left(a\right)-E_j$ in the interval $(X_{(i_j-1)},X_{\left(i_j\right)}]$, and is easily found by the bisection method because $h_j\left(X_{(i_j-1)}\right)<0$, $h_j\left(X_{\left(i_j\right)}\right)>0$, and $h_j\left(a\right)$ is strictly increasing. Note that in the interval $(X_{(i_j-1)},X_{\left(i_j\right)}]$ the function $g\left(a\right)$ may be written as follows:

$$g\left(a\right)=\sum _{l=1}^n{\mathrm{\Lambda }}_{\left(l\right)}(a\wedge X_{\left(l\right)},\widehat{\mathit{\theta }})=\sum _{l=i_j}^n{\mathrm{\Lambda }}_{\left(l\right)}(a,\widehat{\mathit{\theta }})+\sum _{l=1}^{i_j-1}{\mathrm{\Lambda }}_{\left(l\right)}(X_{\left(l\right)},\widehat{\mathit{\theta }}).$$

3.5. Test Statistic

The numbers of observed and expected responses in the interval $I_j=(a_{j-1},a_j]$ are as follows:

$$U_j=\sum _{i:X_i\in I_j}{\delta }_i,e=E_k/k,j=1,...,k.$$

The chi-squared test is based on the random vector

$$\mathit{Z}={(Z_1,\dots ,Z_k)}^T,Z_j={\displaystyle \frac{1}{\sqrt{n}}}(U_j-e),$$

i.e., on the differences between observed and expected values under the GLM model, including the number of responses in the intervals $I_j$.

Set

$${\widehat{A}}_j=U_j/n,{\mathit{C}}_j={\displaystyle \frac{1}{n}}\sum _{i:X_i\in I_j}{\delta }_i{\displaystyle \frac{\partial }{\partial \mathit{\theta }}}ln{\lambda }_i(X_i,\widehat{\mathit{\theta }}),\widehat{\mathit{C}}=({\widehat{\mathit{C}}}_1,\dots ,{\widehat{\mathit{C}}}_k).$$

If s is a dimension of $\mathit{\theta }$, then ${\mathit{C}}_j$ are $s\times 1$ vectors and C is a $s\times k$ matrix. Denote by $\widehat{\mathit{A}}$ the diagonal matrix with diagonal elements ${\widehat{A}}_j$.

The limit distribution of the random vector Z is found by applying the results of Theorems 3.1 and 3.2 of our article [13] (these theorems are also provided in [18] and Appendix A of this article). Note that these theorems can be applied not only for AFT models but also in the case of GLM models, because we can choose various forms of parametric hazard functions ${\lambda }_i(u,\mathit{\theta })$ for different i (see Appendix A).

The proof of the above-mentioned theorems in [13] was obtained by the following steps. At first, the asymptotic properties of the stochastic process,

$$H_n\left(t\right)={\displaystyle \frac{1}{\sqrt{n}}}(N\left(t\right)-\sum _{i=1}^n{\int }_0^t{\lambda }_i(u,\widehat{\mathit{\theta }})Y_i\left(u\right)du),$$

were investigated ([13], Lemma 3.1; Appendix A, and Lemma A1) by applying the central limit theorem (CLT) for martingales under well-known assumptions (see [19]) on the asymptotic properties (consistency and asymptotic normality) of the ML estimator $\widehat{\mathit{\theta }}$ and the assumptions of CLT. Lemma 3.1 implies that the limit distribution of the random vector is Z (see [13], Theorem 3.1; Appendix A, and Theorem A1 ). This distribution is approximated by the normal distribution $N_k(\mathbf{0},\mathit{V}).$ Theorem 3.2 (see Appendix A and Theorem A2) implies that the covariance matrix V is consistently estimated by the matrix:

$$\widehat{\mathit{V}}=\widehat{\mathit{A}}-{\widehat{\mathit{C}}}^T{\widehat{\mathit{i}}}^{-1}\widehat{\mathit{C}},$$

where $\widehat{\mathit{i}}$ is a $s\times s$ matrix (see [18]) that can be written in the following form:

$${\widehat{\mathit{i}}}_{l{l}^{\prime }}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{\delta }_i{\displaystyle \frac{\partial }{\partial {\theta }_l}}ln{\lambda }_i(X_i,\widehat{\mathit{\theta }}){\left({\displaystyle \frac{\partial }{\partial {\theta }_{l^{\prime }}}}ln{\lambda }_i(X_i,\widehat{\mathit{\theta }})\right)}^T,$$

where derivatives of $ln{\lambda }_i$ are provided in (7) and (9) for the gamma regression, and in (11) and (12) for inverse Gaussian regression.

The chi-squared test for the hypothesis $H_0$ is based on the following statistic:

$$Y^2={\mathit{Z}}^T{\widehat{\mathit{V}}}^{-}\mathit{Z},$$

(14)

where ${\widehat{\mathit{V}}}^{-}$ is the general inverse of the matrix $\widehat{\mathit{V}}$. The hypothesis is rejected with an approximate significance level of $\alpha$ if $Y^2>{\chi }_{\alpha }^2\left(r\right)$, where r is the rank of the matrix V.

Note that in the case of the gamma regression, V is a full rank matrix ($r=k$); thus, ${\widehat{\mathit{V}}}^{-}={\widehat{\mathit{V}}}^{-1}.$ In the case of inverse Gaussian regression model, $rank\left(\mathit{V}\right)=k-1.$

4. Simulation Study

The data are simulated by taking two covariates: $z_1$—dichotomous (0—for half of the observations and 1—for the remaining observations) and $z_2\sim U(20,30).$ Different sample sizes n are considered. The Rice rule (see [20]) is used to determine the number of grouping intervals (see Table 1):

$$k=\left[2\sqrt[3]{n}\right].$$

(15)

Table 1. The number of grouping intervals using the Rice rule.

n	30	50	60	70	80	100	150	200
k	6	7	7	8	8	9	10	11
n	300	400	500	600	800	1000	1500	2000
k	13	14	15	16	18	20	22	25

In the assumptions of the limit distribution of the test statistic, it is supposed that k is fixed and the limit distribution is obtained to be chi-squared with k or $k-1$ degrees-of-freedom. Is the approximation accurate if $k=\left[2\sqrt[3]{n}\right]$? Note that if n is fixed, then $k=\left[2\sqrt[3]{n}\right]$ is also fixed. We know that if the size of the sample $n^{*}>k$ is sufficiently large and $k=\left[2\sqrt[3]{n}\right]$, then the chi-squared approximation is accurate. But, taking into account that n is much larger than k (see Table 1), the approximation should also be good for sample size n. Simulations confirm this.

4.1. Simulation Under Hypotheses

The estimated significance levels are obtained using 5000 iterations. Tests with significance levels $\alpha =0.05$ and $\alpha =0.1$ are applied. Table 2 and Table 3 present the results for gamma and inverse Gaussian regression, respectively. Grouping intervals are computed using the Rice rule (15); moreover, different numbers of grouping intervals are considered to see how the convergence speed depends on the number of grouping intervals. The simulation results under the hypothesis demonstrate that the estimated significance levels approach the true value as the number of observations increases.

Table 2. Estimates of the significance level $\alpha$ under the hypothesis, inverse Gaussian regression with log link, ${\beta }_0=-5,$ ${\beta }_1=2,$ ${\beta }_2=0.1,$ $\nu =3$.

n
200	500	1000	1500	2000	$\mathbf{+}\mathbf{\infty }$
$k_{Rice}=11$	$k_{Rice}=15$	$k_{Rice}=20$	$k_{Rice}=22$	$k_{Rice}=25$
0.1150	0.0860	0.0670	0.0678	0.0584	0.05
0.1638	0.1417	0.1204	0.1180	0.1098	0.10
	$k=11$	$k=11$	$k=11$	$k=11$
	0.0735	0.0638	0.0620	0.0520	0.05
	0.1290	0.1139	0.1138	0.1060	0.10
		$k=15$	$k=15$	$k=15$
		0.0684	0.0636	0.0600	0.05
		0.1280	0.1166	0.1190	0.10
			$k=20$	$k=20$
			0.0656	0.0670	0.05
			0.1294	0.1180	0.10

Table 3. Estimates of the significance level $\alpha$ under the hypothesis, gamma regression with log link, ${\beta }_0=-7,$ ${\beta }_1=4,$ ${\beta }_2=0.3,$ $\nu =0.45$.

n
200	500	1000	1500	2000	$+\infty$
$k_{Rice}=11$	$k_{Rice}=15$	$k_{Rice}=20$	$k_{Rice}=22$	$k_{Rice}=25$
0.0910	0.0722	0.0692	0.0672	0.0720	0.05
0.1510	0.1362	0.1304	0.1264	0.1302	0.10
	$k=11$	$k=11$	$k=11$	$k=11$
	0.0788	0.0634	0.0556	0.0562	0.05
	0.1418	0.1206	0.1098	0.1064	0.10
		$k=15$	$k=15$	$k=15$
		0.0726	0.0602	0.0610	0.05
		0.1278	0.1176	0.1128	0.10
			$k=20$	$k=20$
			0.0736	0.0620	0.05
			0.1264	0.1198	0.10

4.2. Simulation Under Alternatives

The data are simulated under various alternatives and values of parameters. For each of the sample sizes considered, we simulate 1000 replications and compute values of the test power. The significance level is 0.05.

In the case of inverse Gaussian regression, the test power under the following alternatives is investigated (see Table 4): gamma regression, log-normal, log-logistic, and Weibull AFT models. For gamma regression, the following alternatives are considered: inverse Gaussian and normal regression, log-normal, and log-logistic and Weibull AFT models, i.e., gamma regression models with shape and scale depending on covariates.

Table 4. Definitions of alternative models.

Model	CDF
gamma regression with log link	(4)
IG regression with log link	(5)
Weibull AFT–log-linear	$1-exp\left\{-{(t/e^{{\mathit{\beta }}^T\mathit{z}})}^{\nu }\right\}$
log-logistic AFT–log-linear	$\mathrm{\Phi }\left((lnt-{\mathit{\beta }}^T\mathit{z})/\sigma \right);\sigma =1/\nu$
log-normal AFT–log-linear	$1-{\left(1+{(t/e^{{\mathit{\beta }}^T\mathit{z}})}^{\nu }\right)}^{-1}$
gamma regression model with shape	(4) with
and scale depending on covariates	$\nu ={\nu }_0+{\nu }_1{z}_1+\dots +{\nu }_j{z}_j$

The results in the case of gamma regression are presented in Table 5. It has become evident that the test power under the IG regression alternative is large even for small sample sizes. The smallest test power values are in the case of the Weibull AFT model alternative, which is reasonable because gamma and Weibull models are very similar for some sets of parameters.

Table 5. Gamma regression. Powers against various alternatives. n: number of observations; k: optimal number of grouping intervals.

Alternative	n; k
	30; 5	50; 6	60; 7	80; 8	100; 9	150; 10	200; 11	250; 12	300; 13
${\beta }_0=1,{\beta }_1=1,{\beta }_2=0.01,\nu =3$
IG with log link	0.706	0.902	0.943	0.987	1	1	1	1	1
Weibull AFT–log-linear	0.282	0.317	0.332	0.346	0.360	0.427	0.493	0.585	0.674
log-logistic AFT–log-linear	0.542	0.569	0.571	0.646	0.682	0.742	0.837	0.897	0.923
log-normal AFT–log-linear	0.361	0.377	0.418	0.439	0.459	0.481	0.487	0.518	0.581
${\beta }_0=1,{\beta }_1=1,{\beta }_2=0.01,\nu =2$
IG with log link	0.813	0.951	0.973	0.995	1	1	1	1	1
Weibull AFT–log-linear	0.241	0.257	0.267	0.272	0.272	0.283	0.300	0.344	0.372
log-logistic AFT–log-linear	0.583	0.651	0.705	0.780	0.838	0.923	0.974	0.986	0.994
log-normal AFT–log-linear	0.415	0.444	0.489	0.513	0.578	0.653	0.709	0.768	0.823
${\beta }_0=1,{\beta }_1=1,{\beta }_2=0.01$, gamma with shape $\nu$
$\nu =1+2z_1$	0.535	0.533	0.569	0.606	0.680	0.781	0.873	0.923	0.968
$\nu =0.7+2z_1$	0.545	0.630	0.651	0.703	0.772	0.910	0.954	0.987	0.994

The results in the case of IG regression are presented in Table 6. It turned out that the test power under all considered alternatives is large even for small sample sizes. The smallest test power values are obtained when the alternative is the log-logistic AFT model.

Table 6. Inverse Gaussian regression. Powers against various alternatives. n: number of observations; k: optimal number of grouping intervals.

Alternative	n; k
	30; 5	50; 6	60; 7	80; 8	100; 9	150; 10	200; 11	250; 12
${\beta }_0=1,{\beta }_1=1,{\beta }_2=0.01,\nu =2$
gamma with log link	0.708	0.816	0.848	0.909	0.946	0.982	0.999	1
Weibull AFT–log-linear	0.739	0.819	0.896	0.922	0.970	0.995	0.999	1
log-logistic AFT–log-linear	0.433	0.544	0.549	0.594	0.664	0.746	0.842	0.891
log-normal AFT–log-linear	0.503	0.616	0.677	0.717	0.805	0.887	0.950	0.978
${\beta }_0=1,{\beta }_1=1,{\beta }_2=0.01,\nu =1.5$
gamma with log link	0.761	0.895	0.903	0.959	0.987	0.996	0.999	1
Weibull AFT–log-linear	0.795	0.878	0.919	0.968	0.993	0.999	0.993	1
log-logistic AFT–log-linear	0.575	0.616	0.665	0.736	0.774	0.849	0.920	0.966
log-normal AFT–log-linear	0.475	0.504	0.534	0.617	0.681	0.802	0.872	0.939

Moreover, the simulation study suggested that in the case of the gamma and inverse Gaussian regression, the Rice rule (15) provides optimal grouping intervals ($k_{opt}=k_{Rice}$) for sample sizes $n\ge 60$, and for smaller samples the number of grouping intervals is $k_{opt}=k_{Rice}-1.$

5. Real Data Examples

Example 1: Failure times (see Table 7) of 76 electrical insulating fluids tested at voltages, ranging from 26 to 38 kV ([21]), are considered.

Table 7. Failure times $T_i$ for 76 electrical insulating fluids tested at voltages $v_i$.

${\mathit{v}}_{\mathit{i}}$ (kV)	Frequency	Failure Times ${\mathit{T}}_{\mathit{i}}$
26	3	5.79 159.52 2323.70
28	5	68.85 108.29 110.29 426.07 1067.60
30	11	7.74 17.05 20.46 21.02 22.66 43.40
		47.30 139.07 144.12 175.88 194.90
32	15	0.27 0.40 0.69 0.79 2.75 3.91 9.88 13.95
		15.93 27.80 53.24 82.85 89.29 100.58 215.10
34	19	0.19 0.78 0.96 1.31 2.78 3.16 4.15 4.67 4.85 6.50
		7.35 8.01 8.27 12.06 31.75 32.52 33.91 36.71 72.89
36	15	36 0.35 0.59 0.96 0.99 1.69 1.97 2.07
		2.58 2.71 2.90 3.67 3.99 5.35 13.77 25.50
38	8	0.09 0.39 0.47 0.73 0.74 1.13 1.40 2.38

The diagnostic methods (see [2]) suggest that the Weibull AFT–power rule model, i.e., $log\left(v_i\right),$ should be used. The results of applying the modified chi-squared test are presented in Table 8. The analysis demonstrated that the Weibull AFT–power rule and gamma regression models are not rejected; however, AIC and BIC are smaller in the case of the gamma regression model. The inverse Gaussian regression model is strongly rejected.

Table 8. Modified chi-squared test, $k=8$. Electrical insulating fluids data.

Model	${\mathit{Y}}^2$	p-Value	AIC	BIC
gamma regression; log link	9.335	0.3149	604.9	611.9
Weibull AFT–power rule model	6.704	0.4604	607.6	614.6
IG regression; log link	88.4	<0.0001	651.3	658.3

Example 2: Hospital cost data (the dataset hospcosts from R package robmixglm) consist of a sample of 100 patients hospitalized at the Centre Hospitalier Universitaire Vaudois in Lausanne during 1999 for “medical back problems”. The response is the cost of stay, and the covariates are as follows: length of stay (in days; the logarithmic transformation was applied), admission type (0: planned; 1: emergency), insurance type (0: regular; 1: private), age (in years), sex (0: female; 1: male) and discharge destination (1: home; 0: another health institution). Data were analyzed in [8] considering the gamma regression and [22] in the Weibull model context.

The results of applying the modified chi-squared test are presented in Table 9. It is clear that the Weibull AFT–power rule and gamma regression models are not rejected. However, AIC is smaller in the case of the Weibull AFT–power rule model. The inverse Gaussian regression model is strongly rejected.

Table 9. Modified chi-squared test, $k=9$. Hospital cost data.

Model	${\mathit{Y}}^2$	p-Value	AIC	BIC
gamma regression; log link	14.57	0.1035	1817.9	1838.8
Weibull AFT	8.82	0.3574	1817.0	1837.8
IG regression with log link	43.42	<0.0001	1866.3	1887.1

Example 3: Table 10 presents the results of an experiment designed to compare the performances of high-speed turbine engine bearings made out of five different compounds (see [2]). Data were fitted using a three-parameter Weibull distribution. The experiment tested 10 bearings of each type, and the times to fatigue failure were measured in units of millions of cycles.

Table 10. Failure times of bearing specimens.

Compound	Failures
I	3.03 5.53 5.60 9.30 9.92 12.51 12.95 15.21 16.04 16.84
II	3.19 4.26 4.47 4.53 4.67 4.69 5.78 6.79 9.37 12.75
III	3.46 5.22 5.69 6.54 9.16 9.40 10.19 10.71 12.58 13.41
IV	5.88 6.74 6.90 6.98 7.21 8.14 8.59 9.80 12.28 25.46
V	6.43 9.97 10.39 13.55 14.45 14.72 16.81 18.39 20.84 21.51

The results using the modified chi-squared test are presented in Table 11. The gamma, Weibull AFT, and inverse Gaussian regression models are rejected. The results do not contradict the results in [2].

Table 11. Chi-squared test, $k=6$. Bearing specimens data.

Model	${\mathit{Y}}^2$	p-Value
gamma regression; log link	13.38	0.0373
Weibull AFT	13.20	0.0216
IG regression with log link	17.37	0.0080

6. Conclusions

The modified chi-squared goodness-of-fit tests were constructed for gamma and inverse Gaussian regression models with possibly censored data. The methodology for grouping intervals was proposed, and practical recommendations based on the simulation results were presented. The results indicated that the test power under various considered alternatives is large even for small sample sizes. Moreover, in the case of the gamma and inverse Gaussian regression the Rice rule (15) provides optimal grouping intervals ($k_{opt}=k_{Rice}$) for sample sizes $n\ge 60$, and for smaller samples, the number of grouping intervals is $k_{opt}=k_{Rice}-1.$ The application of tests was shown using real data. The proposed tests are important in the data modeling process. They are robust to the model structure because in the case of misspecification of the model, the ”expected” number of responses will be far from the observed number of responses, and the test statistic will take large values; therefore, the hypothesis will be rejected. Thus, another model structure could be taken into consideration. The article fills the gap of formal omnibus tests for gamma and inverse Gaussian regression.