Attempts have been made to define new classes of distributions that provide more flexibility for modelling skewed data in practice. In this work we define a new extension of the generalized gamma distribution (Stacy,
Standard lifetime distributions usually present very strong restrictions to produce bathtub curves, and thus appear to be inappropriate for data with this characteristic. The three-parameter generalized gamma (GG) (Stacy, 1962) distribution includes as special models the exponential, Weibull, gamma, and Rayleigh distributions, among others. It is suitable for modeling data with hazard rate function (hrf) of different forms (increasing, decreasing, bathtub and unimodal) and useful for estimating individual hazard functions and both relative hazards and relative times (Cox
and
respectively, where
The GG distribution includes all four more common types of the hrf: monotonically increasing and decreasing, bathtub and unimodal (Cox
Now, we define an extended form of the density function (
where
In order to avoid convergence problems using the maximum likelihood method, Lawless (2002) proposed a re-parametrized density function with new parameters given by
where
The cdf (for
where Φ(·) denotes the standard normal cumulative distribution.
Marshall and Olkin (1997) proposed a method of adding a parameter
where
The MO-F density function, say
where
Survival models with a surviving fraction (also known as cure rate models or long-term survival models) have generated significant interest in the survival analysis literature. Models that accommodate a cured fraction have widely developed. A very popular type of cure rate model is the mixture distribution introduced by Boag (1949) and Berkson and Gage (1952). Basic references on cure rate distributions are the books by Maller and Zhou (1996) and Ibrahim
This paper introduces a new four-parameter model named the Marshall-Olkin generalized gamma (MOGG) distribution by inserting the cdf (
The rest of the paper proceeds as follows. Sections 2–3 formulates the MOGG model and MOGG mixture model. Inference based on maximum likelihood for both models is addressed in Section 4. Two simulation studies are presented in Section 5 to investigate some finite sample properties. In Section 6, our methodology is illustrated on a real data set. Finally, Section 7 presents some concluding remarks.
The MOGG survival function is given by
The corresponding MOGG pdf becomes
Henceforth, we denote by
We can generate a random variable
where
Therefore, the qf of
From
where
In survival and reliability studies, a part of the population may not be susceptible to the event of interest. Maller and Zhou (1996) indicate that it is adequate to consider a two components mixture model, in the sense that one component represents the failure or survival time of susceptible individuals to a certain event (in risk individuals; IR), while the other component represents the survival times of the non-susceptible individuals to the event (out of risk individuals; OR), allowing infinite survival times. An individual belongs to one group (or another) with certain probability. Then, the model formulation is described as follows. Let
All IR individuals will present the event of interest at the same time, i.e., lim
The MOGG distribution in the MOGG mixture model can be interpreted as follows. Suppose that the event of interest in the IR group may be caused by an unknown competing cause leading to latent competing risk scenarios. Let
The MOGG mixture is flexible, because the MOGG distribution is a wider family that contains most commonly used distributions, such as the exponential, Weibull, log normal and gamma models (Table 1).
Let
where
The maximum likelihood estimate (MLE)
We can easily check the adequacy of the fitted GG model by testing the null hypothesis
Let us consider the situation when the failure time
Let
where
We can write the likelihood function for
where
From the likelihood function in (
Hypothesis tests can also be conducted. Let
Alternatively, non-nested models can be compared using the Akaike information criterion (AIC) given by AIC = −2
Here, we evaluate the performance of the MLEs of the parameters of the MOGG model and MOGG mixture model by means of two simulation studies.
From
Table 2 reports the simulation results. We note that the averages of the MLEs of the parameters of the MOGG model are close to the true values. As expected, the SDs and RMSEs decrease as the sample size increases. Table 2 also shows that the CP becomes closer to the nominal value as the sample size increases. Further, we plot the empirical distributions of the MLEs
In this study, we consider the MOGG mixture model given in (
We consider sample sizes of
The data set is obtained from Smith and Naylor (1987) and describe the strengths of 1.5 cm glass fibers, measured at the National Physical Laboratory, England. This data set is of size
The gamma (G), GG, and MOGG distributions are fitted to these data. For comparing the fitted models, we compute the AIC and SBC statistics. Table 4 lists the values of these criteria. According to both criteria, the MOGG and GG distributions are the best models. We also emphasize the gain provided by the MOGG distribution in relation to beta generalized gamma distribution (Cordeiro
The LR statistics for testing the hypotheses
The QQ plot of the normalized randomized quantile residuals (Dunn and Smyth, 1996; Rigby and Stasinopoulos, 2005) in Figure 4 (left panel) suggests that the MOGG model is acceptable. Each point in Figure 4 corresponds to the median of five sets of ordered residuals. The values of the criteria in Table 4, the LR statistics and the QQ plots in Figure 4, reveal that the MOGG model is the best model to these data. The parameter estimates (and 95% asymptotic confidence intervals) for the MOGG distribution are:
In this section, we demonstrate an application of our models described in Section 3 to a well-known dataset on a Phase III cutaneous melanoma clinical trial conducted by the Eastern Cooperative Oncology Group (Kirkwood
The following information were collected from each patient: Observed time (in years, mean = 2.31, SD = 1.93);
We fit the MOGG mixture model. Table 5 presents the MLEs, the standard errors and the
The MLEs of the cure fraction (and standard errors) for patients with tumor thickness of 3.175 mm (median thickness) and stratified by nodal category from 1 to 4 are: 0.4886 (0.0817), 0.3027 (0.0751), 0.1647 (0.0801), and 0.0822 (0.0651), respectively. Standard errors are obtained after application of the delta method. The right panel of Figure 7 shows that the cure fraction decreases more rapidly for patients with a lower nodal category.
We conclude our application dealing with the MLE of the proportion of patients who survived beyond a certain fixed time, which is the practical interest to practitioners. For the sake of illustration, we choose five years. This proportion is estimated from
In this paper, we define a new lifetime model named the
Some special models of the MOGG distribution.
Case | Distribution | Reference | ||||
---|---|---|---|---|---|---|
1 | GG | Cox | ||||
2 | 1 | Weibull | ||||
3 | Gamma | |||||
4 | 1/2 | 1 | Rayleigh | |||
5 | 0 | Log-normal | ||||
6 | −1 | Inverse Weibull | ||||
7 | 1 | 1 | exponential | |||
0 < | 8 | Geometric GG | Ortega | |||
9 | 1 | Geometric Weibull | Barreto-Souza | |||
10 | Geometric Gamma | |||||
11 | 1/2 | 1 | Geometric Rayleigh | |||
12 | 0 | Geometric Log-normal | ||||
13 | −1 | Geometric Inverse Weibull | ||||
14 | 1 | 1 | Geometric exponential | Adamidis and Loukas (1998) | ||
8 | Complementary Geometric GG | |||||
9 | 1 | Complementary Geometric Weibull | Tojeiro | |||
10 | Complementary Geometric Gamma | |||||
11 | 1/2 | 1 | Complementary Geometric Rayleigh | |||
12 | 0 | Complementary Geometric Log-normal | ||||
13 | −1 | Complementary Geometric Inverse Weibull | ||||
14 | 1 | 1 | Complementary Geometric exponential | Louzada | ||
15 | MOGG | |||||
16 | 1 | MO Weibull | Marshall and Olkin (1997) | |||
17 | MO Gamma | |||||
18 | 1/2 | 1 | MO Rayleigh | |||
19 | 0 | MO Log-normal | ||||
20 | −1 | MO Inverse Weibull | ||||
21 | 1 | 1 | MO exponential | Marshall and Olkin (1997) |
MO = Marshall-Olkin; GG = generalized gamma.
Averages of maximum likelihood estimates, SD, RMSE, CP of the parameters of the Marshall-Olkin generalized gamma model
50 | Mean | 0.932 | 0.502 | 2.207 | 0.256 | 1.065 | 0.451 | 2.266 | 1.982 |
SD | 0.242 | 0.143 | 0.660 | 0.112 | 0.215 | 0.139 | 0.568 | 1.244 | |
RMSE | 0.251 | 0.143 | 0.691 | 0.125 | 0.224 | 0.147 | 0.627 | 1.244 | |
CP | 0.913 | 0.918 | 0.921 | 0.920 | 0.934 | 0.931 | 0.925 | 0.957 | |
100 | Mean | 0.988 | 0.477 | 2.297 | 0.227 | 1.052 | 0.462 | 2.158 | 1.971 |
SD | 0.233 | 0.144 | 0.731 | 0.100 | 0.175 | 0.109 | 0.360 | 1.076 | |
RMSE | 0.233 | 0.146 | 0.789 | 0.104 | 0.183 | 0.116 | 0.393 | 1.076 | |
CP | 0.932 | 0.945 | 0.934 | 0.945 | 0.952 | 0.943 | 0.948 | 0.933 | |
200 | Mean | 1.031 | 0.462 | 2.299 | 0.202 | 1.033 | 0.477 | 2.096 | 1.987 |
SD | 0.185 | 0.119 | 0.661 | 0.073 | 0.134 | 0.083 | 0.227 | 0.851 | |
RMSE | 0.187 | 0.125 | 0.725 | 0.073 | 0.137 | 0.086 | 0.246 | 0.851 | |
CP | 0.943 | 0.950 | 0.954 | 0.948 | 0.954 | 0.948 | 0.952 | 0.953 | |
400 | Mean | 1.028 | 0.471 | 2.163 | 0.195 | 1.032 | 0.478 | 2.063 | 1.915 |
SD | 0.136 | 0.086 | 0.396 | 0.050 | 0.093 | 0.057 | 0.149 | 0.548 | |
RMSE | 0.139 | 0.091 | 0.428 | 0.050 | 0.098 | 0.061 | 0.161 | 0.554 | |
CP | 0.948 | 0.946 | 0.951 | 0.946 | 0.951 | 0.945 | 0.952 | 0.947 |
SD = standard deviation; RMSE = square root of mean square error; CP = coverage probability.
Averages of maximum likelihood estimates, SD, RMSE, CP of the parameters of the Marshall-Olkin generalized gamma model
0.2 | 100 | Mean | 0.572 | 1.868 | 0.271 | −0.515 | 0.708 | 0.871 | 0.541 |
SD | 0.138 | 0.428 | 0.132 | 0.222 | 0.310 | 0.257 | 0.223 | ||
RMSE | 0.156 | 0.448 | 0.150 | 0.222 | 0.310 | 0.287 | 0.245 | ||
CP | 0.967 | 0.987 | 0.978 | 0.959 | 0.948 | 0.960 | 0.939 | ||
300 | Mean | 0.548 | 1.933 | 0.255 | −0.512 | 0.710 | 0.899 | 0.518 | |
SD | 0.124 | 0.378 | 0.108 | 0.172 | 0.231 | 0.234 | 0.219 | ||
RMSE | 0.133 | 0.383 | 0.121 | 0.172 | 0.231 | 0.254 | 0.222 | ||
CP | 0.956 | 0.958 | 0.962 | 0.946 | 0.937 | 0.958 | 0.949 | ||
600 | Mean | 0.526 | 1.982 | 0.232 | −0.494 | 0.688 | 0.947 | 0.482 | |
SD | 0.115 | 0.358 | 0.089 | 0.113 | 0.150 | 0.204 | 0.195 | ||
RMSE | 0.117 | 0.358 | 0.095 | 0.113 | 0.150 | 0.210 | 0.200 | ||
CP | 0.954 | 0.948 | 0.952 | 0.946 | 0.937 | 0.959 | 0.949 | ||
2.0 | 100 | Mean | 0.502 | 2.083 | 2.273 | −0.507 | 0.706 | 1.017 | 0.511 |
SD | 0.125 | 0.389 | 1.289 | 0.251 | 0.348 | 0.205 | 0.233 | ||
RMSE | 0.125 | 0.397 | 1.317 | 0.251 | 0.348 | 0.206 | 0.235 | ||
CP | 0.977 | 0.977 | 0.968 | 0.950 | 0.952 | 0.958 | 0.943 | ||
300 | Mean | 0.500 | 2.074 | 2.235 | −0.514 | 0.700 | 1.010 | 0.501 | |
SD | 0.114 | 0.316 | 1.169 | 0.183 | 0.240 | 0.188 | 0.215 | ||
RMSE | 0.114 | 0.325 | 1.192 | 0.183 | 0.240 | 0.188 | 0.216 | ||
CP | 0.960 | 0.948 | 0.960 | 0.953 | 0.948 | 0.951 | 0.954 | ||
600 | Mean | 0.490 | 2.069 | 2.084 | −0.501 | 0.699 | 1.023 | 0.452 | |
SD | 0.092 | 0.248 | 0.908 | 0.132 | 0.172 | 0.151 | 0.115 | ||
RMSE | 0.093 | 0.257 | 0.912 | 0.132 | 0.172 | 0.152 | 0.117 | ||
CP | 0.949 | 0.962 | 0.981 | 0.946 | 0.948 | 0.956 | 0.949 |
SD = standard deviation; RMSE = square root of mean square error; CP = coverage probability.
The AIC and SBC statistics for the fitted distributions
Distributions | Criterion | ||
---|---|---|---|
−2 max | AIC | SBC | |
G | 47.90 | 51.90 | 56.20 |
GG | 29.17 | 35.17 | 41.60 |
MOGG | 24.06 | 32.07 | 40.63 |
AIC = Akaike information criterion; SBC = Schwartz-Bayesian criterion; G = gamma; GG = generalized gamma; MOGG = Marshall-Olkin generalized gamma
MLEs of the parameters for the MOGG model with the covariate treatment
MOGG model | Parameter | Estimate | Standard error | |
---|---|---|---|---|
Complete | −0.082 | 0.165 | - | |
−1.501 | 0.692 | - | ||
−0.198 | 0.434 | - | ||
−2.477 | 0.932 | 0.008 | ||
−0.154 | 0.329 | 0.639 | ||
−0.028 | 0.013 | 0.037 | ||
−0.642 | 0.234 | 0.006 | ||
−0.186 | 0.098 | 0.057 | ||
−2.537 | 2.408 | 0.292 | ||
−0.372 | 0.198 | 0.061 | ||
−0.004 | 0.007 | 0.523 | ||
−0.274 | 0.091 | 0.003 | ||
−0.008 | 0.028 | 0.771 | ||
Reduced | −0.035 | 0.063 | - | |
−2.240 | 1.286 | - | ||
−0.158 | 0.606 | - | ||
−1.560 | 0.816 | 0.056 | ||
−0.789 | 0.311 | 0.011 | ||
−0.257 | 0.134 | 0.055 | ||
−4.329 | 2.857 | 0.130 | ||
−0.422 | 0.185 | 0.022 | ||
−0.283 | 0.088 | 0.001 |
MLEs = maximum likelihood estimates; MOGG = Marshall-Olkin generalized gamma.
Survivor probability of patients after five years for various nodal categories stratified by treatment
Treatment | Nodal category | MLE | Standard error | 95% confidence interval | |
---|---|---|---|---|---|
LL | RL | ||||
Observation | 1 | 0.592 | 0.108 | 0.381 | 0.802 |
2 | 0.443 | 0.088 | 0.271 | 0.615 | |
3 | 0.333 | 0.108 | 0.122 | 0.543 | |
4 | 0.267 | 0.130 | 0.012 | 0.522 | |
Interferon alfa-2b | 1 | 0.627 | 0.136 | 0.361 | 0.893 |
2 | 0.492 | 0.106 | 0.283 | 0.701 | |
3 | 0.391 | 0.157 | 0.084 | 0.699 | |
4 | 0.331 | 0.103 | 0.129 | 0.533 |
MLE = maximum likelihood estimates; LL = lower limit; UL = upper limit.