A four-parameter extended fatigue lifetime model called the odd Birnbaum-Saunders geometric distribution is proposed. This model extends the odd Birnbaum-Saunders and Birnbaum-Saunders distributions. We derive some properties of the new distribution that include expressions for the ordinary moments and generating and quantile functions. The method of maximum likelihood and a Bayesian approach are adopted to estimate the model parameters; in addition, various simulations are performed for different parameter settings and sample sizes. We propose two new models with a cure rate called the odd Birnbaum-Saunders mixture and odd Birnbaum-Saunders geometric models by assuming that the number of competing causes for the event of interest has a geometric distribution. The applicability of the new models are illustrated by means of ethylene data and melanoma data with cure fraction.
The two-parameter Birnbaum-Saunders (BS) distribution is a popular distribution for modeling lifetime data and phenomenon with monotone failure rates that has been applied in several research areas such as engineering, hydrology and survival analysis. However, it does not provide a reasonable parametric fit for non-monotone failure rates such as bathtub shaped and unimodal failure rates, which are common in reliability and biological studies. Bathtub hazard curves have nearly flat middle portions and the associated densities have a positive anti-mode. Unimodal failure rates can be observed in the course of a disease whose mortality reaches a peak after a finite period and then declines gradually.
Many lifetime distributions have been constructed with a view for applications in several areas such as survival analysis, reliability engineering, demography and actuarial studies, and hydrology. The BS model, also known as the fatigue life distribution, is a very popular model which has been extensively used for modeling the failure times of fatiguing materials and lifetime data in the fields cited above. Many generalizations of the BS model have been recently proposed by introducing shape parameters to better explore the tails and other properties of the generated distributions. For example, Cordeiro and Lemonte (2011) discussed the
A random variable
where
where
where
The cdf of the odd Birnbaum-Saunders (“OBS” for short) distribution with an additional shape parameter
where
The BS cdf
Therefore, the parameter
We now consider that the random variable
The cdf and pdf of
and
respectively.
The rest of the paper is organized as follows. In Section 2, we define the odd Birnbaum Saunders geometric (OBSG) model by compounding OBS and geometric distributions. Parameter inference is investigated in Section 3, where two methods (maximum likelihood and Bayesian approach) for parameter estimation are discussed. Further, various simulations are performed for different parameter settings and sample sizes. In Section 4, we introduce two models with long-term survivals and estimate their parameters by maximum likelihood and Bayesian approaches. Applications to two real data sets are addressed in Section 5. The paper provides concludes remarks in Section 6.
Generalizing continuous univariate distributions by introducing additional shape parameters in a distribution is a useful practice to obtain multiple density and hazard rate shapes. Suppose that
Let
The conditional density function of
where
Then, the four-parameter OBSG density function (for
where
Clearly, the OBS distribution is a special case of (
The hazard rate function (hrf) of
Plots of the OBSG density, survival and hazard functions for selected parameter values are displayed in Figures 1
We show some structural properties of the OBSG distribution in the
Several approaches for parameter estimation were proposed in the literature but the maximum likelihood method is the most commonly employed. The maximum likelihood estimators (MLEs) enjoy desirable properties and can be used when constructing confidence intervals and regions as well as in test statistics. The normal approximation for these estimators in asymptotic theory is easily handled either analytically or numerically. Therefore, we consider the estimation of the unknown parameters for the OBSG model by maximum likelihood. Let
where
This log-likelihood can be maximized either directly by using the R (optim function; https://www.rproject.org/), SAS (NLMIXED procedure; SAS Institute Inc., Cary, NC, USA), or Ox (MaxBFGS function; http://www.doornik.com/ox/), or by solving the nonlinear likelihood equations by its differentiation. The score functions for the parameters
where
The MLE
For interval estimation of (
Under standard regularity conditions, the asymptotic distribution of (
Further, we can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain the likelihood ratio (LR) statistics to test some sub-models of the OBSG distribution. For example, the test of
where
An alternative statistical approach is to use the Bayesian method that allows the incorporation of previous knowledge of the parameters through an informative prior distribution. A non-informative prior structure may be considered when previous knowledge is unavailable. In this work we adopt proper prior distributions according to variations of the parametric space, but ensuring non-informativeness according to the fixed hyper-parameters that lead to such a situation. We assume
In the literature, there are various methodologies which intend to analyze the suitability of distribution, as well as select the best fit from among a collection of distributions. One of the most used approaches is derived from the conditional predictive ordinate (CPO) statistic. For a detailed discussion on the CPO statistic and its applications to model selection, see Gelfand
Let denote the full data and denote the data with the deleted
The CPO
Therefore, high CPO
Other criteria, such as the deviance information criterion (DIC) proposed by Spiegelhalter
We simulate the OBSG distribution (for
The results of the Monte Carlo study in Table 1 indicate that the MSEs of the mean of
For censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse may be indicated by a relatively high number of individuals with large censored survival times. Here, the OBS model is modified in two different ways for the possible presence of long-term survivors in the data.
In population-based cancer studies, cure is said to occur when mortality in the group of cancer patients returns to the same level as that expected in the general population. The cure fraction is a useful measure that is of interest to patients analyzing trends in cancer patient survival. Models for survival analysis typically assume that every subject in the study population is susceptible to the event under study and will eventually experience such event if follow-up is sufficiently long. However, there are situations when a fraction of individuals are not expected to experience the event of interest such as when those individuals are cured or not susceptible. Cure rate models have been used to model time-to-event data for various types of cancers, including breast cancer, non-Hodgkin’s lymphoma, leukemia, prostate cancer, and melanoma. The most popular type of cure rate models are often the mixture models (MMs) introduced by Boag (1949), Berkson and Gage (1952), and Farewell (1982). Additionally, MMs allow both the cure fraction and the survival function of uncured patients (latency distribution) to depend on covariates. Longini and Halloran (1996) and Price and Manatunga (2001) have further introduced frailty to MMs for individual survival data. Further, Peng and Dear (2000) investigated a nonparametricMMfor cure estimation, Sy and Taylor (2000) considered estimation in a proportional hazards cure model and Yu and Peng (2008) extended MMs to bivariate survival data by modeling marginal distributions. Fachini
To formulate the odd Birnbaum-Saunders mixture (OBSM) model, we consider that the population under study is a mixture of susceptible (uncured) individuals, who may experience the event of interest, and non-susceptible (cured) individuals, who never will experience the event of interest (Maller and Zhou, 1996). This approach allows to estimate simultaneously whether the event of interest will occur, which is called incidence, and if it does occur then when it will occur, which is called latency. Let
where
Let
Suppose we have data in the form
and the contribution of an individual that is at risk at
The full log-likelihood function for the parameter vector
where
Models with a cure fraction play an important role in survival analysis. Recently, some of these models have been published in the literature. For example, Cooner
For an individual in the population, let
Tsodikov
Equation for
where
We note that
We consider a situation where the time to the event of interest is not completely observed but subjected to right censoring. Let
where
The MLE
where
The Bayesian procedure was used analogously to Section 3.2. We simulate the OBSG cure fraction distribution (for
Analogously to the previous simulation study, 1,000 data sets for classical inference and 200 data sets for Bayesian procedure were generated each one with
Tables 2 and 3 show the results for the OBSGcr model and OBSM model, respectively. Similar results to the previously simulation study can also be observed.
In this section, we provide two applications to real data to demonstrate the flexibility of the OBS, OBSM, and OBSG models. The computations are performed using the NLMixed subroutine in the SAS package.
These data are taken from a study by the University of São Paulo, ESALQ (Laboratory of Physiology and Post-colheita Biochemistry), which evaluated the effects of mechanical damage on banana fruits (genus Musa spp.); see Saavedra del Aguila
Recently, Alizadeh
Table 4 lists the MLEs (and the corresponding standard errors (SEs) in parentheses) of the model parameters and the values for the fitted models of the statistics: Akaike information criterion (AIC), Bayesian information criterion (BIC), and consistent Akaike information criterion (CAIC). The figures in this table indicate that the OBSG model could be selected as the best model because it has the lowest AIC, BIC, and CAIC values among the values for the fitted models. In addition, SEs of the estimates for all fitted models are also quite small.
Formal tests for the extra skewness parameters in the OBSG model can be based on LR statistics described in Section 3. Table 5 list the values of applying the LR statistics to the ethylene data. We then reject the null hypotheses of the two LR tests in favor of the OBSG distribution (Table 5). More information is provided by a visual comparison of the histogram of the data with the fitted density functions. Figure 4 displays the plots of the fitted OBSG, OBS, KwOLLN, and McN densities.
The empirical scaled total-time-on-test (TTT) transform can be used to identify the shape of the hazard function. The TTT plot for the ethylene data given in Figure 5(a) shows a unimodal and bathtub-shaped hrf and therefore indicates the appropriateness of the OBSG distribution as indicated by Figure 3. Further, Figure 5(b) reveals that the estimated hrf of the OBSG distribution can capture both behaviors (unimodal and bathtub-shaped) are presented in the ethylene data. The OBSG distribution provides a closer fit to the histogram of the current data than the other models.
Under a Bayesian approach we also fitted the OBSG, OBS, and BS models for the ethylene data as well as obtained the values DIC, EAIC, EBIC, and LPML criteria to compare these models. Table 6 shows the results that indicate estimates close to those obtained via MLEs; therefore, the OBSG model presented the best fit when using criteria found in Table 6.
Here, we discuss thee application of the OBSM and OBSG models with cure fraction to cancer recurrence with long-term survivors. These data are part of a study on cutaneous melanoma (a type of malignant cancer) for the evaluation of postoperative treatment performance with a high dose of a certain drug (interferon alfa-2b) in order to prevent recurrence. Patients from the years 1991 to 1995 were included in the study with follow-up conducted until 1998. The data were collected by Ibrahim
The values of AIC, CAIC, and BIC statistics (Table 7) and of the DIC, EAIC, EBIC, and LPML statistics (Table 8) are smaller for the OBSM and OBSG models when compared to the values of the Birnbaum-Saunders mixture (BSM) and BSG models thus indicating that the first two models could be chosen as the best models to fit these data.
The LR statistics given in Table 9 indicate that the OBSM and OBSG models are superior to BSM and BSG models in terms of model fitting.
In order to assess if the model is appropriate, Figures 6 and 7 display the empirical survival functions and the estimated OBS, OBSM, BSG, and OBSG survival functions, respectively, from which a significant fraction of survivors can be observed. It can be verified that the OBSM and OBSG models with cure fraction provide good fits to these data.
Many lifetime distributions have been constructed with a view for applications in several areas, in particular, survival analysis, reliability engineering, demography, and actuarial studies, and hydrology. Generalizing continuous univariate distributions by introducing additional shape parameters is an important way to provide very different shapes for the density and hrf of the generated model. We introduce and study a new lifetime model called the OBSG distribution. We derive some general mathematical properties based on a simple mixture representations for its density function. Cure rate models have been used to model time-to-event data for various types of cancers. Based on OBSG distribution, we define two new models with cure rate called the OBSM and OBSG models, where the unobservable number
We are very grateful to a referee and an associate editor for helpful comments that improved the paper. We gratefully acknowledge financial support from CAPES and CNPq.
The necessity and importance of the quantile function, moments and generating function is always important in any statistical analysis such as applied work.
Let
Hence,
where
where
We can write the standard normal qf in terms of the inverse error function, namely
An expansion of the inverse error function erf^{−1}(
Quantiles of interest can be determined from (A.1) or (A.3) by substituting appropriate values for
where
We can use (A.3) for simulating OBSG random variables by setting
For any real
where
Further, the generalized binomial expansion holds
Inserting (A.4) and (A.5) in
where
The ratio of the two power series can be expressed as
where
By differentiating (A.6), we obtain
where
Cordeiro and Lemonte (2011) recently proposed the
The formulae derived from
Some of the most important features and characteristics of a distribution can be studied through moments. Cordeiro and Lemonte (2011) determined the probability weighted moments (PWMs) of the BS distribution since they are required for the ordinary moments of the
where
and
From now on, let
Based on the results by Cordeiro and Lemonte (2011), the moments of
Where
A second formula for
where
The moment generating function of
where
The mean, biases, and MSEs of the OBSG distribution for
Sample size | Parameter (true value) | Classical estimation | Bayesian estimation | ||||
---|---|---|---|---|---|---|---|
Mean | Bias | MSE | Mean | Bias | MSE | ||
0.1238 | 0.0238 | 0.0220 | 0.1227 | 0.0227 | 0.0270 | ||
0.6562 | −0.0438 | 0.0713 | 0.6949 | −0.0051 | 0.0511 | ||
0.7553 | −0.0447 | 0.0588 | 0.8323 | 0.0323 | 0.0453 | ||
2.0410 | 0.0410 | 0.0543 | 2.0377 | 0.0377 | 0.0324 | ||
0.1169 | 0.0169 | 0.0163 | 0.1194 | 0.0194 | 0.0261 | ||
0.6920 | −0.0080 | 0.0513 | 0.6970 | −0.0030 | 0.0399 | ||
0.7953 | −0.0047 | 0.0419 | 0.8234 | 0.0234 | 0.0340 | ||
2.0562 | 0.0562 | 0.0412 | 2.0339 | 0.0339 | 0.0198 | ||
0.1097 | 0.0097 | 0.0129 | 0.1207 | 0.0207 | 0.0256 | ||
0.6952 | −0.0048 | 0.0395 | 0.6874 | −0.0126 | 0.0353 | ||
0.7988 | −0.0012 | 0.0319 | 0.8105 | 0.0105 | 0.0295 | ||
2.0376 | 0.0376 | 0.0293 | 2.0285 | 0.0285 | 0.0140 | ||
0.1080 | 0.0080 | 0.0097 | 0.1218 | 0.0218 | 0.0250 | ||
0.7017 | 0.0017 | 0.0238 | 0.7027 | 0.0027 | 0.0205 | ||
0.8076 | 0.0076 | 0.0191 | 0.8211 | 0.0211 | 0.0179 | ||
2.0328 | 0.0328 | 0.0205 | 2.0228 | 0.0228 | 0.0098 |
MSEs = means squared errors; OBSG = odd Birnbaum-Saunders geometric.
Mean, biases, and MSEs of the OBSGcr distribution for
Sample size | Parameter (true value) | Classical estimation | Bayesian estimation | ||||
---|---|---|---|---|---|---|---|
Mean | Bias | MSE | Mean | Bias | MSE | ||
0.2595 | 0.0095 | 0.0026 | 0.2197 | −0.0303 | 0.0043 | ||
1.3903 | −0.1097 | 0.0845 | 1.5038 | 0.0038 | 0.0816 | ||
0.4737 | −0.0263 | 0.0156 | 0.6379 | 0.1379 | 0.0382 | ||
2.0158 | 0.0158 | 0.0454 | 2.3942 | 0.3942 | 0.3605 | ||
0.2544 | 0.0044 | 0.0015 | 0.2379 | −0.0121 | 0.0016 | ||
1.3790 | −0.1210 | 0.0948 | 1.5196 | 0.0196 | 0.0763 | ||
0.4737 | −0.0263 | 0.0128 | 0.5573 | 0.0573 | 0.0160 | ||
1.9872 | −0.0128 | 0.0194 | 2.1135 | 0.1135 | 0.0688 | ||
0.2533 | 0.0033 | 0.0009 | 0.2443 | −0.0057 | 0.0012 | ||
1.4113 | −0.0887 | 0.0872 | 1.5860 | 0.0860 | 0.0878 | ||
0.4854 | −0.0146 | 0.0104 | 0.5547 | 0.0547 | 0.0119 | ||
1.9966 | −0.0034 | 0.0132 | 2.0483 | 0.0483 | 0.0191 | ||
0.2476 | −0.0024 | 0.0005 | 0.2463 | −0.0037 | 0.0005 | ||
1.4343 | −0.0657 | 0.0767 | 1.5209 | 0.0209 | 0.0824 | ||
0.4844 | −0.0156 | 0.0067 | 0.5252 | 0.0252 | 0.0087 | ||
2.0176 | 0.0176 | 0.0058 | 2.0316 | 0.0316 | 0.0058 |
MSEs = means squared errors; OBSGcr = odd Birnbaum-Saunders geometric cure.
The mean, biases, and MSE of the OBSG mixture distribution for
Sample size | Parameter (true value) | Classical estimation | Bayesian estimation | ||||
---|---|---|---|---|---|---|---|
Mean | Bias | MSE | Mean | Bias | MSE | ||
0.1960 | −0.0040 | 0.0019 | 0.1838 | −0.0162 | 0.0027 | ||
1.8903 | −0.1097 | 0.0837 | 2.2731 | 0.2731 | 0.1187 | ||
0.4820 | −0.018 0 | 0.0074 | 0.6573 | 0.1573 | 0.0291 | ||
1.4378 | −0.0622 | 0.0063 | 1.4437 | −0.0563 | 0.0061 | ||
0.2034 | 0.0034 | 0.0012 | 0.1943 | −0.0057 | 0.0011 | ||
1.9346 | −0.0654 | 0.0946 | 2.1971 | 0.1971 | 0.0844 | ||
0.4891 | −0.0109 | 0.0066 | 0.5926 | 0.0926 | 0.0122 | ||
1.4331 | −0.0669 | 0.0055 | 1.4352 | −0.0648 | 0.0053 | ||
0.2047 | 0.0047 | 0.0008 | 0.1987 | −0.0013 | 0.0008 | ||
1.9164 | −0.0836 | 0.0921 | 2.1151 | 0.1151 | 0.0820 | ||
0.4914 | −0.0086 | 0.0062 | 0.5588 | 0.0588 | 0.0082 | ||
1.4342 | −0.0658 | 0.0051 | 1.4357 | −0.0643 | 0.0048 | ||
0.2024 | 0.0024 | 0.0004 | 0.1992 | −0.0008 | 0.0004 | ||
1.8891 | −0.1109 | 0.0912 | 2.074 | 0.0740 | 0.0838 | ||
0.4815 | −0.0185 | 0.0050 | 0.5425 | 0.0425 | 0.0067 | ||
1.4371 | −0.0629 | 0.0045 | 1.4370 | −0.0630 | 0.0044 |
MSE = means squared error; OBSG = odd Birnbaum-Saunders geometric.
MLEs of the model parameters for the ethylene data, the corresponding standard errors, and the AIC, CAIC, and BIC statistics
Model | Estimates | AIC | CAIC | BIC | ||||
---|---|---|---|---|---|---|---|---|
KwN( | −32.77 (2.55) | 29.40 (0.81) | 13.47 (1.42) | 0.45 (0.03) | 5775.1 | 5775.2 | 5792.9 | |
BN( | −56.17 (2.16) | 32.24 (0.96) | 50.93 (2.57) | 0.41 (0.02) | 5709.9 | 5710.0 | 5727.7 | |
McN( | −186.04 (7.92) | 47.99 (1.77) | 10021.00 (8.85) | 0.46 (0.03) | 4.63 (0.63) | 5638.3 | 5638.4 | 5660.5 |
KwOLLN( | 6.53 (2.77) | 113.18 (14.60) | 2.26 (0.33) | 0.27 (0.01) | 11.29 (2.31) | 5547.0 | 5547.1 | 5569.3 |
BS( | 0.71 (0.02) | 27.47 (0.73) | 5424.0 | 5424.0 | 5432.9 | |||
OBS( | 0.21 (0.01) | 0.28 (0.01) | 27.81 (0.87) | 5425.8 | 5425.9 | 5439.2 | ||
MLEs = maximum likelihood estimators; AIC = Akaike information criterion; CAIC = consistent Akaike information criterion; BIC = Bayesian information criterion; KwN = Kumaraswamy normal; BN = beta normal; McN = McDonald normal; KwOLLN = Kumaraswamy odd log-logistic-normal; BS = Birnbaum-Saunders; OBS = odd Birnbaum-Saunders; OBSG = odd Birnbaum-Saunders geometric.
Likelihood ratio tests
Ethylene | Hypotheses | Statistic | |
---|---|---|---|
OBSG vs. OBS | 173.0 | ||
OBSG vs. BS | 173.2 |
OBSG = odd Birnbaum-Saunders geometric; OBS = odd Birnbaum-Saunders; BS = Birnbaum-Saunders.
Posterior mean, SD, and 95% CI of the parameters for the ethylene data and the DIC, EAIC, EBIC, and LPML statistics
Parameter | Mean | SD | 95% CI | DIC | EAIC | EBIC | LPML | ||
---|---|---|---|---|---|---|---|---|---|
2.50% | 97.50% | ||||||||
OBSG | 0.267 | 0.010 | 0.249 | 0.287 | 5220.931 | 5227.269 | 5245.052 | −2610.026 | |
31.854 | 0.141 | 31.481 | 31.996 | ||||||
0.266 | 0.003 | 0.262 | 0.274 | ||||||
0.305 | 0.012 | 0.276 | 0.320 | ||||||
OBS | 0.248 | 0.002 | 0.244 | 0.250 | 5347.109 | 5353.03 | 5366.368 | −2673.559 | |
27.943 | 0.057 | 27.786 | 27.998 | ||||||
0.287 | 0.001 | 0.286 | 0.290 | ||||||
BS | 27.521 | 0.724 | 26.110 | 28.962 | 5423.964 | 5425.992 | 5434.883 | −2711.748 | |
0.716 | 0.020 | 0.677 | 0.757 |
SD = standard deviation; CI = credible interval; DIC = deviance information criterion; EAIC = expected Akaike information Scriterion; EBIC = expected Bayesian information criterion; LPML = log pseudo marginal likelihood;OBSG = odd Birnbaum- Saunders geometric; OBS = odd Birnbaum-Saunders; BS = Birnbaum-Saunders.
MLEs of the model parameters for the melanoma data, the corresponding standard errors (given in parentheses), and the AIC, CAIC, and BIC statistics
Model | AIC | CAIC | BIC | ||||
---|---|---|---|---|---|---|---|
OBSM | 0.4672 (0.0384) | 7.3671 (3.8938) | 5.9259 (1.1123) | 1.8335 (0.1705) | 1057.2 | 1057.3 | 1073.3 |
BSM | 0.4475 (0.0512) | 1 | 0.9378 (0.0998) | 1.8843 (0.2582) | 1062.5 | 1062.6 | 1074.6 |
OBSG | 0.5511 (0.0495) | 504.560 (29.2503) | 441.570 (33.4233) | 2.9234 (0.6284) | 1058.1 | 1058.2 | 1074.2 |
BSG | 0.6201 (0.0877) | 1 | 1.3107 (0.3799) | 4.6027 (2.6620) | 1069.5 | 1069.6 | 1081.6 |
MLEs = maximum likelihood estimators; AIC = Akaike information criterion; CAIC = consistent Akaike information criterion; BIC = Bayesian information criterion; OBSM = odd Birnbaum-Saunders mixture; BSM = Birnbaum-Saunders mixture; OBSG = odd Birnbaum-Saunders geometric; BSG = Birnbaum Saunders geometric.
Posterior mean, SD, and 95% CI of the parameters for the melanoma data and the DIC, EAIC, EBIC, and LPML statistics
Parameter | Mean | SD | 95% CI | DIC | EAIC | EBIC | LPML | ||
---|---|---|---|---|---|---|---|---|---|
2.50% | 97.50% | ||||||||
OBSM | 0.442 | 0.057 | 0.310 | 0.526 | 1055.365 | 1060.597 | 1076.73 | −527.929 | |
7.063 | 2.076 | 2.597 | 9.886 | ||||||
6.057 | 1.856 | 2.275 | 9.253 | ||||||
1.989 | 0.324 | 1.605 | 2.758 | ||||||
BSM | 0.450 | 0.032 | 0.402 | 0.520 | 1061.477 | 1065.042 | 1077.141 | −531.093 | |
0.954 | 0.081 | 0.808 | 1.123 | ||||||
1.920 | 0.206 | 1.567 | 2.369 | ||||||
OBSG | 0.569 | 0.032 | 0.510 | 0.633 | 1053.903 | 1060.063 | 1076.195 | −527.008 | |
491.035 | 12.154 | 470.983 | 510.809 | ||||||
450.027 | 26.685 | 403.082 | 497.458 | ||||||
3.187 | 0.377 | 2.530 | 3.911 | ||||||
BSG | 0.615 | 0.023 | 0.569 | 0.662 | |||||
1.305 | 0.073 | 1.168 | 1.451 | 1067.451 | 1071.499 | 1083.598 | −534.168 | ||
4.506 | 0.288 | 4.020 | 4.978 |
SD = standard deviation; CI = credible interval; DIC = deviance information criterion; EAIC = expected Akaike information criterion; EBIC = expected Bayesian information criterion; LPML = log pseudo marginal likelihood;OBSM= odd Birnbaum- Saunders mixture; BSM = Birnbaum-Saunders mixture; OBSG = odd Birnbaum-Saunders geometric; BSG = Birnbaum Saunders geometric.
Likelihood ratio tests
Melanoma | Hypotheses | Statistic | |
---|---|---|---|
OBSM vs. BSM | 7.3 | 0.0069 | |
OBSG vs. BSG | 13.0 | 0.0003 |
OBSM = odd Birnbaum-Saunders mixture; BSM = Birnbaum-Saunders mixture; OBSG = odd Birnbaum-Saunders geometric; BSG = Birnbaum Saunders geometric.