Goodness Of Fit Methods Research Articles

A Monte Carlo resampling framework for implementing goodness‐of‐fit tests in spatial capture‐recapture models

Abstract Spatial capture‐recapture (SCR) models provide estimates of animal density from spatially referenced encounter data and has become the most widely adopted approach for estimating density. Despite the rapid growth in the development and application of spatial capture‐recapture methods, approaches for assessing model fit have received very little attention when compared to other classes of hierarchical models in ecology. Here, we develop an approach for testing goodness‐of‐fit (GoF) for frequentist SCR models using Monte Carlo simulations. We derive probability distributions of activity centres from the fitted model. From these, we calculate the expected encounters in the capture history based on the SCR parameter estimates, propagating the uncertainty of the estimates and the activity centre locations via Monte Carlo simulations. Aggregating these test statistics result in count data, allowing us to test fit with Freeman‐Tukey tests. These tests are based on summary statistics of the total encounters of each individual at each trap (FT‐ind‐trap), total encounters of each individual (FT‐individuals) and total encounters at each trap (FT‐traps). We assess the ability of these GoF tests to diagnose lack of fit under a range of assumption violating scenarios. FT‐traps had the strongest response to unmodelled spatial and trap heterogeneity in detection probability (power = 0.53–0.56), while FT‐ind‐traps had the strongest responses to random individual variation in detectability (power = 0.88) and non‐spatial discrete variation in (power = 0.35). The tests, designed to diagnose poor fit in the detection parameters, were insensitive to unmodelled heterogeneity in density (power = <0.001). They also demonstrated low false positive rates (<0.001) when the correct models were fitted; therefore, it is very unlikely that they will provide false indications of poor model fit. We demonstrate that these GoF tests are capable of detecting lack‐of‐fit when unmodelled heterogeneity is present in the detection sub‐model. When used jointly, the combinations of test results are also able to infer the type of lack‐of‐fit in certain cases. Our Monte Carlo sampling methods may be extended to a wider range of GoF tests, thereby providing a platform for developing more GoF methods for SCR.

Analysis of Copula Frailty defective models in presence of Cure Fraction

Introduction: Analyzing long term survivors such as diabetic patients can't be done using the usual survival models. One approach to analyze it is using defective distribution that doesn't force a pre-assumption of cure fraction to the model. To study more than one random variable interacting together, multivariate distributions may be used. However, most of multivariate distributions have complicated forms, which make the computations difficult. Besides, it may be hard to find a multivariate distribution that fits the data properly, especially in health care field. To get over this problem, one can use copula approach. In literature, to the best of our knowledge, only one paper handled copula defective models and didn't consider the effect of covariates. In this paper, we take into consideration not only existed covariates but also unobserved ones by including frailty term. Methods: Two new models are introduced. The first model, used Gumbel copula to take the dependence into consideration together with the observed covariates. The second one take into consideration not only the dependence but also the unobserved covariates by integrating frailty term in to the model. Results: A diabetic retinopathy data is analyzed. The two models indicated the existence of long-term survivals through negative parameters without the need of pre-assuming the existence of it. Including frailty term to the model helped in capturing more dependence between the variables. We compared the results using goodness of fit methods, and the results suggested that the model with frailty term is the best to be used. Conclusion: The two introduced models correctly detected the existence of cure fraction with less estimated parameters than that in mixture cure fraction models. Also, it has the advantage of not pre-assuming the existence of cure fraction to the model. comparing both models, the model with frailty term fitted the data better.

Goodness-of-fit testing for meta-analysis of rare binary events

Random-effects (RE) meta-analysis is a crucial approach for combining results from multiple independent studies that exhibit heterogeneity. Recently, two frequentist goodness-of-fit (GOF) tests were proposed to assess the fit of RE model. However, they tend to perform poorly when assessing rare binary events. Under a general binomial-normal framework, we propose a novel GOF test for the meta-analysis of rare events. Our method is based on pivotal quantities that play an important role in Bayesian model assessment. It further adopts the Cauchy combination idea proposed in a 2019 JASA paper, to combine dependent p-values computed using posterior samples from Markov Chain Monte Carlo. The advantages of our method include clear conception and interpretation, incorporation of all data including double zeros without the need for artificial correction, well-controlled Type I error, and generally improved ability in detecting model misfits compared to previous GOF methods. We illustrate the proposed method via simulation and three real data applications.

Analysis of sample size for friction test of short fiber reinforced polymer composites

AbstractAccording to the recommendations of literature and standards, the minimum sample size for the friction test of thermoplastics is usually 3–5. While this size is valid for the friction test of neat polymer resins with homogeneous structures, its applicability to short fiber reinforced polymer composites with inhomogeneous structures remains unclear because the uneven distribution of discontinuous fibers or fillers in the matrix leads to the performance of the composites being dispersed. In this work, as a typical example, 100 friction samples made of short carbon fiber reinforced polyetherimide (SCF/PEI) composite with a typical content of 10 vol% SCFs are prepared, and their friction coefficients are measured. The maximum likelihood estimation and goodness of fit methods are employed to assess the fitting degree between the overall friction coefficient results and five mathematical distributions. The results show that the overall friction coefficient data distribution is in good agreement with the Lognormal, Generalized extreme value (GEV), and Burr distributions, respectively. Based on the fitting result, the minimum sample size is analyzed by Monte Carlo Simulation. It shows that the minimum sample size strongly depends on the property dispersion parameter (COV: coefficient of variation). Based on this parameter, the minimum sample size is given. In a practical test, the performance dispersion degree of a composite cannot be known in advance, hence a sample size determination method is proposed to yield a reasonable performance description parameter by applying the results of this work, and this method is validated by the overall test data.Highlights Proposed a sample size determination method for practical friction tests. The minimum sample size for the friction test is yielded statistically. The minimum sample size strongly depends on a property dispersion parameter.

Mathematical Model to Calculate the Total Number of Radiation Decays of Radiolabelled-Pembrolizumab in Mice

Immunotherapy with checkpoint inhibitors with Pembrolizumab shows potential to be used as a first-line in cancer treatment. A biodistribution study could be used to maximize efficacy and minimize the risk of the treatment. Therefore, it is necessary to describe the biodistribution of the 89Zr-Pembrolizumab. This study aims to create a mathematical model to explain how 89Zr-pembrolizumab is distributed in the body. Biodistribution data from Biokinetic data of 89Zr-Pembrolizumab in NSG mice engrafted with human lymphocyte peripheral (Hu-PBL-SCID) obtained from literature were used. The organ compartment of the model was divided into three sub-compartments: the vascular, interstitial, and endosomal space. The estimated parameters were the plasma clearance (CL), endocytosis modulation factors (F2), exocytosis modulation factors (F3) in the endosomal space, and modulation factors of the transcapillary flow (MK). According to the visualization of the fitted graphs and the percentage of variation (CV) of the fitted parameters (50%), the unknown parameters were successfully estimated with a goodness of fit method. The estimated value of CL was 2.65x10-5 l/h (CV=7.56%), parameter F2 was estimated for kidney, liver, spleen, and muscle tissue in the range of 0.12 - 0.35 (CV=4.14% - 5.60%), while F3 was estimated in the range of 3.60x10-3 - 0.036 (CV=2.21% - 21.44%), and the modulation factor of the transcapillary flow (MK) was within the range of 8.26 - 46.91 (CV=0.98% - 1.60%). A mathematical model was successfully used to describe the biodistribution of 89Zr-Pembrolizumab in mice.

Interaction between a Cyclically Heated Building and the Ground, for Selected Locations in Europe

The aim of the study was to verify the impact of the location of a cyclically heated building on its energy needs and interaction with the environment. The model building was a large-scale broiler house located in southern Poland. In the examined facility, year-round measurements of selected parameters of the internal and external microclimate were carried out. The tests also covered the temperature of the soil in three measurement columns. The obtained measurement results were used for further computer analyses. A geometric model of the building was made and a calculation model specification was introduced, supported by specialized software for the physics of WUFI®plus structures. The numerical analysis included validation. The validation results were assessed on the basis of the Coefficient of Determination method (R2) and the Goodness of Fit (GOF) method. Due to the lack of normality of the data distribution, a Rang-Spearman correlation analysis was carried out, which showed a very strong data correlation (0.94). The analysis of the R2 coefficient of determination confirmed the high reliability of the model (83%). In the case of the GOF method, a compliance value of 87% was obtained. Differentiated locations were adopted for further analysis, while maintaining the structure of the examined building in reality. Six European locations were selected: Kraków (Poland), Málaga (Spain), Brest (France), Visby (Sweden), Umea (Sweden), and Kiruna (Sweden). The analysis included three variants, in which the length of the technological break was adopted accordingly. A technological break between production cycles was assumed, the three variants of which lasted 3, 7, and 14 days. The analysis of the obtained results showed that the external microclimate significantly determines the energy interaction between the building and the ground. The length of the technological break is very important in a climate dominated by low temperatures (Northern Scandinavia). The south-western part of Europe allows the technological break to be extended without significant differences for heating purposes. The length of the technological break in the range of 3 to 14 days does not significantly affect the intensity of heat exchange with the ground on a yearly basis, regardless of the location of the building. There were differences of no more than 2% between the technological break lasting 3 days and the 14-day break.

Statistical Characteristics of Recycled Aggregate Permeable Concrete

In view of the randomness of mechanical properties and durability of recycled aggregate permeable concrete, a method combining the goodness of fit test with Bayes theory was proposed to analyze its statistical characteristics probability model. Firstly, the optimal probability distribution type of recycled aggregate permeable concrete strength was identified by goodness of fit method. Then, Bayes method was used to estimate the parameters of probability model. Finally, the reliability theory was used to evaluate the durability of recycled aggregate permeable concrete. This method has good theoretical significance and practical value for reasonable evaluation of mechanical properties and durability of recycled aggregate permeable concrete.

Further Development of Studies of Pressure Fields in the Arctic Region of Russia

The results of studies of atmospheric pressure in the Arctic region of Russia in the period from 1948 to 2008 are presented. The analysis of the climatic seasonal variation of the atmospheric pressure fields is carried out. As the main research method, the probabilistic and statistical analysis of the time series of the pressure field 60 years long at fixed points in the region of the Arctic zone of Russia was used. In total, about 90,000 daily (in six-hour increments) pressure values were examined. On the basis of these data, a climatic seasonal variation was constructed as an averaging of the values of a given time series at each point in space and for a fixed date. The characteristics of the seasonal course, its amplitude and phase have been studied. These characteristics were analyzed and their geophysical interpretation was carried out. In particular, the minimum and maximum values ​​of the series were determined for the entire region and the time series of these characteristics were constructed. It is shown that the deviation is asymmetric, this is an unobvious research result. For the maximum and minimum, the best approximations were constructed, and these approximations were tested by known methods of statistical analysis, including maximum likelihood, least squares and goodness of fit methods (tests), in particular, the χ2-criterion. The conducted research has applications both purely physical (allows to explain the nature, genesis and distribution of large-scale atmospheric formations in a climatic year) and prognostic (allows understanding and tracking trends in climate, as well as quantitatively assessing the scale and variability of large-scale atmospheric processes). Numerical calculations were performed on the Lomonosov-2 supercomputer of the Lomonosov Moscow State University.

Comparison of Cox proportional hazards model, Cox proportional hazards with time-varying coefficients model, and lognormal accelerated failure time model: Application in time to event analysis of melioidosis patients

Objective: To compare the prognostic factors of mortality among melioidosis patients between lognormal accelerated failure time (AFT), Cox proportional hazards (PH), and Cox PH with time-varying coefficient (TVC) models.Methods: A retrospective study was conducted from 2014 to 2019 among 453 patients who were admitted to Hospital Sultanah Bahiyah, Kedah and Hospital Tuanku Fauziah, Perlis in Northern Malaysia due to confirmed-cultured melioidosis. The prognostic factors of mortality from melioidosis were obtained from AFT survival analysis, and Cox’s models and the findings were compared by using the goodness of fit methods. The analyses were done by using Stata SE version 14.0.Results: A total of 242 patients (53.4%) survived. In this study, the median survival time of melioidosis patients was 30.0 days (95% CI 0.0-60.9). Six significant prognostic factors were identified in the Cox PH model and Cox PH-TVC model. In AFT survival analysis, a total of seven significant prognostic factors were identified. The results were found to be only a slight difference between the identified prognostic factors among the models. AFT survival showed better results compared to Cox’s models, with the lowest Akaike information criteria and best fitted Cox-snell residuals.Conclusions: AFT survival analysis provides more reliable results and can be used as an alternative statistical analysis for determining the prognostic factors of mortality in melioidosis patients in certain situations.

A generalized linear model for multivariate events

We propose a new generalized linear model for modeling multivariate events (N,X,Y), where N is the duration, X is the magnitude, and Y is the maximum of the event. Such events arise, for example, when a process is observed above or below a threshold. Examples include heat waves, flood, draught, or market growth or decline periods. The model is flexible to include different covariates for different parameters. In addition, we propose a new method for checking the goodness of fit (validation) of the model to data. Our goodness of fit methods are based on distributional fit of appropriately transformed data. We include a data example from finance to illustrate the modeling potential of this new generalized linear model.

Gaussianity Testing as an Effective Technique for Detecting Discontinuous Transmission in 5G Networks

The paper investigates statistical distribution testing-based detection methods in an intermittent signal detection scenario. The relevance of the research is driven by 5G networks based on packet transmission, incorporating the concept of cognitive radio and adapting spectrum detection methods from Long Term Evolution (LTE) Licensed-Assisted Access (LAA). The conducted study refers to the recently proposed methods based on testing goodness-of-fit (GoF) of statistical distributions, which are compared with a conventional energy detector. The authors examine the applicability of well-known GoF methods in intermittent transmission, as they require reconsideration in 5G communication systems, and investigate the behavior of the innovative energy-based GoF. The experiments are carried out for different transmitter activity factors, i.e., channel occupancy and signal-to-noise ratio (SNR), demonstrating the superiority of the GoF-based methods in general and particularly the invented GoF test over other energy-based detectors for discontinuous signals detection.

Bayesian Poisson-gamma and Log-normal models for estimating the relative risks of health insurance claims due to dengue disease in Bandung

Dengue disease is one of the most dangerous diseases and it is commonly found in Bandung, one of the most populous cities in Indonesia. Some patients who need more intensive care at hospital certainly require high fare of treatment. The risk of high fare to spend can be covered by health insurance. As a direct consequence, it is important for insurance companies to anticipate the relative risk of claims filed by health insurance holders due to this disease. In this study, the relative risk of claims that may be filed by health insurance holders is estimated by implementing frequentist approach, i.e. SMR (Standardized Mortality/Morbidity Ratio) model, and Bayesian approach, i.e. Poisson-Gamma and Log-normal models. Based on the implementation on the data of health insurance holders and the number of dengue cases in Bandung in 2013 and 2014 and the analysis of the results, it can be concluded that Bayesian inference is more suitable to be applied than frequentist inference because the estimation of the relative risk of claims in industrial insurance is often related to random variables which have non constant parameters. From one of the goodness of fit methods, i.e. DIC, it can be concluded that Poisson-Gamma model is the best model which suits the data better. From these models, sub-districts that have the highest relative risk estimates are specified to provide an overview for insurance companies to anticipate the possibility of health insurance claims resulting from dengue disease from particular critical sub-districts. More accurate results are expected to be obtained once more complete data can be collected.

Analysis of goodness-of-fit method based on local property of statistical model for airborne sea clutter data

Distribution models are of practical significance to clutter modelling, target detection method and performance evaluation in sea clutter background. The goodness of fit (GOF) between the real sea clutter data and the assumed distribution can be used to choose the proper distribution model. So the great importance of determining the suited statistical model and analysis of goodness-of-fit are highlighted. This paper concentrates on the statistical properties of real L-band airborne sea clutter datasets, which is collected at different grazing angles (30°∼60°). To overcome the shortage of traditional GOF methods, such as Mean Square Difference (MSD) and Modified Mean Square Difference (MMSD), the Variable-window local Mean Square Difference (VW-LMSD) is proposed. By this method, rectangle windows, whose number and position are flexibly selected according to the radar acquirement and sea state, are added to the sea clutter amplitude series to observe the partial GOF results of the statistical distribution models in detail, which provides a multi-angle GOF evaluation to judge the suitable statistical model and establishes a better foundation for clutter modelling, clutter classification and detection criterion selection further.

At Site Flood Frequency Analysis of Baitarani River at Champua Watershed, Odisha

For any development of hydraulic structures and dam modelling, flood frequency analysis is an effective tool to determine the appropriate measures and strategy. Flood frequency analysis has been conventionally used in hydraulic engineering and floodplain management. The present study is an attempt to estimate the expected flood using two probability distributions: Gumbel distribution and Log Pearson III distribution at Champua watershed, Upper Baitarani River Basin, Odisha. The analysis is based on annual maximum flood time series from 1991 to 2018 (28 years) obtained from Water Resources Information System at the Champua gauging station. Three Goodness of fit methods namely Kolmogorov Smirnov, Anderson Darling and Chi Squared tests are used to choose the better model. From the analysis, expected flood for return period 2, 10, 25, 50, 100 and 1000 years are calculated. Gumbel give an expected flood 521.72 cumecs while Log Pearson III give an expected flood of 493.17 cumecs for 2 years return period. It is observed that Gumbel estimated a higher values for all the said return period except for 1000 years where Log Pearson III predicted a much higher values. Goodness of test show inconsistent results. While Chi-squared test indicate Gumbel Method as the better model, the other two tests show that Log Pearson III is the better fitting model for the given dataset. Therefore, Log Pearson III is chosen as the best model. However, the results from both the distributions can be referred for storm management.

Risk Assessment and Its Visualization of Power Tower under Typhoon Disaster Based on Machine Learning Algorithms

For power system disaster prevention and mitigation, risk assessment and visualization under typhoon disaster have important scientific significance and engineering value. However, current studies have problems such as incomplete factors, strong subjectivity, complicated calculations, and so on. Therefore, a novel risk assessment and its visualization system consisting of a data layer, knowledge extraction layer, and visualization layer on power towers under typhoon disaster are proposed. On the data layer, a spatial multi-source heterogeneous information database is built based on equipment operation information, meteorological information, and geographic information. On the knowledge extraction layer, six intelligent risk prediction models are established based on machine learning algorithms by hyperparameter optimization. Then the relative optimal model is selected by comparing five evaluation indicators, and the combined model consisting of five relatively superior models is established by goodness of fit method with unequal weight. On the visualization layer, the predicted results are visualized with accuracy of 1 km × 1 km by ArcGIS 10.4. In results, the power tower damage risk assessment is carried out in a Chinese coastal city under the typhoon ‘Mujigae’. By comparing predicted distribution and similarity indicator of the combined model with those of the other models, it is shown that the combined model is superior not only in quality but also in quantity.

A test on methods for <i>M</i><sub>C</sub> estimation based on earthquake catalog

This study tested five methods widely used in estimating the complete magnitudes (MC) of earthquake catalogs. Using catalogs of observed earthquake properties, we test the performance of these five algorithms under several challenging conditions, such as small volume of events and spatial-temporal heterogeneity, in order to see whether the algorithms are stable and in agreement with known data. We find that the maximum curvature method (MAXC) has perfect stability, but will significantly underestimate MC unless heterogeneity is absent. MC estimated by the b-value stability method (MBS) requires many events to reach a stable result. Results from the goodness of fit method (GFT) were unstable when heterogeneity lowered the fitness level. The entire magnitude range method (EMR) is relatively stable in most conditions, and can reflect the change in MC when heterogeneity exists, but when the incomplete part of the earthquake catalog is dismissed, this method fails. The median-based analysis of the segment slope method (MBASS) can tolerate small sample size, but is incapable of reflecting the missing degree of small events in aftershock sequences. In conditions where MC changes rapidly, such as in aftershock sequences, observing the time sequence directly can give a precise estimation of the complete sub-catalog, but only when the number of events available for study is large enough can the MAXC, GFT, and MBS methods give a similarly reliable estimation.

Integrated meteorological and hydrological drought model: A management tool for proactive water resources planning of semi-arid regions

Conventionally, drought analysis has been limited to single drought category. Utilization of models incorporating multiple drought categories, can relax this limitation. A copula-based model is proposed, which uses meteorological and hydrological drought characteristics to assess drought events for ultimate management of water resources, at small scales, i.e., sub-watersheds. The chosen study area is a sub-basin located at Karkheh watershed (western Iran), with five raingauge stations and one hydrometric station, located upstream and at the outlet, respectively, which represent 41-year of data. Prior to drought analysis, time series of precipitation and streamflow records are investigated for possible dependency/significant trend. Considering semi-arid nature of the study area, boxplots are utilized to graphically capture the rainy months, which are used to evaluate the degree of correlation between streamflow and precipitation records via nonparametric correlations. Time scales of 3- and 12-month are considered, which are used to study vulnerability of early vegetation establishment and long-term ecosystem resilience, respectively. Among four common goodness of fit (GOF) tests, Anderson–Darling is found preferable for defining copula distribution functions through GOF measures, i.e., Akaike and Bayesian information criteria and normalized root mean square error. Furthermore, a GOF method is proposed to evaluate the uncertainty associated with different copula models using the concept of entropy. A new bivariate drought modeling approach is proposed through copulas. The proposed index named standardized precipitation-streamflow index (SPSI) unlike common indices which are used in conjunction with station data, can be applied on a regional basis. SPDI is compared with widely applied streamflow drought index (SDI) and standardized precipitation index (SPI). To assess the homogeneity of the dependence structure of SPSI regionally, Kendall-τ and upper tail coefficient relation is investigated for all stations located within the region. According to results, SPSI similar to nonparametric multivariate standardized drought index (NMSDI) was able to detect both onset of droughts dominated by precipitation as is similarly indicated by SPI and persistence of droughts dominated by streamflow as is similarly indicated by SDI. It also captures discordant case of normal period precipitation with dry period streamflow and vice versa. This makes SPSI a powerful tool for estimating a more practical and realistic drought condition. Finally, combination of severity–duration–frequency (SDF) of drought events through copulas resulted in SDF curves that can be used to obtain the recurrence of extreme droughts and assess drought related ecosystem failure or to aid in optimization of water resources allocation. Results indicated that the newly proposed index (SPSI) is able to represent two main characteristics of meteorological and hydrological drought (drought onset and persistency) and also providing an accurate estimation of the recurrence interval of extreme droughts. The procedures can be used to undertake proactive water resource management and planning to assure water security and sustainable agriculture and ecosystem survival for regions experiencing extreme droughts.

Identification of Statistical Distributions for Cycle Time in Wafer Fabrication

Wafer fabrication is characterized with complex production processes, advanced equipment, and high variability. Cycle time (CT) is a key performance measure in fab operations affecting other measures, such as throughput and cost. Previous studies use analytical and empirical methods in investigating CT. Theoretical statistical distribution of CT have seldom been studied, while in other scientific fields they have been frequently used for analysis and modeling, for example, in economics and medicine. This work investigates the theoretical statistical distribution of CT of a wafer lot in a production step, based on actual data originating at various processes and equipment types of a wafer fab. The data is evaluated with goodness of fit methods for ten distributions describing time. The evaluation results with suggesting Burr and Dagum as the best fitted distributions, known in economics for income. This conclusion differs from the common distributions assumed for CT, as Gamma, Weibull, and their special cases. The results are strengthened with additional analysis of their hazard functions. Log-logistic and lognormal can also be used for mathematical simplicity. The identification of theoretical distributions for CT enables improved statistical analysis and system modeling of fab operation.

NUMERICAL TESTS ON CATALOG‐BASED METHODS TO ESTIMATE MAGNITUDE OF COMPLETENESS

AbstractThis study compares five methods for estimating the completeness magnitude Mc threshold of earthquake catalogs through applying them to synthetic catalogs generated from 3 different models. We have found that the Median‐based analysis of the segment slope (MBASS) method is suitable for catalogs recorded by networks whose detection capability changes rapidly with magnitude and for those with spatiotemporal heterogeneity if the amount of earthquakes is large enough. The Mc by b value stability (MBS) method is optimal in dealing with catalogs recorded by networks whose detection ability improves slowly with magnitude, but it is time‐consuming. The Maximum Curvature technique (MAXC) & Goodness‐of‐fit method (GFT) underestimate Mc and need an Mc criterion. The Mc from Entire Magnitude Range (EMR) method gives a stable and moderate Mc estimation. It is recommended when the amount of events is not large and the tolerance of missing events is relatively high. This study helps us to choose the optimal Mc estimation method in practice to cope with different earthquake catalogs and points out some potential problems caused by these methods.

Application of accelerated failure time models for breast cancer patients' survival in Kurdistan Province of Iran.

Breast cancer is the most common cancer and the second common cause of cancer-induced mortalities in Iranian women. There has been a rapid development in hazard models and survival analysis in the last decade. The aim of this study was to evaluate the prognostic factors of overall survival (OS) in breast cancer patients using accelerated failure time models (AFT). This was a retrospective-analytic cohort study. About 313 women with a pathologically proven diagnosis of breast cancer who had been treated during a 7-year period (since January 2006 until March 2014) in Sanandaj City, Kurdistan Province of Iran were recruited. Performance among AFT was assessed using the goodness of fit methods. Discrimination among the exponential, Weibull, generalized gamma, log-logistic, and log-normal distributions was done using Akaik information criteria and maximum likelihood. The 5 years OS was 75% (95% CI = 74.57-75.43). The main results in terms of survival were found for the different categories of the clinical stage covariate, tumor metastasis, and relapse of cancer. Survival time in breast cancer patients without tumor metastasis and relapse were 4, 2-fold longer than other patients with metastasis and relapse, respectively. One of the most important undermining prognostic factors in breast cancer is metastasis; hence, knowledge of the mechanisms of metastasis is necessary to prevent it so occurrence and treatment of metastatic breast cancer and ultimately extend the lifetime of patients.