Type 1 and type 2 errors in statistics pdf and cdf
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- Introduction to Type I and Type II errors
- Hypothesis Testing for Binomial Distribution
- Type I and Type II Errors
Introduction to Type I and Type II errors
The time to event or survival time usually follows certain skewed probability distributions. These distributions encounter vital role using the Bayesian framework to analyze and project the maximum life expectancy in order to inform decision-making. The Bayesian method provides a flexible framework for monitoring the randomized clinical trials to update what is already known using prior information about specific phenomena under uncertainty. Additionally, medical practitioners can use the Bayesian estimators to measure the probability of time until tumor recurrence, time until cardiovascular death, and time until AIDS for HIV patients by considering the prior information.
However, in clinical trials and medical studies, censoring is present when an exact event occurrence time is not known. The present study aims to estimate the parameters of Gumbel type-II distribution based on the type-II censored data using the Bayesian framework. The maximum likelihood and Bayesian estimators are compared in terms of mean squared error by using the simulation study.
Furthermore, two data sets about remission times in months of bladder cancer patients and survival times in weeks of 61 patients with inoperable adenocarcinoma of the lung are analyzed for illustration purposes. In medical research, data supporting the time until the occurrence of a particular event, such as the death of a patient, are frequently encountered.
Such data are referred to as survival time data which has generally right-skewed distribution, and Gumbel type-II distribution can be used for this purpose. The corresponding cumulative distribution function CDF is. A common feature of lifetime data is that the data points are possibly censored.
In manifold reliability and life-testing studies, experiments are generally windup before failure times of all items are observed. Therefore, adequate information and results on failure times of all objects cannot be obtained. During experimentation, these situations occur due to loss or removal of objects before they fail. Therefore, generally, such experiments are preplanned and purposeful to save time and cost of these testing.
Data obtained from such experiments are called censored. The type-I and type-II censoring are two well-known censoring schemes.
In type-II censoring scheme, the number of failure units are fixed in advanced. For example, the investigator may decide to terminate the study after four of the six rats have developed tumors. There is an enormous literature accessible on estimation of parameters of distributions using type-II censoring, for example, Abbas and Tang [ 2 ] considered ML and least square estimators of Frechet distribution using type-II censored samples.
Okasha [ 3 ] estimated the unknown parameters, reliability, and hazard functions of Lomax distribution under type-II censoring using Bayesian and E-Bayesian estimation. Abu-Zinadah [ 4 ] studied on exponentiated Gompertz distribution based on type-II and complete censored data.
El-Sagheer [ 5 ] studied the generalized pareto distribution under the different censoring schemes. Recently, many authors have worked on Gumbel type-II distribution and Bayesian estimation using different loss functions. Abbas et al. Malinowska and Szynal [ 8 ] also derived Bayes estimators for Gumbel type-II distribution on kth lower record values. Sultana et al. Moreover, Metiri et al. Preda et al.
However, Bayesian estimation of Gumbel type-II distribution based on type-II censoring is not frequently discussed; therefore, we are interested in estimating the unknown parameters of Gumbel type-II distribution under type-II censored data. Including this introduction section, the rest of the paper is arranged as follows: in Section 2 , maximum likelihood estimators MLEs for the parameters are obtained. In Section 3 , Bayesian estimators based on different loss functions by taking noninformative and gamma priors are derived.
The proposed estimators are compared in terms of their mean squared error MSE in Section 4. Section 5 illustrates the applications of proposed estimators with two examples, namely, data set of remission times for bladder cancer and survival times of inoperable adenocarcinoma of the lung.
Finally, conclusions and recommendations are presented in Section 6. It is more convenient to work with log-likelihood. The log-likelihood function is. Equations 6 and 7 cannot be written in closed form. As both parameters are unknown, independent noninformative form of priors can be used. Therefore, the joint posterior density under any loss function is.
Posterior distribution 12 takes a ratio form that cannot be reduced to a closed form. The detail of equation 13 is provided in Appendix. The performance of the proposed Bayesian estimators with their ML counterpart in terms of MSE, different sample sizes, and different values of parameters are considered using Monte Carlo simulation based on prespecified different percentages of failures, i.
The results are reported in Tables 1 — 4 for comparison purposes. From the results of the simulation study, conclusions are drawn regarding the behavior of the estimators, which are summarized as follows: i In terms of MSEs, the ML and Bayesian estimators become closer by increasing the sample sizes.
Therefore, Bayes estimators are much stable than ML estimators. The real data about remission times in months of a random sample of bladder cancer patients presented in Table 5 were reported by Lee and Wang [ 20 ].
A total of patients with different prespecified percentages of events, i. Clearly, Figure 1 confirms that the histogram is slightly skewed to the right and is leptokurtic. Moreover, ML and Bayesian estimates can also be envisioned in Figure 1 , in which the x -axis represents the remission times in months of bladder cancer patients, while the Gumbel type-II density function is taken on the y -axis.
Therefore, it would be appropriate to select positively skewed distributions for describing the behavior of remission times of bladder cancer patients. Amongst the skewed distributions, Gumbel type-II distribution is fitted and the parameter estimates using ML and Bayesian methods are presented in Table 6 for comparison purposes. It is concluded that the proposed estimators of Gumbel type-II distribution fit the data well.
Therefore, it is recommended that the Bayesian estimators can be more beneficial to address the uncertainty in medical-related censored data. The survival times, in weeks, of 61 patients with unoperable lung cancer treated with cyclophosphamide considered in Lagakos and Williams [ 18 ] and in Lee and Wolfe [ 19 ] are presented in Table 7. There are 33 uncensored observations and 28 censored observations, representing the patients whose treatment was terminated because of a devolving condition.
Figure 2 shows the results of different estimation methods and depicts that Gumbel type-II distribution fits the data better, in which x -axis comprises the survival times in weeks of 61 patients with inoperable adenocarcinoma of the lung as the Gumbel type-II density function is taken on the y -axis.
In medical decision-making, Bayesian tools incorporate the state of uncertainty and provide a rational framework for studying such problems. Usually, medical data are generally skewed to the right, and positively skewed distributions can be most suitable for describing unimodal medical data.
It is concluded that ML and Bayesian estimators become closer by increasing the sample sizes and prespecified percentages of failures.
Based on the outcomes of this research study, we may suggest that this study can be further extended by using other skewed distributions considering the Bayesian framework with other loss functions using medical data.
Therefore, the matrix may be defined as. The components of observed FIM are. The observed FIM matrix is rewritten as. This work is mainly a methodological development and has been applied on secondary data, but if required, data will be provided. The authors declare that there are no conflicts of interest regarding the publication of this paper. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Journal overview. Special Issues. Academic Editor: Pritee Khanna. Received 11 Dec Revised 09 Feb Accepted 10 Feb Published 26 Mar Abstract The time to event or survival time usually follows certain skewed probability distributions. Introduction In medical research, data supporting the time until the occurrence of a particular event, such as the death of a patient, are frequently encountered. The corresponding cumulative distribution function CDF is A common feature of lifetime data is that the data points are possibly censored.
Thus, Therefore, the joint posterior density under any loss function is Posterior distribution 12 takes a ratio form that cannot be reduced to a closed form. Simulation Study The performance of the proposed Bayesian estimators with their ML counterpart in terms of MSE, different sample sizes, and different values of parameters are considered using Monte Carlo simulation based on prespecified different percentages of failures, i.
Table 1. Table 2. Table 3. Table 4. Table 5. Remission times in months of a random sample of bladder cancer patients.
Figure 1. Table 6. Table 7. Survival times in weeks of 61 patients with inoperable adenocarcinoma of the lung. Table 8. Figure 2. References E. Abbas and Y. View at: Google Scholar H. Abbas, J. Fu, and Y. Feroze and M. View at: Google Scholar I.
Hypothesis Testing for Binomial Distribution
The methods for making statistical inferences in scientific analysis have diversified even within the frequentist branch of statistics, but comparison has been elusive. We approximate analytically and numerically the performance of Neyman-Pearson hypothesis testing, Fisher significance testing, information criteria, and evidential statistics Royall, This last approach is implemented in the form of evidence functions: statistics for comparing two models by estimating, based on data, their relative distance to the generating process i. A consequence of this definition is the salient property that the probabilities of misleading or weak evidence, error probabilities analogous to Type 1 and Type 2 errors in hypothesis testing, all approach 0 as sample size increases. Our comparison of these approaches focuses primarily on the frequency with which errors are made, both when models are correctly specified, and when they are misspecified, but also considers ease of interpretation. The error rates in evidential analysis all decrease to 0 as sample size increases even under model misspecification. Neyman-Pearson testing on the other hand, exhibits great difficulties under misspecification.
We now give some examples of how to use the binomial distribution to perform one-sided and two-sided hypothesis testing. Determine whether the die is biased. We use the following null and alternative hypotheses:. Example 2 : We suspect that a coin is biased towards heads. When we toss the coin 9 times, how many heads need to come up before we are confident that the coin is biased towards heads? INV 9,. DIST 7, 9,.
Characteristics of the standard normal distribution. The normal distribution is centered at the mean, μ. The degree to which population data values.
Type I and Type II Errors
Two drugs are to be compared in a clinical trial for use in treatment of disease X. Drug A is cheaper than Drug B. Efficacy is measured using a continuous variable, Y, and.
The time to event or survival time usually follows certain skewed probability distributions. These distributions encounter vital role using the Bayesian framework to analyze and project the maximum life expectancy in order to inform decision-making. The Bayesian method provides a flexible framework for monitoring the randomized clinical trials to update what is already known using prior information about specific phenomena under uncertainty. Additionally, medical practitioners can use the Bayesian estimators to measure the probability of time until tumor recurrence, time until cardiovascular death, and time until AIDS for HIV patients by considering the prior information. However, in clinical trials and medical studies, censoring is present when an exact event occurrence time is not known.
In this tutorial, we discuss many, but certainly not all, features of scipy. The intention here is to provide a user with a working knowledge of this package.
Error probabilities and power
The Wrapped package computes the probability density function, cumulative distribution function, quantile function and also generates random samples for many univariate wrapped distributions. It also computes maximum likelihood estimates, standard errors, confidence intervals and measures of goodness of fit for nearly fifty univariate wrapped distributions. Numerical illustrations of the package are given. This is an open access article, free of all copyright, and may be freely reproduced, distributed, transmitted, modified, built upon, or otherwise used by anyone for any lawful purpose. The work is made available under the Creative Commons CC0 public domain dedication. Competing interests: The authors have declared that no competing interests exist.
In null hypothesis significance testing , the p -value [note 1] is the probability of obtaining test results at least as extreme as the results actually observed , under the assumption that the null hypothesis is correct. Reporting p -values of statistical tests is common practice in academic publications of many quantitative fields.