Statistical Modeling of Some Cancerous Diseases Using the Laplace Transform Approach of Basic Life Testing Issues

The purpose of the nonparametric statistical test used in this study is to compare different treatment options by looking at failure behavior in recorded survival data. Patients' survival times are documented after using the proposed approach. The observed data's behavior was assumed to be based on used better than aged in the moment generating function order (UBAmgf) characteristic or a constant failure rate in this study (exponential scenario). Suppose that the survival data is UBAmgf, then the treatment or the machine or system in use produces a better or a higher expected total present value than an older machine governed by an exponential survival function; if the data is exponential, the suggested treatment strategy is ineffective (the recommended treatment approach has neither positive or negative effects on the patients). To guarantee that the suggested statistical test is used correctly, the efficiency and critical values are calculated and compared to those of other tests, and the technique is then applied to medical data.


Introduction
During the past few decades, different classes of life distributions have been introduced in an attempt to model different aspects of aging which has contributed to the development of new highly efficient statistical tests. In addition, the exponential distribution is the most important member of the life distribution classes because it has significant statistical features. Now, we have a dataset that seems to be either UBA mgf or exponential. To support one of the two assumptions, a statistical test is necessary, indicating which claim is right.
The following are the consequences of the most popular classes of life distributions, which include the most of wellknown classes like IFR, UBA, UBAC, and UBA mgf : UBA [13] UBA mgf [14] UBAC [5] The modeling of lifetime data is used in many applications in reliability theory and biostatistics. The time T until some event occurs is the outcome of interest in these applications.
The survivor function is defined as: F has the used better than aged (UBA) property (see Ahmad [13] Definition 1. F has the moment generating function order of used better than aged (UBA mgf ) if, for more details, see Abu Youssef et al. [16].
The primary purpose of this study is to look at how to compare H 0 : F is exponential to H 1 : F is the largest life distribution UBA mgf . The following is the content of the manuscript: using the Laplace transform approach, we give a test statistic for both complete and censored data; for popular alternatives, the Pitman asymptotic efficiency is determined, and selected critical values are listed in Section 2. Finally, in Section 3, we look at several medical science applications to show how important the proposed test is.

The Statistic Tests
Assume X 1 , X 2 , ⋯, X n are random samples from F. Here, we create a test statistic to test if H 0 : F is exponential, against H 1 : F is UBA mgf . Nonparametric testing for classes of life distributions has been considered by many authors (see Abu-Youssef et al. [16,18] and Mahmoud et al. [19].

Complete Data.
Using the Laplace approach, the measure of departure (the Laplace methodology is a good generalization of the Goodness of fit technique β ≠ 1) can be expressed as: Note that under H 0 : δðs, βÞ = 0 and under H 1 : δðs, βÞ > 0.
The test statistic of the proposed UBA mgf class test is given by the following theorem. Assumed that is the moment generating function exists and finite. Theorem 2. Let X be the UBA mgf random variable with distribution function F; then, based on (4), we have where φðsÞ = Ð ∞ 0 e sx dFðxÞ.
Without loss of generality, we assume μð∞Þ is known and equal to one. The empirical estimator of the statistic in (5) can be derived as and the corresponding invariant test statistic can be obtained as The asymptotic normality of the statistic demonstrated in (5) is presented in the following theorem.
It is clear from Table 2 that the statistic δð0:1, 0:2Þ and δð0:4,5Þ perform well for F 1 and F 2 , and it is better than both ξð0:01,5Þ and δ * ð0:01Þ for all cases mentioned above.     Table 3. The asymptotic normality of our test improves as the critical values decrease and the sample size increases (see Figure 1).

Right Censored Data.
One of the most significant developments is due to a unique property of survival data in the life sciences: the data becomes incorrect when certain study participants have not experienced the event of interest at the conclusion of the research or at the time of analysis. Some patients may still be alive or disease-free at the end of the experiment. The subjects' survival time is unknown. This is referred to as censored observations or censored times when individuals are lost to follow-up following a research period.
Using data that has been randomly right censored, the following test statistic is provided to compare H 0 and H 1 .

Computational and Mathematical Methods in Medicine
Let us write the test statistic such as: We can tabulate the upper percentile points for b δ c ðs, βÞ in the same way as before Table 4.
The asymptotic normality of our test improves as the critical values decrease and the sample size increases (see Figure 2).

Applications
To demonstrate the utility of the conclusions in this study, we apply them to certain real-world datasets. Example 1. Take a look at the findings of Table 5 Mahmoud et al.; see [19]. The year-ordered values are as follows.
We calculate the statistic b δ c ð0:4, 5Þ = 1:86, which is greater than the critical value in Table 4. As a result, we infer that this set of data seems to have the UBA mgf characteristic property rather than the exponential. As a response, the treatment strategy chosen is significant.
For the other 51 autologous transplant patients (censored observations), the following are the results of leukemia free-survival times (in months) (see Table 7).
We calculate the statistic b δ c ð0:4, 5Þ = 76:06, which is greater than the critical value in Table 4. As a response, we infer that this set of data seems to have the UBA mgf characteristic property and not exponential.
According to the presented test, the two treatments given to 101 patients have a positive influence (IFR) on their survival lifetime, and so, the results extrapolated from the appropriate sample size can be applied to all patients with advanced acute myelogenous leukemia. However, in this scenario, the proposed test cannot evaluate two distinct treatments because the test results in both cases yielded the same conclusion or choice, with no priority given to selecting the most effective treatment.
Example 3. In this application, we use the data from Hassan [23] which reflect the ages (in days) of 51 liver cancer patients from the Elminia Cancer Center Ministry of Health Egypt (see Table 8), who entered the medical examination in the year 2000. In the investigation, only 39 patients are seen (right-censored), while the remaining 11 individuals are dropped (missing from the investigation).
Example 4. By analyzing the data reported by Abbas et al. [24], which shows the survival times in weeks of 61 individuals with inoperable lung cancer who were cured with cyclophosphamide, the patients whose therapy was terminated due to a devolving state are represented by 33 uncensored observations and 28 censored observations (see Table 9).

Conclusion
In this paper, a statistical test technique has been developed to aid in the quality evaluation of potential treatments for certain cancers. The results of our tests revealed whether the planned treatments had a positive or negative impact on the patients' survival periods. To ensure that the suggested statistical test produces good findings, its efficiency was computed and compared to existing tests. The proposed test can be used to evaluate the efficacy of any treatment approach in any sector of medical research, independent of the type of the treatment method being used. However, as seen in the second application, this test is not advised for comparing two different treatment strategies. But, it is suggested that new nonparametric statistical tests with high efficiency be developed and used to examine the various proposed treatments. It is also suggested that a statistical approach be developed to compare two or more different treatments that cure the same ailment. In addition, the percentage points of the proposed statistics are simulated. The efficacies of our developed tests are compared to El-Arishy et al. [17] and Abu-Youssef et al. [14] based on Pitman asymptotic relative efficiency using some well-known life distributions, namely, linear failure rate family (LFR) and Weibull family. Finally, the findings of the paper are applied to some medical real datasets.

IFR:
Increasing failure rate IFRA: Increasing failure rate average NBU: New better than used NB(W)UC: New better (worse) than used in a convex ordering NBRUL: New better than renewal used in Laplace transform ordering NBRU mgf : New better than renewal used in moment generating function NBU mgf : New better than used in moment generating function NBUCL: New better (worse) than used in a convex Laplace ordering UBA: Used better than age UBAC: Used better than age in convex order UBAC (2): Used better than age in concave order UBAL: Used better than age in Laplace transform UBA mgf : Used better than age in moment generating function RNBU mgf : Renewal new better than used in moment generating function.

Data Availability
The data was mentioned along the paper.

Conflicts of Interest
The authors declare there is no conflict of interest.