Colorectal cancer has been ranked as the fourth most common cancer cause of death and the third most common cancer worldwide [1] [2] . Almost 694,000 deaths and about 1.4 million new cases occurred in 2012. The mortality of patients with metastatic colorectal cancer disease is very high. In point of fact, the necessity of a good and reliable screening method to detect colorectal cancer at an early operable stage is very important. Several techniques including biochemical tests for colorectal cancer (immunochemical FOBT, guaiac faecal occult blood test―gFOBT), sigmoidoscopy, colonoscopy, CT colonography, stool DNA, capsule endoscopy and other methods are used to detect even early stages of colorectal cancer. Colonoscopy is currently the most reliable method for detection of CRC. Sometimes, colonoscopy is uncomfortable for the patients, time consuming and very costly. The management of screen-detected or of symptomatic colon rectal cancer includes high-quality surgery, chemotherapy, adjuvant radiotherapy given either pre-operatively or post operatively and other measures. Very often the outcomes are unsatisfactory. The etiology of colorectal cancer is still not fully understood. Over the past few decades, a number of life style and environmental factors contributing to the occurrence of colorectal cancer have been identified, including low vegetable and fruit intake [3] and tobacco smoking [4] , family history of colorectal cancer [5] , high consumption of red and processed meat [6] , excessive alcohol consumption [7] , diabetes mellitus [8] , inflammatory bowel disease [9] , obesity [10] . However, results remain inconsistent and no single risk factor was identified as being responsible for colorectal cancer. The most of these factors at best confer a very moderate risk for colorectal cancer [11] . Numerous studies have aimed to provide evidence of the presence of infectious agents viral DNA such as human papillomaviruses (HPV), human polyomaviruses, human herpesviruses etc. [12] in colorectal tumor tissues but the evidence has remained inconclusive and is still very limited. The hypothesis that viral infections are involved in the etiology of colorectal cancer is of some public healthy relevance too. However, an impressive systematic review [13] of studies assessing the association between viral infections and colorectal cancer documented that very inconsistent results were observed across the studies analyzed. Overall, there is no published convincing evidence on the role of viral infections in colorectal cancer. Thus far, viral infections do not contribute to the etiology of colorectal cancer. The human intestinal microbiome encompasses at least 100 trillion (10¹⁴) microorganisms and is harbored by more than 1000 species. Some of these species of microorganisms bring about beneficial some other deleterious effects on the host and the gut microbiome is increasingly recognized as having an important role in human health and disease, including colorectal cancer [14] [15] [16] . Recent studies have shown that some harmful microbiota of the huge number of microbial communities which are continuously colonized in the gut may play roles in the development of colorectal cancer [17] [18] . On the whole, accumulating [19] evidence indicates that there is none relationship between a helicobacter pylori infection and colorectal cancer. Regarding the association between the gut microbiome and immunity, a number of studies have shown that Fusobacterium species are somehow related to colorectal cancer. In point of fact, it has been found in former studies that Fusobacterium species particularly Fusobacterium nucleatum as a leading gut bacterium are enriched in colorectal cancer compared to normal tissues or controls [20] - [26] . Fusobacterium species are part of the human oral [27] and intestinal microbiota. But it is still not clear if Fusobacterium nucleatum, an anaerobic gram-negative bacterium, plays an oncogenic role in the development of colorectal cancer.

A systematic literature search and review according to a predefined protocol in PubMed, Google scholar and other sites was conducted to identify relevant studies published while reporting follows the PRISMA statements as much as possible [28] [29] . A combination of different keywords like: review, bacterium, colorectal cancer, virus et cetera has been used in the search filed to search for eligible articles. In addition, the reference lists of the relevant articles including review articles was additionally used as a possible source for identifying studies related to the topic. Titles and abstracts of all identified articles were checked. Studies with potential relevance for the study topic underwent a review only if detailed data information could be extracted without any data access barriers.

Among some discrete distributions like the hypergeometric distribution, the Poisson distribution et cetera the binomial distribution is of special interest. Sometimes, the binomial distribution is called the Bernoulli distribution in honor of the Swiss mathematician Jakob Bernoulli (1654-1705), who derived the same. Bernoulli trials are an essential part of the Bernoulli distribution. Thus far, let us assume two fair coins named as ₀W_t and as _RU_t. In our model, heads of such a coin are considered as success T (i.e. true) and labeled as +1 while tails may be considered as failure F (i.e. false) and are labeled as +0. Such a coin is called a Bernoulli-Boole coin. The probability of success of _RU_t at one single Bernoulli trial t is denoted as

p ( U R t = + 1 ) ≡ p ( U R t ) (1)

The probability of failure of _RU_t at one single Bernoulli trial t is denoted as

p ( U R t = + 0 ) ≡ p ( U _ R t ) ≡ 1 − p ( U R t ) (2)

Furthermore, no matter how many times an experiment is repeated, let the probability of a head or the tail remain the same. The trials are independent which implies that no matter how many times an experiment is repeated, the probability of a single event at a single trial remain the same. Repeated independent trials which are determined by the characteristic that there are always only two possible outcomes, either +1 or +0 and that the probability of an event (outcome) remain the same at each single trial for all trials are called Bernoulli trials. The definition of Bernoulli trials provides a theoretical model which is of further use. However, in many practical applications, we may by confronted by circumstances which may be considered as approximately satisfying Bernoulli trials. Thus far, let us perform an experiment of tossing two fair coins simultaneously. Suppose two fair coins are tossed twice. Then there are 2² = 4 possible outcomes (the sample space), which may be shown as

( [ R U t = + 1 ] , [ 0 W t = + 1 ] ) , ( [ R U t = + 1 ] , [ 0 W _ t = + 0 ] ) , ( [ R U _ t = + 0 ] , [ 0 W t = + 1 ] ) , ( [ R U _ t = + 0 ] , [ 0 W _ t = + 0 ] )

This may also be shown as a 2-dimensional sample space in the form of a contingency table (Table 8).

In the following, the contingency table is defined more precisely (Table 9).

In general it is ( a + c ) = W 0 t , ( a + b ) = U R t , ( c + d ) = W _ 0 t , ( b + d ) = U _ R t and a + b + c + d = N = W R t . Equally, it is W 0 t + W _ 0 t = U R t + U _ R t = W R t = N . Thus far, if one fair coin is tossed n times, we have n repeated Bernoulli trials and an n dimensional sample space with 2ⁿ sample points is generated. In general, when given n Bernoulli trials with k successes, the probability to obtain exactly k successes in n Bernoulli trials is given by

p ( k ) = ( n k ) × p ( U R t = + 1 ) k × ( 1 − p ( U R t = + 1 ) ) n − k (3)

The random variable k is sometimes called a binomial variable. The probability to obtain k events or more (at least k events) in n trials is calculated as

p ( k ≥ X ) = p ( k = X ) + p ( k > X ) = ∑ k = X k = n ( ( n k ) × p ( U R t = + 1 ) k × ( 1 − p ( U R t = + 1 ) ) n − k ) (4)

The probability to obtain less than k events in n Bernoulli trials is calculated as

p ( k < X ) = 1 − p ( k ≥ X ) = 1 − ∑ k = X k = n ( ( n k ) × p ( U R t = + 1 ) k × ( 1 − p ( U R t = + 1 ) ) n − k ) (5)

The formula of the conditio per quam [36] - [54] relationship was derived as

p ( Fusobaceriumnucleatum → Colorectalcancer ) ≡ a + c + d N (6)

and used to proof the hypothesis: if presence of Fusobacterium nucleatum infection then presence of colorectal cancer.

The formula of the conditio per quam [36] - [54] relationship was derived as

p ( Fusobaceriumnucleatum ← Colorectalcancer ) ≡ a + b + d N (7)

and used to proof the hypothesis: without presence Fusobacterium nucleatum no presence of colorectal cancer.

The necessary and sufficient condition relationship was defined [36] - [54] as

p ( Fusobaceriumnucleatum ← → Colorectalcancer ) ≡ a + d N (8)

Scholium.

Historically, the notion sufficient condition is known since thousands of years. Many authors testified original contributions of the notion material implication only for Diodorus Cronus. Still, Philo the Logician (~300 BC), a member of a group of early Hellenistic philosophers (the Dialectical school), is the main forerunner of the notion material implication and has made some groundbreaking contributions [55] to the basics of this relationship. As it turns out, it is very hard to think of the “conditio per quam” relationship without considering the historical background of this concept. Remarkable as it is, Philo’s concept of the material implications came very close to that of modern concept material implication. In propositional logic, a conditional is generally symbolized as “p ® q” or in spoken language “if p then q”. Both q and p are statements, with q the consequent and p the antecedent. Many times, the logical relation between the consequent and the antecedent is called a material implication. In general, a conditional “if p then q” is false only if p is true and q is false otherwise, in the three other possible combinations, the conditional is always true. In other words, to say that p is a sufficient condition for q is to say that the presence of p guarantees the presence of q. In particular, it is impossible to have p without q. If p is present, then q must be present too. To show that p is not sufficient for q, we come up with cases where p is present but q is not. It is well-known that the notion of a necessary condition can be used in defining what a sufficient condition is (and vice versa). In general, p is a necessary condition for q if it is impossible to have q without p. In fact, the absence of p guarantees the absence of q. Example (Condition: Our earth). Without oxygen, no fire. The following table (Table 10) may demonstrate this relationship.

In contrast to such a point of view, the opposite point of view is correct too. Thus far, there is a straightforward way to give a precise and comprehensive

account of the meaning of the term necessary or sufficient condition itself. In other words, if fire is present then oxygen is present too. The following table (Table 11) may demonstrate this relationship.

Especially, necessary and sufficient conditions are converses of each other. Still, the fire is not the cause of oxygen and vice versa. Oxygen is note the cause of fire. In this example before, oxygen is a necessary condition, a conditio sine qua non, of fire. A necessary condition is sometimes also called “an essential condition” or a conditio sine qua non. In propositional logic, a necessary condition, a condition sine qua non, is generally symbolized as “p ¬ q” or in spoken language “without p no q”. Both q and p are statements, with p the antecedent and q the consequent. To show that p is not a necessary condition for q, it is necessary to find an event or circumstances where q is present (i.e. an illness) but p (i.e. a risk factor) is not. On any view, (classical) logic has as one of its goals to characterize the most basic, the most simple and the most general laws of objective reality. Especially, in classical logic, the notions of necessary conditions, of sufficient conditions of necessary and sufficient conditions et cetera are defined very precisely for a single event, for a single Bernoulli trial t. In point of fact, no matter how many times an experiment is repeated, the relationship of the conditio sine qua or of the conditio per quam which is defined for every single event will remain the same. Under conditions of independent trials this implies that no matter how many times an experiment is repeated, the probability of the conditio sine qua or of the conditio per quam of a single event at a single trial t remain the same which transfers the relationship of the conditio sine qua or of the conditio per quam et cetera into the sphere of (Bio-) statistics. Consequently, (Bio) statistics generalizes the notions of a sufficient or of a necessary condition from one single Bernoulli trial to N Bernoulli trials. However, in many practical applications, we may by confronted by circumstances which may be considered as approximately satisfying the notions of a sufficient or of a necessary condition. Thus far, under these circumstances, we will need to perform some tests to investigate, can we rely on our investigation.

Many times, for some reason or other it is not possible to study exhaustively a whole population. Still, sometimes it is possible to draw a sample from such a population which itself can be studied in detail and used to convince us about the properties of the population. Roughly speaking, statistical inference derived from a randomly selected subset of a population (a sample) can lead to erroneous results. The question raised is how to deal with the uncertainty inherent in such results? The concept of confidence intervals, closely related to statistical significance testing, was formulated to provide an answer to this problem.

Confidence intervals, introduced to statistics by Jerzy Neyman in a paper published in 1937 [56] , specifies a range within a parameter, i.e. the population proportion p, with a certain probability, contain the desired parameter value. Most commonly, the 95% confidence interval is used. Interpreting a confidence interval involves a couple of important but subtle issues. In general, a 95% confidence interval for the value of a random number means that there is a 95% probability that the “true” value of the value of a random number is within the interval. Confidence intervals for proportions or a population mean of random variables which are not normally distributed in the population can be constructed while relying on the central limit theorem as long as the sample sizes and counts are big enough (i.e. a sample size of n = 30 and more). A formula, justified by the central limit theorem, is known as

p C r i t = p C a l c ± ( z A l p h a / 2 × ( 1 N × p C a l c × ( 1 − p C a l c ) 2 ) ) (9)

where p_Calc is the sample proportion of successes in a Bernoulli trial process with N trials yielding X successes and N-X failures and z is i.e. the 1 - (Alpha/2) quantile of a standard normal distribution corresponding to the significance level alpha. For example, for a 95% confidence level alpha = 0.05 and z is z = 1.96. A very common technique for calculating binomial confidence intervals was published by Clopper-Pearson [57] . Agresti-Coull proposed another different method [58] for calculating binomial confidence intervals. A faster and an alternative way to determine the lower and upper “exact” confidence interval is justified by the F distribution [59] .

Furthermore, an approximate and conservative (one sided) confidence interval was developed by Louis [60] , Hanley et al. [61] and Jovanovic [62] known as the rule of three. Briefly sketched, the rule of three can be derived from the binomial model. Let p_U denote the upper limit of the one-sided 100 × (1 − α)% confidence interval for the unknown proportion when in N independent trials no events occur [62] . Then p_U is the value such that

π U = ( − ln ( α ) n ) ≈ ( 3 n ) (10)

assuming that α = 0.05. In other words, an one-sided approximate upper 95% confidence bound for the true binomial population proportion p, the rate of occurrences in the population, based on a sample of size n where no successes are observed (p = 0) is 3/n [62] or given approximately by [ 0 < π < ( 3 / n ) ] . The rule of three is a useful tool especially in the analysis of medical studies. The following table (Table 12) will illustrate this relationship.

Under conditions where a certain event did not occur [60] in a sample with n subjects (i.e. p = 0) the interval from 0 to (−ln(α)/n) is called a 100 × (1 − α)% confidence interval for the binomial parameter for the rate of occurrences in the population.

Another special case of the binomial distribution is based on a sample of size n where only successes are observed (p = 1). Accordingly, the lower limit of a one-sided 100 × (1 − α)% confidence interval for a binomial probability p_L, the rate of occurrences in the population, based on a sample of size n where only successes are observed is given approximately by [ ( 1 − ( − ln α / n ) ) < p < + 1 ] or (assuming α = 0.05)

π L = 1 − ( − ln ( α ) n ) ≈ 1 − ( 3 n ) (11)

The following table (Table 13) may illustrate this relationship.

To construct a two-sided 100 × (1 - (α))% interval according to the rule of three, it is necessary to take a one-sided 100 × (1 − (α/2))% confidence interval. In this study, we will use the rule of three [63] too, to calculate the confidence interval for the value of a random number.

A test statistics of independent and more or less normally distributed data which