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

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

This may also be shown as a 2-dimensional sample space in the form of a contingency table (Table 3).

In the following, the contingency table is defined more precisely (Table 4).

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 [22] - [35] relationship was derived as

p ( EBV ​ DNA → Hodgkin ’ slymphoma ) ≡ a + c + d N (6)

and used to proof the hypothesis: if presence of EBV infection (EBV DNA) then presence of Hodgkin’s lymphoma.

The formula of the conditio per quam [22] - [35] relationship was derived as

p ( EBV ​ DNA ← Hodgkin ’ slymphoma ) ≡ a + b + d N (7)

and used to proof the hypothesis: without presence of EBV infection (EBV DNA) no presence of Hodgkin’s lymphoma.

The necessary and sufficient condition relationship was defined [22] - [35] as

p ( EBV ​ DNA ← → Hodgkin ’ slymphoma ) ≡ 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 [36] 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. Table 5 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. Table 6 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 [37] , specifies a range within a parameter, i.e. the population proportion π, 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 Alpha / 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 [38] . Agresti-Coull proposed another different method [39] 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 [40] .

Furthermore, an approximate and conservative (one sided) confidence interval was developed by Louis [41] , Hanley et al. [42] and Jovanovic [43] known as the rule of three. Briefly sketched, the rule of three can be derived from the binomial model. Let π_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 [43] . Then π_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 π, the rate of occurrences in the population, based on a sample of size n where no successes are observed (p = 0) is 3/n [43] or given approximately by [0 < π < (3/n)]. The rule of three is a useful tool especially in the analysis of medical studies. Table 7 will illustrate this relationship.

Under conditions where a certain event did not occur [41] 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 π_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)) < π < +1] or (assuming α = 0.05)

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

Table 8 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 [44] 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 follow a chi-squared distribution is valid as with many statistical tests due to the central limit theorem. Especially, with large samples, a chi-squared distribution can be used. A sample is considered as large when the sample size n is n = 30 or more. With a small sample (n < 30), the central limit theorem does not apply and erroneous results could potentially be obtained from the few observations if the same is applied. Thus far, when the number of observations obtained from a population is too small, a more appropriate test for of analysis of categorical data i.e. contingency tables is R. A. Fisher’s exact test [45] . Fisher’s exact test is valid for all sample sizes and calculates the significance of the p-value (i.e. the deviation from a null hypothesis) exactly even if in practice it is employed when sample size is small. Fisher’s exact test is called exact because the same uses the exact hypergeometric distribution to compute the p-value rather than the approximate chi-square distribution. Still, computations involved in Fisher’s exact test can be time consuming to calculate by hand.

The hypergeometric distribution, illustrated in a table (Table 9), is a discrete probability distribution which describes the probability of a events/successes in a sample with the size ₀W_t, without replacement, from a finite population of the size N which contains exactly _RU_t objects with a certain feature while each event is either a success or a failure. The formula for the hypergeometric distribution, a discrete probability distribution, is

p ( a ) = ( U R t a ) × ( N − U R t W 0 t − a ) ( N W 0 t ) (12)

The hypergeometric distribution has a wide range of applications. The Hypergeometric distribution can be approximated by a Binomial distribution. The elements of the population being sampled are classified into one of two mutually exclusive categories: either conditio sine qua non or no conditio sine qua non relationship. We are sampling without replacement from a finite population. How probable is it to draw specific c events/successes out of ₀W_t total draws from an aforementioned population of the size N? The hypergeometric distribution, as shown in a table (Table 10) is of use to calculate how probable is it to obtain c = (₀W_t − a) events out of N events.

A statistical hypothesis test is a method to extract some inferences from data. A hypothesis is compared as an alternative hypothesis. Under which conditions does the outcomes of a study lead to a rejection of the null hypothesis for a pre-specified level of significance. According to the rules of a proof by contradiction, a null hypothesis (H₀) is a statement which one seeks to disproof. The related specific alternative hypothesis (H_A) is opposed to the null hypothesis such that if null hypothesis (H₀) is true, the alternative hypothesis (H_A) is false and vice versa. If the alternative hypothesis (HA) is true then the null hypothesis (H₀) is false. In principle, a null hypothesis that is true can be rejected (type I error) which lead us to falsely infer the existence of something which is not given. The significance level, also denoted as α (alpha) is the probability of rejecting a null hypothesis when the same is true. A type II error is given, if we falsely infer the absence of something which in reality is given. A null hypothesis can be false but a statistical test may fail to reject such a false null hypothesis. The probability of accepting a null hypothesis when the same is false (type II error), is denoted by the Greek letter β (beta) and related to the power of a test (which equals 1 − β). The power of a test indicates the probability by which the test correctly rejects the null hypothesis (H₀) when a specific alternative hypothesis (H_A) is true. Most investigator assess the power of a tests using 1 − β = 0.80 as a standard for adequacy. A tabularized relation between truth/falseness of the null hypothesis and outcomes of the test are shown precisely within a table (Table 11).

In general, it is 1 − α + α = 1 or (1 − α − β) + α = 1− β. Figure 1 may illustrate these relationships.

The mathematical formula of the causal relationship k [22] - [35] defined as

k ( U R t , W 0 t ) ≡ ( ( N × a ) − ( U R t × W 0 t ) ) ( U R t × U _ R t ) × ( W 0 × W _ 0 t ) 2 (13)

and the chi-square distribution [46] were applied to determine the significance of causal relationship between a EBV and HL. A one-tailed test makes it much easier to reject a null hypothesis (no causal relationship) while a two-tailed test makes it more difficult to reject a null hypothesis and is more conservative on this account. For this reason, in causal relationship testing, a two-tailed test is preferred. In general, a p value of less than 0.05 is considered as significant. In this context, what is the necessary connection between a cause and effect? What

ties a cause and its own effect together? Is there a necessary connection between a cause and effect at all? Theoretically, it is neither justified nor necessary to reduce causation as such to an act of observation or measurement. Sill, case-control studies, experiments, observations et cetera can help us to recognize cause effect relationships. In this context it is necessary to stress out that every single event (effect) has its own cause, which is the logical foundation of the mathematical formula of the causal relationship k. It is therefore entirely clear that this is the fundamental difference to Pearson’s methodological approach. Obviously, although under some certain specified circumstances Pearson’s product-moment correlation coefficient [47] or Pearson’s Phi [48] coefficient can yield the same numerical result as the mathematical formula of the causal relationship k, there is nothing truly exciting about such a coincidence. Nevertheless, when conducting experiments and analyzing data, views in which correlation and causation are brought very close together are incorrect and worthless. The mathematical formula of the causal relationship k is neither identical nor can the same mathematical formula be reduced to Pearson’s product-moment correlation coefficient [47] or to Pearson’s Phi [48] Coefficient (Mean Square Contingency Coefficient). In contrast to Pearson’s product-moment correlation coefficient and to Pearson’s Phi Coefficient (Mean Square Contingency Coefficient) the mathematical formula of the causal relationship k is defined and valid at every single Bernoulli trial t or at every single event. Sir Austin Bradford Hill (1897-1991), an English epidemiologist, proposed 1965 a set of nine criteria (Strength, Consistency, Specificity, Temporality, Biological gradient, Plausibility, Coherence, Experiment, Analogy) [49] to establish epidemiologic evidence of a causal relationship (Bradford Hill criteria). In point of fact, Bredford’s “fourth characteristic is the temporal relationship of the association” [49] and in last consequence the “post hoc ergo propter hoc” logical fallacy. Causation cannot be derived from the “post hoc ergo propter hoc” [35] logical fallacy. Consequently, the Mathematical Formula of the causal relationship k can neither be reduced to the Bradford Hill criteria nor is the same just a mathematization of Bradford Hill criteria.

The chi-squared distribution [46] is a widely known distribution and used in hypothesis testing, in inferential statistics or in construction of confidence intervals. The critical values of the chi square distribution are visualized by Table 12.

A chi-square goodness of fit test can be applied to determine whether sample data are consistent with a hypothesized distribution. The chi-square goodness of fit test is appropriate when some conditions are met. A view of these conditions are simple random sampling, categorical variables and an expected value of the number of sample observations which is at least 5. The null hypothesis (H₀) and its own alternative hypothesis (H_A) are stated in such a way that they are mutually exclusive. In point of fact, if the null hypothesis (H₀) is true, the other, alternative hypothesis (H_A), must be false; and vice versa. For a chi-square goodness of fit test, the hypotheses can take the following form.

H₀: The sample distribution agrees with the hypothetical (theoretical) distribution.

H_A: The sample distribution does not agree with the hypothetical (theoretical) distribution.

The X² Goodness-of-Fit Test can be shown schematically as

χ 2 ≡ ∑ t = + 1 t = + N ( ( Observed t − Expected t ) 2 Expected t ) (14)

The degrees of freedom are calculated as N − 1. If there is no discrepancy between an observed and a theoretical distribution, then X² = 0. As the discrepancy between an observed and a theoretical distribution becomes larger, the X² becomes larger. This X² values are evaluated by the known X² distribution.

The original X² values are calculated from an original theoretical distribution, which is continuous, whereas the approximation by the X² Goodness of fit test we are using is discrete. Thus far, there is a tendency to underestimate the probability, which means that the number of rejections of the null hypothesis can increase too much and must be corrected downward. Such an adjustment (Yate’s correction for continuity) is used only when there is one degree of freedom. When there is more than one degree of freedom, the same adjustment is not used. Applying this to the formula above, we find the X² Goodness-of-Fit Test with continuity correction shown schematically as

χ 2 ≡ ∑ t = + 1 t = + N ( ( | Observed t − Expected t | − ( 1 2 ) ) 2 Expected t ) (15)

When the term (|Observed_t − Expected_t|) is less than 1/2, the continuity correction should be omitted.

The theoretical (hypothetical) distribution of a sufficient condition is shown schematically by the 2 × 2 table (Table 13).

The theoretical distribution of a sufficient condition (conditio pre quam) is determined by the fact that b = 0. The X² Goodness-of-Fit Test with continuity correction of a sufficient condition (conditio per quam) is calculated as

χ 2 ( IMP ) ≡ ( ( | a − ( a + b ) | − ( 1 2 ) ) 2 ( a + b ) ) + ( ( | ( c + d ) − ( c + d ) | − ( 1 2 ) ) 2 ( c + d ) ) = ( ( | a − ( a + b ) | − ( 1 2 ) ) 2 ( a + b ) ) + 0 (16)

or more simplified as

χ 2 ( IMP ) ≡ ( ( | − b | − ( 1 2 ) ) 2 ( a + b ) ) + 0 (17)

Under these circumstances, the degree of freedom is d.f. = N − 1 = 2 − 1 = 1 .

The theoretical (hypothetical) distribution of a necessary condition is shown schematically by the 2 × 2 table (Table 14).

The theoretical distribution of a necessary condition (conditio sine qua non) is determined by the fact that c = 0. The X² Goodness-of-Fit Test with continuity correction of a necessary condition (conditio sine qua non) is calculated as

χ 2 ( SINE ) ≡ ( ( | ( a + b ) − ( a + b ) | − ( 1 2 ) ) 2 ( a + b ) ) + ( ( | ( d ) − ( c + d ) | − ( 1 2 ) ) 2 ( c + d ) ) = 0 + ( ( | d − ( c + d ) | − ( 1 2 ) ) 2 ( c + d ) ) (18)

or more simplified as

χ 2 ( SINE ) ≡ ( ( | − c | − ( 1 2 ) ) 2 ( c + d ) ) + 0 (19)

Under these circumstances, the degree of freedom is d.f. = N − 1 = 2 − 1 = 1 .

The theoretical (hypothetical) distribution of a necessary and sufficient condition is shown schematically by the 2 × 2 table (Table 15).

The theoretical distribution of a necessary and sufficient condition is determined by the fact that b = 0 and that c = 0. The X² Goodness-of-Fit Test with continuity correction of a necessary and sufficient condition is calculated as

χ 2 ( NecessaryANDSufficient ) ≡ ( ( | ( a ) − ( a + b ) | − ( 1 2 ) ) 2 ( a + b ) ) + ( ( | ( d ) − ( c + d ) | − ( 1 2 ) ) 2 ( c + d ) ) (20)

or more simplified as

χ 2 ( NecessaryANDSufficient ) ≡ ( ( | − b | − ( 1 2 ) ) 2 ( a + b ) ) + ( ( | − c | − ( 1 2 ) ) 2 ( c + d ) ) (21)

Under these circumstances, the degree of freedom is d.f. = N − 1 = 2 − 1 = 1 .