The 2 × 2 table in this article is defined [15] - [37] in general more precisely (Table 3) as follows.

In general it is ( a t + b t ) = A t , _{( c t + d t ) = A _ t} , ( a t + c t ) = B t , ( b t + d t ) = B _ t and a t + b t + c t + d t = N t . Equally, it is B t + B _ t = A t + A _ t = N t . In this context, it is p ( a t ) = p ( A t ∩ B t ) , p ( A t ) = p ( a t ) + p ( b t ) or in other words p ( A t ) = p ( A t ∩ B t ) + p ( b t ) = p ( A t ∩ B t ) + p ( A t ∩ B _ t ) while p(A_t) is not defined as p(a_t). In the same context, it is p ( B t ) = p ( a t ) + p ( c t ) = p ( A t ∩ B t ) + p ( c t ) and equally that p ( B _ t ) = 1 − p ( B t ) = p ( b t ) + p ( d t ) . Furthermore, the joint probability of A_t and B_t is denoted by p ( A t ∩ B t ) . Thus far, it is p ( A t ∩ B t ) = p ( A t ) − p ( b t ) = p ( B t ) − p ( c t ) or in other words it follows that p ( B t ) + p ( b t ) − p ( c t ) = p ( A t ) . Define Λ = p ( b t ) − p ( c t ) , the famous Einstein’s term under conditions of probability theory and we obtain p ( B t ) + Λ = p ( A t ) . In general, it is p ( a t ) + p ( c t ) + p ( b t ) + p ( d t ) = 1 . These relationships are viewed by the table (Table 4) as follows.

In the case of independence of A_t and B_t it is

p ( A t ∩ B t ) ≡ p ( A t ) × p ( B t ) (1)

The mathematical formula of the necessary condition relationship (conditio sine quam non) [15] - [37] of a population was defined as

p ( A t ← B t ) ≡ a t + b t + d t N t ≡ + 1 (2)

and used to proof the hypothesis: without A_t no B_t . In particular it is

p ( A t ← B t ) ≡ p ( a t ) + p ( b t ) + p ( d t ) p ( A t ← B t ) ≡ p ( A t ∩ B t ) + p ( B _ t ) p ( A t ← B t ) ≡ p ( A t ∩ B t ) + ( 1 − p ( B t ) ) p ( A t ← B t ) ≡ + 1 (3)

Scholium.

The study design and other factors can have an impact on bias with respect to the necessary condition. A different question worth asking concerns the relationship between the independence of an event A_t (a condition) and another event B_t (conditioned) and the necessary condition relationship. A fundamental question worth considering at this stage is whether is it possible that an event A_t is a necessary condition of event B_t an even if the event A_t (a necessary condition) is independent of an event B_t (the conditioned). In this context, the conditio sine qua non was defined as

p ( A t ← B t ) ≡ p ( A t ∩ B t ) + p ( B _ t ) ≡ + 1 (4)

or as

p ( A t ← B t ) ≡ p ( A t ∩ B t ) + ( 1 − p ( B t ) ) ≡ + 1 (5)

Under conditions where an event A_t is independent of an even B_t it is equally true that

p ( A t ∩ B t ) ≡ p ( A t ) × p ( B t ) (6)

Rearranging equation before it is

p ( A t ) × p ( B t ) + ( 1 − p ( B t ) ) ≡ + 1 (7)

or

p ( A t ) × p ( B t ) ≡ p ( B t ) (8)

or

p ( A t ) ≡ + 1. (9)

Only under conditions where p(A_t) = 1, theoretically it is possible to treat A_t as a necessary condition of B_t even if A_t is independent of B_t and vice versa, otherwise not. In other words, it is very difficult to treat a statistically significant conditio sine qua non relationship as very convincing if at the same time an event A_t is independent of and event B_t and vice versa. While discussing the statistical significance of results with respect to a necessary condition (or a sufficient condition or a necessary and sufficient condition), such or similar arguments should be considered. Due to an inappropriate study design or other sources of possible bias, the statistical significance of a conditio sine qua non relationship should be treated with very great cautious if evidence is provided that at the same time the same investigated parameters are independent of each other.

Under some circumstances, the rule three and other methods can be used to test the significance of a necessary condition. In this publication, the chi-square [38] goodness of fit test was used to determine whether sample data are consistent with a hypothesized (theoretical) distribution of a necessary condition. In particular, the hypotheses can take the following form.

H₀: The sample distribution does agree with the hypothetical (theoretical) distribution of a necessary condition.

H_A: The sample distribution does not agree with the hypothetical (theoretical) distribution of a necessary condition.

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

χ 2 ≡ ∑ t = + 1 t = + N ( ( Observed t − Expected t ) 2 Expected t ) (10)

The degrees of freedom are calculated as N − 1. Interestingly, if there is no discrepancy between an observed and a theoretical distribution at all, then the value of the calculated X² = 0. As the discrepancy between an observed and the theoretical distribution of a necessary condition becomes larger, the X² becomes larger. This X² values are evaluated by the known X² distribution. An adjustment (Yate’s correction for continuity) can be used 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 ) (11)

Under circumstances, where the term (|Observed_t − Expected_t|) is less than 1/2, the continuity correction should be omitted. The theoretical (hypothetical) distribution of a necessary condition is shown schematically by the 2 × 2 table (Table 5).

The theoretical distribution of a necessary condition (conditio sine qua non) is determined by the fact that c_t = 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 ) ) (12)

or more simplified as

χ 2 ( SINE ) ≡ ( ( | − c | − ( 1 2 ) ) 2 ( c + d ) ) + 0 (13)

Under these circumstances, the degree of freedom is d.f. = N − 1 = 2 − 1 = 1 . The conditio sine qua non model can be used widely and is one of the new and appropriate methods of analysis of binary outcome variables. In this context, meta-analysis and systematic reviews aims to combine effects estimated from several studies to achieve greater precision of the conclusions drawn and can provide us with more convincing and reliable evidence of some special aspects of medicine. In meta-analysis the heterogeneity between the studies can be modelled via the additive properties of the chi square distribution too. In general, let X_t denote n independent random variables which follow a chi-square distribution. The sum of these independent chi-square variates is itself a chi-square variate which is known as the additive property of independent chi-squares. There may be disadvantages in the use of the chi-square-goodness-of-fit test. Still, the chi square distribution, a continuous probability distribution, is related to the standard normal distribution and is a simple and good measure of model adequacy. However, a particular concern with the use of the chi-square-goodness-of-fit test is a priori justified if expected cell frequencies of a 2 × 2 table are too small (all are less than one).

Huxley [38] and Darwin [39] claimed more than a century ago that humans share recent common ancestors with the African apes. Modern molecular methods have spectacularly confirmed their prediction. Genomic divergences between humans and other hominoids and especially our closest living evolutionary relatives the common chimpanzee (Pan troglodytes) and bonobo (Pan paniscus or pygmy chimpanzee) are very small but not zero. Ebersberger et al. [40] , Fujiyama et al. [41] and other sequenced the chimpanzee genome. According to Ebersberger et al. “the chimpanzee genome were sequenced and compared to corresponding human DNA sequences ... the average sequence difference is low (1.24%)” [40] . The Chimpanzee Sequencing and Analysis Consortium calculated “the genome-wide nucleotide divergence between human and chimpanzee to be 1.23%” [42] and confirmed results from other and more limited studies. In other words, the difference between chimpanzee genome and compared to corresponding human DNA sequences is very small. Still there is a difference and this very small difference makes the difference. A chimpanzee is not a human being, a human being is not a chimpanzee. Even if both are similar and “relatives” both are equally not the same. The relationship between the mathematical formula of the causal relationship k [15] - [37] and the closest existing mathematical relatives, Pearson’s measures of relationships, is similar to the circumstances aforementioned. In contrast to Pearson’s product-moment correlation coefficient [43] or to Pearson’s Phi [44] Coefficient (Mean Square Contingency Coefficient et cetera), the mathematical formula of the causal relationship k [15] - [37] is defined at every single event, at every single Bernoulli trial t, as

k ( U R t , W 0 t ) ≡ ( p ( U R t × W 0 t ) − ( p ( U R t ) × p ( W 0 t ) ) ) ( p ( U R t ) × p ( U _ R t ) ) × ( p ( W 0 ) × p ( W _ 0 t ) ) 2 (14)

where _RU_t denotes the cause and ₀W_t denotes the effect while the chi-square distribution [45] can be applied to determine the significance of causal relationship k. This small difference makes the difference. Only under conditions where the probability of events is constant from trial to trial, we can extrapolate from one Bernoulli trial to N Bernoulli trials with some consequences one of which is that

k ( U R t , W 0 t ) ≡ N × N × ( p ( U R t × W 0 t ) − ( p ( U R t ) × p ( W 0 t ) ) ) N × N × ( p ( U R t ) × p ( U _ R t ) ) × ( p ( W 0 ) × p ( W _ 0 t ) ) 2 (15)

or that

k ( U R t , W 0 t ) ≡ ( N × N × p ( U R t × W 0 t ) − ( N × p ( U R t ) × N × p ( W 0 t ) ) ) ( N × p ( U R t ) × N × p ( U _ R t ) ) × ( N × p ( W 0 ) × N × p ( W _ 0 t ) ) 2 (16)

or at the end

k ( U R t , W 0 t ) ≡ ( ( N × a t ) − ( U R t × W 0 t ) ) ( U R t × U _ R t ) × ( W 0 × W _ 0 t ) 2 (17)

where N is the sample size, a t = N × p ( U R t ∩ W 0 t ) , R U t = N × p ( R U t ) , R U _ t = N × p ( R U _ t ) , 0 W t = N × p ( 0 W t ) , 0 W _ t = N × p ( 0 W _ t ) . Several factors can have an impact on the calculated causal relationship k with the potential of bias.

Scholium.

Firstly, the relationship between condition and cause has an impact on the causal relationship k. A proper and deeper analysis of the relationship between cause and condition is beyond the scope of this article and can be found in literature [15] - [37] . We will be concerned with the latter sort of entity in this article from a pragmatically point of view. In the hope of casting light on the tricky problems of the relationship between condition and cause, the concept of independence is of use too. The question whether an event A_t can be a (necessary, sufficient, necessary and sufficient) condition of an event B_t even if both are independent of each other, is already answered few lines before. Still, under which circumstances can we treat an event as a cause or as the cause of another event? Can an event be a cause of another event without being a (necessary, sufficient, necessary and sufficient et cetera) condition of the same event? The concept of this article is restricted on its capacity to bring high degrees of conceptual exactness and rigour to questions like these but not incapable. Most authors who have written on the question of the relationship between condition and cause came to different conclusions. Currently still worthy of consideration is the remark of von Bar.

“Die erste Voraussetzung, welche erforderlich ist, damit eine Erscheinung als die Ursache einer anderen bezeichnet werden könne, ist, daß jene eine der Bedingungen dieser sein. Würde die zweite Erscheinung auch dann eingetreten sein, wenn die erste nicht vorhanden war, so ist sie in keinem Falle Bedingung und noch weniger Ursache. Wo immer eine Kausalzusammenhang behauptet wird, da muß er wenigstens diese Probe aushalten… Jede Ursache ist nothwendig auch eine Bedingung eines Ereignisses; aber nicht jede Bedingung ist Ursache zu nennen.” [46]

Translated into English:

“The first requirement, which is required, thus that something could be called as the cause of another, is that the one has to be one of the conditions of the other. If the second something had occurred even if the first one did not exist, so it is by no means a condition and still less a cause. Wherever a causal relationship is claimed, the same must at least withstand this test… Every cause is necessarily also a condition of an event too; but not every condition is cause too.”

From this statement, it could appear that there is a gap between what is a cause and what is a condition. Is it possible to generalize this finding? In probabilistic approaches to causation, it is obvious that a cause of an event is equally a condition of the same event too. Clearly, the same relationship must not be given the other way too. A condition of an event must not equally be a cause of the same event. In summary, the objections build on the contradiction between condition and cause are no longer justified. A cause is a condition of an event too but not necessarily vice versa. A condition of an event must not be equally the cause of the same event. Thus far, like other fundamental concepts, the concepts of necessary conditions, the concepts of sufficient conditions and the concepts of necessary and sufficient conditions can be one of the handy tools to determine precisely whether a causal relationship is significant or not. A study which provides evidence of a significant causal relationship k without at the same time providing evidence of a significant necessary condition, or of a significant sufficient condition or of a significant necessary and sufficient condition should be treated with some cautious.

Secondly, a proper study design is necessary to use the mathematical formula of the causal relationship k with confidence otherwise bias is possible. For example, the probabilities of two events within a population are know precisely and shown schematically by the 2 × 2 table (Table 6).

The causal relationship k (Table 6) is calculated as k = +0.314800094. Using the conditional probabilities, we obtain the following picture (Table 7).

Now we perform a study A with a sample size of n = 10,000. The verum group is n Verum = n / 2 = 5000 and the placebo group is n Placebo = n / 2 = 5000 . We do expect that the probabilities within the sample are the same like in the population. Under these conditions we obtain the following picture (Table 8).

Thus far, this study has provided the following data (Table 9).

Calculating the causal relationship k under these conditions, we obtain

k ( U R t , W 0 t ) ≡ ( ( ( ( 10000 ) × ( 4955 ) ) ) − ( ( ( 5000 ) × ( 4955 ) ) ) ) ( ( 4955 ) × ( 5045 ) ) × ( ( 5000 ) × ( 5000 ) ) 2 = + 0 .991031209 (18)

a causal relationship k of k = +0.991031209. The study with the sample size n = 10,000 should have obtained a causal relationship k = +0.314800094 while the same obtained a causal relationship k = +0.991031209. This example demonstrates that the study design as such can be a source of bias if inappropriate measures are taken. To reduce the bias, it makes sense to consider the prevalence of a factor/an event within the population as much as possible as an essential part of study design. According to the data above, the prevalence of a cause within the population is p(_RU_t) = 0.999. The sample size of the study is still n = 10,000. Under these assumptions, the sample size of the verum group should be considered as n Verum = p ( R U t ) × n = 0.999 × 10000 = 9900 while the sample size of the placebo group is n Placebo = p ( R U _ t ) × n = 0.001 × 10000 = 1 (Table 10).

The causal relationship k (Table 10) is calculated as k = +0.314800094. The situation does not really change if a case control study is regarded. If the same problem is analysed within a case control approach, this should not have any systematic influence on the probabilities (Table 11) of the sample studied.

Let us assume that the sample size of this case control study B is n = 1000, with 500 cases and 500 controls. We obtain the following data (Table 12).

The data of this case control study (Table 12) are demanding that the causal relationship k is equal to k = +0.229415734 while the same causal relationship k should be equal to k = +0.314800094 what motivates us to affirm the following conclusion. Inappropriate study design can lead to severe bias. Given the difficulty of the problems as associated with study design it is useful to adopt a strategy of extreme caution under conditions when the data of a study provide evidence of a significant cause effect relationship but fails in the same respect to provide some evidence of a significant necessary condition relationship or of a significant sufficient condition relationship or of a significant necessary and sufficient condition relationship otherwise conclusions may run into difficulties.

The chi-squared distribution [38] 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 13.

Under some circumstances the chi-square [45] goodness of fit test can be used to

test the significance of a causal relationship. Under conditions where the probability of events is constant from trial to trial, we expect a constant causal relationship k_t. In other words, at each Bernoulli trial t it is

| k ( U R t , W 0 t ) | ≡ | 1 | (19)

Performing N Bernoulli trials (Sample size N), the basic relationship will not change. It follows that

N × | k ( U R t , W 0 t ) | ≡ N × | 1 | (20)

or that

N × | k ( U R t , W 0 t ) | − N × | 1 | = 0 (21)

Simplifying equation we obtain

N × ( | k ( U R t , W 0 t ) | − | 1 | ) = 0 (22)

Multiplying equation by itself it is

N × ( | k ( U R t , W 0 t ) | − | 1 | ) × N × ( | k ( U R t , W 0 t ) | − | 1 | ) = 0 × 0 (23)

or

N 2 × ( | k ( U R t , W 0 t ) | − | 1 | ) 2 = 0 (24)

Dividing equation by N * |1| = N, we obtain

N 2 × ( | k ( U R t , W 0 t ) | − | 1 | ) 2 N = 0 N = 0 (25)

or

N × ( | k ( U R t , W 0 t ) | − | 1 | ) 2 = 0 (26)

or the X² value as

χ 2 = N × ( | k ( U R t , W 0 t ) | − | 1 | ) 2 = 0 (27)

The chi square (X²) statistic can be used to investigate whether the observed distribution of the causal relationship differ from the theoretical expected distribution of the causal relationship. The table (Table 8) contains the critical values of the chi-square distribution (degrees of freedom, df = 1). Upper-tail and lower-tail critical values of the chi-square distribution with v degrees of freedom are provided by software packages.