² This is the basic limitation of all considerations of this paper. If we analyze the variables which have more than two values (“or continuous variables”) both the typology of their causal relations and the statistical relations corresponding to different types are quite different than in the case of dychotomic i.e. “yes or no” attributes.

³ T. Kotarbinski defines the cause as “the essential component of a sufficient condition” i.e., such a component without which a given set of events or variables cannot be a sufficient condition for the given effect. [T. Kotarbinski, Traktat o dobrej robocie (“Treatise on good work”), Warszawa 1957, p. 27.]

⁴ Or the equivalent: p a/b = 1, or: p b/A = 0, or: p A/b = 0. We use here the symbol A to indicate “the occurrence of A” and symbol a to indicate “the occurrence of non-A”. (the same with other “attributes” or “events” as B, C, S ...)

⁵ Or: p a/B = 0, or p A/B = 1, or: p b/a = 1.

⁶ Or more simply: .

⁷ We omit starting from here other equivalent formulas. Moreover we must introduce here another limitation of this consideration, i.e. that they are valid only for these causal relations between variables which are really acting in producing B in the population under analysis. If, for example, A is a necessary condition for B but the supplementary factor S forming together with A a sufficient condition for B does not occur, then, of course, formula IV is not true, because in this case p B/A = 0. The same thing holds for formula V.

⁸ Let us suppose that after introducing an additional predictor S₁ into the relationship AB and stating that the obtained probability p (S₁A → B) is greater than p (A → B) we are interested not only in the statistical aspect of the analysis (rising prediction) but we are also trying to answer the question whether S₁ is a “real” stimulus producing B or S₁ is only positively associated with an unknown “real” stimulus. Equation VIII is an effort to answer to this type of question.

⁹ To give a trivial example, we may say that the probability of death of a tubercular patient is usually higher than is determined by “pure mechanisms” of tuberculosis. It is also determined, e.g., by the probability of a tubercular patient's dying in a traffic accident. And when the “alternativeness” of causal relations is a rather general “phenomenon”, this means that we rather rarely have “pure one-cause” statistical relationships.

¹⁰ Here and in the following equation (XIV) we assume, for the sake of simplicity, that S₁, S₂, and A₂ are mutually exclusive.

¹¹ It would be more correct to say that relation AB is more direct than CB, instead of introducing dychotomic classification. It is obvious that for any causal relation we may find another one which is more direct than it.

¹² This is a rather rare case which might be called “an isolated chain of causal transformations” which is the fulfilling of Laplace's deterministic ideas (even when limited in time and space).

¹³ If we compare this with equation VIII, we may see that the difference between two links of the causal chain and two components of the same cause is not only terminological but is also expressed in different statistical relationships between variables. This differentiation will be the point of departure of our analysis of typology of “elaborations” proposed by P. Kendall and P. Lazarsfeld (see note 16, part II).

¹⁴ This also is true only on the assumption that A₁ and A₂ (or E and H) are mutually exclusive.

¹⁵ Let us give an example to clarify this scheme. Suppose that in order to fulfill a difficult combat task (B) at the time t ₃ it is necessary for a soldier at the time t ₂ to have both courage (C₂) and good orientation in the situation (A). But in order to be courageous at the time t ₂ it was necessary to be both courageous (C₁) and not shocked by the enemies' artillery (E) at the time t ₁1. And in order to be oriented in the situation (A) it was necessary to be both courageous enough (C₁) to look for information and to be in a favorable place where this information might be found (D) at the time t ₁.

¹⁶ The problems of elimination of spurious correlation was introduced into the methodology of social research by P. Kendall and P. Lazarsfeld in their essay “Problems of Survey Analysis” published in the volume: “Continuities in Social Research” (R. K. Merton and P. Lazarsfeld: eds. Glencoe 1950). Proposed in this paper typology of basic statistical operation for the causal interpretation was further developed by P. Lazarsfeld in the paper: “Interpretation of Statistical Relationships as a Research Operation” published in the Volume: “Language of Social Research” (P. Lazarsfeld and M. Rosenberg—eds. Glencoe 1955). See also: Herbert Hyman: Survey Design and Analysis. Glencoe 1955, and Herbert Simon's chapters on causality and spuriousness in his “Models of Man” N.Y. 1957.

¹⁷ See: P. Kendall and P. Lazarsfeld: “Problems of Survey Analysis”. According to the authors, the correlation is spurious when after introducing a “test variable” (C) being antecedent in the time order to the independent variable (A), we obtain the independence values in the partial correlation AB when C is held constant.

¹⁸ See: P. Lazarsfeld: “Interpretation of Statistical Relationships as a Research Operation”.

¹⁹ Let us come back to the example given above in note 9. Suppose that there exists a certain probability of death, for a tubercular patient, which is determined both by physiological mechanisms and by the actual situation in medical science at the present. This probability is now rather low. Suppose moreover that for a person there is a certain (now rather high) probability of his being killed in a traffic accident. If we then suppose that tuberculosis implies being in a sanatorium and (consequently) having a very low change of being killed in a traffic accident—then tuberculosis will be negatively correlated with death although nobody doubts their causal relations.

²⁰ This is obvious, because here will always have p B/A > p B/a. This implicit assumption seems to be the basis of the general practice of inferring causal propositions from correlational statements, but the assumption does not always have to be true, and it seems to be true only in rather rare situations.

²¹ See above: note 16. We have changed the symbols here in order to use symbols from the paper analyzed. Thus the effect is indicated by (y), the independent variable by (x) and the “test variable” by (t). (xt = 0) means that x and t are statistically independent, while (xt ≠ 0) means that between x and t there is a (positive or negative) correlation. The symbols (t;xy = 0) and (t;xy ≠ 0) refer to the existence or non existence of the partial correlation between x and y when t is held constant.

²² The following analysis is based on one assumption which is needed in order to avoid reduction ad infinitum although, (as we stated above) it does not always have to be true. The assumption concerns the relationships between the independent variable x and the test variable t: we assume namely that the mutual independence of x and t (tx = 0) is not a spurious independence but means that they are not related causally. We assume also that the correlation between t and x (tx ≠ 0) is not a spurious one but reflects their mutual causal connection. Of course, after finishing the three-variables (txy) analysis, the validity of these assumptions may be verified by applying another test variable (z) and then analyzing the relationships of the new set of variables, namely ztx.

²³ See above: note 11.

²⁴ Although this seems to be an artifical—i.e., purely logical possibility, relationships of this type were found, for example, in a recent survey on social attitudes of Warsaw students. It was found that the degree of anti-egalitarianism of students (y) is determined both by professional status of his family (t) and by the income of the family (x). We also found a strong correlation between t and x, which was fairly obvious because the profession of the father determined with high probability the income of the family. But when we introduced the scheme of partial correlation we found that the relationship x;ty did not disappear, nor did the relationship t;xy. Thus we were compelled to state that t determines y both directly and indirectly (by the intermediary of x). In substantive language, we found that the professional group of the family determines a student's egalitarianism both indirectly by determining his family's income and directly by being a student's reference group in this type of attitude.