Abductive Analysis for Quantitative Variables

Conditions of Transparency and Methodological Implications

This scientific note is dedicated to the exploration of the fundamental conditions for the analysis of sets of quantitative variables, such as the economic Euro-parameters. The primary objective consists in determining the necessary and sufficient conditions for any reliable analysis, with particular emphasis on establishing the priorities and magnitude of interventions concerning location, dispersion, and correlation.

Definition of Transparent Variable and the Concept of Abnormality.

A quantitative variable is defined as "transparent" (according to Stoppa, 2024). "frank" and "informative" are considered synonyms. A variable is transparent when it can be interpreted, within its context, as a ranking of merit. Conversely, a variable that does not satisfy the criterion of transparency is qualified as "irregular". This distinction is of crucial importance, as this note proposes to demonstrate that the transparency of the variables involved in a given situation constitutes the necessary and sufficient condition for the validity of any reliable analysis.

Abductive Reasoning and the Explanation of Non-Transparency

An emblematic example of an abnormal variable is the Deficit (B) among the five Euro-27 parameters (GDP, Deficit, Debt, Inflation, and Employment). To explain the non-transparency of the Deficit, abductive reasoning (Peirce, 1977) is employed. It is postulated that, if an "extra" magnitude (E) existed, capable of explaining the abnormality of the Deficit (B), then the existence of E would be inferable. This inferential process, proceeding "backward" from effect to cause, suggests that E – presumably an external variable influencing the economic policy of countries – constitutes a valid explanatory hypothesis.

The validity of this hypothesis is subsequently verified through a second abductive reasoning. If the magnitude E, as an effect, is capable of distinguishing "progressive" countries (Deficit ≤ 0) from "conservative" countries (Deficit > 0), then the existence of E can be accepted. This empirical distinction confirms the necessity of transparency.

Implications of Transparency and Solution for Irregular Variables

The discovery of a possible influence of an external variable (E) reveals that its consequences manifest in two distinct types of countries, which precisely coincide with the two segments of the Deficit: progressives and conservatives.

Concomitantly, transparency is sufficient for a meaningful analysis. In the absence of transparency, the evaluation of the trend of other variables in relation to the ambiguous growth of the variable in question would lose its meaning. The Deficit, for instance, is ambiguous, being interpretable as a ranking of merit in one segment and of penalty in another. Consequently, it is meaningless to evaluate the trend of other variables in the presence of an ambiguous Deficit.

To resolve the problem of non-transparency, it is proposed to split the analysis, separating the 15 progressive countries from the 12 conservative countries. In this manner, the Deficit is transparent for the former, and acquires transparency for the latter by changing its sign. In summary, the necessary and sufficient conditions for a rigorous analysis of quantitative variables are twofold: the existence of a real situation and the transparency of the variables themselves.

Methodological Application and Analysis of Euro-Parameters

The analysis focuses on the principal statistical summaries: location, dispersion, and correlation, considered essential for guiding and quantifying interventions at the European level. For the year 2007, the zeta values ($\zeta_i$), obtained by standardizing the parameters (GDP, Deficit, Debt, Inflation, and Employment), are intrinsically transparent with respect to the context, both for progressive and for overall countries. For conservative countries, the Deficit requires a sign change to acquire transparency.

Medians and ranges are employed as measures of location and dispersion respectively. The medians of the standardized variables are comparable, as they all represent rankings of merit. The useful statistical summaries for the three groups (progressive, conservative, and overall) are available in Tables 1, 2, and 3 in the appendix.

For the assignment of weights, , proportional to the medians, , of the zeta points, the formula

is used, with a preference for

For inversely proportional weights, as in this specific case, the sign of the medians is inverted.

Intervention Priorities and Correlations

The intervention priorities based on location for the progressive and conservative groups differ significantly from those for the overall group. Similarly, the priorities related to dispersion vary among the groups. Specifically, the solution for the overall group is considered decidedly misleading. The most relevant partial correlations confirm the distinctions emerging from the disaggregated analysis. For example, for progressive countries, strong correlations are observed between GDP and Employment (0.83) and between Employment and Debt (0.71); for conservative countries, between GDP and Inflation (0.79) and a negative correlation between Debt and Inflation (-0.78). These contrast with the correlations observed for the overall data, such as Deficit and Employment (0.43) and Deficit and Debt (-0.42).

Conclusion

This note demonstrates, for the Euro-parameters, that, based on two distinct abductive reasonings, the presence of a non-transparent variable indicates the existence of an important two-level qualitative external variable, without which any analysis is invalidating, as verified.

Bibliographic references

Peirce C.S., 2023, in N. Bosco, 1977, Antologia degli scritti di C. S. Peirce. Giappichelli, Stoppa G., 2024, Blog: blog.gabriele.pro