This is not an analysis focused on predicting age at death as a function of dietary food groups and/or other determinants. The purpose is only to see whether using dietary food groups with a numerical approach different from the traditional one yields similar findings. Age at death is simply used as the endpoint chosen for this purpose.

In the traditional way to search for a diet, a dietary score, or similar tools consisted of defining a diet from the available data and then seeing whether it is associated with good or bad outcomes of any kind of events in the study population, a procedure that we arbitrarily call the “direct” approach. The definition of a dietary score can be made a priori, that is, defining its composition using the available food groups and/or nutrients, on the basis of the opinions of the investigators, or a posteriori, assigning this task to computer procedures and models that take this decision on the basis of statistical interplays and procedures, largely independent from the investigators’ opinions. Then, the selected diet is statistically associated with the cohort events using predictive models to determine whether the diet, or subclasses of the diet, are related to the occurrence of events and/or with different probabilities.

Actually, it has been shown that a priori and a posteriori procedures perform in similar ways [1], although the latter is used very rarely. These have been the traditional approaches that have identified several diets or dietary scores, such as the Mediterranean Diet [2] and several others. The number of these proposals is rather large. We may just mention some of them as examples, like the WHO diet proposal [3], the Mediterranean diet pyramid [4], the Healthy Eating Index [5], the DASH dietary rules against hypertension [6], some major Greek scores [7,8], the Mediterranean Adequacy Index [9], two reports from the EPIC project [10,11], the NIH-AARP Diet [12], an Italian a-posteriori index [13], the Alternative Healthy Eating Index [14], and many more. In a review of a priori Mediterranean scores, 30 were mentioned and commented on [15]. All these dietary scores, following various statistical approaches and variables, such as the length of follow-up, have shown their ability to predict all-cause mortality risk, often indicating that a diet rich in vegetables and fish is protective. In contrast, a diet rich in animal foods and sugar products does increase the risk. Clearly, many other papers adopting different choices reported the role of dietary habits in relation to all-cause mortality, although none reached the cohort's extinction. Actually, the literature is full of contributions of this type, and mention of many others would be redundant and useless, because describing their accomplishments has nothing to do with the main purpose of the present analysis, which deals only with technical issues.

We were stimulated to try a different approach, that is, an “inverse” approach, to validate the conclusions of studies conducted with the “direct” approach. This means to identify, in a population study, a subgroup of individuals who, after a sizable period of time, were exempt from a defined endpoint or carrying the disease or anyhow in a better or worse shape than others in terms of disease or death, and to describe their diet based on actual food groups used by people with versus those without events or with different ones.

Both “direct” and “inverse” approaches require the availability of dietary analysis at baseline and of events at the end of follow-up. The difference is the direction of the study, i.e. in the “direct” approach the diet goes to search for events, while in the “inverse” approach the procedure goes the other so that the events go back to search for their original diets (Figure 1). Reviewing randomly some major studies of this type, we had the impression that such an “inverse” approach was not previously considered systematically, although it is quite difficult to assure that this is true.

This analysis had technical purposes, and the meaning focused on the attempt to see whether a numerical approach, different from the traditional one in studying the relationship of diet with disease, may reach similar conclusions. The concept of a “direct” dietary approach means adopting the characteristics of a dietary pattern and applying them to a given population, with the (probabilistic) aim of determining whether they may predict different risk levels for a given morbidity or other events. The “inverse” dietary approach, on the opposite side, starts from a population sample and measures the dietary characteristics among groups carrying or not carrying a given morbidity or outcome. In the latter case, the approach is descriptive and not probabilistic.

The material of this analysis was already largely exploited in terms of “direct approach” [18,19,20,21]. For example, a contribution in 2022 [20] provided a multiple linear regression with AD as the endpoint and including as predictors the same (“direct”) dietary score plus 32 other variables, including social data, traditional risk factors, behavioral lifestyle habits, clinical signs, and several major prevalent diseases. The dietary score, subdivided into 3 tertiles, showed strong predictive power in the presence of many potential confounders, but without multicollinearity. The study population in this analysis is highly specific, as it is limited to middle-aged men from rural Italy in the 1960s and excludes women. From this point of view, it is unknown whether the conclusions would extend to women, different age groups, different cultural and socio-economic situations, and nowadays lifestyles. The cohort size was also quite limited, though partly compensated for by the extremely long follow-up, which reached extinction and therefore allowed coverage of the entire life span of AD. We do not claim any particular utility of this analysis, since it started as an attempt to answer a curiosity dealing with a methodological problem. The analysis seems unique since we could not find in the current literature other analyses using AD as an endpoint in an extinct population sample.

In general, our impression is that the “inverse” approach and its relationships with the “direct” one should not depend on the characteristics mentioned above, but this must be confirmed by further analyses. In addition, using larger denominators of the study population, it might be possible to replicate this analysis using a single disease as an endpoint, if it represents a great proportion of morbid/fatal events. Among other further tests, it would be interesting to replicate the comparison of “direct” versus “inverse” approaches using artificially created data files.

Findings of the procedure defined as “direct” approach show nothing new compared with previous analyses of the same material, except the beneficial role of olive oil that could be tested here for the first time since we were able to dissociate it from seed oils. Distributing the several dietary variables into 3 AD classes was the job of the “inverse” procedure, which offered a new vision of the problem. Out of 19 food groups, 15 were significantly different across the 3 tertiles of factor score in the “direct” approach, while only 7 of them did so in the 3 AD classes by the “inverse” approach. It should be again clarified that the “direct” approach needs the construction of a dietary score. Still, there is no difference between the a priori and a posteriori procedures; the only difference is the greater difficulty in using the latter. Moreover, these two procedures likely do not interfere with the “direct” versus the “inverse” approaches analyzed in the present contribution.

Apparently, there are no contradictions between these approaches, but the “inverse” approach seems more selective despite the use of the same data, simply handled and distributed in a different way. The “direct” approach is, in fact, based on probabilistic procedures, while the “inverse” one is more oriented towards a descriptive procedure. From a different perspective, at least in our experience, the “direct” approach is more sensible since it selects more variables (food groups). In comparison, the “inverse” approach is more specific, selecting fewer variables with large weights in association with the events, which points to the core of the problem. The most evident example concerns the beneficial intake of olive oil, which is clearly greater in the “inverse” than in the “direct” approach. Nevertheless, some of the differences could be attributed to the interference of the arbitrary definition of the 3 levels of dietary habits based on the PCA factor score subdivided into equal-numerical groups derived from its rank distribution, and to the arbitrary classification of AD into 3 classes in the “inverse” approach. Actually, the latter derives from the common way of classifying a person’s age, as adult (41-69 years), old (70-79 years), and old-old (80-104 years). Only after that, we discovered that the 3 classes had a rather similar size, almost corresponding to the 3 tertiles of the distribution, a condition that may represent an advantage.

Actually, the “direct” approach creates a predictive model that can be applied to other populations. In contrast, the “inverse” approach simply describes some facts, namely the characteristics of the dietary habits of individuals who carry or do not carry an event. As a consequence, its major role is to confirm the value of the traditional “direct” approach. This “inverse” approach cannot be compared with other similar experiences, as they are simply absent in the current literature. For the moment, we can only say that it tends to confirm the findings of the traditional “direct” approach, clearly viewed from an opposite point of view.

The idea to run the “inverse” approach partly comes from areas such as engineering, statistics, and mathematics, where an “inverse problem” is an inference or estimation problem [23,24]. The data are finite in number and contain errors, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The data are only indirectly related to the unknown, an additional complication in an “inverse problem”. Although the optimal determination of a model from a finite data set which generalizes Backus-Gilbert theory for nonlinear effects may enable to evaluate to what degree the reconstructed model resembles the true model [24], so far, a simple and clear procedure to compare the performance of the performed approaches, “direct” versus “inverse”, was not found and much more work is needed to find an acceptable solution. This extends to the present analysis in the sense that a formal comparison between our “direct” versus “inverse” approaches cannot be adequately performed.

The innovative “inverse” procedure proposed here for showing the relationship between a dietary habit and an event (AD in this case) produced findings comparable with the traditional procedure obtained by a “direct” approach, despite some small differences. More work is needed to demonstrate its real value. For the moment, we preliminarily conclude that the “inverse” approach could serve as a confirmatory procedure. It is too soon to say whether other important advantages exist. Testing on larger, more diverse population samples is needed to confirm their apparent roles. However, using AD as an endpoint will likely be much more difficult, as studies examining cohort extinction are extremely rare. However, large groups of coherent morbid conditions might be enough for the purpose, and certainly other disciplines from engineering, statistics, and mathematics may contribute to formal comparisons and individualization of optimal pre-test conditions for useful reciprocal applications.