The FRAT-up risk-assessment algorithm is based on probability contributions from single risk factors. In the following, we show how we extract probabilities from odds ratios by means of a few mathematical steps.

Initially, we assume that each risk factor is dichotomous; we explain this further in the section on how to generalize to cases with scalar and synergy risk factors. Let E ₀, E ₁,…, E _n be n + 1 dichotomous random variables with values in {0;1}, and E=(E ₀, E ₁,…, E _n ). We say that the i ^th risk factor is present if E _i =1. Let d ₀, d ₁,…, d _n be n + 1 events. We assume the following conditional independence relations:

Equation 1: d _i | E _i ⊥ d _j , E _j ∀j ≠ i

We call d _i a fall event specific to risk factor E _i . Assumptions from Equation 1 can be phrased saying that risk factor–specific falls are mutually independent conditional on their associated risk factor. We define the event d as the union of the factor-specific events, d _i ’s (Figure 2). That is, d is verified if at least one of the d _i ’s is verified. This is an assumption of causal independence where the “causes”, E ₀, E ₁,…, E _n, contribute independently to the probability of the effect d; for a complete formal definition see [40]. In our case study, d is the presence of at least one fall event during a given time span (if there is no fall, it is not verified), while E is an observation of the risk factor exposures of a subject before the time span.

The conditional probability of d given E can then be calculated as in Figure 3, by De Morgan laws and assumptions in Equation 1. This function models the probability of an event given a set of possible causes and is known as noisy-OR gate [41] (in this case OR refers to the logical operator). We make the assumption in Figure 4. C _i is a quantity yet to determine. C _i is the contribution to the probability of the effect d given by the exposure to the risk factor E _i . A method to assign values to the contributions C _i is introduced in the following. Using the equation in Figure 4, the equation in Figure 3 becomes the one depicted in Figure 5. Since we want to model a minimum probability of the adverse event that is applied even in the absence of any observation-specific exposures, we assign P(E ₀=1)=1. C ₀ is the risk that is present in this case. To assign values to the contributions of the exposures, we start from the OR. The OR relative to risk factor E _i , with i=1,…, n, is defined as in Figure 6. Note that the condition E ₀=1 is always true and is highlighted above just for convenience.

There is no single way of translating odds ratios to probabilities, since an exact function would require more information than what is conveyed by the odds ratios alone, so some assumptions are needed. We present a possible set of assumptions that leads to a univocal way of computing exposure contributions.

We assume that OR _i may be approximated as in Figure 7. Informally, Assumption (a) states that the odds ratio computed on the whole population is similar to the odds ratio computed restricting the population to subjects having at most one exposure. This assumption is obviously true in models where each subject has at most one exposure; otherwise there is a difference in the two values. This has not been quantified yet; the quality of the approximation will be experimentally compared with other methods as a future development.

Given the assumptions in Equation 1 and Figure 4, the derivation depicted in Figure 8 follows. Substituting the equation in Figure 8 in the equation in Figure 7 and solving for C _i , we finally get the equation depicted in Figure 9. We substitute it in the equation in Figure 5, with a result that is depicted in Figure 10.

We assume to know C ₀, which was calculated by leaving it as a free parameter and then learning it with an equation-solving algorithm. In particular, we used the bisection method, imposing the reported number of total falls from [1].

This model requires that we know for every risk factor if it is present or not. In the following section, we present the way FRAT-up deals with missing values.

For a general reference on how to get relative risk from odds ratio and the incidence of the outcome of interest in the unexposed group, see [42].

LPADs are logic programs [43] where the head of a clause is a disjunction of annotated atoms. The clauses are of the form:

h ₁: p ₁v … v h _n : p _n ← b ₁∧ … ∧ b _m ∧ ⌉c ₁∧ … ∧ ⌉c _l

where h ₁,…, h _n are the atoms, and p ₁,…, p _n are the probabilities related to each disjunct. Each atom h _i has probability p _i if the body is true, and the atom does not appear in the head of any other clause. When it does, the intended semantics are the distribution semantics as in [44], with the bodies contributing independently to the probability of the atom [40]. The probabilities p ₁,…, p _n should sum up to 1, with an implicit “null” atom when the explicit probabilities sum up to less than 1.

Roughly speaking, for each clause containing a disjunction in its head, different instances are generated, each containing the clause with exactly one disjunct. The probability of a query would be given by the sum of all the probabilities of the instances whose models contain it.

We adopt the syntax of the cplint [45] implementation. Note that the disjunction in the head of clauses is indicated with the symbol “;”, while the conjunction is indicated as usual in Prolog with “,”. The equation in Figure 5 can be easily implemented with LPAD rules (Code 1 LPAD template with computed fall probability contributions):

fall(X) : c0.

fall(X) : c1 :- e1(X).

fall(X) : c2 :- e2(X).

...

Where c0≡C ₀, c1≡ C ₁, e1(X)≡ (E ₁=1), c2≡ C ₂, e2(X)≡ (E ₂=1) ...

The assessment tool should provide reliable information even when part of the subject’s data is missing. Missing values may arise when a test has not been (or cannot be) performed or the involved clinical professional does not consider its outcome decisive and reliable. In these cases, we have used the prevalence of the risk factors extending Code 1 as follows:

fall(X) : c0.

e1(X) : p1 :- u1(X).

fall(X) : c1 :- e1(X).

e2(X) : p2 :- u2(X).

fall(X) : c2 :- e2(X).

...

where u1(X), u2(X)… is true when the existence of the factor 1, 2… for subject X is not determined.

A scalar factor, with exposure levels from 0 (no exposure) to m (maximum exposure), is implemented similarly to a set of m dichotomous factors, one for each exposure level starting from level 1. The LPAD rule related to level k fires if the scalar risk factor has a level of k or higher.

Positive synergies (eg, comorbidities) between risk factors are well documented in the scientific literature. Since this would violate the causal independence assumption made before, we adjusted the model, following the Deandrea meta-analysis [29], introducing synergy factors.

A synergy factor, representing the potential synergies between S dichotomous risk factors, is implemented similarly to a scalar risk factor having a maximum possible level of S - 1 where, having a number of exposures equal to q, with 0 ≤ q ≤ S, the level is 0 if q=0 v q=1 and is q - 1 otherwise. So the risk starts increasing when there is a synergy between at least two factors.

The methodology that leads from risk factor odds ratio to LPAD rules is fully automatized. A working prototype has been produced and tested in the Java programming language (version 1.7); it may read risk factor odds ratios from an ontology or another source and outputs an LPAD program directly usable for risk assessment.

Synthetically (see Figure 11), risk factors data complete with odds ratio are read from an ontology or other data source; a data structure containing odds ratios is created and then transformed (by means of the equation in Figure 9) in another containing probability values. Finally LPAD rules are compiled: these rules are applied to a subject to give their probability of falling in a given time span.

FRAT-up discriminative performance and calibration have been tested on the InCHIANTI dataset (NCT01331512), where 1453 persons have been initially enrolled (1150 subjects aged 65 or more) and have undergone four consecutive visits globally covering a 9-year follow-up. It is a population-based epidemiologic study conducted in the Chianti region of Italy in two sites: Greve in Chianti (Area 1; 11,709 inhabitants; >65 years: 19.3%) and Bagno a Ripoli (Village of Antella, Area 2, 4704 inhabitants; >65 years: 20.3%). This study investigates age-related decline in mobility [46].

The InCHIANTI study started in September 1998 with the baseline assessment (first wave), which was completed in March 2000. Every 3 years, a follow-up assessment was performed. So, 3-year and 6-year follow-up assessments were performed respectively in 2001-2003 and 2004-2006 (second and third wave). A 9-year follow-up was then performed in 2007-2009 (fourth wave). The fifth wave is now ongoing.

At each wave, subjects were asked about the occurrence of any fall in the previous 12 months. In addition, clinical evaluation of the subjects was performed to collect information on fall-risk factors (other clinical variables were also collected, which are not of interest for this work [46]).

Our study used the information about risk factors from the first three waves, considering only subjects aged 65 or up. By doing so, we obtained 2319 samples from 977 subjects (every subject can have up to three samples).

At each wave, the risk factors of each subject were used prospectively to calculate their risk of falling at the subsequent wave (eg, the risk factors from the clinical evaluation at baseline were used to calculate the future risk of falling, which was compared with the recorded information on the occurrence of any falls in the 12 months before follow-up 1, and so on).

The estimators present in the InCHIANTI dataset and the algorithms to derive the risk factors from them are listed in Multimedia Appendix 1.

The discriminative ability and calibration of FRAT-up were validated by means of receiver operating characteristic (ROC) curve, area under the ROC curve (AUC), Brier score, and the Hosmer-Lemeshow test [47]. Since FRAT-up requires no training of the algorithm based on the available data, these metrics were computed by using all the available data as the test set.