When assessing an individual, a clinician is more interested in the symptoms the individual presents rather than in the score he/she obtains on an instrument. The FPA produces an attempt to provide the clinician with exhaustive information about a patient’s specific set of clinical issues, to identify the situation in a both quantitatively and qualitatively rigorous way. What allows the FPA to carry out this critical task are its mathematical and theoretical foundations. In fact, the FPA is a conjoint application into the clinical framework of two theories from mathematical psychology: The Knowledge Space Theory [10, 11], and formal concept analysis [12]. A very trivial intuition behind the FPA’s conceptualization is the possibility of describing each item of an instrument in terms of the clinical-diagnostic issues it explores [8, 13]. In this way, it is possible to identify which clinical issues are investigated by any item of an assessment tool for some specific disorder. In the FPA, each item is named a clinical object, while each clinical-diagnostic issue (e.g., DSM diagnostic criteria) is named a clinical attribute. Starting from these basic definitions it is possible to build a matrix where each row is an object and each column is an attribute. Each cell ij of such a matrix, named clinical context, contains either a 1, whenever the item i investigates the attribute j, or a 0 otherwise. The clinical context displays and represents the object-attribute assignment. Notice that, once the set of attributes is specified, any item can be analysed with respect to the presence/absence of each attribute; this means that the same matrix could even contain items from different tools. Furthermore, this representation allows the clinician to discriminate between response patterns having the same numerical score. This aspect is particularly relevant because a patient could reach a given score by satisfying different sets of symptoms: each item is therefore important not for the score it provides, but for the attributes it investigates. This possibility is completely lost in traditional tools, and it represents one of the most relevant improvements provided by the FPA that differentiates patients not based on their score, but on the clinical symptoms they present.

Starting from the clinical context, through some formal passages [13] a clinical structure can be built. This structure uses the information included in the context to derive the prerequisite relations among the items of the context. This passage is crucial in the construction of an adaptive algorithm for psychological assessment [7]. In fact, the structure defines the admissible response patterns (also named the clinical states) that are defined by the item-attribute assignment of the context. For instance, if an item i₁ investigates attributes {a₁, a₂} and a second item i₂ investigates attributes {a₁, a₂, a₃}, any pattern including item i₂ but not i₁ will be excluded from the structure given the item-attribute assignment.

So far, the deterministic skeleton of the FPA has been described. In the deterministic case no error is assumed during the assessment phase; in other words, the answer to a specific item is assumed not to be affected by any kind of error, and therefore the collected information is treated as certain. It is clear how such an assumption does not adequately describe reality for two main reasons. On the one hand, two main errors could occur in the assessment phase; for instance, in the case of the ADOS-2, the clinician could report a certain behaviour even if it did not occur, or could miss a behaviour that did occur. These two kinds of errors are respectively a “false positive” and a “false negative” [8, 11]. On the other hand, the clinical states (i.e., the different clinical conditions in which a patient could be) could present different prevalence rates in the population. For these reasons, the introduction of a probabilistic model is needed. In the FPA, a model that accounts for both error rates and probability of the states is the Basic Local Independence Model (BLIM) [10]. Within the BLIM, all the responses to the items are locally independent given the state of the individual; moreover, a probability value is associated with each state. Formally, the clinical structure becomes a probabilistic clinical structure (Q, , p) where (Q,) is the clinical structure and p is a probability distribution of the states of . Given (Q,, p), each response pattern has a probability value: This last one is defined by means of a response function which assigns each response pattern its conditional probability given a state K (for all states K in [13, 14]:

The conditional probability p (R, K) is determined by the two error parameters β and η, respectively the false negative and false positive errors related to each item, as expressed by the following equation:

The model parameters are the two error rates η_q and β_q for each item, and a probability value π_K for each state. In this way, it is possible to estimate the probability of observing the real clinical state of a patient.

The first step in applying the FPA is the construction of the clinical context that depicts the item/attribute assignment. This operation first requires the specification of the attributes that must be included in the context. Recalling that the clinical attributes can be symptoms, diagnostic criteria or theoretical considerations related to the disorder at hand, this operation has been conducted by referring to the DSM-5 (APA, 2013) and other information from the literature [1, 6]. Table 1 lists the main attributes considered when assessing ASD symptoms.

The second operation conducted to build the context was the selection of the items. In the current application, all items of the ADOS-2 were included in the context. To have a clearer picture of the instrument, the five main parts of the tool were analysed separately. Therefore, the items of the ADOS-2 were subdivided into five different contexts (one for each part). Finally, six experts (four psychologists and two child psychiatrists) in the field of ASD and in the administration of the ADOS-2 independently performed the item/attribute assignment. Disagreements among raters were solved through direct discussion about the specific criticisms.

Whenever a clinical context is built, five important and informative configurations may occur due to the item-attribute assignment. First, some empty rows may occur (i.e., rows of the matrix that contain only 0s), meaning that the specific item does not investigate any of the selected attributes. Second, some empty columns may occur, indicating that none of the items of the tool investigate that specific attribute. This is crucial because it indicates that the assessment tool at hand does not investigate some of the selected (and thus necessary) attributes. Third, equal columns may occur (i.e., columns having 1s in the same positions) indicating that it is impossible to distinguish between the presence/absence of either of the equivalent attributes in the assessment. Fourth, equivalent rows may occur, which indicate equivalent items (i.e., items that convey, from a clinical perspective, the same information). These items are redundant and thus, in terms of efficiency, could be collapsed. Fifth, some rows are present that are included in other rows (i.e., the 1s in one row are all present in the other row that also has some other 1s). This last case is important because the included items represent so-called prerequisites of the included ones. In other words, in this case there is one item that investigates some attributes and another item that investigates all the attributes of the first item, plus some other attributes. Therefore, from a clinical perspective, an affirmative answer to this last item implies an affirmative answer to the first one as well.

The last step of the application of the FPA is the statistical testing of the robustness of the model in describing the collected data. To this aim, the BLIM parameters have been estimated starting from a dataset of patients evaluated by means of the ADOS-2. Parameter estimation has been iteratively carried out based on the Expectation-Maximization (EM) algorithm [9, 13, 15]. Through this procedure, we were able to obtain an estimate of the two error parameters for each item (false negative and false positive) and the probability of each clinical state in the structure. The procedure computes a fit statistic based on Pearson’s Chi-square; this statistic, due to the sparseness of this kind of data matrix, is not reliable. Therefore, the obtained value has been tested by means of a parametric bootstrap (with 5000 replications) to check for the reliability of the obtained value. Of course, the estimate of the error rates for each item can be viewed as fit indices. It has been observed [9] that, in general, such values must be as low as possible. More precisely, the following condition must hold true for every item: η_q < 1- β_q. In other words, a positive answer to an item is more likely to occur if the patient has its attributes, rather than as the result of a false positive on the item. In the next section, the results obtained from the construction and analysis of the contexts are displayed.

The BLIM was defined and tested on the collected data.

The clinical structure is the formal and graphical representation of the relations among the items of the context and, furthermore, it can display the relations between sets of items and sets of attributes. It is usually represented by a graph in which every node is a collection of items (i.e., the clinical state) and the corresponding set of attributes (i.e., the set of criteria satisfied by a patient having that clinical state). The structure is built according to both the relations among items and the attributes depicted in the clinical context and the prerequisite relations depicted by the same context. Some of the crucial configurations regarding the rows and columns of the context have precise counterparts in the structure. For instance, an empty row (i.e., an item that does not investigate any of the selected attributes) will result in the absence of that item in the structure. The same happens to empty columns. When two rows are equivalent, the corresponding items will be included systematically in the same states. Finally, the prerequisite relation occurring whenever the attributes investigated by an item i are a subset of those investigated by j in terms of states in the structure means that there will not be any state including j that does not also include i.

The clinical structure of both the T module and module 1 were tested using the BLIM. The fit of the model parameters was estimated by means of the EM procedure [9, 15]. To test the reliability of the observed Chi-square, a parametric bootstrap with 5000 replications was conducted for each of the two modules [13]. Concerning the T module, the results showed that the model adequately fit the data (x²(260,952) = 757.04; p > .05; number of states = 1,156). The reliability of this result was confirmed by the parametric bootstrap (bootstrap p > .05). Moreover, the estimates of the error rates for the T module, reported in Table 7, satisfy the condition of acceptability (i.e., β_q + η_q < 1) for every item.

On the other hand, for module 1, the model fit the data quite well (x²(65, 280) = 17, 585; p > .05; number of states = 224) even though the reliability of the observed results is not supported by the bootstrap (bootstrap p > .05). Nonetheless, as can be seen in Table 8, even for this module the error rate estimates are acceptable for every item.

In general, it can be concluded that the two models accurately explain the observed data. The estimates can then be implemented into an algorithm constituting an adaptive tool for the assessment of the diagnostic issues for ASD. Such a tool is derived from the ADOS-2 by removing its redundancies and focusing on the specific issues useful for diagnosis.