Three-Tier Phenomenological Assessment System

• The three-tier phenomenological assessment system is a systematic framework that decomposes clinical judgment into qualitative, quantitative, and evaluation-based analyses.
• It integrates subjective clinician judgments with mathematically validated 3WD theory, ensuring transparent and reproducible diagnostic insights.
• The framework partitions candidate disorders into high, medium, or low tiers, enhancing clinical decision-making alongside conventional methods.

A three-tier phenomenological assessment system is a structured methodology for psychiatric diagnosis that formalizes the clinicians’ subjective approach (CSA) using three-way decision (3WD) theory. The system decomposes clinical judgment into three interdependent analytic layers—qualitative analysis, quantitative analysis, and evaluation-based analysis—and systematically integrates their outputs to partition candidate mental disorders into three clinically meaningful strata. This framework is intended as a transparent, complementary adjunct to manual-based methods such as DSM/ICD in mental disorder classification (Wang et al., 2022).

1. Structural Components: Qualitative, Quantitative, and Evaluation-Based Analysis

The assessment system operationalizes CSA by delineating it into three analytic modules:

1. Qualitative Analysis: Encodes expert pairwise comparisons among disorders as a binary “preference” relation, imposing the trichotomy property (for each pair, a strict preference, indifference, or the inverse must hold) and transitivity (preferences propagate). The objective is to generate a total, partial, or weak order over the set of candidate disorders.
2. Quantitative Analysis: Assigns a real-valued weight to each disorder. Weights are computed either through an eigenvector-derived procedure applied to a positive reciprocal comparison matrix (clustered in a hierarchical fashion to mitigate inconsistency for large sets) or via mapping to a clinician-calibrated ordinal importance scale that is itself weighted by pairwise comparison.
3. Evaluation-Based Analysis: Transforms the resulting ranked list or weight vector into a tripartite partition—high, medium, or low importance—by trisecting the values with two adaptively chosen thresholds. These tiers correspond to clinically actionable decision regions: “accept,” “defer,” or “reject.”

This modular separation enables both transparent articulation of clinician subjectivity and systematic integration with algorithmic triage.

2. Mathematical Framework and Decision Logic

Let $D = \{d_1, d_2, \ldots, d_n\}$ be the set of disorders under consideration.

2.1 Pairwise Preferences and Ranking Generation

For each $(d_i, d_j)$ pair, clinicians express $d_i \gg d_j$ (strict preference), $d_i \sim d_j$ (indifference/tie), or $d_i \ll d_j$ (inverse preference). The preference relation $\gg$ imposes:

• Trichotomy: Exactly one relation holds for any pair.
• Transitivity: $d_i \gg d_j$ and $d_j \gg d_k \implies d_i \gg d_k$.

This yields either a total order or, in the presence of incompleteness/ties, a weak order. When needed, quantitative ranking is derived via the Evaluation Status Value (ESV):

$v(d) = \frac{|\{ x \in D \mid d \gg x \}|}{|D|}$

The ESV facilitates ordering even in non-total preference structures.

2.2 Eigenvector-Based Quantitative Weighting

For sets $n > 9$, disorders are clustered hierarchically to at most nine clusters per level. A comparison matrix $M = [m_{ij}]$ is constructed:

• $m_{ij} > 0$, $m_{ij} = 1/m_{ji}$, representing relative clinical importance.
• The principal eigenvector $w$ satisfies $M w = \lambda_{\max} w$.
• Consistency is evaluated as $C.R. = \frac{\lambda_{\max}-n}{(n-1) R.I.}$, where $R.I.$ is the random index for $n$.
• $C.R. < 0.10$ is required for admissibility; weights are normalized to $\sum w_i = 1$.
• The process recurses through the hierarchy, resulting in a unique $w(d)$ for each $d$.

2.3 Importance Scale Assignment

Alternatively, clinicians select $K$ qualitative intensity levels (e.g., “significantly matched” to “not matched”). Pairwise comparison and eigenvector analysis on the intensity labels yield weights $s_1, \ldots, s_K$, with the same $C.R.$ consistency requirement. Each disorder is mapped to the corresponding weight based on its assigned qualitative class.

2.4 Trisection into Three Tiers

Given a weight vector $\{w(d_i)\}$, introduce thresholds $h, \ell$ such that:

• $H = \{ d \mid w(d) \ge h \}$
• $M = \{ d \mid \ell < w(d) < h \}$
• $L = \{ d \mid w(d) \le \ell \}$

Thresholds are chosen either by:

• Percentile method: For $0 < \beta < \alpha < 100$, $h = v_{\lceil \alpha n / 100 \rceil}$, $\ell = v_{\lfloor \beta n/100 \rfloor}$.
• Mean and standard deviation: $h = \mu + k_1 \sigma$, $\ell = \mu - k_2 \sigma$ for selected $k_1, k_2 \ge 0$.

3. Clinical Tiers: Interpretation and Action Mapping

After trisection, each disorder is classified as:

• High (H): Disorders with strong subjective evidence, meriting immediate investigation or action.
• Medium (M): Disorders with moderate suspicion, for which further data gathering or deferral is indicated.
• Low (L): Disorders unlikely on current evidence, to be rejected or deferred.

This mapping renders the gradation of clinical suspicion explicit and actionable within the diagnostic workflow.

4. Consistency Validation and Empirical Demonstration

Throughout each phase, consistency of pairwise comparisons is monitored via the computed $C.R.$ values. In all reported numerical examples, $C.R.$ remains well below the 10% threshold (often below 1%), demonstrating the practical feasibility of eliciting coherent subjective judgments from clinicians in this structured format. While explicit sensitivity or specificity assessments against diagnostic ground truth are not reported, mathematical consistency is maintained at all levels (Wang et al., 2022).

5. Worked Examples: Matrix Construction and Three-Tier Partition

A representative example involves six disorder-clusters with matrix $M$ (partial form):

$$\begin{pmatrix} 1 & 1/2 & 1/3 & 1/4 & 1/2 & 1/9 \ 2 & 1 & 1/7 & 1/9 & 1/5 & 1/6 \ 3 & 7 & 1 & 1/8 & 1/6 & 1/7 \ 4 & 9 & 8 & 1 & 1/2 & 1/3 \ 2 & 5 & 6 & 2 & 1 & 1/2 \ 9 & 6 & 7 & 3 & 2 & 1 \end{pmatrix}$$

Solving $M w = \lambda_{\max} w$ yields

$w = (0.140, 0.041, 0.290, 0.038, 0.071, 0.420)^T,$

with $C.R. \approx 0.76\% < 10\%$.

Weights are propagated within clusters to individual disorders. For intensity-scale classification, five qualitative levels are compared, resulting in $s = (0.450, 0.277, 0.147, 0.081, 0.046)^T$, $C.R. = 0.53\%$. Disorders are mapped to these levels before trisection (e.g., using $\mu \pm \sigma$ as thresholds) and assigned to H/M/L accordingly.

6. Significance and Methodological Implications

The system’s core methodological contribution is unifying subjective expert judgment with rigorously checked, mathematically grounded ranking and weighting mechanisms, embedded in the 3WD framework. It enables:

• Transparent articulation and audit of subjective preference structures.
• Quantification of disorder importance through eigenvector and intensity-scale approaches.
• A principled, data-compatible three-way triage that integrates clinical reasoning with systematic statistical validation.

This suggests the system offers a reproducible route to combine manual and computational modes of diagnosis without discarding clinician expertise. A plausible implication is broader applicability of such tiered assessment—beyond psychiatry—to domains requiring nuanced, interpretable categorization of subjective or semi-structured domain knowledge (Wang et al., 2022).