# Aligning the Goals Hybrid Model for the Diagnosis of Mental Health Quality

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## Abstract

**:**

## 1. Introduction

## 2. Problem Definition and Optimization Model

**“Multi-criteria Model for Diagnosis of Mental Health”**, the application of the AHP method with geometric mean is due to the acceptance of the method by the decision-maker. In other words, the questions presented to the decision-maker made sense to him/her, and he/she had confidence in answering them.

#### 2.1. Machine Learning

**“A” or “B”**? These are models developed from supervised learning, which is the construction of a model for the prediction or estimation of an

**“output”**based on one or more

**“inputs”**[15].

**inputs**) can be continuous, categorical, or both in classification models. The dependent variable (

**output**) is categorical and usually dichotomous (

**“A” or “B”**). Depending on the classification problem, an output that is not dichotomous can be converted to be so. Model development includes steps such as training and evaluation or performance tests. During training, the goal is to obtain data that satisfactorily contemplate the problem’s complexity so that an accurate predictive model can be developed from logistic regression, Naïve Bayes, and other classifiers based on machine learning. During the evaluation stage, the predictive capacity of the built model is finally tested using new data not associated with the previous training stage [15].

#### 2.1.1. Logistic Regression

**logit**) that relates the linear predictors of the model (systematic component) to the expected values of the response variable (random component). The determination of the parameters (regression coefficients) is performed using the maximum likelihood method, which generates values that maximize the likelihood function and generally present consistent mathematical properties [16].

**y**) in logistic regression is dichotomous; two values are attributed to it: one for the event of interest, called a success, and zero for the complementary event, called failure. The probability of success is provided by

**π**, and 1 − πi provides the probability of failure. Considering a series of independent random variables,

_{i}**x**and a vector,

_{1}, x_{2}, x_{3}, …, x_{n,}**β = β**formed by unknown model parameters, the probability of success is provided by

_{0}, β_{1}, β_{2}, …, β_{p}**deviance**,

**D**). A model’s deviance is defined as the deviation of this model regarding the saturated model, in which all parameters fit perfectly to all observations, according to the definition.

**β**, the deviance is used to measure the discrepancy of an intermediate model of

_{0}**p**parameters regarding the saturated model. The smaller the deviance, the better the fit of the model. Among nested models, tests such as the likelihood ratio can assess the significance of a fit difference via deviance. This test is also used to verify the significance of the contribution of each predictor variable for the fit in a given model [16].

**Akaike Information Criterion**(AIC). This criterion considers the model’s fit and simplicity, penalizing models with more variables. Since AIC is a measure related to the loss of fit of a given model, the smaller this value is, the better the fit of the model [17].

**AIC = −2log(L**

_{p}) + 2(p)#### 2.1.2. Random Forest

#### 2.1.3. Naïve Bayes

**B**, occurring on the condition that

**A**has already occurred, as defined by

**P(B|A)**is the conditional probability of

**B**occurring given that

**A**occurred;

**P(B)**and

**P(A)**are the probabilities of the occurrence of

**B**and

**A**, respectively; and

**P(B|A)**is the conditional probability of

**A**occurring given that

**B**occurred [24].

**P(y**is the posterior probability, which is the probability of a given observation, with its respective predictor variables

_{i}|x)**(x)**, belonging to class

**y**;

_{i}**P(x|y**is the conditional probability (likelihood), which is the probability of verifying observations that belong to a given class. It is decomposed into probabilities concerning each of the model’s predictor variables, which are multiplied among themselves, and in this case, P (x

_{i})_{1}| y

_{1}) x…x P(x

_{i}|y

_{i}); P(y

_{i}) is the probability of the referred class (prevalence); and P(x) is the occurrence probability of the predictor variables under discussion. Ultimately, the denominator can be ignored since it is the same for all classes. Unlike logistic regression, in Naïve Bayes, there is no post-training variable selection phase [15,24].

#### 2.2. Analytic Hierarchy Process (AHP)

- (a)
- STRUCTURING: This deals with the formulation of the problem and the identification of objectives. This phase aims to identify, characterize, and organize the relevant factors in the decision-support process.
- (b)
- EVALUATION: This allows for the subdivision of a subphase partial evaluation of actions (alternatives) according to each point of view (criteria) and an overall evaluation subphase considering the various partial evaluations.
- (c)
- RECOMMENDATION: In this phase, sensitivity and robustness analyses are carried out to verify whether changes in the parameters of the evaluation model interfere with the final result. It is a fundamental phase that contributes to generating knowledge about the problem, increasing the confidence of the decision-maker in the obtained results.

#### 2.2.1. Multi-Criteria Methodology

- It is easy to use for non-specialists, preferably transformed into a computer program that is as user-friendly as possible, featuring visual graphic resources;
- It constitutes a logical and transparent method;
- It enables freedom from ambiguity for input data interpretations;
- It encompasses both quantitative and qualitative criteria;
- It values judgments;
- It allows the decision-maker to have algorithms that enable the use of criteria that are independent of each other, such as algorithms that help in problems in which the evaluation criteria are interdependent, and, similarly, it can deal with alternatives that are independent of each other;
- It incorporates human behavior issues into decision-making processes.

- Considering the subjectivity of decision-makers, that is, the individual perceptions and envisioning involved in the aspects of problems, decision-makers find it most challenging to explain their perceptions;
- Structuring the problem according to the shared vision;
- Identifying common points of view;
- Knowing where decision-makers are inconsistent;
- Checking what can be changed and for what reason.

- Define and structure the problem;
- Define the set of criteria or attributes or both, that will be used to rank the alternatives;
- Choose whether to use discrete or continuous methods; in cases of opting for discrete methods (conceived to work with a finite number of alternatives), it must favor the use of methods either from the French School or the American School;
- Identify the preference system of the decision-makers;
- Choose the aggregation procedure.

- The choice of alternatives;
- The construction of criteria and information aggregation;
- The classification of the alternatives in which the dominance of the groups is identified;
- The ordering of a classification hierarchy among the alternatives.
- The structuring phase of a problem can be divided as follows [32]:
- The structure and composition of the components;
- The analysis;
- The synthesis of information.

_{1}, P

_{2,}or P

_{3}. The methods that generate an ordering of alternatives according to the decision-maker’s preference are associated with P

_{1}; the methods whose output data only indicate the favorite final solution to the problem are associated with P

_{2}; and the methods that sort alternatives into predefined categories are associated with P

_{3}. However, it is interesting to remember that the methods of category P

_{1}can also be used to determine a single final solution to the problem by selecting the best alternative in the ranking generated by it.

**Decision-maker**: The agent has the power and responsibility to ratify the decision, assuming the consequences for this act, whether positive or negative. The decision-maker can be an individual or a group of people who establish the limits of the problem, specify the objectives to be achieved, and issue judgments. Not all decision-makers have decision-making power. Therefore, it is essential to distinguish the degree of influence of decision-makers in the decision-making process [44,45].

**Analyst**: This is the agent who interprets and quantifies the decision-makers’ opinions, structures the problem, elaborates the mathematical model, and presents the results to the decision-maker. Constant dialog and interaction with decision-makers in a steady learning process are necessary [46].

**Model**: This is a simplified representation of reality through rules and mathematical operations that transform the decision-maker’s preferences and opinions into a quantitative result [46].

**Alternative**: This is a potential action and constitutes the decision’s object or the one directed to support it. It is identified at the beginning of the decision-making process or during it. It may become a solution to the problem under study [44].

**Criterion**: This is a function,

**g**, defined in a set,

**A**, that assigns ordering values from set

**A**and represents the preferences of the decision-maker from a certain point of view [47]. According to Morais [46], a problem with several criteria will be defined as g

_{1}, g

_{2}, …, g

_{j}, …, g

_{n}. The evaluation of an action, “a”, according to the criterion, “j”, is represented by g

_{j}(a). Representing different points of view (aspects, factors, characteristics) with the help of a criteria family, F = {g

_{1}, …, g

_{j}, …, g

_{n}}, constitutes one of the most delicate parts of formulating decision problems. The criteria are classified according to the verified preference structure, as shown in Table 2.

**Criterion:**This is a function,

**g**, defined in a set,

**A**, which assigns ordering values from the set,

**A,**and represents the preferences of the decision-maker from a certain point of view [47]

**.**

**“b”,**belong to set A, in the way that

**”a”**dominates

**”b”**(

**aDb**) if, and only if,

**g**being that

_{1}(a)≥ g_{1}(b),**j = 1, 2, …n,**in which at least one of the inequalities is strictly preferred. It can be noted that the dominance ratio of

**”a”**in

**”b”**is characterized by being a strict partial order, that is, an asymmetric and transitive relation. If

**”a”**dominates

**”b”**,

**”a”**is superior to

**”b”**in all problem criteria [47].

**Efficient Action**: This is an action (or alternative);

**”a”**is considered efficient if, and only if, there is no other action from set

**A**in which

**a**dominates. The set of efficient actions of

**A**can be

**A**itself if the dominance relation is empty. It is generally considered a set that contains engaging actions to be analyzed in greater depth, even if there are no good reasons to disregard the inefficient ones [47].

**Decision Matrix**: Also called the evaluation matrix, in the decision matrix, each line expresses the measures of the evaluations of the alternative,

**i**, concerning

**n**considered criteria. Each column, in turn, expresses the evaluation measures of m alternatives concerning criterion

**j**. Assuming that

**a**represents the evaluation of the alternative (or action),

_{ij}**A**, it belongs to the set of potential actions:

_{i}**A**, [

**a**]; according to criterion

_{ij}**g**, a matrix similar to Table 3 below can be constructed [46]:

_{j}#### 2.2.2. The Classic Analytic Hierarchy Process (AHP) Method

**A**element (from the row) is more critical than the

_{i}**A**element (from the column), any value from 1 to 9 must be entered (Table 5). Otherwise, if

_{j}**A**is less important than

_{i}**A**, an inverse number to values 1 to 9 is inserted, i.e., 1/2, 1/3, and so on. In square matrices, we have

_{j}**a**.

_{ij}, para i = 1, 2, …, n e j = 1, 2, …, n**A**is consistent if and only if

**λ máx ≥ n.**

**A**is eigenvector

_{vi}**i**,

**P**is priority vector

_{i}**i**, and

**n**is the number of alternatives.

**λmax**and

**n**, it is necessary to calculate the consistency ratio (

**RC**), represented by the following formula:

**the**consistency index (

**IC**) is represented by

#### 2.2.3. Analytic Hierarchy Process (AHP)—Average of Normalized Values Method

- (a)
- Normalization by the sum of each column’s elements:$${W}_{i}\left(Mj\right)=\frac{{a}_{ij}}{{\sum}_{i=1}^{m}{a}_{ij}}j=1,\dots ,n$$
- (b)
- The sum of elements of each normalized line, divided by order of the matrix:$${W}_{i}({M}_{i})={\sum}_{j=1}^{m}{W}_{i}\left(Mj\right)/n\hspace{1em}\hspace{1em}\forall i=1,\dots ,m$$
- (c)
- Calculation of the eigenvalue associated with the calculated vector in the previous item:$$M\xb7W=\lambda max\xb7W\hspace{1em}\hspace{1em}\lambda max=\frac{1}{n}{\sum}_{i=1}^{n}\frac{{\left[AW\right]}_{i}}{wi}$$

#### 2.2.4. Analytic Hierarchy Process (AHP)—Geometric Mean Method

- (a)
- The product of the elements of each row raised to the inverse of the order of the matrix:$$Wj({M}_{i})=\sqrt[n]{{\prod}_{j=1}^{n}aij}i=1,\dots ,n$$
- (b)
- Normalizing the obtained priority vector and calculating the eigenvalue associated with the calculated vector will produce an identical result to the λmax of the average normalized values method.

## 3. Results

#### 3.1. Method of Preparing the Database

- Your mood;
- Your self-confidence;
- Your interest in life;
- Your ability to endure difficult situations.

- Your eating habits;
- Your energy (willingness to do things);
- Your sleep;
- Your physical health (pain, tremors, malaise);
- Your sexuality (sexual satisfaction).

- Your coexistence with your family (the one you live with);
- Your coexistence with friends;
- Your coexistence with other people;
- Your financial conditions for family support.

- Your interest in working/studying;
- Your leisure activities (the things you like to do);
- Your ability to fulfill obligations;
- Your household tasks (cooking, cleaning the house, shopping, fixing things);
- Your interest in engaging in other activities.

#### 3.2. Description of the Proposed Hybrid Model Development

#### Hybrid Algorithms

## 4. Discussion

## 5. Conclusions and Future Studies

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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MCDM | MCDA |
---|---|

Existence of a well-defined set, A; the existence of a decision-maker, D | The border of A is diffuse and can be modified during the process. There is no decision-maker, D, but rather a set of actors participating in the decision-making process |

Existence of a well-defined preference model in the mind of the decision-maker, D | Preferences are rarely well-defined, which are uncertainties, partial knowledge, conflict, and contradictions |

Unambiguous data | It recognizes the data’s ambiguity, often inaccurately or arbitrarily defined |

Existence of an optimal solution to a well-defined mathematical problem | It is impossible to determine whether a solution is good or bad by considering only the mathematical model since cultural, pedagogical, and situational aspects affect the decision |

Criterion | Preference Structure | Description |
---|---|---|

True criterion | Complete preorder (traditional model) | Any difference implies a strict preference |

Quasi-criterion | Semi-ordered (threshold model) | There is a constant indecision zone between indifference and strict preference |

Interval criterion | Interval order (variable threshold model) | There is a variable indecision zone between indifference and strict preference over the scale |

Pseudo-criterion | Pseudo-order (double threshold model) | A sudden shift from indifference to strict preference is avoided, with a hesitation zone represented by weak preference |

Criteria | g_{1} | g_{2} | …………. | g_{J} | …………. | g_{m} |

Limits | q_{1}, p_{1} | q_{2},p_{2} | …………. | q_{j}, p_{j} | …………. | q_{n}, p_{n} |

Alternatives | ||||||

A_{1} | a_{11} | a_{12} | …………. | a_{1j} | …………. | a_{1n} |

A_{2} | a_{21} | a_{22} | …………. | a_{2j} | …………. | a_{2n} |

………… | …………. | …………. | …………. | …………. | …………. | …………. |

A_{i} | a_{i}_{1} | a_{i}_{2} | …………. | a_{ij} | …………. | a_{in} |

……….. | …………. | …………. | …………. | …………. | …………. | …………. |

A_{m} | a_{m}_{1} | a_{m}_{2} | …………. | a_{mj} | …………. | a_{mn} |

A_{1} | a_{11} | a_{12} | …………. | a_{1j} | …………. | a_{1n} |

A = | Criterion 1 | Criterion 1 | Criterion 2 |

1 | Numerical Evaluation | ||

Criterion 2 | 1/numerical evaluation | 1 |

Scale | Numerical Evaluation | Reciprocal |
---|---|---|

Extremely preferred | 9 | 1/9 |

Between very strong and extremely | 8 | 1/8 |

Very strongly preferred | 7 | 1/7 |

Between strong and very strong | 6 | 1/6 |

Strongly preferred | 5 | 1/5 |

Between moderate and strong | 4 | 1/4 |

Moderately preferred | 3 | 1/3 |

Between equal and moderate | 2 | 1/2 |

Equally preferred | 1 | 1 |

Critérios | Definições dos Critérios |
---|---|

Cr1: Emotions and feelings | Related aspects to possible changes in emotions and feelings |

Cr2: Physical health | Related aspects to possible changes in physical health |

Cr3: Interpersonal relationships | Related aspects to possible changes in interpersonal relationships |

Cr4: Routine | Related aspects to possible changes in daily behavioral routine |

Alternatives |
---|

a1: Worse than before |

a2: No change |

a3: Better than before |

AUTOVETOR 2020 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AHP Classic | AHP Normalized Values | AHP with Geometric Mean | ||||||||||

CR1 | CR2 | CR3 | CR4 | CR1 | CR2 | CR3 | CR4 | CR1 | CR2 | CR3 | CR4 | |

WORSE THAN BEFORE (%) | 0.30 | 0.28 | 0.32 | 0.34 | 0.32 | 0.28 | 0.32 | 0.34 | 0.28 | 0.32 | 0.34 | 0.34 |

Same as BEFORE (%) | 0.48 | 0.42 | 0.44 | 0.46 | 0.46 | 0.42 | 0.44 | 0.46 | 0.42 | 0.44 | 0.46 | 0.43 |

Better THAN BEFORE | 0.21 | 0.20 | 0.14 | 0.12 | 0.27 | 0.20 | 0.14 | 0.12 | 0.20 | 0.14 | 0.12 | 0.13 |

Consistency Index Results—The Year 2020 | ||||
---|---|---|---|---|

Index/Alternatives | Criterion: CR1: Emotions and Feelings | Criterion: CR2: Physical Health | Criterion: CR3: Interpersonal Relationships | Criterion: CR4: Routine |

IC | 0.18 | 0.22 | 0.29 | 0.33 |

Eigenvector 2021 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

AHP Classic | AHP Normalized Values | AHP with Geometric Mean | ||||||||||

CR1 | CR2 | CR3 | CR4 | CR1 | CR2 | CR3 | CR4 | CR1 | CR2 | CR3 | CR4 | |

WORSE THAN before (%) | 0.34 | 0.56 | 0.03 | 0.25 | 0.56 | 0.56 | 0.03 | 0.25 | 0.56 | 0.03 | 0.25 | 0.37 |

SAME AS BEFORE (%) | 0.43 | 0.35 | 0.46 | 0.53 | 0.35 | 0.35 | 0.46 | 0.53 | 0.35 | 0.46 | 0.53 | 0.40 |

Better THAN BEFORE | 0.13 | 0.06 | 0.09 | 0.10 | 0.06 | 0.06 | 0.09 | 0.10 | 0.06 | 0.09 | 0.10 | 0.14 |

Consistency Index Results—The Year 2021 | ||||
---|---|---|---|---|

Index/Alternatives | Criterion: CR1: Emotions and Feelings | Criterion: CR2: Physical Health | Criterion: CR3: Interpersonal Relationships | Criterion: CR4: Routine |

IC | 0.18 | 0.23 | 0.27 | 0.32 |

Model | Train Time (s) | Test Time (s) | AUC | CA | F1 | Precision | Recall | Log Loss | Specificity |
---|---|---|---|---|---|---|---|---|---|

Random Forest | 0.034 | 0.004 | 0.999 | 0.980 | 0.970 | 0.980 | 0.960 | 0.044 | 0.990 |

Naïve Bayes | 0.007 | 0.008 | 0.975 | 0.887 | 0.828 | 0.837 | 0.820 | 0.183 | 0.920 |

Logistic Regression | 0.042 | 0.001 | 0.997 | 0.973 | 0.959 | 0.940 | 0.940 | 0.120 | 0.990 |

Model | Train Time (s) | Test Time (s) | AUC | CA | F1 | Precision | Recall | Log Loss | Specificity |
---|---|---|---|---|---|---|---|---|---|

Random Forest | 0.028 | 0.004 | 1.000 | 0.995 | 0.995 | 0.995 | 0.995 | 0.044 | 0.990 |

Naïve Bayes | 0.009 | 0.001 | 0.992 | 0.911 | 0.915 | 0.931 | 0.911 | 0.339 | 0.972 |

Logistic Regression | 0.046 | 0.001 | 0.971 | 0.895 | 0.883 | 0.889 | 0.895 | 0.284 | 0.841 |

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© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Costa, W.S.; Pinheiro, P.R.; dos Santos, N.M.; Cabral, L.d.A.F. Aligning the Goals Hybrid Model for the Diagnosis of Mental Health Quality. *Sustainability* **2023**, *15*, 5938.
https://doi.org/10.3390/su15075938

**AMA Style**

Costa WS, Pinheiro PR, dos Santos NM, Cabral LdAF. Aligning the Goals Hybrid Model for the Diagnosis of Mental Health Quality. *Sustainability*. 2023; 15(7):5938.
https://doi.org/10.3390/su15075938

**Chicago/Turabian Style**

Costa, Wagner Silva, Plácido R. Pinheiro, Nádia M. dos Santos, and Lucídio dos A. F. Cabral. 2023. "Aligning the Goals Hybrid Model for the Diagnosis of Mental Health Quality" *Sustainability* 15, no. 7: 5938.
https://doi.org/10.3390/su15075938