A Mathematical Model of Suicidal-Intent-Estimation in Adults

Retrospective assessment of suicidal intent is important to prevent future attempts. The objective of the study is to mathemat ically model the method of suicidal intent estimat ion. Real-life data of 200 suicide attempters has been collected according to Beck’s suicide intent scale (BSIS), which is composed of three constructs and 20 indicators to assess the suicidal intent as ‘low’, ‘medium’ o r ‘high’. Each indicator possesses three preconditions for intent scoring. For conventional scoring first 15 indicators are used. The collected data has been analysed to note its distribution, reliability and mining significant indicators. Three Multilayer Feed Forward Neural Net (MLFFNN) classifiers have been developed. MLFFNN-1 is developed with first fifteen indicators to mimic the conventional way of scoring. MLFFNN-2 has been designed with all twenty indicators to note whether the network could better classify with more in formation. Significant (or quality) indicators, obtained through Multiple Linear Regressions and the Principal component analysis (PCA) are finally used to construct the MLFFNN-3. It is to see whether high quality information better influence the classification task. Performances of the neural nets are then compared and validated with the scorings performed by a group of psychiatrists (who are the human experts) and the regressions analysis. The paper observes that MLFFNNs have outperformed the human experts and regressions in terms of both speed and accuracy. MLFFNN-1 is found to be the best of the lot. It concludes that BSIS could efficiently be mapped onto neural networks.


Introduction
According to World Health Organization (WHO), suicide is an act of self-killing[1]. However, it could be prevented by judging the 'intent' at an appropriate t ime [2].Yearly about 1 million people co mmit suicide [3]. It is one of the first 20 leading cause of death [3]. The rate of suicide is quite high, estimated one in every 40 seconds [4]. It has been noted through series of studies that in the last 45 years suicide rates have been increased by 60% across the world [4]. It is the third leading cause of death among people aged between 15 and 44 years and second leading cause of death among those who are 10-24 years of age [4]. Attempting suicide is 20 times higher than actually co mmitting it [4]. Cu mulat ively suicide caters about 1.8% of total global burden of disease in 1998 and the predicted value would be 2.4% by the end of 2020 [4]. Therefore, suicide has a well recognized significance fro m the public health perspective.
Mental illnesses, such as dep ression , sch izoph ren ia, personality d isorders, alcoho lis m are so me o f the most do min ating 'intent ' facto rs o f su icide [3]. So me ot her important causes are unemployment [5], poverty [6], ch ronic illnesses [7] [8], familial tendencies [9], substance abuse [10] and so forth. Usually the subjects are treated for these disorders. In this context it is important to note that there is no concrete evidence to prove that medications and psychological treat ments given to these populations actually reduces the vulnerability of suicide or not. While surveying the literature, it may be noted that most of the randomized control trials related to suicide prevention have failed to yield co mplete informat ion about the suicidal trends [11]. It is because of the fact that these studies are targeted to a specific population co mposed of possibly a ho mogeneous group of subjects [12]. Researchers also believe that randomized control trial of suicide-prone individuals to a treatment condition might be unethical as it could be too much risky [12]. Moreover, there are no available valid measures of suicidal behaviour or ideation. Hence, the research scopes in suicide study are mostly confined to 'interventions' using standardized assessment instruments, which are able to measure the variations in the course of suicidality. Based on the nature of the course, treatments might be better planned. The most important of all is that the early recognition of the 'intent' which is reflected through emotional and physical sign-symptoms. However, early recognition of 'intent' is a difficu lt task. This is because of the facts that the onset of new symptoms or changes in the earlier sympto ms often remains unnoticed and immeasurable. Therefore, routine screening of vulnerable population is the only way to mine such risk and its level/severity.
There are several assessment tools for the screening of suicide [13]. Screening is conducted by interviewing the subjects. The correctness of the interpretation depends on the quality and quantity of the information and the experience level of the interv iewers. It is important to note that the experience and the perceptual abilities of the interviewer are completely individualized phenomena and because of it some degrees of variations might be seen in the final 'intent' scoring. Another issue is that, such interpretations could be biased as biasness is another considerable hu man factor that may influence the scoring process. Due to these prevailing issues, suicide is often under or over treated. As a result, the incidences are increasing in the population despite of commendable progress in mental healthcare.
Given this scenario, the objective of this study is to mathematically model and automate the suicidal 'intent' classification task using artificial neural network. Beck's Suicide Intent Scale (BSIS) [14] has been used to construct and validate the networks. It has three major constructs and under these constructs there are total 20 indicators. Each indicator has three preconditions, which are individually assessed during the interview and scored. The first, second and third preconditions are scored as '0', '1', and '2', respectively. Final scoring is the sum of all scores. BSIS structure has been discussed in detail in section 2. The paper also aims to investigate the influence of the 'number' (i.e., quantity) and the 'significance' (i.e., the quality) of the indicators on suicidal 'intent' classificat ion using neural networks. Su mmarily, the paper attempts to fit BSIS and its interpretation into neural netwo rks. It is discussed in detail under section 3.
Rest of the paper is organized as follows. The structure and interpretation of BSIS has been elaborated in section 2. Real-life data co llect ion, techniques used for the analysis of the data, and the development of Multilayer Feed forward Neural Netwo rks (M LFFNN) to automate the suicide 'intent' classification task are described in section 3. Sect ion 4 shows the experiments, related results and its interpretations. Finally, the paper concludes and shows some future directions in section 5.

Beck's Suicide Intent Scale (BSIS)[14]
Aron T.  developed a scale based on the seriousness of suicidal 'intent' or will[15] [16]. It is administered to the suicide attempters through personal interviews to assess the verbal and nonverbal behaviours prior to and during the most recent suicide attempts. The scale is structured into three major constructs, such as 'Objective Circumstances Related to Suicide Attempt' (which covers the objective circumstances reflecting the preparation), 'Self report' (wh ich measures the lethality of the acts), and 'Other aspects' (which reflects the passive influences, e.g., drugs, alcohols etc., subjects' reactions following an attempt and so forth). Each construct is composed of several indicators or item o r factors, shown in Table 1. In this table, column headings are the major constructs. Under each construct there are the indicators (italicized). The load of each indicator is then assessed with three preconditions (which are star marked) and the corresponding score or ratings of each precondition are assigned within parenthesis to quantify the 'intent' loads [17].
Under  (see table 1). Actual scoring is performed with these fifteen indicators under the first two constructs due to sufficient inter-rater reliability with Cronbach's alpha between 0.81 to 0.91 [18]. Ho wever, there is a third construct, called 'Other aspects' possesses five indicators, such as Reaction to attempt (RA), Visualization of death (VD), Number of previous attempts (NPA), Relationship between alcohol intake and attempt (RAA), and Relationship between drug intake and attempt (RDA), which are usually excluded fro m the scoring as all these are passive or indirect features of the 'intent'.
Final scoring: Final scoring is done by summing up the scores of the preconditions under the indicators. For examp le, all the first preconditions are scored as '0', second conditions as '1', and the th ird conditions as '2' (see Tab le 1). Scoring is done during interviewing the subjects. Sum scores from 15-19 denote 'Low Intent', while 20-28 refer to 'Medium Intent', and score more than 29 points towards 'High Intent' of co mmitting suicide. The intent usually increases with the numbers of attempts. In this dataset, it has been estimated that 60.5% patients have 'Low' intent (i.e. there is one attempted suicide), 37.5% have 'Medium' intent (i.e. having history of two suicidal attempts), and 2% possesses 'High' intent (i.e. having mo re than 2 suicidal attempts). Between any two consecutive attempts, the average time span is 6.7 months for this dataset.

Materials and Methods
This section discusses various processes adopted in this work. These are as follows.
1) Data collection 2) Data analysis, and 3) Develop ment of Multilayer Feed forward Neu ral Networks (M LFFNN) to auto mate the 'intent' classification task.

Data Collection
Capturing warning signs and symptoms (i.e. the data) of suicide 'intent' is the most important task to understand the possibility of suicide attempts [19] [20]. In this study, the informat ion has been captured fro m patients' data sheets (n = 200) and structured using BSIS (described in section 2.2 in table 1). The subjects have been interviewed by experienced med ical doctors (mean level of experience is about 6.8 years).  [14] 'Objective circumstances related to suicide attempt' 'Self report' 'Other aspects'
Precautions against discovery/intervention (PADI): *No precautions (0) *Passive precautions (as avoiding other but doing nothing to prevent their intervention; alone in room with unlocked door) (1) *Active precautions (as locked door) (2)

Conception of method's lethality (CML):
*Did less to self than he or she thought would be lethal (0) *Wasn' t sure if what s/he did would be lethal (1) *Equalled or exceeded what s/he thought would be lethal (2) 3. Number of previous attempts (NPA): *None (0) *One or two (1) *Three or more (2) 4. Acting to get help during/after attempt (AH): *Notified potential helper regarding attempt (0) *Contacted but did not specifically notify potential helper regarding attempt (1) *Did not contact or notify potential helper (2) 4. Seriousness of attempt (SA): *Did no seriously attempt to end life (0) *Uncertain about seriousness to end life (1) *Seriously attempted to end life (2) 4. Relationship between alcohol intake and attempt (RAA): *Some alcohol intake prior to but not related to attempt; reportedly not enough to impair judgment, reality testing (0) *Enough alcohol intake to impair judgment; reality testing and diminish responsibility (1) *Intentional intake of alcohol in order to facilitate implementation of attempt (2)

Relationship between drug intake and attempt (RDA):
*Some drug intake prior to but not related to attempt; reportedly not enough to impair judgment, reality testing (0) *Enough drug intake to impair judgment; reality testing and diminish responsibility (1) *Intentional intake of drug in order to facilitate implementation of attempt (2) 6. Active preparation for attempt (APA); *None (0) *Minimal to moderate (1) *Extensive (2)

Conception of medical rescuability (CMR):
*Thought that death would be unlikely if he received medical attention (0) *Was uncertain whether death could be averted by medical attention (1) *Was certain of death even if he received medical attention (2) 7. Suicide Note (SN): *Absence of note (0) *Note written, but torn up; note thought about (1) *Presence of note (2) 7. Degree of premeditation (DP): *None (0) *Impulsive suicide contemplated for three hours of less prior to attempt (1) *Suicide contemplated for more than three hours prior to attempt (2  The reason for choosing BSIS is that it is reliable, easy executable, having high inter-rater reliability, and possesses higher predictive value.
In this work, suicide 'intent' levels are assigned as 'low', 'medium', and 'high' according to the number o f suicidal attempts. Low intent refers to one attempt, while medium and high intents denote two and more than two attempts, respectively with an average time span of slightly higher than 6 months between any two consecutive attempts.
Appropriate ethical measures have been taken to collect and store the data preserving the privacy of the patients and the sources. Two senior psychologists have helped to organize the data. Time taken to collect the data is approximately 1 year.
Two hundred males and females each between 18 to 50 years of age has been considered in this study as this age group poses to be the most predominant threat. The collected data has finally been checked for redundancies, missing values and errors. Table 2 d isplays the overview of data collection.

Data Analysis Using Mi nitab-14
Data analysis is broadly composed of data preprocessing and processing techniques. It actually helps model abstraction fro m the dataset. By data analysis valid and interesting patterns within the data could be noted within the abstracted model. Such patterns might be helpful in designing the knowledge-enabled systems for the decision making. In this study, the collected data has been carefully checked for errors, redundancies, and missing values. Cronbach's alpha [21] has been measured to check the reliability of the data. The threshold of alpha is 0.7 [22]. After the initial check, the follo wing tasks are performed to analyze it using Minitab-14 [23].
• Examining data distribution by measuring means, standard deviations, and skewness (see table 2 in section 3).
• Viewing the distributions of intent scores among males and females with a scatter plot.
• Analysis of Variance (ANOVA) (see table 3 in section 3) for checking inter and intra group differences of the BSIS indicators and therefore the fidelity of the model.
• The main effects of each BSIS indicator on 'intent' score plots have also been tested (see figure 2 in section 3). The constants and the coefficients obtained through regressions are then used to validate the model with the test data (30% of the sample).
• Multiple Linear Regressions (MLR) to investigate the effects of the indicators on the intent score and mining the significant indicators. It has also been used to predict the intent level using the coefficients of each indicator and the constant, obtained during scoring.
• Principal component analysis (PCA) has been performed to extract significant indicators further. In this study, Eigen values more than '1.5' are considered as principal co mponents. The experimental results could be seen in figure 4 and table 5.

The Propose d MLFFNN Models
The key objective of this work is to mathemat ically model and automate the 'intent' classification task using BSIS structure. The study proposes three multilayer feed forward neural networks (M LFFNN). These are developed using These two types of nets will test the quantitative influence of the indicators on the 'intent' classification task. The study assumes that the misclassification must be less with MLFFNN-2 due to more information feeding to the hidden layer of the net. The third net i.e. M LFFNN-3 has been developed by the number of 'significant' or 'quality' indicators, engineered through MLR and Principal component analysis (PCA) using Minitab 14. The samp le matrix would be 200 x Y, where 'Y' denotes the total number of significant indicators, obtained by the regressions and PCA. Being statistically 'significant', such indicators could be considered as the 'quality' indicators to reflect the 'intent' level. Based on this argument the paper hypothesizes that these significant or quality indicators should be able to handle the 'intent' classification task in a more simp le way with much less computational complexity.
The reasons for choosing neural network for modelling the BSIS-based classification task are that neural networks can handle nonlinearity much efficiently[25] [26][27] [28]. It may be assumed that the neural nets map the ways human processes the inputs from the environment. The inputs are perceived by using the initial knowledge state (i.e. the weights of inter nodal connectors), which are updated iteratively to min imize the diagnostic error. The perceived informat ion is further processed in the hidden nodes using the medical logic, wh ich is co mposed of intuitions, beliefs, and clinical confidence. Activation function mimics such med ical logic. The differences in the logic could be represented by different activation functions.

Structure of the Network
Input layer: The nu mber o f nodes in this layer is equal to the number of attributes chosen for each MLFFNN, i.e., 15 for M LFFNN-1, 20 for M LFFNN-2 and the number of significant attributes obtained with regressions and PCA is considered for MLFFNN-3. Its activation function has been considered as linear as the input data must not be man ipulated to preserve its consistency with BSIS scoring system.
Hidden layer: The numbers of nodes in this layer are chosen equal to the number of nodes in the input layer. This is due to the fact that in case the number of nodes in the hidden layer is chosen less than the number of nodes in the input layer, the convergence accuracy may suffer. On the other hand, more number of nodes requires more p rocessing time and thereby increases the complexity of the network. Hence, by doing so, both the adversities could be avoided. The activation function in this layer has been chosen as Sig mo idal (see equation 1) for all the M LFFNN models. In this equation, 'y' is the net input to the node and 'Z' is the nodal output. The notation 'λ' is the slope of the function.
Author assumes that the activities in the hidden layer mimic the perceptual processing of the medical doctors during the 'intent' scoring process. The weights between the input-hidden-output nodes are the doctors' existing knowledge. The knowledge is updated over a period of time based on the learning capacity, which depends on the rate of learning and the mo mentu m. The med ical logic (wh ich is a composition of a doctor's belief, intuitions, and judging capacity) might be represented by the activation function.
Output layer: In all the proposed networks, it contains single node which co mputes the output. The activation function set for this layer is also Sigmo idal. Th is layer mimics the interpretation and the decision making process of the medical doctors.
The computed output (y) for each case is finally co mpared with the target output (t). Mean squared error (MSE) between 't' and 'y' be then estimated using equation 2.
In this equation 'n' denotes the sample size. Using scaled conjugate gradient backpropagations the networks are trained. It is worth noting that all the M LFFNNs are t rained with 70% o f the sample (i.e., t rain ing data size=140). Remaining 30% (i.e., 60 cases) is used for testing of the MLFFNNs. A 10-fo ld cross validation is performed in the training-testing process to randomly choose the training and test cases for avoiding experimental bias. Training automatically stops when generalizat ion ceases to improve further.
The A generic M LFFNN model could be viewed in Fig.1. In this figure, 'input layer' contains 'n' number of nodes. 'Hidden layer' possesses 'n' number of nodes, where 'n' refers to the number of indicators the net is handling. Inter nodal weights between the input and hidden layer are denoted by 'w ik '. On the other hand, 'v k ' denote the connectors' weights between the hidden nodes and the output node. The output layer consists of single node to compute the output of the neural net. The notation d(t,y) refers to the deviations or errors between the computed output (y) and the target output (t). 'E' is the squared error and MSE is the mean of the squared errors. The notations 'η' and 'λ' refer to learn ing rate and mo mentum constant.

Experiments and Analysis of Results
In this section, experimental results have been shown sequentially and illustrated as follows.

Data Statistics
The distribution of intent score according to the gender could be seen in Fig.2. In this figure 'SCORE-FM ' and 'SCORE-M' denote the intent scores obtained for female and male subjects, respectively. The plot shows that the 'high' risk of suicidal intent is more in males (i.e. three cases) compared to females (i.e. one case); almost equal distribution for the 'med iu m' risk; and female predo minance (three vs. one) in the 'lo w' risk zone.  The nature of data and its distribution have been examined. Table 2 shows the mean, standard deviation, variance and skewness of the indicators. It may be seen that the mean score is ma ximu m fo r the indicator 'APA' and minimu m for 'CML', which mean that the overall effects on 'intent' scoring is mostly contributed by 'APA' values. The indicator 'I' shows the most spread data distribution and so the highest variance. A mong twenty, 15 indicators show positive skeweness (mean: 0.23, maximu m: 0.92 for the case of 'CML'). On the other hand, 'APA' presents the most negative skewness.
Reliability measure with Cronbach's alpha shows the alpha value as 0.798, wh ich is significant [22].

Data Analysis
Under this section, the results of the following experiments have been displayed.
1) A NOVA (refers to table 3) 2) Main effect of each indicator on the SCORE (could be seen in Fig.3) 3) M LR (could be seen in table 4, equation 6, and Fig.4) 4) PCA (refers to table 5 and Fig.5), and 5) Performances of the M LFFNNs (could be seen in table 6).

The Analysis of Variance (ANOVA)
ANOVA has been performed to test whether the groups of indicators are same or different fro m each other. Table 3 has shown the ANOVA result. In this table the 'F' statistics denotes a ratio of inter group variation and intra group variation. Therefore, a larger 'F' statistics value indicates higher inter group and intra group variations. Here, the 'p' value less than 0.05 supports that the experimental model is statistically significant. The main effect plot shows the average outcome for each value of each indicator, combining the effects of the other indicators if and only if all indicators are independent. From ANOVA , it is seen that there is a considerable difference within and between groups, hence main effect plots could be useful to note the influence of the average scoring of the indicators on 'intent' assessment. In figure 3, the mean score is plotted against the value of the indicators.
In this figure the term 'SCORE' denotes the suicidal 'intent' score. In these plots it may be observed that for indicator 'I', the score '2' averages of the most, while '0' is the lowest and '1' in between to influence the overall scoring of suicidal 'intents' for this dataset. Table 4 shows the summary of the average score loading/effects on 'intent' scoring. For th is case 50% of the indicators have highest average score loading of '2', while scores '0' and '1' contribute 25% each.  (see table  7).
Significant indicators are then extracted with the help of paired-t-test. Table 5 shows the significant indicators, such as 'T', 'PADI', 'SN', and 'AP' with p-values <0.05 (CI 95%). Average effect score loading The residual plots for the intent score could be seen in figure 3, wh ich is a four-in-one plot to have a more concise view o f the model. The top left and right plots show that the residuals (i.e. the deviations of the observed responses from the predicted responses) are normally distributed across the normality line. Ho wever, deviations could be noted for four cases having values higher than +10. The shape of the histogram plot resembles a normal probability distribution.

Principal Co mponent Analysis
After obtaining the measures on the observed indicators PCA has been conducted for extracting more nu mber of significant indicators that would account for the most of the variance in the observed indicators. It is performed using Minitab 14. One aim o f this study is also to reduce the number of indicators based on the principal/significant indicators. Principal co mponents are linear comb inations of optimally -weighted observed indicators. Table 6 has shown the corresponding 'proportion' and 'cumulative' values. The term 'Proportion' denotes how much percentage of variance a variable has in the data. For example, 'I' accounts for 12.4% of variance in the data.

Results of the Performances of MLFFNN-1, 2 and 3
Performances of all the nets have been tested on a P-IV (Intel dual core) co mputer with 2 GB RAM and 3.0 GHz processor speed and compared among themselves and ten other human experts (mean experience 4.3 years). The performance results are shown in table 7   While co mparing with the interpretations given by ten human experts, it is seen that the average time taken to complete one fu ll task with 15 indicators is around 15 minutes. It is 25 minutes and 10 minutes with 20 and 6 indicators. The MSE are 5% with 15 and 20 indicators. MSE increases with six indicators probably due to less number of informat ion.

Conclusions and Future Scopes
The study draws the following conclusions.
• The attributes (i.e. BSIS indicators) are independent of each other as shown in the results of ANOVA (F-statistics 1.61).
• The main effect plot (refer to table 4 and figure 2) shows that the mean scores of 50% of the indicators are the highest, i.e. '2' and accordingly influence the final 'intent' scoring in this dataset.
• By M LR and PCA, total six indicators such as 'I', 'T', 'PADI', 'AH', 'SN' and 'AP' are found statistically significant (i.e., considered as the quality indicators) in influencing the final intent scoring. It corroborates with the earlier results obtained through factor analysis [29] [30]. These are used to design and develop the MLFFNN-3.  • The opposite picture is seen in case of M LFFNN-3 having the least number of informat ion and hence taking more t ime (i.e. 0.07 sec) and iterations (i.e. 273) to converge.
The contribution of the paper lies on (i) systematic collection of 'intent' data fro m the real-life subjects and its analysis and (ii) designing and developing the neural networks to model the co mplete informat ion of BSIS to map the way doctors make the scoring using their medical knowledge and logic.
The extension of this work could be the develop ment of a complete GUI for its practical deployment in the clin ics for the doctors to use as an assistive tool. Also the MLFFNNs could be trained with different datasets to enrich its maturity and robustness. However, the GUI-based tool has to be standardized prior its deployment.