Exploring intertemporal decision-making dynamics through functional data analysis: investigating variations in different discount function's dimensions

Abstract

Intertemporal preferences are closely related to essential aspects of an individual’s emotional and cognitive domains. Discount functions are used to quantify these preferences, which can help us understand conditions such as addiction, depression, and Attention Deficit Hyperactivity Disorder (ADHD). However, traditional parametric models are limited when dealing with intertemporal preferences, mainly when behavioural biases are involved. This study exploits Functional Data Analysis (FDA) to investigate the properties of discount functions in intertemporal choices comparing people suffering from the Hikikomori pathology (a condition that involves social withdrawal) and normal people. Notably, the goal of this research is to look for statistically significant differences in the dynamics of intertemporal decision-making according to different gravity of the Hikikomori condition through the magnified FDA lens on different functional dimensions; the distinctive curves of discount functions categorised by Hikikomori scores prompted a more profound investigation via the so-called augmented functional analysis of variance. The original curves and their derivatives, and the discount rates and their first derivatives provide the different functional dimensions explored. This original approach of analyzing differences between subgroups according to decision-making behaviours is exciting from a methodological and practical perspective.

1 Introduction

Intertemporal choice refers to decisions that are spread over multiple periods. The Discounted Utility Model furnishes an essential framework for comprehending how decision-makers behave in such circumstances. According to this model, the utility of the result decreases over time, and the discount function measures its decrease. A higher discount function suggests that the individual has a greater perception of the uncertainty of the future. Two key concepts define intertemporal preferences: the discount rate and impatience. The discount rate estimates the psychological factors that contribute to the diminution of the present utility of the outcome. At the same time, impatience quantifies how much utility one is willing to lose to receive a reward in the current moment. Although Samuelson’s model has been a reference point in intertemporal choices, its predictions often diverge from empirical behaviour. Therefore, modelling and behavioural perspectives have been used to investigate anomalous preferences, i.e., those that do not conform to the classical rationality assumed by DU model. In cognitive psychology, the notion of limited rationality and systematic cognitive distortions have underlined that decision-makers could be less rational than they think. Over the years, hyperbolic models have reduced the gap between empirical and normative models by assuming a discount rate that diminishes over time and a non-constant degree of impatience.

Previous investigations have examined the interaction between impatience and discount factor about anomalies in the DU model and non-rational attitudes (Ventre et al. 2023a) and temporal dispersion, which explores how time is perceived (Takahashi et al. 2008; Ventre et al. 2023b). Noting the link between behaviour and intertemporal preferences, research has expanded to the study of decision-making and pathological habits about obesity (Schiff et al. 2016), attention deficit hyperactivity disorder (ADHD) (Fassbender et al. 2014), schizophrenia (Zhang et al. 2017) and addictions (Li et al. 2019).

Intertemporal preferences and discount functions' interpretation are complex notions. They are often based on parametric models, which may have inherent limitations. The decision-makers behaviour is contextualized in these models, but discount function shape assumptions are often simplified to facilitate analysis. Therefore, these models may only partially capture the variety of individual preferences. Complex dynamics are often present in intertemporal behaviour, which goes beyond simple parameter modelling. Accordingly, it is necessary to research more flexible and empirical alternatives that can better grasp the diversity of human behaviour in the context of intertemporal choices. Indeed, parametric approaches often lead to classical inconsistencies in intertemporal choices. Flexible approaches in data analysis are needed to overcome these limitations, and Functional Data Analysis (FDA) can be an ingenious tool to study the properties of intertemporal preference.

This study focuses on intertemporal choices through the use of the FDA magnifying glass that allows us to capture additional features of discount functions without complying with assumptions that do not reflect reality. Having introduced the methodological part, the approach is applied to the study of the Hikikomori syndrome (Kato et al. 2011) and allows for a sophisticated description of the case study, overcoming the limitations of a rigidly parametric approach. The main emphasis of the application is on the extension of the so-called augmented functional analysis of variance proposed by Maturo and Porreca (2022) to the context of intertemporal choices. The goal is to capture possible differences between groups of curves relating to the Hikikomori condition according to different dimensions. Hence, the shapes of different characteristics of the discount functions, differentiated by Hikikomori scores, prompted a more in-depth examination through the augmented functional analysis of variance, which is an extension of the well-known fANOVA (see, e.g. Ramsay and Silverman 2005; Di Battista et al. 2014; Maturo et al. 2018). Specifically, the original curves, their derivatives, and discount rates and their first derivatives delineate various functional dimensions under scrutiny. This novel method of dissecting decision-making behaviours proves intriguing, offering insights from both methodological and interpretative standpoints.

In a previous work by Ventre et al. (2024), no statistically significant differences between Hikikomori and non-Hikikomori were discovered, even on dividing the groups according to the severity of the disease. The possible limitation of the latter study is that Ventre et al. (2024) focused only on the original curves, neglecting all the functional dimensions that may include information other than that contained in the initial discount functions. Therefore, the primary objective of this study is to investigate the problem by trying to understand whether additional functional features can lead to different conclusions. In addition, this study also focuses on some details of the experimental phase, such as increasing the number of permutations and erasing functional outliers that may have led to biased analysis in Ventre et al. (2024).

The paper proceeds with Sect. 2, introducing the fundamental concepts of intertemporal choices models, FDA and augmented functional analysis of variance. Section 3 provides an application for the Hikikomori dataset. Section 4 presents the discussion, conclusions, and future research directions.

2 Material and methods

2.1 Background

In the literature, researchers investigate inter-temporal choices from various psychological, behavioural and modelling perspectives to understand how individuals make decisions. The discounted utility model is the standard mathematical representation of intertemporal choices, which supposes that the decision-maker prefers the option that furnishes the highest level of utility.

Fixed $x_i \in {\mathbb {R}}$ the available results and $t_i \in {\mathbb {R}}^{+}$the points of time, the DU model states that:

$$\begin{aligned} U\left( x_1, t_1; \ldots ; x_n, t_n\right) =\sum _i^n U\left( x_i\right) f\left( t_i\right) \end{aligned}$$

(1)

where $\left( x_1, t_1; \ldots ; x_n, t_n\right)$ is an intertemporal prospect. It means that if the prospect is accepted by the DM then the outcome $x_i$ will be received at time $t_i$; $U\left( x_1, t_1; \ldots ; x_n, t_n\right)$ is the utility associated with the intertemporal prospect; $U\left( x_i\right)$ is the cardinal utility of $x_i$; $f:[0,+\infty ) \rightarrow [0,+\infty )$ is the discount function characterized by the following properties: $\quad f(0)=1$; $\quad f(t)$ is monotonous decreasing; and $\lim _{t \rightarrow \infty } f(t)=0$.

Crucial information is also given by the discount rate that indicates “the proportional variation of f(t) in a standard period” (Read and Roelofsma 2003):

$$\begin{aligned} \rho (t)=-\frac{f^{\prime }(t)}{f(t)} \end{aligned}$$

(2)

In simpler terms, Eq. 2 reveals the extent to which the value of f(t) diminishes in relation to time, delivering a measure of the rate at which future outcomes are discounted or devalued. This concept is fundamental in understanding time’s impact on decision-making and assessing future outcomes in economic or decision-theoretic contexts.

The derivative of the discount rate $\rho (t)$ with respect to t can be obtained by applying the differentiation rules. The derivative of the discount rate formula given by Eq. 2 is:

$$\begin{aligned} \frac{d\rho (t)}{dt} = \frac{f''(t)f(t) - [f'(t)]^2}{[f(t)]^2} \end{aligned}$$

(3)

Eq. 3 reflects how the discount rate changes instantly in response to slight variations in time. Here, $f^{\prime }(t)$ is the first derivative of the discount function, and $f^{\prime \prime }(t)$ is the second derivative. The interpretation of the derivative of the discount rate is related to the concavity or convexity of the discount function. If $\frac{d \rho (t)}{d t}>0$, it indicates that the discount rate is increasing over time, indicating concavity in the discount function. Conversely, if $\frac{d \rho (t)}{d t}<0$, it implies that the discount rate is decreasing over time, indicating convexity in the discount function. Comprehending the behaviour of the discount rate and its derivatives is essential for analyzing how individuals or entities adjust their preferences for present versus future outcomes, grasping nuances in decision-making processes over time.

Impatience is a key concept in this field and refers to the preference for immediate rewards over future ones. It is defined as “the value at 0 of a $1 reward available at instant t”. The Delay Discounting (DD) task offers two options: a smaller amount of money obtainable immediately (SS) and a more considerable amount available later (LL). The normative model doesn’t specify a particular choice as it depends on the individual’s status of impatience and perception of future uncertainty. Those who choose SS over LL are considered more impatient and are more likely to discount future rewards. The existence of temporal inconsistency implies that preferences change over time, creating a gap between empirical evidence and the normative model.

The degree of impatience associated to the interval $\left[ t_i, t_j\right]$ is given by:

$$\begin{aligned} I_{\left[ t_i, t_j\right] }=1-\frac{f\left( t_i\right) }{f\left( t_j\right) } \; \;\; \; \;\;\;\; \forall t_i, t_j \in [0,+\infty ). \end{aligned}$$

(4)

Equation 4 represents the amount of money that the agent is willing to lose in exchange for anticipating the availability of a $\$ 1$ reward (Cruz Rambaud and Muñoz Torrecillas 2016).

Previous studies assumed that the discount function is exponential, which results in a constant discount rate and degree of impatience over time, in line with the assumptions of a rational decision-maker. However, empirical evidence has shown that people systematically violate the predictions of the exponential model (Rao and Li 2011). As a result, alternative functions have been developed to provide a more accurate empirical description. The main difference between the linear, exponential, and hyperbolic models in Table 1 is that while the exponential and linear functions show a constant discount rate for all future periods, the hyperbolic models present a discount rate that changes over time.

Table 1 Types of discount functions in intertemporal choice by Read and Roelofsma (2003)

Table 1 displays the limitations of models used to understand how people discount over time. These models are based on an inflexible structure that assumes a specific form for the discount function over a restricted set of parameters. They may fail to capture individual dissimilarities, as human behaviours are highly variable over time. Although hyperbolic discounting provides a helpful framework, it is essential to acknowledge its limitations. For the above reasons, we suggest using FDA as illustrated in Sect. 2.2.

2.2 The study of non-rationality in intertemporal preferences via functional data analysis

To fit available information to a discounting model, we often face the challenge of not being able to detect changes in inconsistency or the so-called magnitude effect. To overcome this problem, we suggest using FDA to fit data to a wide range of discount functions that reflect different levels of inconsistency and non-separability. By doing so, we can treat each vector of indifference pairs as a function in the time domain, rather than a finite-dimensional vector, which allows us to represent sequences of individual discrete observed data as functions and analyse them as single entities (Ventre et al. 2024).

One common method to approximate the functional form of the data is to assume that the sample paths lie in a finite-dimensional space that is spanned by a basis. This means that the function f(t) can be represented as a linear combination of basis functions as follows:

$$\begin{aligned} f_i(t)=\sum _{k=1}^K a_{i k} \phi _k(t) \;\; \;\;\;\ \; i=1, \ldots , n \end{aligned}$$

(5)

where $f_i(t)$ is the reconstructed function for the ith unit (time points could be different for each unit); $\phi _k(t)$ are linearly independent and known functions, called basis functions; and $a_{i k}$ is the coefficient that links each basis function together in the representation of $f_i(t)$. For a b-spline approximation to be considered a discount function, it must satisfy certain conditions: $f(0)=1$; $f(t)>0$; and finally, f(t) is strictly decreasing.

In order to address certain restrictions, we can use the monotone smoothing approach suggested by Ramsay and Silverman (2005). When we need to impose constraints on functions, such as monotonicity and positivity, it can be challenging to enforce those constraints using linear combinations of basis functions. Ramsay proposed a solution to this problem by transforming it to one in which the estimated curves are unconstrained. In this case, we require a fitting function f(t) that is monotonically decreasing, even if the observations do not exhibit perfect monotonicity:

$$\begin{aligned} y_j=b_0-b_1 f\left( t_j\right) +e_j \end{aligned}$$

(6)

To solve this problem, we can express Df(t) as the exponential of an unconstrained function w(t) to obtain:

$$\begin{aligned} D f(t)=\exp [w(t)]. \end{aligned}$$

(7)

Since w(t) can be positive or negative and is not constrained, we can expand w in terms of a set of basis functions. After integrating both sides of this equation, we obtain:

$$\begin{aligned} f(t)=\int _{t_0}^t \exp [w(u)] d u \end{aligned}$$

(8)

where $t_0$ represents the fixed origin for the range of t-values that the data is being fitted. The intercept term $b_0$ is equivalent to the value of the approximating function when $t=0$, i.e. $f(0)=1$. To enable the use of monotonically decreasing functions, Ramsay and Silverman (2005) suggested keeping $b_1$ separate and normalizing w(u) for numerical stability. Upon substitution, the resulting equation is:

$$\begin{aligned} y_j=b_0-b_1 \int _{t_0}^{t_j} \exp [w(u)] d u+e_j \end{aligned}$$

(9)

where w(u) is now the logarithm of the data-fitting function$.$

2.3 The augmented functional analysis of variance (fANOVA) extended to the study of intertemporal choices

In the FDA context, the functional analysis of variance (fANOVA) is used to determine if there is a statistically significant difference between sets of independent curves. Numerous statistical methods have been proposed to evaluate the null hypothesis of equal functional means. In this article, we will focus on the Ramsay and Silverman technique (Ramsay and Silverman 2005). The analysis is based on the assumption of a single factor, which involves V distinct groups $(v=1,2, \ldots , V)$ and $N=$ $\sum _{i=1}^V n_v$ observations, with $n_v$ observations within each group v.

The functional ANOVA model for the ith observation $(i=1,2, \ldots , N)$ in the vth group is expressed as:

$$\begin{aligned} \textrm{y}_{i v}(t)=\mu (t)+\gamma _v(t)+u_{i v}(t) \end{aligned}$$

(10)

Equation 10 describes the functional response of the ith curve in the vth group. The response is denoted by $y_{iv}(t)$. The grand mean function is represented as $\mu (t)$. $\gamma _v(t)$ denotes the functional effect of being in a specific treatment v. The residual function, which is the unexplained variation for the ith observation within the vth group, is denoted by $u_{iv}(t)$.

The null hypothesis posits that the groups share identical functional means, while the alternative one suggests the existence of a difference among them as follows:

$$\begin{aligned} \left\{ \begin{aligned}&H_0: \mu _1(t)=\mu _2(t)=\cdots =\mu _V(t) \\&H_1: \mu _{v'}(t) \ne \mu _{v''}(t) \quad \text {for at least one couple } (v', v'') \text { with } v' \ne v'' \end{aligned} \right. \end{aligned}$$

(11)

If ${\text {SSA}}(t)$ represents the variance between functional groups and ${\text {SSE}}(t)$ is the functional variance within groups, the point-wise functional F-statistics is given by:

$$\begin{aligned} F(t)=\frac{\frac{S S A(t)}{V-1}}{\frac{S S E(t)}{N-V}} \end{aligned}$$

(12)

where $\textrm{V}-1$ and $\textrm{N}-\textrm{V}$ are the degrees of freedom allowing us to compute MSA(t) and MSE(t).

In the same way as classical ANOVA, a high F(t) value suggests that the variance explained by the model is greater than the unexplained variance. However, the key difference between this method and traditional univariate or multivariate ANOVA is that the $\textrm{F}(\textrm{t})$ value is not constant but varies throughout the entire range (Ramsay and Silverman 2005; Di Battista et al. 2014; Di Battista et al. 2016).

The classical significance level was initially designed for testing a single hypothesis rather than a continuous spectrum. Consequently, we need to be cautious while making claims within this range. To tackle this problem, we can use the permutation test, which is functionally equivalent to the univariate F-test statistic. This notion involves computing the Fisher test statistic as a function derived from a series of point estimates for each domain point. But, to formally examine the null hypothesis of no relationship between the functional variables, we require a unique test statistic. We can derive the null hypothesis distribution by repeatedly estimating the test statistic while permuting the curves randomly and embracing the maximum value from the observed F-statistics function. In the first step, we evaluate the observed F-statistic function and obtain the observed functional F, whose maximum is essential for estimating the test’s p value. In phase two, we randomly re-label the curves with different group numbers and compute the F-statistics at each time domain point to determine the maximum of these functions. We repeat this re-labelling process multiple times, and for each iteration, we estimate the pointwise F-statistic function along with its maximum. In step three, we identify the pointwise 0.05 critical value within the null distribution at each domain point and calculate the 95th percentile of the F-statistic values corresponding to that point. This effort enables us to ascertain the critical threshold value without requiring conventional statistical tables for the F distribution. The final step furnishes us with the maximum 0.05 critical value from the null distribution, obtained by computing the 95th percentile of the distribution generated via the permutations in phase two. This technique can enable us to address the problems that arise while analysing the null hypothesis of no relationship between the functional variables. By employing the permutation test, we can create a null hypothesis distribution to help us evaluate the critical threshold value without depending on conventional statistical tables.

The approach described above can be applied to discount functions but the latter information overlooks important features that are contained in the data. Thus, this paper proposes an extension to the fANOVA approach by utilizing additional information on discount function characteristics. The original functions neglect the additional information, such as the speed, acceleration, and discount rate of the phenomenon. Therefore, we propose to extend the augmented fANOVA model proposed by Maturo and Porreca (2022) to the field of intertemporal choices.

Let the functional derivative of order r for the ith smoothed discount function be given by:

$$\begin{aligned} f_i^{(r)}(t)=\sum _{j=1}^K c_{i j r} \theta _j^{(r)}(t) \; \; \; \; \; \; \quad j=1, \ldots , K \end{aligned}$$

(13)

where $c_{i j r}$ is the coefficient of the ith curve and jth b-spline and $\theta _j^{(r)}(t)$ is the rth derivative of the jth basis function. The subscript r indicates the estimated value of the coefficient $\textrm{c}_{\textrm{ijr}}$ for a specific rth derivative. As highlighted by Ramsay and Silverman (2005), selecting the appropriate basis system is crucial for accurately estimating derivatives. The basis for representing the object must be able to handle the order of the derivative we wish to compute. In the case of b-spline bases, this requires that the order of the spline is at least one greater than the order of the derivative we intend to calculate. Using order four b-splines allows us to deal with continuos first and second derivatives.

When extending Eq. 10 to derivatives, we may also test the null hypothesis that functional groups have the same functional r-derivative means against the alternative hypothesis that there are some differences:

$$\begin{aligned} \left\{ \begin{aligned}&H_0: \mu _1^{(r)}(t)=\mu _2^{(r)}(t)= \cdots =\mu _V^{(r)}(t) \\&H_1: \mu _{v'}^{(r)}(t) \ne \mu _{v''}^{(r)}(t) \quad \text {for at least one couple } (v', v'') \text { with } v' \ne v'' \end{aligned} \right. \end{aligned}$$

(14)

Extending Eq. 10 to discount rates involves considering the null hypothesis that the functional groups have the same functional discount rates means against the alternative hypothesis that there are some differences:

$$\begin{aligned} \left\{ \begin{aligned}&H_0: \rho _1(t)=\rho _2(t)= \cdots =\rho _V(t) \\&H_1: \rho _{v'}(t) \ne \rho _{v''}(t) \quad \text {for at least one couple } (v', v'') \text { with } v' \ne v'' \end{aligned} \right. \end{aligned}$$

(15)

Consequently, we can express the point-wise functional F statistic related to the rth derivative as follows:

$$\begin{aligned} F^{(r)}(t)=\frac{\frac{S S A(t)^{(r)}}{V-1}}{\frac{S S E(t)^{(r)}}{N-V}} \end{aligned}$$

(16)

where $S S A(t)^{(r)}$ and ${\text {SSE}}(t)^{(r)}$ are the sum of squares among groups and the sum of squares of residuals based on the groups composed of the curves expressed via the rth derivatives or discount rates or even their derivatives.

If we limit our attention to $r=1$, $r=2$, and to discount rates and their first derivative, we can get specific cases of Eq. 13, which are the point-wise F-statistic based on the velocity of the curves, denoted as $F^{(1)}(t)$, the point-wise F-statistic based on the acceleration, denoted as $F^{(2)}(t)$, the point-wise F-statistic based on the proportional variation of a discount function f(t) within a standard period, and finally the point-wise F-statistic based on how the discount rate itself is changing concerning time. All this information leads to the possibility of understanding on what dimensions there are potential statistically significant differences between groups of subjects relative to intertemporal choices.

3 Application

3.1 The Hikikomory condition and the questionnaire

Hikikomori state pertains to self-imposed confinement for a minimum of 6 months. It is a condition that was initially noticed in Japan but has since spread to other countries (Kato et al. 2011; Chauliac et al. 2017; Ranieri and Luccherino 2018; Amendola et al. 2022). Official reports from Italy document the phenomenon’s occurrence when a 28-year-old man had been living in complete isolation for a period extending to 10 years (De Michele et al. 2013). The reasons for this phenomenon are complex, ranging from social and educational factors to family context and personal traits such as susceptibility, difficulty in relationships, and attachment to parents. The HQ-25 is a tool created and approved in Japan to gauge self-imposed social isolation, as defined by Teo et al. (2018). It classifies individuals into three groups: Hikikomori, pre-Hikikomori and non-Hikikomori. The tool comprises 25 questions that relate to the respondent’s experiences in the last 6 months. The questions range from a minimum score of 0 (to indicate strong disagreement) to a maximum score of 4 (to indicate total agreement). The Italian version we used in this study is based on the earlier questionnaire validated by Amendola et al. (2022) for adults aged between 18 and 50 (see the Table 3).

The model has three factors: socialisation, isolation, and emotional support, which are positively correlated with psychoticism, personality dysfunction and Internet use. The total score is the sum of the responses, with a maximum of 100 points. Amendola et al. (2022) showed that with a sensitivity of 94% and a specificity of 61%, a cutoff score of 42 discriminated between subjects at risk of Hikikomori and those not at risk and therefore, this previous study divided the sample into two categories concerning the latter cut-off.

In the literature, some of these disorders have been investigated for inter-temporal discounting and found significant results for disorder discrimination (Steinglass et al. 2017; Nicolai and Moshagen 2017; Lopez-Guzman et al. 2019). However, a previous investigation found no significant differences between Hikikomori groups concerning the decision-making dynamics involved in the inter-temporal choices (Ventre et al. 2024). We are driven to investigate this relationship further to understand whether significant differences exist between the various groups at different dimensions. At the same time, we aim to make the anlysis more robust by deleting functional outliers before assessing the different functional dimensions.

The questionnaire was carried out via the vercel.app platform, from which the data was subsequently downloaded and converted into an Excel file. The users who experienced were anonymous and only provided their gender and age (18 years or older). The questionnaire consists of two parts, preceded by an introductory paragraph explaining how to answer. The first part regards calculating the individual discount functions, while the second aims to determine the Hikikomori scores (see Table 2). The questionnaire link was shared on digital platforms and social channels, targeting Italian individuals willing to fill it in voluntarily to diversify data collection as much as possible. After opening the link, people were asked to accept informed consent before participating in the investigation. The first section of the questionnaire consists of two alternating inquiries: the first query collects the values needed to form the empirical discount functions, and the second question is intended to confound the respondent. The second section of the questionnaire consists of 25 questions aimed at defining the Hikikomori scores. Respondents scored each question from 0 to 4, and the Hikikomori score is obtained by counting up three sub-scores: socialisation, isolation, and emotional support. Some questions have a reverse scoring scale. The sample is composed of 271 individuals, of whom 74 are men (27.31%), 195 are women (71.96%), and two individuals with a different gender identity (0.74%). The average age is 21.56 years, and the average Hikikomori score is 35.09. Employing a score threshold of 42, as indicated by Amendola et al. (2022), 169 individuals (62.36%) are classified as non-Hikikomori/not at risk of Hikikomori (NH), while 102 individuals (37.64%) are classified as Hikikomori/at risk of Hikikomori (H).

Table 2 Questionnaire used for the interpolation of the individual discount functions

3.2 Descriptive statistics of the Hikikomori dataset

The following is a summary of the results obtained from the data collected. As seen in Fig. 1, the median of the distributions assessed at each time instant decreases over time, following a hyperbolic trend. In Fig. 1, we observe how the asymmetry of the distributions increases as time progresses. The data concentrates more on smaller values of the discount function, which is also what we expect when dealing with intertemporal choices. Nevertheless, Fig. 1 shows that a survey without functional data analysis provides limited insights. To gain a deeper understanding, the FDA is applied.

Fig. 1

Violin plots of f(t) values on different days

Figure 2 illustrates the original smoothed discount functions, the centred discount functions, and the first and second derivatives. Each of these graphs can provide additional information that the others cannot capture with the same intensity. Figure 2b shows the discount functions centred to the functional mean. Figure 2b is particularly interesting because it allows us to understand how each individual behaves in the same intertemporal period as differences from the average of the sample. The graphs of the first derivatives (Fig. 2c) and the second derivatives (Fig. 2d) of the discount functions instead allow us to gather additional information on the speed of the discount functions in the different parts of the time domain and on their acceleration. Naturally, based on this information, the behaviour of each individual’s intertemporal choices can be evaluated in different time windows. Figure 2e and 2f show the discount rate and discount rates’ first derivatives for each subject.

Table 3 The Italian version of the 25-item Hikikomori questionnaire (Amendola et al. 2022)

Fig. 2

Discount functions, centred discount functions, first and second derivatives, discount rates, and discount rates’ first derivatives

At this point, we focus on the intertemporal choice behaviour of groups of individuals characterized by a similar Hikikomori score. The severity score of each individual was categorized into many classes ranging from three to five to analyze the discount functions. This purely descriptive analysis actually allows us to have a preliminary idea of whether there may be differences between groups based on the severity of the physical mortality score established by the questionnaire. Naturally, this is only a preliminary descriptive analysis because a statistical test is needed to establish whether there are statistically significant differences.

Figure 3 proposes three charts. Three types of groupings are proposed based on different thresholds for each dimension, according to the severity of the Hikikomori condition assessed by the test score. The first subdivision (Fig. 3a) is based on three classes, which are divided as follows: the first group is composed of those who have a score lower than 42, the second class is composed of those who have a score between 42 and 60, and finally the third class is made up of those who have a score above 60. Instead, Fig. 3b considers a subdivision into four groups. The first group is made up of those who have a score lower than 42 while the second group is made up of respondents with a score between 42 and 60; the third group has a score between 61 and 70 while the last group is composed of those with a score greater than 70. Figure 3c is based on a five-group grouping where the thresholds are 20, 40, 60 and 70. From Fig. 3a, it can be observed that the High category changes its concavity twice: the first time within 14 days and the second one at around 90 days. This means that the impatience of this group could vary, alternating between intervals in which impatience is increasing and others in which impatience decreases. By increasing the number of classes, in Fig. 3b, a distinction emerges between the High class and Very High class, which is characterised by a greater steepness in the first period. The discount function associated with this class decreases to about 0.2 in the first seven days, while the other functions reach the same no earlier than 45 days. This could indicate a very pronounced impatience of the Very High class. Finally, Fig. 3c does not add any particular information to what has been discussed so far, confirming the idea that the peculiarities of the curves emerge as the degree of Hikikomori syndrome increases.

Fig. 3

Discount functions represented in different groups according to the Hikikomori score

Figures 4 and 5 propose the same type of reasoning but focus on the first and second derivatives. The first derivatives of the discount function are always negative because the discount function is always decreasing. However, the second derivatives can be both positive and negative because the change in concavity is possible using FDA. In these figures, it is evident that the functional means take on different forms as the Hikikomori condition increases. However, establishing whether these differences are statistically significant is not possible without a statistical test. It can be observed that even concerning the first derivative, the peculiarities of the curves increase as the division into classes increases. In Fig. 4c, the High and Very High classes are characterised by their trends, especially in the first part of the period. On the other hand, there is no symmetrical behaviour for the second derivative between the High and Very High classes, but the first maximum point increases as the Hikikomori score increases.

Figure 6 proposes three graphs which, as previously, provide three types of groupings based on the severity of the pathology, but this time, the object of interest is the discount rate. Interestingly, all types of grouping on this dimension are characterised by high variability of the individual curves and very different shapes of the functional averages if compared to the original functions and the derivatives comparison. In Fig. 6b and 6c, there are two distinct groups present. In particular, the blue and green groups in Fig. 6b and the light blue and blue groups in Fig. 6c have functional forms that are noticeably different from the other groups. When we focus on the discount rates and look at 4- or 5-group classification, it seems that individuals with a high level of Hikikomori have less marked short-term fluctuations. In Fig. 7, we can observe some peculiarities similar to those noted for the discount rates. Two distinct groups are very different from the others. For instance, the green and blue curves in Fig. 7c display a much wider oscillation in the average of the first derivative of the discount rate when compared to the other groups. The observations made concerning the discount rate in Figs. 6 and 7 combined with information observed in Fig. 3 allow us to consider two fascinating phenomena: a change in the concavity of the High class discount function occurs on the first peak of the discount rate; and there is a greater decrease in the impatience of the Very High class in a period in which fluctuations in the discount rate of this class are less pronounced than for the other classes.

Fig. 4

First derivatives represented in different groups according to the Hikikomori scores

Fig. 5

Second derivatives represented in different groups according to the Hikikomori scores

Fig. 6

Discount rates represented in different groups according to the Hikikomori scores

Fig. 7

Discount rates’ first derivatives represented in different groups according to the Hikikomori scores

3.3 Results of the augmented functional analysis of variance

To determine whether the differences observed in Sect. 3.2 are statistically significant and interpret the results, it is necessary to use a non-parametric statistical test such as A-fANOVA.

Figure 8 presents the results of the A-fANOVA model applied to the groups shown in Figure  3, that is, in cases where we have a subdivision into three, four or five groups based on the severity of Hikikomori syndrome evaluated through the questionnaire. The analysis of variance on the original curves already gives slightly different results than the study conducted by Ventre et al. (2024). This result is the simple consequence that the functional outliers that could compromise the statistical test results have been eliminated. In addition, the number of permutations has been increased to 400. As can be seen, when considering the three groups, the A-fANOVA results show that the functional F statistic crosses the critical point-wise level in the period between four days and seven days. This indicates a significant difference between groups in that part of the domain, as shown in Fig. 3a. In other words, the functional mean of those with high Hikikomori (green curve) differs from the functional averages of the other two groups. However, if we aim to decide based on the whole domain, we cannot state that there is a significant difference overall because the F-test statistic does not exceed the maximum critical value level. The four and five-group analyses highlight that the observed functional F statistic lies below the point-wise critical level and maximum one. In all three cases, we have enough evidence in favour of the null hypothesis. However, looking locally at Fig. 3a, the interval in which the difference is significant coincides with the interval in which the High class curve reaches a maximum distance from the other two groups before intersecting. Thus, the intersection in Fig. 8 indicates that the intertemporal preferences expressed by the High class are different from those expressed by the other groups in that small part of the time domain.

Figure 9 presents the results of the A-fANOVA model applied to the groups shown in Figs. 4 and 5, that is, in cases where we have a subdivision into three, four or five groups based on the first and second derivatives. The study of the differences between groups based on a subdivision into three groups and using the derivatives of the discount functions provide very interesting results because the functional F-statistic crosses the point-wise critical level in two parts of the domain after two days and in the period ranging from 14 to 20 days. The circumstance is associated with what we observed for the original functions from the fourth to the seventh day. Trying to interpret this result, also based on the functional means presented in Fig. 4a, we notice that, in the first part of the domain, the green curve representing those who have a high level of pathology is more negative than that of the other groups because the discount functions decrease faster than the others. Starting from the fourth day, on the other hand, the rate of decrease of the discount functions distinguished by the first derivative tends to slow down and becomes less marked than the rate of decline of the other groups (see the green functional mean in Fig. 4a). The latter indicates that the discount functions of people with a high Hikikomori score make faster oscillations. However, the analyses conducted using four and five groups and all the A-fANOVA that utilized the second derivatives indicated that the observed functional F statistics were below the point-wise critical level and the maximum one. There was sufficient evidence to support the null hypothesis in the three cases.

The results of the A-fANOVA model applied to the groups presented in Figs. 6 and 7 are shown in Fig. 10. These groups are subdivided into three, four, or five sets based on the discount rates and their first derivatives. The study on discount rates turns out to be particularly interesting. As can be seen in the three charts in Fig. 6 relating to the discount rate, the functional F (blue curve) crosses the critical threshold (dotted red curve) in the period from the second day to the tenth day. To understand this difference, we can look at the three images in the first column of Fig. 10. It is clear that regardless of whether the grouping is into three, four, or five, some groups have functional forms that appear to be significantly different from those of the others. For example, discount rate curves of different pathology levels show a unique trend compared to other groups. In the first part of the time domain, the Hikikomori individuals have a higher discount rate than the other groups. In the central part of the domain, they have a lower discount rate than the other groups, and in the final part of the domain, they have a higher discount rate than other groups. Also, in this case, if we want to test the whole domain, we would still have to accept the null hypothesis; nevertheless, it can be stated that there is a significant difference in some parts of the time domain. However, it is exciting to detect that a range of significance for the discount rate is identified for all three subdivisions. In particular, as the number of classes increases, the interval narrows and concentrates around D4. Observing the representation of the discount rates in Fig. 6c, the significance in that part of the time domain confirms a symmetrical attitude in the High and Very High classes, as opposed to the various oscillations of the other groups. On the other hand, as far as the study of the derivatives of the discount rates is concerned, there are no particular differences between the groups (see the second column of charts in Fig. 10).

Fig. 8

Permutation functional F-tests on the original curves for three, four, and five groups based on the Hikikomori scores

Fig. 9

Permutation functional F-tests on the first and second derivatives for three, four, and five groups based on the Hikikomori scores

Fig. 10

Permutation functional F-tests on the discount rates and first derivatives of the discount rates for three, four, and five groups based on the Hikikomori scores

4 Discussion and conclusions

This study aimed to tackle the issue of intertemporal choices by using FDA. The reason for combining FDA with the study of the discount function was to use more detailed methods of obtaining the individual discount functions without strict assumptions and improve the understanding by investigating the additional features obtained by an FDA approach. In other words, an essential advantage of the proposed approach is that, unlike a classic multivariate ANOVA, no assumption should be respected on the distribution, such as normality and homoscedasticity. As a result, the data in this study provides a more detailed knowledge and interpretation of individuals’ behavioural and cognitive aspects.

This study proposes to extend the previous research conducted by Ventre et al. (2024) that found no statistically significant differences between Hikikomori and non-Hikikomori at the original curves level. In particular, building upon the aforementioned idea and recognizing the connections between the phenomenon of Hikikomori and psychiatric disorders (Pozza et al. 2019), depression (Kato et al. 2011), and Internet addiction (Tateno et al. 2019), our objective is to delve deeper into investigating the behaviour of the discounting function in different groups of subjects based on the gravity of their Hikikomori condition and different functional dimensions. This pursuit is motivated by the extensive documentation in the scientific literature regarding the correlation between the behaviour expressed in the delay discounting task and conditions such as addiction (MacKillop et al. 2011), depression (Takahashi et al. 2008), and ADHD (Wilson et al. 2011; Scheres et al. 2008, 2013). The starting point of this paper is that the study conducted by Ventre et al. (2024) may have a limitation as it only focused on the original curves, ignoring additional functional dimensions that may contain information beyond that present in the initial discount functions. Therefore, the primary objective of this study is to investigate whether additional functional features can lead to different conclusions when assessing the role of the Hikikomori pathology in intertemporal choices. Additionally, this study focused on some details of the experimental phase, such as increasing the number of permutations and removing functional outliers that may have led to biased analysis in Ventre et al. (2024).

The additional characteristics considered in this study are the first derivatives of the discount functions, the second derivatives, the discount rates, and the derivative of the discount rates. All these components in the time domain have an exact interpretation in the context of intertemporal choices, and for this reason, they are of predominant interest. Experimental analysis through the A-fANOVA shows exciting results that could not be found using the original functions alone. The comparison between groups based on the first derivatives and discount rates shows significant differences in some parts of the domain. This circumstance indicates that Hikikomori individuals in some parts of the time domain have behaviour that differs from individuals not affected by this pathology. However, this difference is not statistically significant across the whole domain. The peculiarities found in some dimensions are pretty exciting and deserve more in-depth analysis by medical specialists. In particular, some interesting situations emerge in this study. In practice, from the analysis of the functional means of the groups in each of the five dimensions, differences in the patterns are immediately noticeable. Still, the A-fANOVA never produces statistically significant results across the entire domain. To fully understand this phenomenon, we must ask ourselves about the test to establish whether the differences between groups are statistically significant. Indeed, the functional F statistic, as in the classical case, is composed of a numerator which indicates the amount of variability between the groups and, therefore, involves the differences between the functional means of the groups and the general mean. Instead, in the denominator of the F statistic, we have the functional variability in the individual groups, which involves the differences between the functions in the respective groups and the functional average of the particular groups taken individually. Therefore, high functional variability in one or more groups contributes to a high value of the denominator of the F statistic. For this reason, although there are visible differences between the functional means of the groups, which we find in a high numerator of the F statistic, an equally high value of the denominator due to high variability in the groups contributes to contrasting the numerator and thus over the entire domain, the test is not significant although some differences are noted at the local level.

Our results remark that the variation in the value assigned to the future by the Hikikomori group compared to other groups with respect to the discount rate curves is very interesting and worth future investigations. At the beginning of the time domain, the higher discount rate of high-level Hikikomori subjects could indicate a preference for immediate gains or difficulty planning for the future. This may be related to their tendency to isolate themselves from society. In the central part of the time domain, the discount rate is lower than in the other groups, which could indicate an increase in consideration for the future or a change in values and perspectives. There may be some sort of reflection or emotional growth that leads to greater consideration for the future or simply confusion. Finally, at the end of the time domain (after 45 days), the discount rate functional mean of people in the more severe Hikikomori class is once again higher and could be due to a series of factors, such as a feeling of loss of control over the future or a renunciation of hope for an improvement. This interpretation could suggest an evolution in how Hikikomori individuals view time and the future, with variations in their attitude over time. However, it is essential to underline that this is only one possible interpretation and that understanding the Hikikomori phenomenon would require more detailed and in-depth research in psychology and sociology.

The present work made it possible to highlight features of the discount function and, in general, intertemporal choices that a-priori could not easily be questioned. In particular, the graphical representation of the discount rate in Fig. 6a shows that in the interval in which the difference between the three classes is significant, the discount rate of the High class is characterised by smaller fluctuations. This observation, combined with the information described in Fig. 3a opens the way for possible future studies that address an important issue that this paper has highlighted: the relationship between a discount rate with smaller fluctuations in a period and an impatience that could change its sign.

The study is fascinating both from a methodological and practical perspective. It’s worth noting that the data collection process relies on collecting information from individuals who have volunteered to participate online. As a result, it’s important to remember the potential for self-selection bias, which could impact the accuracy of the results obtained through statistical tests used in the study. Future studies could focus on experimental research that, through randomisation, can eliminate bias due to possible confounding variables. Another very interesting line of research is definitely trying to build models of forecasting the profiles of investors identified with discount functions treated with the FDA through statistical learning techniques and provide exciting interpretability and explainability metrics (see, e.g. Maturo and Verde 2022, 2023, 2024).