Mathematical Analysis and Modeling of the Factors That Determine the Quality of Life in the City Councils of Chile

Abstract

The quality of life index is an indicator published yearly since 2010 by the Institute on Urban and Territorial Studies and the Chilean Chamber of Construction, involving 99 municipalities and communes from the national territory. This research provides an approach to understanding how various dimensions and variables interact with quality of life in Chilean communes considering multiple factors and perspectives through information from public sources and social indicators. For the research, variables were analyzed considering demographic, sociodemographic, economics and urban indicators, where the model developed allows for an understanding of how the variables are related. In addition, it was discovered that education, own incomes, municipal spending and green areas directly relate to quality of life, while overcrowding and municipal funds negatively affect rates of communal welfare. Moreover, the variables chosen as explanatory variables allow for the development of an efficiency model. For this purpose, Cobb–Douglas and trans-logarithmic forms were tested, and it was found that Cobb–Douglas fits better to the data set and structures of the variables. The results of the efficiency model show that education, municipal funds and own incomes significantly affect efficiency, with a mean value of approximately 47%, minimum values close to 30% and maximum values of approximately 60%. Finally, a cluster analysis was developed through k-means, k-medoids and hierarchical clustering algorithms, where, in all cases, the results were similar, suggesting four groups with differences and variations in analyzed variables, especially in overcrowding, education, quality of life and wellness.

1. Introduction

Towns and their municipalities are elements that, when taken all together, form a nation or a state, each one having certain levels of independence and autonomy with respect to the central government. The focus pursued by local municipalities is to look after the needs and problems that afflict the citizens, understanding that, given the diversity in various factors of a nation, not all citizens are afflicted by the same needs at the same time within their daily lives [1,2].

Not only is it necessary to understand the problems and needs from the population, but also the design of projects and solutions should be focused on looking after an optimal usage of resources, with smart planning and the allocation of supplies. Given both situations, the solution design should be accurate with a proper understanding of the gaps in which the resources must be allocated, increasing and optimizing the outcomes and urban welfare [3,4]. Due to the above, tools that allow for an understanding of the gaps and situations of a population were designed, allowing policy design making in an optimal and accurate way.

One way to understand and measure the problems and gaps that afflict towns is through a wellness assessment in citizens, by considering the variables interacting with people within their daily routines [5,6]. This kind of analysis helps in understanding the differences among geographical zones and regions, developing policies and projects that make sense with various contexts and situations that each sector of the nation are afflicted by at one point.

Chile is a nation with a strong centralist tendency, in part due to having centralized political power, which means that most of the policies are established according to the needs of the Metropolitan Region and its capital city, Santiago [7,8], without considering other regions and regional branches of government that require solutions according to their particular and unique problems in terms of the enormous variety of geographic and demographic characteristics of the country.

The main goal of this analysis is to determine the factors that affect the urban wellness in Chilean municipalities considering indicators of the quality of urban life, which comprises approximately 100 municipalities across the country, involving around 80% of the Chilean population. It is worth mentioning that, currently, the publication and reports on quality of life developed for various councils and towns in Chile are focused on measuring and developing the index, but not on analyzing interactions among indexes and variables involved with the population’s daily life. Due to the above, it is hard to know which are the dimensions that are most important for explaining or improving quality of life.

2. Literature Review

2.1. Quality of Life

Quality of life and urban wellness are concepts that have received special attention over time, including studies that relate these concepts to international development, health benefits, education and social interactions in towns for a better understanding of different phenomena analyzed. These kinds of studies have a direct impact and implications on policy development, especially in terms of smart planning in urban environments, considering the effectiveness and efficiency in resources and the supply distribution for problems and needs of citizens [9,10].

Studies and assessments on quality of life are reported across a wide range of regions throughout various organizations whose purpose is to analyze and study indexes related to urban welfare. In Europe, the European Commission develops an annual report on quality of life considering approximately 83 countries from the European Union, the analysis of which includes variables related to satisfaction levels with the city, insecurity, job opportunities, purchasing power, mobility through the city, culture, parks, health and perception on public management [11]. Each one of these variables is weighted in order to obtain a level on the quality of urban life according to (1).

$${\mathrm{QoL}}_i=\sum _{j=1}^n{p}_{i,j}\times {\alpha }_{i,j}\times 100.$$

(1)

Other indexes on quality of life are published by organizations such as Numbeo, which is a platform whose purpose is to deliver information about the cost of living, urban quality and other characteristics related to urban life in various countries and cities. The way of making the assessment on urban welfare is similar to the previous formula seen in (1), considering variables such as purchasing power, real state development, living cost, security perception, mobility, pollution and environmental indicators. These variables are weighted in a similar way to (1) in order to obtain an index on the level of urban welfare and quality of life.

In general, the assessment of quality of life is a subject with no clear consensus. Plenty of research on this topic can be found with different variables and dimensions considered in the analysis. Puertas et al. (2020) [12] developed this index for cities in 23 European countries, taking into account variables such as alcohol consumption, education, happiness, GDP, public investment and environmental investment. Macke et al. (2018) [13] developed and analyzed the interaction in variables related to urban welfare for cities in the south of Brazil, considering variables such as security, efficiency in public services, urban development, environmental factors, green areas, air quality and various economical factors. Somarriba and Zarszosa (2019) [14] studied the quality of life in 28 European countries, looking into the understanding of the interactions among variables, including in their studies variables related to gender, population age, children, salaries, education, child mortality and factors related to relationships in family environments. Although the variables considered seem to be different for each study, some of them are repeated throughout the analysis, where education, job opportunities and environmental conditions appear more frequently and are more considered for researchers while studying this kind of phenomenon.

2.2. Quality of Life in Chile

From 2010, the Chilean Chamber of Construction and the Institute of Urban and Territorial Studies UC developed an index for the quality of urban life considering various aspects and elements from Chilean towns, its main purpose being to look into opportunities and gaps in the perception of quality of life and urban welfare across the cities, working as a tool for improving the design and development of projects and policies for the needs of Chilean citizens. The index considers variables from dimensions such as job opportunities, sociocultural conditions, mobility, health, natural environment, real state development and urban conditions on cities, where all of these variables are obtained from surveys in communes involved. Later, through principal component analysis, the amount of variables is reduced and weighted into six main dimensions that are weighted the same for the calculation of the quality of life index [15]. For the year 2021, the results on the quality of life are shown in

Table 1. Level of perceptions on quality of life—QoL Chile 2021.

Satisfaction Level	Propotion (%)
High	13.2%
Mid—High	24.2%
Mid—Low	18.2%
Low	44.4%

Source: Report on Quality of Life Index, 2021.

, with approximately 44% of towns showing a low level of perception on urban welfare and only 13% of towns having a high level on this index.

Figure 1 shows the quality of life index for different Chilean regions: Figure 1a shows communes of northern Chile, where the levels of quality of life are approximately 45%; Figure 1b shows communes of central Chile—excluding the Metropolitan Region, which is shown in Figure 1c—where the values of quality of life are approximately 50%, which is slightly better than northern Chile; Figure 1c shows values for the Metropolitan Region and its capital city (Santiago), where this region has the highest amount of communes included in the quality of life index and, moreover, is the only region that includes communes with a quality of life close to 80 points; Figure 1d shows the index for southern Chile, which is similar to central Chile, where the values are approximately 50%, with fewer communes included in the quality of life index.

2.3. Green Areas

Green spaces, parks or green areas in urban environments are considered as a resource necessary not only for measuring the quality and welfare of urban life, but also as a way of determining the development level of a city [16]. The impact that green areas have on quality of life can be considered from different points of view: some researchers focus on benefits in health for citizens through the interaction among people in parks, considering these spaces as a source of social interaction, whereas others consider the impact on climate change and urban development for wellbeing [17,18].

The COVID-19 breakout led the population to live in lockdown for long periods, with short periods of social interaction, the above derived from the measures that the government developed in order to avoid SARS-CoV-2 spreading among people. During this period, some studies focused on analyzing how these spaces contributed to the mental health of the population. In Krakow, Poland, data from surveys showed that approximately 75% of participants considered that parks and green areas had a significant impact on a reduction in stress and anxiety levels generated from isolation and lockdowns [19]. Researchers from the University of Bari Aldo Moro in Italy showed how the perception of quality of urban life is directly related to green areas in urban environments [20], concluding that small increments on green areas or parks can significantly improve the perception on urban welfare, especially in people with low incomes.

2.4. Population Density

In recent years, it is common to find studies that relate population density to quality of life. Some researchers are focused on analyzing how density relates to environmental quality and, therefore, its relationship with urban welfare. Ahmad et al. (2005) [21] and Zuhri (2014) [22] developed studies in Indonesia and Pakistan, respectively, finding out that population density is related to pollution and air quality, negatively affecting environmental indicators and decreasing perceptions on urban welfare. Fassio et al. (2012) [23] developed a research in north Italy with the purpose of measuring the relationship between population density and quality of life. The initial hypothesis is based on various previous analyses, where, in Norway, people living in cities with a high population density tend to have lower perceptions in various dimensions of their daily lives [24], whereas, in Italy, people living in cities with a low population density show higher levels of quality of life and mental health [25].

2.5. Domestic Violence

Domestic violence is a phenomenon that afflicts families from any country, religion, belief or economical condition, its impact affecting not only the relations between those who are involved directly, but also those who are witnesses of these acts. Lucena et al. (2017) [26] showed how high levels of quality of life correlate to high levels of security and family insertion and lower odds of suffering from domestic violence according to surveys conducted for women in Brazil. Alsaker et al. (2018) [27] analyzed data from surveys for women in Norway, finding a significant correlation between domestic violence indexes and quality of life. The results agree with previous studies developed in Norway with women who suffer from any kind of domestic violence in their homes [28].

2.6. Overcrowding

The population is growing at a fast pace: it is estimated that, for the year of 2050, the population will reach 9.7 billion inhabitants in the world [29]. This number represents different levels of threats for different regions, Africa being one of the continents with the highest population growth and most populated cities in the world. Overcrowding is defined by the World Health Organization (WHO) as the average living area per person within a house, whereas other organizations, such as Eurostats, define overcrowding as the number of inhabitants per room within a house [30]. Both definitions are part of the same concept but consider different calculation methods.

Overcrowding has been studied for explaining and understanding relationships among communities and families from various countries. Researchers from the University of Vanderbilt in USA showed that overcrowding is directly related to bad relations among people living in the same house [31]. For children in Nigeria, overcrowding is related to high levels of stress, a bad performance in schools and aggressive behavior with their friends or classmates [29,32]. Ruiz and Urria (2020) [33] conducted research with Chilean families, looking for a correlation between overcrowding and mental health and depression. The results show that there are dynamic relations noted by the variables, with behavior similar to being unemployed or having a chronic illness.

2.7. Education

According to the International Network for Education in Emergency Situations, the level of education of a person is determined by a set of courses and study programs that a person has completed, an academic degree being a way of showing dexterity and knowledge in a particular topic [34]. Through the years, education has become one of the main predictors that explains variables for people’s development, which is a factor that determines job opportunities, income and social status, and is a strong predictor for wellbeing and quality of life [35,36,37]. Pascarella and Terenzini (2005) [38] showed a strong relation between education and job success, which leads to rewards in terms of status, higher incomes and higher perceptions of quality of life and social status.

2.8. Employability

Employment is a key element for the healthy development of citizens within societies and communities. This factor constitutes a general basis for the economy. In addition, employment works as a cohesive force in society, allowing people to interact with social systems, delivering opportunities for all people in communities. Thangiah et al. (2020) [39] developed research with data from semi-rural towns in Malaysia and found that quality of life directly correlates with employment and job opportunities, its results being similar to other studies conducted in the UK and Germany that analyze interactions between employment and different dimensions of quality of life [40,41]. Magnano et al. (2019) [42] showed how employment directly correlates to quality of life for people in Italy. Moreover, employability tends to lower stress levels and concerns about being unemployed for significant periods of time.

2.9. Sources of Municipal Incomes and Municipal Expenses

Public spending and social projects referring to health, education, environment or social development play a fundamental role in the economy of a nation, since these elements allow us to distribute and assign resources to citizens from various social statuses. In many countries, this spending represents approximately 30% of the economic demand [43], it being a main actor in policies whose purpose is to improve quality of life and urban welfare. Haile and Niño-Zarazúa (2018) [44] showed how public spending and social policies have a causal effect on human development and human welfare, while Mafrolla and D’Amico (2016) [45] studied the impact of policies related to the leisure time of citizens in Italy, showing that these kinds of projects have a direct impact on quality of life while lowering stress levels.

The sources of municipal income are various and depend on the country and its policies about resources allocation. For instance, the United Kingdom concentrates its sources of municipal income in government grants, municipal fees and commercial taxes [46]. Spain concentrates its municipal income in local fees and government grants, with other sources of income from self-assets and taxing autonomous towns and local entities [47]. Chile has its sources of municipal income split into own incomes, municipal common funds, transferred payments and other incomes [48]. The first one corresponds to incomes derived from activities developed within the towns, especially the ones related to economics and industrial terms, whereas the second one corresponds to a common fund that is distributed among towns, working as a way of reaching a certain level of budget when its own sources of income are scarce or not enough.

2.9.1. Own Incomes

Own incomes correspond to incomes derived from economic activities within the region, the sources of which are various, such as finding taxes, patents, property rights, driving licenses, penalty fees and other permissions granted [49]. According to the Public Studies Center, own incomes correspond to approximately 46% of municipal resources. The most expensive activities are municipal patents (13%), territorial taxes (12%) and transfers among local governments (12%) [50]. It is necessary to mention that these kinds of incomes are regulated by law N°18695, which establishes what kind of income can be part of municipal assets.

2.9.2. Municipal Common Fund

The municipal common fund corresponds to a fund established in 1979. The main purpose of this fund is to distribute the resources earned by various municipalities through the country in an equal way, working as a tool for reducing the fiscal and financial gaps among towns [51]. This fund is regulated by Law N°20237, which is about municipal rents. In addition to the regulation previously mentioned, the distribution of this common fund is established by DFL N°1 2006, which establishes an order for the distribution of the fund with proportional weights depending on needs and gaps of each municipality.

2.9.3. Cluster Analysis

Various researchers have been working with these kinds of algorithms, focused on finding clusters in regions per continent, country or particular zones within regions given a set of features. Zmuk (2015) [52] developed an analysis with hierarchical clustering methods with variables such as quality of life, purchasing power, security, health indexes, CPI, real estate development and pollution in order to find similarities and groups in European Countries, and it was found that old European Union members seem to have a higher quality of life level on average than other countries in the region. Hirschberg et al. (1991) [53] analyzed groups in various countries from different regions, considering a total of 23 variables, taking into account dimensions such as political rights, civil freedom, life expectancy, literacy levels, education and GDP. Differences and groups with significant variations among variables considered were found.

3. Methodology

Studies, research and analyses of various phenomena are usually carried out through econometric theory and its modeling tools, especially those useful for variable selection and for understanding the relations and behavior among variables. Efficiency modeling is carried through an economic perspective, specifically from a microeconomic one, considering aspects for measuring the efficiency: inputs (independent variables) and the output (dependent or response variables).

Figure 2 shows a proposed methodology for the analysis, where the first step on the literature review is developed in Section 2, exploratory data analysis and preprocessing are conducted in Section 3.1, focusing on univariate behaviors, variable transformations and correlation from predictors and response variables, Section 3.2, Section 3.3, Section 3.4, Section 3.5, Section 3.6 and Section 3.7 correspond to model development, multivariate regression analysis, efficiency and cluster analysis and, finally, Section 4 focuses on conclusions and a comparative analysis on the results.

It is worth noting that Chile has 346 towns in its territory and that the analysis was applied to 99 of those towns. Although the proportion of towns used for carrying the analysis is approximately 29%, in terms of population, the towns analyzed represent 80% of the Chilean population due to the fact that the quality of life index is developed for towns with approximately 50,000 inhabitants. Another reason for choosing the 99 towns, besides the population, is the change in sources of information related to variables for each town, making it difficult to track the characteristics and variables from public sources in time. Moreover, there is a lack of formal periodicity and data for publishing reports, especially for towns far from metropolitan areas. It is worth mentioning that, besides the 99 towns analyzed, all of the regions are involved in the research, meaning that there is at least one town per region.

The response variable corresponds to the quality of urban life index (QoL) interacting with 19 independent variables. The data are extracted from various sources of information corresponding to reports from the Library of the National Congress of the Republic, Statistics Institute, Urban Development Minister, Center for Studies and Analysis of Crime and the results of the CASEN survey, all published in 2022 but related to characteristics of 2021 due to a delay in report publishing; thus, the quality of life and variables for each town correspond to the same period of time. The variables considered for the analysis were the quality of life index (response variable), population density, poverty, municipal common fund, own incomes, green areas, green area per inhabitant, % of students with more than 450 points on PDT (education), health centers, inhabitants per health center, domestic violence, domestic violence per 100,000 inhabitants, overcrowding and other variables related to municipal expenses corresponding to municipal activities, self-management, cultural programs, leisure programs, social programs and community services.

The model development stage was carried out through variable selection criteria, considering AIC, BIC and stepwise regression with backward and forward iterations and a mixture of those two. Once the variables are selected the model must be validated in terms of its structure and variable behavior considering the significance of the model, significance of each ${\beta }_i$, collinearity among regressors estimating ${\mathrm{VIF}}_i$ and ${\mathrm{TOL}}_i$ and residual analysis for validating the output.

With the variables chosen, the efficiency model was developed by testing Cobb–Douglas and trans-logarithmic functional forms through an LR test. As previously mentioned, the efficiency was developed considering that quality of life is the output of the functional or production form, while the features chosen for the econometric model (explanatory variables) were considered inputs for the production function.

3.1. Exploratory Data Analysis

Exploratory data analysis was conducted considering four groups from the variables. The first one corresponds to a response variable (quality of life), the second one corresponds to demographical features, the third one corresponds to sociodemographic variables and the fourth group corresponds to municipal expenses and public budget.

Sources of information are various: the quality of life index was obtained from a report on the quality of life 2021; poverty, municipal common fund, own incomes and all variables related to municipal expenses (municipal activities, self-management, cultural programs, leisure programs, social programs and community services) were obtained from urban development minister reports; population density, overcrowding and health centers were obtained from the Library of the National Congress of the Republic; domestic violence was reported by the Center for Studies and Analysis of Crime; education was reported with results on the CASEN Survey; and population data were obtained from the Statistics Institute.

In some figures or tables, variables are abbreviated only for visualization; thus, sometimes, the quality of life will be referred to as QoL, population density as density, municipal common fund as MCF, own income as OI, municipal expenses as MEs, self-management expenses as Se.Es, cultural expenses as Cu.Es, leisure expenses as LEs, social expenses as So.Es, community services as Co.Es, green areas as GAs, inhabitants per green area as IGAs, % of students with more than 450 points on PDT as education or Ed, health centers as HCs, inhabitants per health centers as IHCs, domestic violence as DV, rate of domestic violence as RDV and overcrowding as Oc.

3.1.1. Quality of Urban Life

The quality of life index is represented as a value greater than 0, where 0 represents the worst urban welfare. There is no established maximum value for the index, but the higher the value, the better the quality of life. The quality of life (QoL) shows a mean of 50 points and a median of 49 points, with minimum values of approximately 39 points and maximum values close to 80 points. According to histogram (a) in Figure 3, it is possible to appreciate the presence of outliers in the right tail of the curve, which are shown in the boxplot (b) of Figure 3. The distribution of values is concentrated between 40 and 60 points.

3.1.2. Variable Transformation

Before analyzing the independent variables, it is necessary to transform some of them in order to fix the behavior of the distributions. The transformation chosen for this purpose corresponds to a logarithmic transformation shown in (2), beside other ones existing, such as root square, inverse transformation or Box–Cox transformation. One of the benefits of choosing log-transformation is the fact that it allows us to center the distribution of the variables adding symmetry to the form, especially in right-skewed distributions, obtaining properties close to the ones required for a better fit of some regression models [54].

$$X_i^{{}^{\prime }}=\ln \left(X_i\right).$$

(2)

The variables transformed with logarithmic transformation are the population density, municipal common fund, own income, municipal expenses, self-management expenses, cultural expenses, leisure expenses, social expenses, community expenses, green areas, health centers and inhabitants per health center. Figure 4 shows distributions on variables before and after log-transformation, where distributions are centered by applying such a transformation, adding symmetry and correcting the initial right-skewed form.

3.1.3. Demographical Variables

Table 2. Demographic variables summary statistics.

Variable	Log Population Density	Log Green Areas	Log Green Area per Inhabitant	Log Health Center	Log Inhabitants per Health Center
Minimum	2.08	6.14	∼0.00	1.10	7.37
$Q_1$	4.72	12.45	3.08	2.49	8.36
Median	6.26	13.11	4.37	2.83	8.84
Mean	6.39	12.99	4.62	2.95	8.81
$Q_3$	8.40	13.90	5.81	3.54	9.30
Maximum	10.06	15.02	16.30	4.39	10.14

shows a summary for variables related to demographic characteristics, where all of them were adjusted through logarithmic transformation. It is possible to appreciate how the transformation centers the variables by adding similarity between the mean and median and among measures of the position according to histograms in Figure 5.

Correlations among quality of life and independent variables shown on Figure 5 are approximately 0.21, where, in all cases except for inhabitants HC (inhabitants per health center), the correlation values are positive. It is worth noting that there are high values in correlations among independent variables, such as cases for green areas with health centers or population density with inhabitants HC.

3.1.4. Sociodemographic Variables

Table 3. Sociodemographic variables summary statistics.

Variable	% Poverty	% Students with More Than 450 Points PDT (Education)	Domestic Violence per 100 k Inhabitants	Domestic Violence	Overcrowding
Minimum	2.79%	43.84%	141	144.3	2.90%
$Q_1$	8.87%	58.40%	587	607.3	12.80%
Median	10.68%	63.85%	845	735.7	14.40%
Mean	10.59%	64.34%	1142	744.3	14.58%
$Q_3$	12.61%	68.47%	1467	846.3	16.90%
Maximum	18.87%	94.13%	4388	1808.6	23.40%

summarizes variables of the sociodemographic kind, which were not adjusted through logarithmic transformation due to their initial distributions and showed symmetry in terms of maximum and minimum values with respect to their means and medians. This can be seen in histograms from Figure 6.

Figure 6 shows correlations among quality of life and sociodemographic variables. In this case, the correlations show higher values compared to those shown in Figure 5: poverty shows a negative correlation of −0.69, education has the highest level of correlation with quality of life among explanatory variables, with a correlation of 0.76, the rate of domestic violence (Rate.DV) shows a negative correlation of −0.52, and overcrowding has similar levels with poverty, with a correlation of −0.62. Some of the independent variables show strong correlations among them, such as the case for poverty and education, with a correlation of −0.64, and poverty and overcrowding, with a correlation of 0.56.

3.1.5. Municipal Expenses Variables

Variables related to municipal expenses and municipal budget were all adjusted through logarithmic transformation. It is possible to appreciate how the variables are centered and show symmetry, as well as similar values in their means and medians, as shown in

Table 4. Municipal expenses and public budget summary statistics.

Variable	Municipal Common Fund	Own Incomes	Municipal Expenses	Self-Management Expenses	Cultural Expenses	Leisure Expenses	Social Expenses	Community Expenses
Minimum	14.40	14.24	4.04	15.27	5.18	4.78	8.00	14.24
$Q_1$	15.42	15.34	10.57	15.95	11.43	11.41	13.45	15.31
Median	15.88	16.02	11.32	16.56	12.00	11.95	13.97	15.91
Mean	15.88	16.13	11.18	16.63	12.00	11.91	14.02	16.08
$Q_3$	16.32	16.88	12.09	17.16	12.86	12.97	14.70	16.82
Maximum	18.03	18.91	14.54	19.26	15.79	15.18	17.08	18.17

and the histograms in Figure 7.

The correlation among quality of life and municipal expenses and public budget is shown in Figure 7, where there are some variables that show high levels of correlation with the response variable, own incomes (OIs) show a correlation level of 0.7 with quality of life, which is one of the highest among explanatory variables, self-expenses show the second highest correlation, with a level of 0.59 with quality of life, and the rest of the variables show correlation levels of around 0.30. Some of the explanatory variables show high levels of correlation, with own incomes (OIs) showing a correlation of 0.83 with self-expenses and a correlation of 0.79 with community expenses.

3.1.6. Correlation Analysis

The entire matrix of correlations is shown in

Table 5. Correlations among variables.

QoL	1	0.2	−0.7	−0.4	0.7	0.2	0.6	0.3	0.2	0.3	0.4	0.3	0.3	0.8	0.3	0.0	0.0	−0.5	−0.6
Density		1	−0.1	0.0	0.2	0.0	0.4	0.1	0.1	0.2	0.3	0.0	−0.1	0.0	−0.2	0.6	0.1	−0.5	0.1
Poverty			1	0.3	−0.6	−0.1	−0.4	−0.3	−0.2	−0.2	−0.4	−0.2	−0.3	−0.6	−0.2	0.0	0.0	0.5	0.6
MCF				1	0.0	−0.1	0.2	0.0	0.2	0.2	0.3	0.2	0.0	−0.2	0.4	0.0	0.6	0.2	0.2
OI					1	0.2	0.8	0.4	0.4	0.4	0.8	0.5	0.2	0.6	0.6	0.1	0.5	−0.3	−0.3
ME						1	0.2	0.1	0.0	0.0	0.1	0.1	0.1	0.1	0.1	−0.1	0.0	0.0	0.0
Se.E							1	0.4	0.4	0.5	0.7	0.5	0.2	0.5	0.6	0.1	0.5	−0.3	−0.2
Cu.E								1	0.5	0.4	0.3	0.2	0.1	0.3	0.4	−0.1	0.2	−0.2	−0.2
LE									1	0.4	0.3	0.2	0.1	0.3	0.3	0.0	0.3	−0.1	0.0
So.E										1	0.3	0.3	0.1	0.3	0.3	0.1	0.3	−0.2	−0.1
Co.E											1	0.4	0.1	0.3	0.6	0.2	0.6	−0.3	−0.2
GA												1	0.6	0.2	0.5	−0.1	0.4	−0.1	−0.1
IGA													1	0.3	0.2	−0.3	0.0	−0.1	−0.3
Ed														1	0.3	−0.1	0.0	−0.4	−0.6
HC															1	−0.6	0.6	0.1	−0.2
IHC																1	0.1	−0.3	0.3
DV																	1	0.3	0.1
RDV																		1	0.3
Oc																			1

. The values were calculated through the Pearson correlation index due to the distribution and adjustments that the data received during previous steps. It is possible to notice mixed correlations between the independent and response variables, with values of $\rho$ close to 0.7, such as poverty ($\rho =-0.69$), own incomes ($\rho =+0.7$) and education ($\rho =+0.7$). It is worth noting that there is a correlation between independent variables because there are some strong parallels with high correlation indexes, such as own incomes and self-management expenses ($\rho =+0.83$), own incomes and community expenses ($\rho =+0.79$) or education and overcrowding ($\rho =-0.62$), which suggest that there might be problems with multicollinearity in the model fit stage.

For verification purposes, a sphericity test was developed with a Bartlett test returning a p-value of less than 5% for a chi-squared value of 1.47, indicating that the correlation matrix shown in Table 5 is significantly different from the identify matrix, which means that the correlations are significantly different from zero: ${\rho }_{i,j}\ne 0$.

3.2. Model Identification

The model identification step was performed through iterative methods, which allow us to test a combination of variables for different models to fit. The selection of this method is due to the number of variables in the analysis. Trying all possible combinations of models with 19 variables returns approximately 260,000 models to test, which is humanly and computationally impossible to carry out. The algorithm to identify the model is stepwise regression with iterations in the form of forward, backward and stepwise (mixture) regression. The results of the iterations process are shown in

Table 6. Results from stepwise regression.

Criterion	Model	Metric (AIC or BIC Value)	Number of Variables	Coefficient of Determination	F-Test Value	F-Test Statistic	MAPE	MSE
AIC	Backward	194.64	9	86.0%	76.22	≤$0.05$	3.86%	11.48%
	Forward	196.28	10	85.9%	87.19	≤$0.05$	3.87%	11.44%
	Stepwise	196.28	10	85.9%	87.19	≤$0.05$	3.87%	11.44%
BIC	Backward	217.11	7	85.1%	86.01	≤$0.05$	4.04%	12.54%
	Forward	221.32	7	84.5%	85.41	≤$0.05$	4.09%	13.03%
	Stepwise	221.32	7	84.5%	85.41	≤$0.05$	4.09%	13.03%

. All models are significant according to the Fisher criterion, with p-values for this test of less than 5%, meaning that all models are significant with the existence of at least one ${\beta }_i$ significantly different from 0. The coefficients of determination are approximately 85% and, in terms of errors, the mean absolute percentage error (MAPE) is approximately 3.9%, whereas the mean squared error (MSE) is approximately 12%.

The model chosen from the stepwise selection was the model developed with backward iterations under the AIC criterion. The parameters after fitting the variables into a multivariable regression are shown in

Table 7. Parameters from multivariate regression.

Variable	$\widehat{{\mathit{\beta }}_{\mathit{i}}}$	Standard Error	t-Value	Significance p (≥|t|)
Intercept	40.96	7.57	5.41	≤$0.05$
Population Density	0.30	0.12	2.37	≤$0.05$
Municipal Common Fund	−3.91	0.42	−9.28	≤$0.05$
Own Incomes	1.02	0.52	1.96	≤$0.05$
Self-Management Expenses	2.81	0.66	4.24	≤$0.05$
Social Expenses	0.47	0.25	1.86	≤$0.05$
Inhabitants per Green Area	0.25	0.10	2.45	≤$0.05$
Education	11.39	4.56	2.49	≤$0.05$
Overcrowding	−0.39	0.10	−6.38	≤$0.05$

, where all variables are significant under a 5% level of confidence and the errors from the model show some symmetry, with minimum values of approximately −6.5 and maximum values of 6.4. Furthermore, the mean and median of the residuals are approximately −0.32. The results of the parameter estimations show mixed effects on the correlation among the response and independent variables, following the correlations shown in Table 5.

The results of analyzing variance inflation factors and tolerance are shown in

Table 8. Variance inflation factors and tolerance values for collinearity measuring.

	Population Density	Municipal Common Fund	Own Incomes	Self-Management Expenses	Social Expenses	Inhabitants per Green Area	Education	Overcrowding
VIF	1.23	1.39	4.21	4.17	1.42	1.17	2.82	1.73
TOL	0.81	0.72	0.24	0.24	0.71	0.86	0.35	0.58

, where all VIFs values are under 5, that being the threshold for considering problems on the collinearity of ${\mathrm{VIF}}_i\ge 10$ or ${\mathrm{VIF}}_i\ge 5$ [55]. The tolerances are all greater than 0.1, working in the same way as ${\mathrm{VIF}}_i$, but, in this case, the threshold for these values are ${\mathrm{TOL}}_i\le 0.1$ or ${\mathrm{TOL}}_i\le 0.2$ [56]. With the previous results, it can be concluded that the model has no problems with multicollinearity, which means that there is no information shared among independent variables; hence, OLS is appropriate for estimating the model coefficients.

3.3. Outlier Analysis

Although the model has shown no problems in collinearity and there is significance in terms of the regression model and its variables, it is necessary to analyze outliers in terms of the response and explanatory variables. It is worth mentioning that a value could be considered as an outlier due to a different behavior in terms of the response variable, known as a leverage point, or in terms of explanatory variables, known as a discrepancy point. In case of meeting both conditions (leverage and discrepancy), the point is known as an influential point, meaning that the point is a significant outlier in terms of its response and independent values, affecting the parameter estimation through OLS in the model itself.

3.3.1. Leverage Values

Leverage is a measure that works as a tool for estimating how far a predictor is from the rest of predictors. Its form of calculation is as follows:

$$h_i=\frac{1}{n}+\frac{{(X_i-\overline{X})}^2}{{\sum }_{j=1}^n{(X_j-\overline{X})}^2}\in \left(\frac{1}{n},1\right),$$

(3)

with a mean value

$$\overline{h}=\frac{p+1}{n}.$$

(4)

The values of (3) are known as hat values. The threshold for the analysis is $h_{ii}=0.20$, and Figure 8 show six values that are out of the threshold indicated by the red line, which means that those values differ from other points in working as a predictor for the response variable.

3.3.2. Discrepancy Values

A value or entry that does not have a proper fit in the model is considered as an outlier. A way to measure this discrepancy is by analyzing the residuals from the fitted model. Studentized residuals are used for measuring the previous condition, which is obtained as follows:

$$E_i^{\ast }=\frac{E_i}{S_{E(-i)}\sqrt{1-h_i}}.$$

(5)

Figure 9 shows the histogram of studentized residuals, which are calculated through ${ϵ}_i=\widehat{y_i}-y$. It is possible to appreciate how there are some values out of the boundaries generated by the 95% confidence interval.

3.3.3. Influential Points

According to the above, influential points are those that can be considered as leveraging points and discrepancy points at the same time. A way to measure both features is by using the DFFITS index, which is calculated as follows:

$${\mathrm{DFFITS}}_i=E_i\times \sqrt{\frac{h_i}{1-h_i}}.$$

(6)

The index shown in (6) has a threshold calculated as follows in (7), which returns a value of 0.67. Figure 10 shows the index calculated for each data point. It is possible to appreciate how eight points stay outside the boundary shown in red in the horizontal line in the Y-axis.

$$|{\mathrm{DFFITS}}_i|>2\times \sqrt{\frac{k+1}{n-k+1}}.$$

(7)

Looking at values above the horizontal red line in Figure 10, it can be appreciated how four out of eight values that stay outside the boundary are from the Metropolitan Region (Huechuraba, Providencia, Ñuñoa, Vitacura). All of them are communes within the highest levels of own incomes (above third quartile) and lowest levels of municipal common fund (below first quartile), all of them except for Huechuraba appear below the first quartile in overcrowding levels, where Huechuraba is between the mean and third quartile, Vitacura has the highest value of green area per inhabitant compared to all communes analyzed and, in terms of the response variable (quality of life), they appear above the third quartile, with high levels of urban welfare.

3.4. Model Robustification

OLS estimation is a widely used method for regression analysis. One of its pitfalls is the fact that it is sensitive to outliers and forms on the data points that it is applied on [57], making the usage of methods for estimating the parameters and coefficients of the model in a robust way necessary. The robustification of the model is performed through a robust linear model, which corresponds to a methodology derived from M-estimators, whose application is not as sensitive as traditional OLS. This method involves a balance between least angle regression (LAD) with the efficiency of OLS [58]. The method is represented in the formulation that follows:

$$\min \sum _{i=1}^n\rho \left({ϵ}_i\right)=\min [\rho (Y_i-X_i^{{}^{\prime }}\times \widehat{\beta })].$$

(8)

The results on robust estimation are shown in

Table 9. Parameters from robust regression.

Variable	$\widehat{{\mathit{\beta }}_{\mathit{i}}}$	Standard Error	t-Value	Significance p (≥|t|)
Intercept	38.29	6.65	5.75	≤$0.05$
Population Density	0.33	0.17	1.97	≤$0.05$
Municipal Common Fund	−3.83	0.39	−9.82	≤$0.05$
Own Incomes	1.22	0.55	2.20	≤$0.05$
Self-Management Expenses	2.77	0.71	3.87	≤$0.05$
Social Expenses	0.49	0.27	1.83	≤$0.05$
Inhabitants per Green Area	0.24	0.09	2.54	≤$0.05$
Education	9.30	4.51	2.06	≤$0.05$
Overcrowding	−0.64	0.13	−4.77	≤$0.05$

, returning a coefficient of determination $r^2$ of approximately 86%. The coefficients of parameters are sightly similar to those obtained through traditional OLS estimation, with differences in the intercept and education. Moreover, the order of relations among variables is the same, which means that direct and inverse correlations remain among the response and independent variables.

3.5. Model Validation

The model validation was developed by analyzing residuals from the model. A Shapiro–Wilk test and Anderson–Darling test do not reject the null hypothesis, which means that, given the distributions of residuals, it cannot be rejected that the values come from a normal distribution. The Breusch–Pagan test returns homoscedasticity in residuals, which means that variances are constantly in the form ${\sigma }_{ϵ}$ for each ${ϵ}_i=\overline{y}-y$. The Durbin–Watson test shows that we do not reject independence in errors. Finally, a Durbin–Wu–Haussmann test was developed in order to test endogeneity in the model, measuring exogeneity among regressors and features [59]. The results do not reject the null hypothesis, meaning that OLS is proper and there is no need for other types of estimation, such as two-stage least squares (2LSL) and instrumental variables. A summary of the tests applied on the model is shown in

Table 10. Model validation.

Test	Statistic	p-Value	Conclusion
Shapiro–Wilk	W = 0.98	≥$0.05$	Do not reject normality on residuals
Anderson–Darling	A = 0.54	≥$0.05$	Do not reject normality on residuals
Breusch–Pagan	BP = 15.30	≥$0.05$	Do not reject homoscedasticity on residuals
Durbin–Watson	DW = 2.16	≥$0.05$	Do not reject independence on errors
Durbin–Wu–Haussmann	WH = 9.18	≥$0.05$	No need for 2LSL

.

Figure 11 shows fitted values (red) against actual values of quality of life (blue). The curve below fitted and actual values (gray curve) corresponds to residuals from the model in the form of ${ϵ}_i=\widehat{y}-y$, showing no trend or any form in the plot, followed by the results in Table 10, which validates the results obtained through the tests applied.

3.6. Efficiency Analysis

Traditional economic theory, specifically in microeconomics, mentions that productors are technically efficient, implying that they are capable of making an optimal usage of resources in their processes and systems. In practice, this does not happen in all cases because not all productors are capable of minimizing their costs and production systems, with gaps between the desired and current results.

Stochastic frontier analysis is a tool that allows us to analyze and measure technical efficiency in contexts of production systems. For our case, each one of the municipalities correspond to a productor i. A base assumption is that all municipalities (productors) develop their functions under the same conditions in terms of technology and opportunities [60]. The model that allows us to measure the technical efficiency was developed by Cobb and Douglas [61], it being one of the most used models in micro-economic theory, production and efficiency assessment.

$$y_i=f(X_i,\beta )\times {\mathrm{TE}}_i\stackrel{}{\to }{\mathrm{QoL}}_i=f(X_i,\beta )\times {\mathrm{TE}}_i,$$

(9)

In (9), ${\mathrm{QoL}}_i$ corresponds to the quality of urban life in each municipality i, where $i=1,\dots ,N$, $X_i$ is a vector of predictors that explain the quality of life for each council i, $f(X_i,\beta )$ is the production frontier and $\beta$ are coefficients from the estimated model.

$${\mathrm{TE}}_i=\frac{{\mathrm{QoL}}_i}{f(X_i,\beta )},$$

(10)

In (10), if ${\mathrm{TE}}_i=1$, the index reaches the maximum level of efficiency given $f(X_i,\beta )$. In cases where ${\mathrm{TE}}_i<1$, there is a gap between the maximum or desired production and current levels of production. Adding stochastic terms to the model in (9), we obtain the following expression:

$${\mathrm{QoL}}_i=f(X_i,\beta )\times {\mathrm{e}}^{{ϵ}_i}\times {\mathrm{TE}}_i,$$

(11)

In (11), an error term is added to the model and $f(X_i,\beta )\times e^{ϵ}$ corresponds to a stochastic frontier with $f(X_i,\beta )$ deterministic and an error $ϵ$ in the form of ${ϵ}_i=v_i-u_i$, where $v_i$ corresponds to a stochastic noise and $u_i$ is a term corresponding to technical inefficiency. An assumption related to the error term is that v has a symmetric distribution and is independent of u, while, when $u>0$ with a distribution truncated in the form $u\sim N^{+}(0,{\sigma }_u^2)$, the restrictions of the truncated form of u allows each municipality to limit their outcome to the frontier, meaning that it is impossible to obtain an efficiency of over 100%.

3.6.1. Selection of Functional Form

The selection of a functional form is developed using two approaches: one with the traditional Cobb–Douglas form and the other with the trans-logarithmic form, the second one being an extension of the traditional form developed by Fuss McFaffen in 1978 [61].

$$\mathrm{Cobbs}-\mathrm{Douglas}:\ln \left({\mathrm{QoL}}_i\right)=\sum _{i=1}^N{\beta }_i\times \ln \left(X_i\right)+{ϵ}_i,$$

(12)

$$\mathrm{Translog}:\ln \left({\mathrm{QoL}}_i\right)={\beta }_0+\sum _{i=1}^N{\beta }_i\times \ln \left(X_i\right)+\frac{1}{2}\sum _{i=1}^N\sum _{j=1}^N{\beta }_{i,j}\times \ln \left(X_i\right)\times \ln \left(X_j\right)+{ϵ}_i,$$

(13)

The contrast between forms in (12) and (13) is developed through an LR test with the following form:

$$\mathrm{LR}=-2\times \ln \left(\frac{\mathrm{L}\left({\mathrm{H}}_0\right)}{\mathrm{L}\left({\mathrm{H}}_1\right)}\right)=-2\times (\ln \left(\mathrm{L}\left({\mathrm{H}}_0\right)\right)-\ln \left(\mathrm{L}\left({\mathrm{H}}_1\right)\right)),$$

(14)

The hypotheses for the LR test shown in (14) are the following:

${\mathrm{H}}_0$: Cobbs and Douglas Functional Form.

${\mathrm{H}}_1$: Translogarithmic Functional Form.

The test was developed with a critical value of ${\chi }_{k,\alpha }^2=2.71$. The LR test returned a value of 0.75, which is less than 2.71 (critical value for rejection of null hypothesis); therefore, the null hypothesis cannot be rejected and the functional form selected for measuring efficiency is the Cobb–Douglas form shown in (12).

The results of the efficiency model are shown in

Table 11. Parameter estimation in efficiency model.

	Intercept	Population Density	Municipal Common Fund	Own Incomes	Social Expenses	Inhabitants per Green Area	Education	Overcrowding
${\beta }_i$	45.47	0.35	−2.55	2.93	0.43	0.14	9.12	−7.76
Rate	≤$0.01$	0.72	0.01	≤$0.05$	0.66	0.88	≤$0.01$	≤$0.01$

. The municipal common fund, own incomes, education and overcrowding show significance with respect to efficiency. In addition, the $\gamma$ parameter returns a value of 0.22, which is different from 0, under a 5% level of confidence.

3.6.2. Efficiency Measurement

The mean efficiency can be obtained using (15), which returns a mean value of 46.7%, the horizontal red line in Figure 12 corresponds to the mean and the median is approximately 47%, with minimum values of 27% and maximum values of 62%.

$$E\left[{ϵ}^{-\left(u_i\right)}\right]=2\times \left(1-\mathrm{\Phi }\left({\sigma }_s\sqrt{\gamma }\right)\right)\times e^{-\gamma {\sigma }_s^2/2}.$$

(15)

Table 12. Efficiency and quality of life per council ranked from highest to lowest efficiency.

i	Council	QoL	Efficiency	i	Council	QoL	Efficiency
1	Concón	61.68	61.62%	51	Rengo	47.21	46.29%
2	Calama	52.97	60.81%	52	Quinta Normal	48.68	46.66%
3	Coyahique	52.05	60.75%	53	Lampa	49.58	46.59%
4	Providencia	76.27	59.76%	54	Las Condes	72.76	46.00%
5	Ñuñoa	68.25	58.40%	55	Los Ángeles	48.13	45.74%
6	Lota	43.24	57.17%	56	San Miguel	56.47	45.46%
7	Coronel	50.28	56.94%	57	La Florida	51.06	45.36%
8	Padre Las Casas	45.85	56.73%	58	La Granja	43.82	45.14%
9	Santiago	61.68	56.42%	59	Castro	47.39	44.89%
10	Alto Hospicio	44.23	56.06%	60	Constitución	44.77	44.59%
11	Independencia	52.13	55.78%	61	San Ramón	41.98	44.54%
12	Antofagasta	51.15	55.56%	62	San Javier	43.29	44.37%
13	Estación Central	53.67	55.55%	63	Puerto Montt	49.90	44.29%
14	La Serena	54.01	55.30%	64	La Calera	47.35	44.02%
15	Lo Prado	45.34	54.38%	65	Limache	44.92	43.81%
16	Coquimbo	50.62	54.38%	66	Penco	44.41	43.67%
17	Chillán Viejo	46.12	53.94%	67	Villa Alemana	49.06	43.56%
18	Angol	45.93	53.94%	68	Peñaflor	46.47	43.51%
19	Padre Hurtado	49.38	52.76%	69	Lo Espejo	39.70	43.50%
20	Hualpén	51.84	52.67%	70	La Pintana	39.98	43.41%
21	Chiguayante	51.50	52.62%	71	San Vicente	46.96	43.40%
22	Valdivia	51.57	52.61%	72	Rancagua	53.16	43.30%
23	Tomé	45.86	52.47%	73	El Bosque	43.47	43.11%
24	Maipú	53.15	52.29%	74	Quillota	49.47	42.52%
25	Quilicura	56.23	52.18%	75	Villarrica	46.79	42.05%
26	San Joaquín	50.73	52.09%	76	San Felipe	47.66	42.02%
27	Los Andes	50.62	51.97%	77	Machalí	53.11	41.68%
28	Concepción	56.58	51.90%	78	Molina	41.60	41.67%
29	Arica	48.12	51.84%	79	Talagante	45.95	41.32%
30	Valparaíso	47.03	51.83%	80	Conchalí	43.85	41.27%
31	Macul	57.48	51.61%	81	Puerto Varas	52.79	40.73%
32	Temuco	52.07	51.69%	82	Renca	47.33	40.70%
33	Chillán	50.77	50.68%	83	Colina	51.49	40.61%
34	Quilpué	50.98	50.65%	84	Talcahuano	51.68	40.61%
35	San Pedro	53.59	50.32%	85	Copiapó	47.09	39.38%
36	P. Aguirre Cerda	46.43	50.24%	86	La Reina	59.79	38.46%
37	Punta Arenas	55.09	49.57%	87	Melipilla	44.73	38.31%
38	La Cisterna	53.39	49.24%	88	Vallenar	42.87	38.04%
39	Maule	41.90	49.06%	89	Puente Alto	44.92	37.39%
40	Lo Barnechea	62.41	48.76%	90	Ovalle	42.65	36.55%
41	Talca	51.70	48.03%	91	Buín	46.93	34.89%
42	Linares	46.16	47.97%	92	Cerrillos	46.83	34.68%
43	Curicó	49.86	47.90%	93	Osorno	47.19	34.53%
44	San Antonio	48.44	47.58%	94	Paine	45.06	34.44%
45	San Carlos	45.13	47.51%	95	San Bernardo	43.62	32.07%
46	Vitacura	78.09	47.47%	96	San Fernando	45.23	30.68%
47	Cerro Navia	41.01	47.44%	97	Huechuraba	52.09	29.54%
48	Iquique	51.74	47.39%	98	Viña del Mar	51.21	29.31%
49	Pudahuel	49.15	47.15%	99	Recoleta	45.54	26.80%
50	Peñalolén	49.09	47.06%

shows municipalities ranked per efficiency (highest to lowest). It can be seen how municipalities with a low QoL index show high values of efficiency, where municipalities with low QoL are ranked in the top ten municipalities. These cases include Lota ($QoL=45.85$), with an efficiency = 57.17% (top 6), Padre Las Casas ($QoL=45.85$), with an efficiency = 56.73% (top 8), and Alto Hospicio ($QoL=44.23$), with an efficiency = 56.06% (top 10).

3.7. Cluster Analysis

With the quality of life regression analysis and efficiency measured per municipality, an analysis of clustering was developed in order to find differences and similarities among municipalities through hierarchical and non hierarchical clustering. The variables used in this analysis are the same as the the regression analysis, including the results of the efficiency index developed in Section 3.6.

3.7.1. Data Preprocessing

Depending on the algorithm used for the analysis, types and structures of data, the wrong results could be yielded, especially with those that are sensitive to outliers or extreme values. Lopes and Gosling (2020) [62] showed how the results of various classical techniques on clustering depend on the form, structure and distributions of the data, which indicates how important data exploration and preprocessing are regardless of the chosen model.

The first step of clustering analysis is data normalization. Virmani et al. (2015) [63] proposed techniques for data normalization and scaling before the application of algorithms in clustering analysis, specifically for those whose results depend on central tendency measures or position measures such as k-means or k-medoids.

Data normalization was developed through the min-max technique, whose results ensure that all variables have similar scales between 0 and 1.

$$X_i^{{}^{\prime }}=\frac{X_i-\min \left(X_i\right)}{\max \left(X_i\right)-\min \left(X_i\right)}.$$

(16)

3.7.2. K-Means Algorithm

Given a set of observations $X_i$ and the number of groups k, with $k<n$ and $S=S_1,S_2,\dots ,S_k$, the k-means algorithm works according to the following:

$${\mathrm{argmin}}_S\sum _{i=1}^k\sum _{X\in S_i}\left|\right|X-{\mu }_i{\left|\right|}^2={\mathrm{argmin}}_S\sum _{i=1}^k\left|S_i\right|\times \sigma \left(S_i\right),$$

(17)

The minimization model in (17) depends on parameters $\mu$ and $\sigma$, and, due to the behavior and properties of the means and variances, they are sensitive to scales and outliers in data distributions [63]. Due to the above, the idea of applying data normalization prior to applying the algorithm for clustering is reinforced.

3.7.3. K-Medoids Algorithm

For robustification and contrasting results from k-means, a k-medoids algorithm was developed. The main difference between these algorithms lies in that k-medoids is based on a different set of steps and rules for creating groups, depending less on means from the data in each cluster [64]. The steps for this algorithm are the following:

1. 

Initiation: k number of clusters and D set of points for n observations and i characteristics.

2. 

Output: Set of k groups that minimize the sum of differences among nearest objects to the nearest center.

3. 

Method: Arbitrary selection of k groups for D.

4. 

Repeat: Randomly select an object that is not near to the center and compute S as a measure of the change between $O_j$ and $O_{random}$.

5. 

Rule: if $S<0$, change $O_j$ to $O_{random}$ in the new center.

6. 

Rule: Repeat until there are no changes.

3.7.4. Selection of k-Clusters

The selection of clusters k is developed through WSS and silhouette indexes. The first one focuses on minimizing within the sum of squares (WSS) for different values of k. A pitfall for this index is that its optimal value occurs when $k=n$, which means that each point is a cluster itself (due to the variance being equal to zero). Therefore, it is necessary to contrast WSS with the silhouette as a way of ensuring that the selection is carried out in a proper and objective manner.

One advantage of this method for selecting the number of clusters is that both indexes have a low computational complexity and cost. Moreover, both have simple interpretation rules and it is pretty intuitive to determine what they mean while changing the number k in the analysis [65].

$$\mathrm{WSS}=\sum _{i=1}^k\sum _{C_i\in k}(X_c-\overline{X_i}),$$

(18)

$$\mathrm{Silhouette}=s\left(i\right)=\frac{b\left(i\right)-a\left(i\right)}{\max \left(a\right(i);b(i\left)\right)}\stackrel{}{\to }|C_i|>1\&s\left(i\right)=0\stackrel{}{\to }|C_i|=1,$$

(19)

$$a\left(i\right)=\frac{1}{|C_i|-1}\sum _{j\in C_i,i\ne j}\mathrm{d}(i,j),$$

(20)

$$b\left(i\right)={\min }_{k\ne i}\frac{1}{|C_k|}\sum _{j\in C_k}\mathrm{d}(i,j),$$

(21)

Figure 13 shows the results of the WSS and silhouette indexes for the clustering selection. Within the sum of squares (WSS) shown in Figure 13a,c, the same results under the elbow method on the curve suggest $k=4$ clusters, while the silhouette index in Figure 13b,d suggests that $k=4$ groups is their optimal value (vertical red line in Figure 13).

3.7.5. Hierarchical Clustering

Another method developed to lead clustering analysis is hierarchical clustering. This method is a combination of rules that allow for a visual inspection of the groups and their relations among data points [66]. The steps for generating the groups are as follows:

1. 

$P_N=C_i\forall i\in (1,\dots ,N)$.

2. 

$C_j=O_j\forall i\in (1,\dots ,N)$.

3. 

$k=1$ all elements in the same group.

4. 

Rule: $N-k>1$.

5. 

Rule: Selection of $C_i,C_j\in P_{N-k+1}$ through local criterion.

6. 

$C_{N+k}=C_i\cup C_j$.

7. 

$P_{N+k}=(P_{N-k+1}\cup C_{N+l}):(C_i,C_j)$.

8. 

$k=k+1$.

9. 

Decision: while $N-k>1$.

The results of hierarchical clustering are shown in Figure 14, with a respective dendrogram for the analysis. The groups generated are plotted with squares in different colors for different values of height in the vertical axis.

3.7.6. Clustering Results

The results from k-means, k-medoids and hierarchical clustering algorithms generate four groups: $k=4$.

Table 13. Summary on features and variables per cluster.

Variable	${\mathit{C}}_1$	${\mathit{C}}_2$	${\mathit{C}}_3$	${\mathit{C}}_4$
Elements in Cluster	34	21	7	37
Quality of Life	50.9	47.4	68.5	46.9
Population Density	1443	10797	8739	214
Municipal Common Fund	16.36 M	8.98 M	3.08 M	6.73 M
Own Incomes	23.32 M	8.94 M	77.06 M	5.34 M
Self-Management Expenses	31.14 M	16.85 M	90.87 M	9.18 M
Social Expenses	2.98 M	1.31 M	8.37 M	1.03 M
Green Areas	5.04	3.64	7.01	4.33
Education	67.9%	57.1%	84.8%	61.3%
Overcrowding	14.3%	17.1%	9.5%	14.4%
Efficiency	47.2%	46.1%	50.8%	45.8%

shows the results for each one of the clusters in terms of the mean value of each variable. There are differences in all variables, especially in education, overcrowding, own incomes, public expenses and public budget. Cluster 3 shows the highest values of the quality of life, own incomes, social expenses and education. With it having the lowest values of the municipal common fund and overcrowding, it could be said that this is the group with the features best explaining quality of life. Cluster 2 works in the opposite way to cluster 3, with the lowest values of quality of life, expenses and education, while having the highest values in overcrowding and population density. This group could be considered as the one with the worst features in terms of quality of life and its associated variables. The other two clusters (1 and 4) have a mixture in terms of variables. Their overcrowding rates are similar, but cluster 1 has a higher index of quality of life while cluster 4 has a lower population density.

Figure 15 shows clusters plotted onto a map for the Metropolitan Region (Chilean capital territory). The reason for choosing this region is, due to the number of municipalities involved in the analysis, this region has been compared to others, so it is a better way to visualize the differences and similarities among groups in a map. It can be seen from Figure 15 how municipalities are distributed on the region. Municipalities in cluster 3 (group with better features in terms of quality of life and variables) are distributed all together in the region. Being neighbors to one another, this zone corresponds to wealthier areas in the region, which contrasts with the other municipalities within the region, which are more distributed and involve other groups from the analysis.

4. Conclusions

There are various measures and forms that many researchers and specialists focus their efforts on when explaining quality of life, and they depend on the purpose and focus of their analysis. However, it is possible to appreciate how some characteristics and features are repeated among studies. Such is the case for climate and environmental indexes, health conditions, education and job opportunities [67]. These variables can be considered as a basis for measuring and considering if a determined person lives in a healthy environment with proper urban welfare.

Multivariate regression analysis works as a tool for not only estimating values given a set of features, but also for understanding behavior among variables, especially in terms of correlations and levels among a response and a set of variables or characteristics. According to what was stated in Section 1, currently, the reports on quality of life focus on making an assessment for the index, but there is no analysis of how the characteristics interact and correlate with quality of life. At first sight, the model and analysis developed allow us to obtain a level of the response of quality of life with each one of the variables chosen, without focusing on coefficients from the regression analysis. Looking at Table 9, it can be said that the municipal common fund and overcrowding negatively affect quality of life, whereas own incomes, self-management expenses, social expenses, green areas per inhabitant, education and density positively affect the quality of life; thus, there is a way to focus on resources with a proper understanding on how different dimensions affect quality of life and urban welfare.

4.1. Quality of Life Model Analysis

The analysis was conducted with a total of 19 variables that work as predictors for the quality of urban life with various and mixed dimensions and contexts. The variables were separated into three main groups, considering demographic features, sociodemographic features and public expenses and budget per council. The model identification was developed through iterative methods with stepwise regressions, returning eight variables explaining approximately 86% of the variability from the response variable, without problems of collinearity and validating the output through residual analysis.

The coefficient analysis model developed showed mixed effects in terms of correlations among the response and independent variables. Overcrowding has a positive $\widehat{\beta }$ coefficient, meaning that an increase of 1 point in overcrowding is expected to produce a decrease of 0.64 in quality of life. The municipal common fund also shows a negative $\widehat{\beta }$ coefficient of −3.83. In this case, the variable is log-transformed, meaning that a 1% increase in the municipal common fund is expected to decrease the quality of life by 3.8%. On the other hand, the population density, own incomes, self-management, social expenses and inhabitants per green area show positive $\widehat{\beta }$ coefficients, meaning that an increase of 1% of each one of these variables produces 0.33%, 1.22%, 2.77%, 0.49% and 0.24% increases in quality of life, respectively, Education has a positive $\widehat{\beta }$ coefficient with no transformation, meaning that an increase of 1 point in this variable produces an expected increase of 9.3 points in quality of life.

The analysis of coefficients on the linear model shows mixed effects in terms of the correlation among independent and response variables. Overcrowding and the municipal common fund show a negative correlation with quality of life, while the others directly relate to the response variable. The behavior on overcrowding agrees with studies developed by Gove et al. (1979) in USA, Gray (2001) in New Zealand and Makinde et al., (2016) in Nigeria, whose studies show how overcrowding is related to negative relations at home [31], a bad performance, aggressiveness and antisocial behavior with classmates [29,32], decreasing perceptions of quality of life and urban welfare. The municipal common fund is a variable that shows an inverse correlation with quality of life. Although studies and research tend to show how variables related to public budget and municipal expenses directly relate to quality of life [44,45], the municipal common fund responds to other kinds of behaviors and effects. Looking at which municipalities show the lowest levels of this variable, we find Vitacura, Providencia, Las Condes and Lo Barnechea. These municipalities in the Metropolitan Region are one of the wealthiest zones of the region and country, with the highest levels of economic development and highest number of companies located in their territory, so they do not need to depend on these kinds of funds to reach proper levels of budget for municipal expenses. The previous behavior is similar but contrary to the one presented by own incomes, which directly relates to quality of life and the response to the inverse conclusion of the municipal common fund.

Results on social and self-management expenses agree with the analysis developed by Mafrolla and D’Amico (2016) and Haile and Niño-Zarazúa (2018), whose results show that municipal expenses generate better levels of quality of life and urban welfare [44,45] and citizens tend to report better urban welfare indexes and living happier lives when the municipal expenditure is focused more on providing public and social projects and goods [68] or on dimensions such as education, health or leisure according to data from people in USA.

Education is one of the variables that is most used in research and analyses that aim to try and explain quality of life and human development. The results on this variable agree with others studies, whose findings show how education levels directly affect and improve quality of life, human development and other dimensions, such as job opportunities, income and mental health [38,69]. Moreover, education is related to better human capital, creates valuable consumption amenities, has a significant relationship with overall life satisfaction and has positive effects in terms of long-term health according to results from research conducted in Australia [70,71].

Green areas results are similar to studies whose purpose was to explain the correlation between quality of life and various forms of green areas development in urban environments for different cities in various countries. This shows how parks and green environments tend to increase the urban welfare, which is similar to the results of studies developed in cities from China, Poland and other metropolitan cities from various countries [16,18,19], especially after the COVID-19 pandemic altered perceptions on the impact of green development and its importance for the urban quality of life according to data from people in UK [72].

With all previous findings, correlations can be set and measured between variables, characteristics and quality of life. This allows for a better understanding of how to focus resources on aspects expected to produce a determined change, either increasing or decreasing urban welfare. This is also the case for allocating and distributing resources and time to factors with a significant impact on the response, such as education, overcrowding or public expenses.

4.2. Efficiency

During the course of the analysis, a special emphasis was placed on studying the correlation between quality of life and various factors associated to the municipalities involved in the modeling. The European Union has been focused on the roles that public and urban policies play and the importance of them with respect to covering gaps in needs and problems in citizens. List and Sturm (2006) [73] show that an improvement in the quality of life of a community can be seen as a beneficial and responsible act from local governments toward their communities. The results of the efficiency analysis show how, through the stochastic frontier, the mean efficiency is approximately 47%, with minimum values of 30% and maximum values of approximately 60%, without a correlation between quality of life and efficiency due to the fact that some municipalities with a high quality of life index show low results in the efficiency measurement. The results agree with the analysis developed by Cordero et al. (2016) [74], whose results show significance in terms of municipal expenses, with efficiency in quality of life and urban welfare. An aspect to consider is that the efficiency model does not try to understand the relationship between the response and independent variables, but instead works as a tool for understanding which aspects should be considered in order to minimize the waste of resources or which gaps should be taken into account at the time of looking for improvements in quality of life.

4.3. Cluster Analysis

A cluster analysis is considered as a technique for exploratory data analysis and is used in various cases in the first steps of dealing with data and information. The main reason for developing a cluster analysis is to find groups with similarities and differences in the variables chosen for explaining quality of life. K-means, k-modes and hierarchical clustering return similar results, suggesting four groups with significant differences among characteristics in towns analyzed.

A clustering analysis can be found in various studies that aim to make characterizations of cities, which makes it easier to understand variables associated with urban environments. The characterization method depends on the kind of variable related to the analysis. Studies can be found that suggest grouping in terms of historical features, demographic and human development index characteristics and economic indicators among regions. An analysis developed in cities and regions from the UK suggests three groups using the k-means algorithm, the first one being historic and university cities, the second being industrial cities and the third being larger cities [75]. Another research developed a cluster analysis through k-means in regions from Indonesia, suggesting three groups in terms of human development indexes (HDIs), making the clusters from the highest to lowest HDI [76]. An analysis of regions from Latvia developed a k-means clustering algorithm, suggesting four groups with significant differences in economic and social indexes. The results characterize the first group as the one with the lowest level of unemployment and highest level of entrepreneurship, the second group as having the largest share of people at working age and highest unemployment rate, and the other two clusters as a mixture of the first and second one [77]. In all cases, the results are suggestions and mechanisms for supporting decision making on social project development and understanding regions and cities regarding their various characteristics.

The four groups show part of the differences among regions and municipalities in the study. The fact that cluster 3 has seven municipalities with the highest incomes in the Metropolitan Region and the country shows gaps among the towns clarifies the difference in realities among municipalities placed in the same region with no significant geographical distance. The result of cluster 2 are the opposite to cluster 3. While it shows the wealthiest municipalities, the first one is composed of towns with the worst attributes, such as the lowest levels of education, the highest levels of overcrowding, the lowest amount of green area and the second lowest quality of life index. Cluster 4 is mostly composed of towns far from metropolitan areas, with the lowest levels of the population density and moderate levels of public expenses, whereas cluster 1 works as the opposite of cluster 4, and is mostly composed of big cities that can be considered as regional capitals of their provinces, with mean levels of quality of life, the second highest level of education and the second lowest population density.

4.4. Suggestions

The main focus when developing this study was on the usage of cross-sectional data, meaning that all variables were taken from a particular year. This kind of analysis considers data from a period of time without involving movements in time from the variables. It also considers the fact that quality of life is an index published annually by its authors. An efficiency analysis with panel data can be performed in order to analyze the evolution and movements that the process of quality of life and its explanatory variables have had.

The variables chosen for the analysis were various and depended on different contexts. Sources of information such as the Statistics Institute or Library of the National Congress of the Republic deliver not only the data presented in this study, but there are many other kinds of variables that can be chosen to explain quality of life and urban welfare. Such are the cases of political and voter turnout, security perception, territorial management and the usage of resources in urban development and connectivity, the latter being a variable that has a special emphasis on towns living in regions far from national territories, such as Coyhaique or Punta Arenas.

The main limitation while conducting the research was the availability of a quality of life index that is developed by external entities and used as a response in the analysis. According to the evolution of the criterion for developing this index, in further analyses like this one, more councils and towns could be added to the analysis in order to have a better representation of different realities from the Chilean population, especially with ones living in areas far from metropolitan cities, which are not included in the quality of life index report due to the difficulty in tracking their data, lack of a public report of indexes and population per council.