using non-linear models for a complexity approach to ... · using non-linear models for a...

DOI 10.1007/s11135-006-9032-8Quality & Quantity (2008) 42:1–21 © Springer 2006

Using Non-linear Models for a ComplexityApproach to Psychological Well-being

MONICA GONZALEZ∗, GERMA COENDERS and FERRAN CASASFaculty of Economics, Research Institute on Quality of Life Studies, University of Girona,Campus Montilivi, 17071 Girona, Spain

Abstract. Psychological well-being in adolescence is an increasing field of study. The liter-ature identifies a large number of dimensions of psychological well-being. However, evenwhen considering all these dimensions, the explanatory power of most models is rather low.Complexity theories can be a productive alternative, at the theoretical but especially themethodological level, to the limitations more traditional approaches to psychological well-being have. In this paper, we suggest a structural equation modelling approach to complexitythat focuses on the non-linearity property. Given the large number of dimensions, the modelis estimated in two steps as described by Joreskog [(2000) Latent Variable Scores and TheirUses. Lincolnwood IL: Scientific Software International] First, a confirmatory factor analy-sis is fitted and Anderson and Rubin’s factor scores are saved. Then all possible products andsquared terms of the factor scores are computed and are used as predictors of the depen-dent variable using an ordered logit model. The results from a sample of 968 Catalan adoles-cents show that a non-linear model including interaction effects among the eight dimensions,age and gender, has a higher explanatory power to predict satisfaction with life as a whole,compared to a linear model. Important consequences for the study of psychological well-being in adolescence emerge from the methodological procedure we have followed, which canbe used to study any type of complex psychological and psychosocial phenomenon.

Key words: non-linearity, complexity sciences, adolescence, psychological well-being,satisfaction with life as a whole.

1. Introduction

Psychological well-being, as a component of quality of life, has been afield of important developments during the last two decades. Its study inrelation to childhood and adolescence has been, comparatively speaking, muchmore limited, though during the 1990s an increase of interest towards thedevelopment of adequate instruments has taken place (Casas et al., 2000).

∗Author for correspondence: Monica Gonzalez, Faculty of Economics, ResearchInstitute on Quality of Life Studies, University of Girona, Campus Montilivi, 17071 GironaSpain. E-mail: [email protected]

2 MONICA GONZALEZ ET AL.

Nowadays, the most important limitations for the study of psychologi-cal well-being in adolescence are of an epistemological nature. First of all,there is a relative lack of consensus both at the definition and explana-tory theory levels (see, for instance, Pollard and Lee, 2003). A review of themost relevant explanatory theories about psychological well-being (see, e.g.,Diener, 1994) shows that they are generally centred on specific aspects witha low connection between them.

There are many variables which are considered to be part of or relatedto psychological well-being: personality dimensions, self-esteem, perceptionof control, life objectives, social support, values and so on. However, themajority of results that have been obtained up to now explain with diffi-culties the relationships among these variables (Mathews et al., 1999; Eidand Diener, 2004). This is so to the extreme that the strength of the inter-relationships among those variables varies depending on which of them aretaken into account when studying psychological well-being, as the researchby Cha (2003) shows.

Mathews et al. (1999) think that obtaining these scarce results (in termsof low correlations and reliability problems) in adult and non-adult pop-ulations alike, is due to the fact that dynamic systems are been studiedthrough linear techniques. Clair (1998), Luce (1999) and also Cummins(1996, 1998, 2000), highlight that several behaviours do not let themselvesto traditional linear analyses. In fact, linear mathematical–statistical toolshave and continue being applied when the fulfilment of certain characteris-tic by data often do not happen. In Klein and Stoolmiller’s words (2003),within the context of social sciences, a linear model only offers a question-able representation of reality.

More and more pieces of research go in the direction of a paradigmbased on complexity as an alternative of a paradigm based on simplicity,which has been the ruling tendency during many years. As a new episte-mology many authors defend it is, complexity sciences offer a new way ofapprehending reality. They provide a less reductionistic approach of basicmechanisms of behaviour and social reality (Munne, 1995). The qualita-tive motivation that distinguishes these new sciences fits together with thetraditional spirit of social and human sciences, which have been tradition-ally more likely than natural sciences to avoid reductionistic approaches.Complexity theories can contribute, in fact, to “re(wake-up)” psychologists’interest towards the complexity and indeterminism of human behaviourand thought (Vallacer and Nowala, 1994; in Hernandez and Valera, 2001).

Despite the existence of some discrepancies among complexity research-ers, there is a common conceptual body many authors agree on (Weaver,1948; Flood et al., 1988, in Allegrini et al., 2004; Riofrıo, 2001). It is thefollowing: (1) Complex systems generally have, although not necessarily, ahigh number of elements; (2) The elements or aspects of a complex system

COMPLEXITY APPROACH TO PSYCHOLOGICAL WELL-BEING 3

interact in a dynamical way and change through time; (3) The nature of therelationships among the elements of a complex system is characterised byits high degree of connectiveness: one element influences and is influencedby a high number of other elements; (4) These relationships are non-linear:small causes generate big effects and the other way round (this is the socalled sensitivity to initial conditions); (5) Relationships are short-termed;(6) There exists positive and negative feedback as well; (7) A complex sys-tem has a history: it evolves as time goes by and, so, its present state isdetermined by the past; and (8) It is difficult to establish borders within acomplex system: the observer’s perspective influences the definition of theselimits as they are frequently a consequence of his/her descriptive objectives.

Taken into account the limitations of the study of psychological well-being we have mentioned before, we think it is possible to find a produc-tive alternative in complexity theories. The properties of complexity areshared by many psychosocial phenomena, and this includes psychologicalwell-being.

The complexity paradigm is nowadays understood as being composedby different theories (chaos theories, fractal theory, theory of catastrophes,and theory of fuzzy sets) (Munne, 1995, 2004). Despite the fact thesetheories have different characteristics, as they were formulated in differ-ent moments in time by different authors and within different disciplines,they share some features. Among these features the non-linearity propertyis one of the most remarkable. In fact, all complexity theories are, with noexception, non-linear theories. The properties of non-linear phenomena aredefined in opposition to linear ones and, according to Munne (1993), theyare the following: (1) Lack of direct proportionality between cause andeffect; (2) Presence of undetermination, (3) Unpredictability of the phe-nomenon to be explained, and (4) Discontinuity in the processes of change.

What we have made in this article is, in short, an attempt to studypsychological well-being in adolescence from a non-linear perspective,although the availability of cross-sectional data alone did not make itpossible to include the fourth property related to change discontinuity.

In this article we use a relatively large number of psychosocial con-structs considered to be closely related to psychological well-being (seeGonzalez, 2006 for more details). They are the following: satisfaction withlife as a whole and with specific domains in life, self-esteem, perceivedsocial support, perception of control and values. There exists a high con-sensus in considering that the exploration of the above elements is of firstneed to deepen into the structure of psychological well-being (Diener andLucas, 1992). However, they have generally been studied in an isolatedway, although there are some attempts of theoretical integration (see, forinstance, Cummins, 2000; Cummins et al., 2002).


Many strategies for non-linear modelling of variables measured witherror have been suggested (for the latest developments see for instanceBollen and Paxton, 1998; Schumacker and Marcoulides, 1998; Klein andMoosbrugger, 2000; Schumacker, 2002; Marsh et al., 2004). The largenumber of variables makes it unfeasible to use one-step full informationmaximum likelihood methods which are generally considered to be thebest. Instead we suggest using the two-step approach of Joreskog (2000) tonon-linear structural equation models.

2. Approaches to Non-linear Modelling of Variables Measured with Error

Non-linear regression models can be specified by including interactionterms constructed as products between pairs of variables (Jaccard et al.,1990). This is the so called MRA or moderated regression analysis. It isalso possible to add quadratic terms. Equation (1) includes all possible firstorder (β1 and β2) main effects, and second order (β3, β4 and β5) interac-tion and quadratic effects:

E(y)=β0 +β1x1 +β2x2 +β3x1x2 +β4x21 +β5x

22 (1)

In Equation (1) the slope of y in relation to x1 is β1 +β3x2 +β4x1 andthus depends on the initial values of the rest of the independent variablesincluded in the model and on the own initial value of x1. As opposed to alinear model, direct proportionality between cause and effect is no longerassumed, but the effects may change depending on the initial conditions.

If variables are measured without error, such a regression model can beestimated by ordinary least squares (OLS). This approach is frequent in thesocial and behavioural sciences (Cohen and Cohen, 1983; Jaccard et al., 1990;Irwin and McClelland, 2001; Conner et al., 2002; Davis-Blake et al., 2003; New-som et al., 2003). If the explanatory variables are measured with error, thisapproach leads to biased estimates. This bias persists even if summated rat-ing scales (SRS, e.g. Spector, 1992) are used as regressors because SRS are notperfectly reliable except for an infinite number of items. The remaining biasis especially relevant for interaction effects that are usually of low magnitude(second order effects) and may easily go undetected if biased.

The most widely used methods to model variables measured with error arestructural equation models (SEM, e.g. Bollen, 1989; Raykov and Marcoulides,2000), partial least squares or PLS (e.g. Wold, 1975; Fornell and Cha, 1994;Chin, 1998; Chin and Newsgted, 1999), and two stage least squares TSLSregression (Bollen, 1996).

Kenny and Judd (1984) proposed a possible specification for modellinginteraction effects with SEM. Kenny and Judd’s approach requires eachlatent variable to relate to at least two indicators and implies the formation


of multiple indicators based on the products of the main effect indicators.These products are then used as indicators of the latent interaction.

Different alternatives have been proposed for developing Kenny andJudd’s (1984) SEM approach using full information maximum likelihood esti-mation. It is not our aim to provide a comprehensive presentation (see forthis purpose Joreskog, 1998; Li et al., 1998; Schumacker and Marcoulides,1998; Cortina et al., 2002; Moulder and Algina, 2002; Batista-Foguet et al.,2004a; Marsh et al., 2004) but we can cluster these approaches by:

• The number of product indicators of the latent interaction used, fromnone (Klein and Moosbrugger, 2000; Schermelleh-Engel et al., 1998maximise the likelihood of the dependent and main effect variablesonly) through one (Joreskog and Yang, 1996; Batista-Foguet et al.,2004b), through non-overlapping pairs (Batista-Foguet et al., 2004a;Marsh et al., 2004), to all (Jaccard and Wan, 1995, 1996; Algina andMoulder, 2001).

• The use (Jaccard and Wan, 1995, 1996; Joreskog and Yang, 1996;Algina and Moulder, 2001; Batista-Foguet et al., 2004b) or failure touse (Schermelleh-Engel et al., 1998; Klein and Moosbrugger, 2000;Batista-Foguet et al., 2004a; Marsh et al., 2004) complex non-linearparameter constraints.

• The assumption of normality (Jaccard and Wan, 1995, 1996; Joreskogand Yang, 1996; Schermelleh-Engel et al., 1998; Klein and Moosbrug-ger, 2000; Algina and Moulder, 2001; Batista-Foguet et al., 2004b) orits absence (Batista-Foguet et al., 2004a; Marsh et al., 2004).

• The treatment of the mean structure from inclusion (Joreskog andYang, 1996; Schermelleh-Engel et al., 1998; Klein and Moosbrugger,2000), through inclusion with centered main effect indicators (Alginaand Moulder, 2001; Marsh et al., 2004), to exclusion (Jaccard and Wan,1995, 1996; Batista-Foguet et al., 2004a,b).

• The application to one-equation (Jaccard and Wan, 1995, 1996; Joreskogand Yang, 1996; Algina and Moulder, 2001; Marsh et al., 2004) orsimultaneous equation models (Batista-Foguet et al., 2004a,b)

Except those of Batista-Foguet et al. (2004a) and Marsh et al. (2004),maximum likelihood SEM approaches are very demanding in that theyeither require computationally intensive methods or a great degree ofexpertise from the user. More importantly for the purposes of this paper,all these methods are unworkable for very large models. Complex systemsgenerally have a high number of elements. In our case, the model includes10 main effects, so that all possible second order interactions amount to45, and thus the number of latent variables would be 55 and the numberof indicators should preferably be equal or larger than 110, and in no caselower than 65.


TSLS is only marginally more complex than OLS and can deal with largemodels. It has the disadvantage of using a limited information estimator,which leads to poorer estimates (Schermelleh-Engel et al., 1998; Moulderand Algina, 2002). In particular, an arbitrary choice must be made on whichindicator is used in the second stage and which indicators are used as instru-mental variables in the first stage. The results depend heavily on this choice.

PLS also makes it possible to fit very large models. PLS constitutes arather complex procedure that is in fact not so far from OLS on SRS, fromwhich it differs by the fact that the weights of the indicators are not equalbut computed from the optimization of certain criteria, which bring themclose to being principal component weights. In fact, the results are reportedto be quite stable for different weights of the indicators (McDonald, 1996)so that MRA on SRS and PLS tend to give nearly identical results. Onthe positive side, PLS shares with regression the property of providing opti-mal predictions and can successfully be applied for predictive purposes orwhenever the aim of the analysis is exploratory, the theory is weak, or thenumber of variables is too large for formal modelling (Joreskog and Wold,1982). Wold (1982) introduced the term “soft modelling” to refer to thesesituations. However, on the negative side, and in the same way as MRAon SRS, PLS has the limitation that it does not eliminate measurementerror bias, as it is consistent only under perfect reliability or with an infi-nite number of items per dimension (Wold, 1982; Dijkstra, 1983; Fornelland Cha, 1994; Hulland, 1999; O’Loughlin and Coenders, 2004). Never-theless, it must be admitted that there has been growing interest in PLS(even if the technique dates back to the 1970s, nearly half of its applica-tions reported in the Social Sciences Citation Index in March 2005 werepublished in 2001 or later). This is probably due to a mystification of the“soft modelling” term (McDonald, 1996). In fact, many of these applica-tions of PLS are for non-predictive purposes, for which the presence of biasis a fundamental drawback. In this paper, the purpose is theory estimationand testing, for which unbiasedness is indeed a key requirement.

Half way between SEM and PLS there is the use of OLS on factorscores (FS) (Joreskog, 2000; Schumacker, 2002). OLS on FS is easy toimplement and has been reported to give very similar estimates to maxi-mum likelihood SEM (Schumacker, 2002) though standard errors are onlyapproximate. A related method which is suited for small samples is dis-cussed in Coenders et al. (2003). To perform OLS on FS one must proceedas follows:

• In a first step, a confirmatory factor analysis model or CFA model(a particular case of SEM) is fitted only to the main effect indicators,which reduces modeling burden. If the dependent variable has multipleindicators it may also be included in the model. This model has the


twofold objective of assessing construct validity of the indicators and ofobtaining appropriate FS for the main effect variables. A wealth of cri-teria for computing FS has been defined. Among them is the Andersonand Rubin (1956) method which corrects for measurement error bias asthe correlations among factor scores are constrained to be equal to thefactor correlation parameters of the CFA model.

• In a second step, main effect FS can be conveniently multiplied orsquared. Observed variables measured without error can also be addedto the factor score file. Then an appropriate regression model can befitted with general statistical software.

The procedure is flexible in that it can be extended to accommodate anytype of dependent variable. When the dependent variable is binary, a logisticregression model may be estimated by maximum likelihood (Hosmer andLemeshow, 1989). In this kind of regression, the obtained coefficients canbe interpreted as the change in the dependent variable that has been trans-formed into logits (Menard, 1995). Logit(y) is defined through Equation (2):

Logit(y)= ln(

p(y =1)

1−p(y =1)

)=β0 +β1x1 +β2x2 +β3x1x2 + β4x

21 +β5x

22 ,

(2)

where the x variables may be Anderson and Rubin FS if measured witherror. Computing the logit implies a non-linear transformation of the leftpart of the equation. Products and polinomial terms can be included at theright part of the equation and have a similar interpretation to Equation(1) (Pampel, 2000). From here on, when we refer to linear or non-linearmodels, we will be explicitly referring to the right part of the equation.Logistic regression can be extended to ordinal dependent variables withmore than two categories (Menard, 2002) or even to the nominal case.

Ordinal indicators of the regressor variables will usually require no spe-cial treatment due to the robustness of CFA point estimates to ordinalmeasurement (see Coenders et al., 1997 and references therein). However,since ordinal data are not normally distributed, robust test statistics arerequired (e.g. Satorra and Bentler, 1994).

In this article, the logit variant of MRA on FS method is used. TheLISREL 8.54 program (Joreskog and Sorbom, 1993; du Toit and du Toit,2001) was used for the first step CFA model and SPSS Version 12.0 wasused to fit an ordered logit model for the second step.

3. Subjects and Methods

The sample belonged to high schools throughout the province ofGirona (Catalonia, N.E. of Spain). The students could be considered as


representative of the characteristics of the majority of middle class fam-ilies living in each municipality. A sample of 929 subjects was obtained,473 (50.9%) of them being girls and 456 (49.1%) boys. Their mean age was14.02 years (SD=1.25). The size of the final sample can be considered ade-quate for CFA models (Boomsma and Hoogland, 2001).

The instrument included the following measures related to psychologicalwell-being:

1. Satisfaction with life as a whole and with specific domains in life. Eightitems were included to measure satisfaction with school performance,with learning, time use, enjoying time, preparation for future, family,friends, and with sport activities. An item of satisfaction with life as awhole was also incorporated.

2. Rosenberg’s Self-Esteem Scale (1965).3. Two sub-scales (family and friends) from Vaux et al.’s Social Support

Appraisals (1986).4. Pearlin and Schooler’s Mastery Scale (1978).5. Values. Adolescents were asked to what extend they would like to be

appreciated by other people at the age of 21 for each of the follow-ing 12 values: intelligence, technical skills, social skills, knowledge ofcomputers, professional status, family, sensitivity, sympathy, money,power, knowledge about the world and image.

4. Confirmatory Factor Analysis

As a previous step to the CFA model, a principal component analysis(PCA) with varimax rotation was applied to each of the groups of variablesdescribed in the previous section. The dimensions thus identified were sub-mitted to the CFA model in a next step.

Indicators with very low standardised loadings or those which led tohigh-residual covariances were rejected, one by one, as they were consid-ered invalid. It can also be the case that correlations among items whichhave been formulated negatively and among those which have been for-mulated positively (all of them supposedly measuring the same construct)are higher in comparison to correlations between pairs of items formulatedwith a different sign. This could be indicating the existence of a contentfactor and a style of response factor, rather than two latent factors withdifferent content (Billiet and McClendon, 1998). Uncorrelated style fac-tors were added where appropriate. The same was done when a constructappeared to have several dimensions in which we were not interested froma substantive viewpoint.

The modification process was stopped when we obtained satisfactoryvalues of the fit indices (standardized root mean squared residual = 0.050,


Figure 1. Path diagram of the global final model (POSITIVE: Positiveself-esteem/NEGATIVE: Negative self-esteem/SSFAMILY: Perceived social sup-port from family/SSFRIEND: Perceived social support from friends/SAT.ENJO:Satisfaction with enjoying time/SAT.LEAR: Satisfaction with learning/SESTEM:Self-esteem/SSUPPORT: Perceived social support/CONTROL: Perception of con-trol/MATERIA: Material values/CAPACIT: Capacities and knowledge values/INTERPE: Interpersonal relationships values) (With the objective of simplifying thediagram the error terms are not shown).

Tucker and Lewis Index = 0.960, comparative fit index CFI = 0.967 androot mean squared error of approximation RMSEA<0.0363) (Figure 1).

Although one indicator had a non-significant negative error varianceand some others had not very high standardized loadings, dropping moreitems would have led to a too large a loss of content, and so we retained


this model. In Table I we can see all standardized loadings and the list offinal items in each dimension. The estimates of this model were used tocompute FS by the Anderson and Rubin method.

5. Logistic Regression Model Selection

In what follows we are going to describe the particularities of the modelswe have fit with the objective of discriminating whether a non-linear modelconstructed from the dimensions of the CFA model, age and gender, has ornot a better explanatory power than a linear model. The fitted models are:

(a) Linear.(b) Non-linear with interaction terms constructed as the products of all

possible pairs of variables.(c) Non-linear with interaction and quadratic terms.

These models are nested and can be compared by means of a likelihoodratio test constructed by subtracting their respective χ2 deviance values.

In Table II we can see that Nagelkerke’s pseudo R2-values are verydifferent between models a and b and that the difference in deviance χ2 issignificant. Thus, it can be concluded that the effects of some of the vari-ables depend on the value of other variables.

None of the quadratic terms is statistically significant in model c. Thedifference in deviance χ2 between models c and b is also non-significantand the gain in Nagelkerke’s pseudo R2 is negligible. Thus, it cannot beconcluded that the effects of some of the variables depend on their ownvalues and we maintain model b for the sake of parsimony.

6. Results of the Final Model

The significant main effects for the selected model b are shown in Table IIIand are the following: satisfaction with enjoying time and satisfaction withlearning, both with a positive effect. As the numeric variables are mean-centred, these effects refer to the mean value of all the remaining variablesand to the group of boys (reference group for the gender dummy variable).The non-significant main effects that intervene in a significant interactionmust also be considered relevant variables in the model (Hartmann andMores, 1999; Irwin and McClelland, 2001).

The significant interactions are perception of control-capacities andknowledge values (negative), perception of control-satisfaction with learn-ing (negative), self-esteem-capacities and knowledge (positive), self-esteem-material values (negative), self-esteem-gender (positive), perceived socialsupport-interpersonal values (negative), perceived social support-satisfaction


Table I. Factors and their indicators and standardized loadings of the final specifica-tion of the psychological well-being model

Factors and their indicators Standardizedloadings

SAT.ENJO Satisfaction with enjoying timeSAT.ENJO Satisfaction with amusing 0.663SAT.FRIE Satisfaction with friends 0.715

SAT.LEAR Satisfaction with learningSAT.SCHO Satisfaction with school performance 0.614SAR.LEAR Satisfaction with learning 0.686SAT.FUT Satisfaction with preparation for future 0.676

SESTEEM Self-esteemSESTEEM1 On the whole. I am satisfied with myself 0.567SESTEEM2 At times I think I am no good at all −0.471SESTEEM3 I feel that I have a number of good qualities 0.406SESTEEM4 I am able to do things as well as most other people 0.389SESTEEM6 I certainly feel useless at times −0.514SESTEEM7 I feel that I’m a person of worth at least on anequal plane with others

0.334

SESTEEM9 All in all, I’m inclined to feel that I am a failure −0.617

SSUPORT Perceived social supportSS.FAM1 My family cares for me very much 0.437SS.FAM2 My family holds me in high esteem 0.378SS.FRIE1 I can rely on my friends 0.508SS.FAM4 I am loved dearly by my family 0.479SS.FRIE5 My friends look out for me 0.510SS.FAM7 My family really respects me 0.404SS.FRIE6 My friends and I are important to each other 0.496SS.FRIE7 My friends and I have done a lot for one another 0.475

CONTROL Perception of controlCONTROL2 I cannot solve my problems −0.593CONTROL4 I often feel defenceless in connection with solvingmy own problems

−0.750

CONTROL 5 Now and then I feel life is passing very slowly −0.751

MATERIA Material valuesVAL.MONE Value to money 0.798VAL.POWE Values to power 1.003

CAPACIT Capacities and knowledge valuesVAL.INT Intelligence value 0.676VAL.TECH Technical skills value 0.663

INTERPE Interpersonal relationships valuesVAL.SSKI Social skills values 0.698VAL.SIMP Value to sympathy 0.609


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Table III. Estimates of model b (with interaction terms)

Estimate s.e. p-value

CONTROL 0.325 0.205 0.113SESTEEM 0.177 0.311 0.569SSUPPORT −0.387 0.250 0.121CAPACIT −0.001 0.197 0.995INTERPE −0.174 0.197 0.376MATERIA 0.069 0.079 0.380SAT LEAR 1.047 0.152 0.000SAT ENJOY 0.668 0.226 0.003AGE −0.140 0.083 0.091GENDER −0.192 0.154 0.212CONTROL * SESTEEM 0.047 0.104 0.650CONTROL * SSUPPORT 0.457 0.244 0.061CONTROL * CAPACIT −0.600 0.226 0.008CONTROL * INTERPE 0.031 0.234 0.895CONTROL * MATERIA 0.155 0.137 0.260CONTROL * SAT LEAR −0.396 0.165 0.017CONTROL * SAT ENJOY −0.138 0.255 0.588CONTROL * AGE 0.097 0.118 0.410CONTROL * GENDER −0.257 0.300 0.392SESTEEM * SSUPPORT −0.299 0.307 0.330SESTEEM * CAPACIT 0.690 0.298 0.021SESTEEM * INTERPE 0.293 0.336 0.384SESTEEM * MATERIA −0.423 0.193 0.028SESTEEM * SAT LEAR 0.253 0.214 0.236SESTEEM * SAT ENJOY −0.454 0.339 0.181SESTEEM * AGE −0.134 0.173 0.437SESTEEM * GENDER 0.893 0.446 0.045SSUPPORT * CAPACIT 0.024 0.264 0.927SSUPPORT * INTERPE −0.552 0.274 0.044SSUPPORT * MATERIA 0.236 0.121 0.051SSUPPORT * SAT LEAR 0.300 0.204 0.142SSUPPORT * SAT ENJOY 0.285 0.134 0.034SSUPPORT * AGE −0.312 0.140 0.026SSUPPORT * GENDER −0.318 0.359 0.376CAPACIT * INTERPE −0.150 0.077 0.053CAPACIT * MATERIA 0.179 0.129 0.168CAPACIT * SAT LEAR −0.045 0.172 0.793CAPACIT * SAT ENJOY 0.250 0.224 0.264CAPACIT * AGE −0.072 0.108 0.505


Table III. continued

Estimate s.e. p-value

CAPACIT * GENDER −0.082 0.279 0.766INTERPE * MATERIA −0.225 0.130 0.083INTERPE * SAT LEAR −0.288 0.184 0.118INTERPE * SAT ENJOY −0.028 0.205 0.891INTERPE * AGE 0.079 0.107 0.460INTERPE * GENDER 0.296 0.285 0.298SAT LEAR * SAT ENJOY −0.094 0.176 0.591SAT LEAR * AGE −0.009 0.081 0.913SAT LEAR * GENDER −0.377 0.221 0.089SAT ENJOY * AGE 0.308 0.121 0.011SAT ENJOY * GENDER 0.496 0.306 0.106AGE * GENDER 0.135 0.121 0.266

with enjoying time (positive), perceived social support-age (negative), andsatisfaction with enjoying time-age (positive). We provide some exampleson how these interactions should be interpreted depending on whether themain effects are or fail to be significant:

• Perception of control-satisfaction with learning (negative). This interac-tion implies that for large values of perception of control, the effect ofsatisfaction with learning on overall life satisfaction is lower than themain effect, while for low values of perception of control it would behigher than the main effect. For very large values of perception of con-trol, the effect of satisfaction with learning on overall satisfaction caneven be negative.

• Self-esteem-capacities and knowledge (positive). As neither of the vari-ables has a significant main effect, this interaction implies that for largevalues of self-esteem, the effect of capacities and knowledge values onoverall satisfaction is positive. For high values of capacities and knowl-edge the effect of self-esteem on overall satisfaction is positive. For lowvalues of either variable, the effect of the other would be negative.

7. Discussion

Our data, which have been obtained through a cross-sectional study witha sample of adolescents of a particular context, do not make it possible toexplore psychological well-being from all theories of complexity (chaos the-ories, fractals, catastrophe theory and fuzzy sets theory). However, a meth-odological and epistemological approach from one of the most important


properties of the complexity paradigm, non-linearity, has been achievable.From here on, defending the existence of non-linear relationships amongsome of the elements well-being is composed of, has a series of relevantconsequences for its study.

First of all, this means that we cannot talk of direct proportion betweencauses and effects when studying psychological well-being. The fact of adopt-ing a non-linear model instead of a linear one leads to consider that linearrelationships among the elements psychological well-being is formed by areonly part of the model (only satisfaction with learning and satisfaction withenjoying time have main effects on satisfaction with life as a whole). The restis explained through interaction effects among different latent factors.

The interaction between two independent variables reflects the relation-ships that one of the variables has with the dependent variable (satisfac-tion with life as a whole) moderated by the effect of the other independentvariable. A positive sign means that the effect of one variable is higher forhigh values of the other variable, whereas a negative sign means the otherway round. These interactions can often lead to meaningful interpretations.For instance, the effect of satisfaction with enjoying time on overall lifesatisfaction is larger for older adolescents. As adolescents grow up, theymight perceive a greater freedom to spend their free time (this includesboth activities and friends). The effect of satisfaction with enjoying time onsatisfaction with life as a whole is greater for also high values of perceivedsocial support from friends. The effect of self-esteem on satisfaction withlife as a whole seems to be higher for girls than for boys. It is often high-lighted in the literature that girls’ worry about their image is still highercompared to boys’ in nowadays societies.

The comparison between our results and the scientific literature cannotbe done in a strict sense, as the majority of studies have used a linear meth-odology of analysis. All in all, it might be of interest to take into accountthe results obtained with such different methodologies. However, and as thiswould exceed the objectives of this work, we refer the reader to the followingpieces of research: Ben-Zur (2003), Bostic and Ptacek (2001), Casas et al.(2004a), Casas et al. (2004b), Huebner (2004), Huebner et al. (2001), Furn-ham et al. (2002), Garaigordobil et al. (2003), Kasser and Ahuvia (2002),Park and Kim (1998), Polce-Lynch et al. (2001), Sagiv and Schwartz (2000)(these and other studies are widely commented in Gonzalez, 2006).

One of the most important consequences of adopting a linear modelwould be the rejection of an important number of latent factors thatdo not have a main effect on satisfaction with life as a whole. Thesevariables are: perception of control, self-esteem, perceived social support,capacities and knowledge values, interpersonal values, material values, ageand gender. Adopting a linear model would also have driven to wronglyconclude that the only relationships that exist among the different elements


of psychological well-being are fixed. Differently to the linear model, withinthe non-linear, the effect of a first variable on a second one depends on athird variable and even the sign of the effect may change according to thevalues of that variable.

Another condition for non-linearity, apart from sensibility to initial con-ditions, is the lack of proportion between input and output. That is, smallchanges in one of the variables might have as a result large changes in thedependent variable for certain values of a third variable.

In relation to unpredictability when studying psychological well-being,we have seen that all studied variables are related in one or another way tosatisfaction with life as a whole. Psychological well-being in adolescence is,in fact, a fuzzy concept and, thus, it is not possible to know all dimensionsor aspects it is composed of. Our best model attained a R2-value of just0.5. This means probably many other variables could be also related (e.gpersonality and family variables, life objectives and so on). This opinion isalso shared by Eid and Diener (2004).

There exists an important debate about whether the study of com-plex phenomena can be carried out without having panel data, which issomething very common within social and psychosocial studies. In rela-tion to this, Byrne (2002) defends the interest that focusing instead onemergent structures might have for social and psychological sciences in gen-eral. Complexity is usually understood, in fact, as an emergent phenom-enon (Corning, 2002) and, so, complexity laws would explain, apart fromthe processes of change themselves, the emergence of a macro-phenomenon(psychological well-being in this case) from micro-phenomena (for instance,the elements psychological well-being is formed by or related with). Thus,and following Byrne’s recommendation, the emergency of the former hasbeen supposed to take place in this work as a consequence of the patternof non-linear interactions which happen among the latter.

It is worth mentioning that the majority of statistical techniques devel-oped within complexity theories have not been originally designed toanalyse social and psychological phenomena. All in all, non-linear modelshave been developed in last years with the objective of obtaining a betterfit of the statistical analysis of the phenomena studied by the social andhuman sciences (see, as an example, Klein and Stoolmiller, 2003). We alsoagree with the opinion of Jorge (2002) that some instruments and tech-niques which are being utilised nowadays can be used for the objectives ofcomplexity theories, with the condition that the assumptions for its appli-cation, utilisation and afterwards interpretation are revised under the ideasof the complexity paradigm.

Psychological well-being in adolescence is a complex phenomenon andstudying it through non-linear techniques has a lot of potentialities. Theapproach followed in this work has been basically methodological. The use


of non-linear regression analysis on factor scores computed by the Ander-son and Rubin method has proven feasible for quite a large model, as inany case required by complexity theories. Other known strategies to non-linear modelling of variables measured with error suffer from serious draw-backs for models of this size (maximum likelihood SEM), produce poorestimates (TSLS) or fail to correct measurement error bias (PLS). The useof this strategy also makes it very easy to include a non-linear relationshipin the left side of the equation, for instance, by means of a logistic trans-formation and can thus be recommended on a general basis. Of course, thisapproach is not limited to the study of psychological well-being but can beused to study many other complex psychological phenomena.

The strategy we have adopted in this article of trying to get to the wholesystem (psychological well-being) through the comprehension of the inter-actions among the elements it is composed by or related with is, moreover,very common for complex approaches and is grounded on a totality prin-ciple, which according to Munne (1994) should inspire any research abouthuman-being’s behaviour.

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