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Abstract The purpose of this study was to test an integrative model in which peer relations at different levels of social complexity (friendship quality, social preference, and proximity prestige) are associated with children's loneliness, with children's self-perceived social competence acting as a mediator of these associations. A middle childhood sample of 509 Chinese children (233 girls and 276 boys; 3rd to 6th grade) completed a battery of sociometric and self-report questionnaires. Bootstrap analysis showed that self-perceived social competence mediated the relations between each peer variable and loneliness. In the integrative model tested with SEM, the mediating effect of self-perceived social competence in the relation between friendship quality and loneliness and between social preference and loneliness remained significant. However, self-perceived social competence no longer mediated the association between proximity prestige and loneliness, when considering the simultaneous influences of the three peer variables (friendship quality, social preference, and proximity prestige). The whole model accounted for 56% of the variance in loneliness. These findings suggest that self-perceived social competence played an important role in children's loneliness, that the quality and the quantity of direct peer relations (friendship quality, social preference, and part of proximity prestige) were associated with loneliness, and that indirect friends had a relatively lower but significant influence on children's loneliness. The results are discussed in terms of their implications for preventing children's loneliness.

. Introduction As a social species, human beings innately have the need for social connection. An absolute or relative lack of social connection can result in the painful emotion of loneliness (Peplau and Perlman, 1982 and Weiss, 1973). Loneliness is a common emotional experience among children. As many as 80% of children report having experienced loneliness at school (Berguno, Leroux, McAinsh, & Shaikh, 2004). One study of third-grade children found that 23% had a moderate level of loneliness, a feeling that steadily increased during the next two years (Jobe-Shields, Cohen, & Parra, 2011). Substantial evidences show that the feeling of loneliness in childhood not only is associated with children's current life quality but also predicts future maladjustment (Masi et al., 2011, Rotenberg, 1999 and van Dulmen and Goossens, 2013). Lonely children are more likely to experience low self-esteem (Sletta, Valås, Skaalvik, & Søbstad, 1996), increased levels of social anxiety and social avoidance (Vanhalst, Goossens, Luyckx, Scholte, & Engels, 2012), poorer academic performance (Benner, 2011), higher risk of dropping out of school and of delinquency, and more mental and physical health problems (Harris et al., 2013 and Heinrich and Gullone, 2006). Loneliness in childhood is a valuable predictor of depressive symptoms in adolescence (Qualter, Brown, Munn, & Rotenberg, 2010). Two theories have systematically illuminated the nature of loneliness and the factors that influence it: the social needs theory and the cognitive processes approach (Terrell-Deutsch, 1999). The social needs theory claims that loneliness is a response to unmet needs for social connection or to unsatisfactory interpersonal relationships. Consistent with this theory, the majority of research has focused on peer relations to understand children's loneliness, and during the past three decades, peer relations have been found to be a critical factor in children's loneliness (Asher & Paquette, 2003). In contrast to the social needs theory, cognitive processes theory suggests that loneliness is not a result of unmet inherent social needs but of dissatisfaction with one's perceived social relationships. In other words, it is the cognitive awareness of a deficiency in either the quality or the quantity of one's social relationships that leads to the discomfort of loneliness (Peplau and Perlman, 1982 and Terrell-Deutsch, 1999). However, there is little research examining the effects of individuals' internal cognitive representations of social relations (e.g., perceived social competence) in the link between peer relations and children's feelings of loneliness. 1.1. Peer relations and loneliness Experiences with peers constitute an important developmental context for children. Children's peer experiences can be divided into several levels of analysis—individual characteristics, social interactions, dyadic relationships, and group membership and composition. The latter three levels of peer system reflect social participation at different interwoven orders of social complexity (Hinde, 1987 and Rubin et al., 2006). Previous research has extensively explored the relations between loneliness and multiple types of peer relations, including peer relations at the dyadic and group levels (Margalit, 2010). Acceptance by the peer group and friendship have been found to be important factors for understanding children's experiences of feeling lonely (e.g., Asher and Paquette, 2003 and Margalit, 2010). The preponderance of research on children's loneliness has focused on the possible influence of acceptance versus rejection by peers (Asher & Paquette, 2003). The group's acceptance of a child refers to the degree to which the child is liked or disliked by group members, and group acceptance is an indicator of the child's social status in the group (Ladd, 1999). Peer acceptance is typically assessed using peer nominations, a sociometric method in which children identify (or “nominate”) group members they like most and like least. Acceptance is measured by the number of “like most” nominations a child receives, standardized according to class size, and rejection is the standardized number of “like least” nominations received. These two scores can be combined to create a social preference score (the difference between the acceptance and rejection scores), reflecting the child's relative standing in terms of acceptance by the peer group (Hymel, Vaillancourt, McDougall, & Renshaw, 2002). Previous studies have shown that children who have higher levels of peer acceptance or are more socially preferred are less likely to report feeling lonely (e.g., Mouratidis and Sideridis, 2009, Shin, 2007, Yu et al., 2005 and Zhou et al., 2005). Friendship lies at the dyadic level of peer experience. A “friend” is defined as a person you know well and like, usually not a member of your family (Hornby, 2010). Friendship is the relationship between friends. The definition of friend suggests that friendships may vary in their degree of mutual knowledge and affection, characteristics which constitute friendship quality. Researchers have consistently found a high negative correlation between friendship quality and children's loneliness (e.g., Hoza et al., 2000, Nangle et al., 2003, Parker and Asher, 1993 and Sun et al., 2009). Specifically, studies have indicated that companionship and support from friends were conducive to lessening or eliminating children's loneliness (Parker and Asher, 1993 and Uruk and Demir, 2003). Moreover, longitudinal research has shown that friendship quality in middle childhood significantly and negatively predicted loneliness two years later (Zhou, Zhao, Sun, & Ding, 2006). Nonetheless, relatively few studies have examined the link between the quantity of friends children have and children's loneliness, and the findings of these studies were inconsistent. Ladd, Kochenderfer, and Coleman (1997) found that although the quantity of friends did not predict children's loneliness as they made the transition to kindergarten, it positively predicted other aspects of children's adjustment (such as academic readiness and school involvement). Parker and colleagues (Parker and Asher, 1993 and Parker and Seal, 1996) have shown that in middle childhood having one friend is predictive of lower loneliness, but they did not explore the effect of having more than one friend. Recent research in middle childhood samples has suggested that the number of friends is important in predicting loneliness. With a sample of third- through sixth-grade students, Nangle et al. (2003) demonstrated that the number of both best friends and good friends was important in predicting children's positive adjustment, including a lower level of loneliness. The number of early mutual friends at the ages of 9–10 has been shown to significantly and negatively predict loneliness two years later (Zhou et al., 2006). Although research consistently suggests that both friendship processes and acceptance by peers can predict children's feelings of loneliness, these variables as typically measured (e.g., by friendship nominations and social preference scores) still leave much of the variance in loneliness unexplained (Asher & Paquette, 2003). One possible explanation is that not all friends are equally important in protecting a child from being lonely. Indeed, a child's friendships and social status in the peer group are embedded in a larger social network of peer relations and experiences (Gifford-Smith & Brownell, 2003). The peer group is a network in which most children connect with several other children by friendship. However, each of these friends has a distinct influence on the child, and one reason for this variation is that a child's friends may vary in social status. For example, friends with many connections in the peer group might provide more information or resources to the child than more isolated friends could. Thus one way to assess a child's connectedness in the peer group is to take into account not just the number of friends, but also the number of “friends of friends.” Even if a child has few direct friends, a high number of indirect friends might reduce the risk of loneliness. A useful index of the quantity of indirect friends is called proximity prestige, a measure that developed from social network analysis. In recent years, social network analysis, a method used to study the structure and characteristics of social networks (Scott, 2000), has been identified as an appropriate method for studying the peer context of child and adolescent social behaviors and emotions, such as aggression (Faris & Ennett, 2012), substance use (Ennett et al., 2006, Mercken et al., 2012 and Mundt, 2011), prosocial behaviors (Gest, Graham-Bermann, & Hartup, 2001), depression (Okamoto et al., 2011), and loneliness (Chamberlain, Kasari, & Rotheram-Fuller, 2007). This method generates a visual depiction of the complex inter-relations in a social network. Each actor (e.g., a child) is represented by one node, and the relationships (e.g., friendships) between any two actors are represented by lines. Some actors are linked directly, and some are linked indirectly through relationships with other actors. In the visual representation of these relationships, the distance between two actors is one unit if the actors are linked directly, two units if they are linked indirectly through one other actor, and so on. The shorter the distance is between actors, the closer the relationships are between them. An actor's influence domain is the number or percentage of all actors linked to him or her directly and indirectly in a closed group. However, the intensity of the actor's influence varies depending on the closeness of the relationship. The farther away that neighbors are from an actor, the smaller the actor's influence on them. Using this type of information about a social network, Lin (1976) first proposed a measure of prestige called proximity prestige. According to Wassermann and Faust (1994), the proximity prestige of an actor is the proportion of all other actors directly or indirectly connected to the actor in a closed group (influence domain) divided by the average distance these other actors are from the actor (the mean of influence intensity). This means that when relationships are defined by friend nominations, then the relationships are weighted differently depending on whether the link is direct (one child is nominated by another) or indirect (one child is nominated by another, who in turn is nominated by someone else). In a friend nomination network, the influence domain is the quantity of direct and indirect friends. The influence intensity represents the closeness of these relationships. Therefore, proximity prestige is a composite representation of the quantity and the closeness of children's direct and indirect friends. Meanwhile, this index, which is calculated based on friend nominations by others, also reflects children's objective friendship status in the peer group. In social network analysis, one widely used index is centrality, which is the proportion of nominations one actor receives from other actors in a group. The actors who receive many positive nominations are considered to be prominent and to occupy a central position in the peer group. However, the centrality index makes no distinction in terms of the popularity of the nominators. For example, two actors who receive the same number of nominations, but who differ in how active they are in the larger network, would have the same centrality index score. Proximity prestige extends the concept of centrality by taking the popularity of the nominators into consideration. It is important to distinguish proximity prestige from other concepts of peer relationships (e.g., peer acceptance, social preference), especially because the number of direct and indirect friends in the social network may highly coincide with the number of “like most” nominations, the basis of peer acceptance and social preference scores. But naming a specific child as a friend involves affirmation of a special dyadic relationship, whereas peer acceptance and social preference are unilateral constructs representing the general affective inclination of the group toward an individual. Even though these three constructs (peer acceptance, social preference, and proximity prestige) all lie at the group level of peer experience, proximity prestige is based on the social network of friendships and thus represents greater social complexity than the other two indices. Friendship quality, social preference (a measure of acceptance by the group) and proximity prestige lie at the dyadic and group level of peer relations, and they reflect different facets of children's experience with peers. Therefore, the first goal of the present study was to test whether variables representing peer relationships at different levels of social complexity were associated with children's loneliness, and it was hypothesized that the proximity prestige index would provide extra explanatory power above that of friendship quality and social preference in predicting children's loneliness. 1.2. Self-perceived social competence and loneliness Measures of peer relations are often objective measures of social status, which may not correspond with children's subjective experience of loneliness. According to cognitive processes theory, it is a child's cognitive appraisal of peer relations that gives rise to the feeling of loneliness (Terrell-Deutsch, 1999). Self-perceived social competence is the internal cognitive estimation of one's own social competence. Children who perceive higher social competence are more likely to be satisfied with their peer relations, and less likely to feel lonely. Sun et al. (2009) discovered that self-perceived social competence was more predictive of loneliness than was objective social status. In addition, children's social success may lead to positive social evaluations of their own competencies (Cole, 1991); in turn, positive self-perception may further lead to less loneliness. It is supposed that self-perception of social competence is a mediator of the link between peer relations and loneliness. Studies with third- to sixth-grade children have shown that self-perceived social competence mediates the association between direct peer relations (peer acceptance and friendship quality) and loneliness (Sun et al., 2009 and Zhou et al., 2005). However, these studies did not take indirect peer relations into consideration. Therefore, the second goal of the present study was to test whether proximity prestige (a reflection of both direct and indirect friendships and objective friendship status) in the peer network is associated with children's loneliness through the mediator of self-perceived social competence. 1.3. Peer relations, self-perception, and loneliness Previous studies have extensively explored different levels of peer relations as predictors of children's loneliness; however, the studies have tended to concentrate on each predictor independently rather than integratively. Based on an extensive critical review, Gifford-Smith and Brownell (2003) emphasized that children's peer relations are multifaceted, intersecting and overlapping systems, and it is necessary and valuable to attend to multiple aspects of the peer experiences simultaneously from an integrative perspective. Hinde (1987) also emphasized that events and processes at each level of peer experience are constrained and influenced by events and processes at other levels (Rubin et al., 2006). Additionally, researchers have pointed out that peer relations constitute just one facet of the contexts in which peer interactions occur and develop. These contexts determine the particular skills, perceptions, and cognitions that will be most effective for children's successful functioning in the peer group (Brownell and Gifford-Smith, 2003 and Sheridan et al., 2003). Thus, the third goal of the present research was to test a conceptual model in which peer experience at multiple levels of social complexity (namely friendship quality, social preference, proximity prestige, and self-perceived social competence) jointly predicts children's loneliness; in this model, children's self-perceived social competence is also tested as a mediator, representing a possible underlying mechanism in the association between peer relations and children's loneliness. It was noted that the level of social interaction was not represented in the conceptual model. Social interaction encompassed different kinds of behaviors and behavioral tendencies (Rubin et al., 2006). The theoretical and substantial evidences showed that social behaviors were the antecedents or predictors of peer relations in childhood (Chen et al., 1992 and Ladd et al., 2002). According to the concept and theory of loneliness, it was highly emphasized that loneliness was the response to the lack of social connection (peer relations). The present study mainly focused on the link between peer relations and the feeling of loneliness in the theoretical framework of loneliness. Thus, the antecedents of peer relations (e.g., the factors of social interaction) were not included in the analyses. Both the social needs theory and the cognitive processes approach of loneliness indicate that the satisfaction or perceived satisfaction of interpersonal relationships could lower or eliminate feeling lonely (Terrell-Deutsch, 1999). The interpersonal relationships mentioned by the theories are the positive relations what individuals want to obtain (contrary to negative relations, such as peer victimization). Most of the research on children's loneliness has focused on the great contributions of two kinds of positive peer relations, acceptance by peer group and the friendship (Asher and Paquette, 2003 and Margalit, 2010). The research has consistently found that children who are better accepted by peers or have a high-quality friendship with a best friend report experiencing less loneliness than other children. On the contrary, as one facet of friendship, the quantity of friends children has been occasionally examined its relation with loneliness, and the results are not consistent. Meanwhile, the research has not taken indirect friendships and the statuses of friends into consideration. Therefore, the index of proximity prestige is introduced to represent the number of indirect friends and the statuses of friends. The positive peer relations in the peer group can be represented by the total of proximity prestige (composite representation of the quantity and the closeness of children's direct and indirect friends), friendship quality (the relationship with best friend) and social preference (the affective inclination of the group toward an individual). The present study aimed to explore the simultaneous influence of the three kinds of peer relationships (friendship quality, social preference, and proximity prestige) on children's loneliness, especially the unique contribution of proximity prestige on loneliness. 1.4. Culture and loneliness Most research on children's loneliness has been conducted in Western cultures, especially North American, which are typified by an individualistic orientation. Individualist cultures value individualization, autonomy, and privacy. Asian countries including China are collectivistic cultures that prioritize relational bonds and group cohesion (Oyserman, Coon, & Kemmelmeier, 2002). The Western cultural orientation of individualism has been found to be associated with higher loneliness among adults, even in Western societies (Rokach and Bauer, 2004 and Seepersad et al., 2008). Multinational studies of the loneliness of 3rd to 6th grade children conducted in Canada, Brazil, China, and Southern Italy revealed no mean differences in self-reported loneliness across these four samples. However, the overall patterns of relations between social behaviors (such as aggression and shyness-sensitivity) and loneliness differed across samples. For example, shyness is viewed as maladaptive in Western cultures (Fox, Henderson, Marshall, Nichols, & Ghera, 2005), and indeed, results of the multinational study showed that shyness was positively associated with loneliness in Canadian, Brazilian and Italian children, but not associated with loneliness in Chinese children. These results suggested that the nature of children's loneliness may be affected by the broad socio-cultural context (Chen, He, De Oliveira, et al., 2004). In contemporary China, one major mission of schooling is to help children develop collectivistic ideologies (Chen, 2000). Children are encouraged to participate in group activities, develop positive attitudes toward the group, and learn social skills that facilitate group functioning. The emphasis on social connections results in an interdependent sense of self (Markus & Kitayama, 1991). In this context, social alienation and loneliness are more likely to be viewed as maladaptive or problematic (Chen, 2000). Chinese children's self-perceived social competence is also likely to be influenced by cultural context. Self-evaluation has been encouraged traditionally in Chinese culture (e.g., Luo, 1996). In Chinese schools, children are regularly required to engage in self-evaluation, in the belief that children who are conscious of their strengths and weaknesses are likely to perform and regulate their behavior better according to social norms and values. A particular emphasis is placed on self-awareness of the negative aspects of one's social and moral character (Chen, He, & Li, 2004). Therefore, self-perceived social competence would be expected to have close links with peer relations and feelings of loneliness in Chinese children. 1.5. Study hypotheses Researchers consider peer relations to be a multidimensional social network structure (Gifford-Smith and Brownell, 2003 and Margalit, 2010), and many studies have explored the effects of one or two facets of peer relations on children's loneliness. However, little research has examined multiple facets of peer relations in relation to children's loneliness, and the underlying mechanisms of these associations. The goal of the current study was to explore multiple indices of peer relations as predictors of loneliness in middle childhood, and to test children's self-perceived social competence as a mediator of these links in the context of Chinese culture. Based on the empirical research and associated conceptual models to date, a new conceptual model was developed and tested in the current study (see Fig. 1). It was hypothesized that: (a) friendship quality, social preference, proximity prestige, and self-perceived social competence would be negatively correlated with children's loneliness; (b) proximity prestige would explain unique variance in loneliness scores; and (c) self-perceived social competence would mediate the association between peer relations and loneliness in both separate and integrated analyses. The mediation model of peer variables, self-perceived social competence and ... Fig. 1. The mediation model of peer variables, self-perceived social competence and loneliness. The independent variables were friendship quality, social preference, and proximity prestige.

3. Results 3.1. Descriptive statistics Descriptive statistics for the observed variables are presented by gender and grade in Table 1. Compared to boys, girls reported lower loneliness, higher perceived social competence and friendship quality; girls also had higher social preference scores and higher proximity prestige. The only significant grade difference was in proximity prestige. The proximity prestige of sixth graders was higher than that of the other grades, and the proximity prestige of fourth and fifth graders was higher than that of third graders, but there was no significant difference between fourth and fifth graders' proximity prestige. Fig. 2 shows students' proximity prestige within one class. In this figure, students (called actors in the language of social network analysis) are situated within the network as a function of direct and indirect friendship nominations. The students who received more friendship nominations (direct and indirect) had higher proximity prestige (e.g., v2, v3). In addition, some students received few nominations, but the nominators themselves had many nominations, so that the students indirectly had comparatively high proximity prestige. For example, v30 and v35 each had one direct nomination, but their proximity prestige differed because the proximity prestige of their nominators differed: the child who nominated v30 (i.e., v16) received more nominations than the child who nominated v35 (i.e., v44). Table 1. Means and standard deviations of the research variables by gender and grade. Variable Gender T Grade F Boys Girls Third Fourth Fifth Sixth n = 276 n = 233 n = 125 n = 127 n = 129 n = 128 M (SD) M (SD) M (SD) M (SD) M (SD) M (SD) Loneliness 1.88 (0.78) 1.56 (0.60) 5.17⁎⁎⁎ 1.76 (0.70) 1.70 (0.73) 1.73 (0.76) 1.74 (0.70) 0.16 Self-perceived social competence 2.78 (0.54) 2.93 (0.50) − 3.37⁎⁎⁎ 2.85 (0.53) 2.85 (0.57) 2.86 (0.52) 2.84 (0.49) 0.05 Friendship quality 2.89 (0.75) 3.11 (0.73) − 3.34⁎⁎⁎ 2.95 (0.63) 2.86 (0.89) 3.07 (0.75) 3.06 (0.68) 2.24 Social preference − 0.53 (1.78) 0.63 (1.40) − 8.20⁎⁎⁎ 0.00a (1.65) 0.00a (1.76) 0.00a (1.72) 0.00a (1.75) 0.00 Proximity prestige 0.41 (0.12) 0.44 (0.08) − 3.90⁎⁎⁎ 0.35 (0.11) 0.43 (0.10) 0.42 (0.08) 0.49 (0.09) 45.07⁎⁎⁎ Note. N = 509. a Values are Z scores. ⁎⁎⁎ p < .001. Table options The sociogram of one class (class size=63), depicting each child's proximity ... Fig. 2. The sociogram of one class (class size = 63), depicting each child's proximity prestige. Each node denotes one child in the class; the line denotes one child (sender) who nominates another child (receiver) as a friend. The size of the nodes denotes the children's proximity prestige, and the bigger the node is, the more prestigious the child is. The sociogram was drawn with Pajek. Figure options 3.2. Correlations among peer relations, self-perceived social competence, and loneliness Table 2 shows that, as expected, loneliness was negatively and significantly associated with self-perceived social competence (the hypothesized mediator) and with friendship quality, social preference, and proximity prestige (the hypothesized predictors of children's self-reported loneliness). Self-perceived social competence was positively and significantly correlated with the three indicators of peer relations (friendship quality, social preference, and proximity prestige). This pattern of correlations indicates that it would be appropriate to test the proposed mediation model. Table 2. Correlations among the research variables. Variables 1 2 3 4 5 1. Loneliness 2. Self-perceived social competence − .69⁎⁎ 3. Friendship quality − .41⁎⁎ .34⁎⁎ 4. Social preference − .44⁎⁎ .29⁎⁎ .31⁎⁎ 5. Proximity prestige − .37⁎⁎ .23⁎⁎ .26⁎⁎ .60⁎⁎ Mean 1.73 2.85 2.99 0.00a 0.42 Standard deviation 0.72 0.53 0.75 1.72 0.10 Note. N = 509. a Values are Z scores. ⁎⁎ p < .01. Table options 3.3. Mediating effect of self-perceived social competence Table 3 presents the results of the bootstrap analyses for each of the separate mediation models. The 99% bias-corrected bootstrap confidence interval excluded zero for each of the three peer variables. In regard to model fit, the three models are all just-identified which have df = 0, CFI = 1, TLI = 1.000, RMSEA = 0, and SRMR = 0. The results suggested that self-perceived social competence was the mediator of the relations between friendship quality and loneliness, social preference and loneliness, and proximity prestige and loneliness, respectively. Table 3. The mediated effects of self-perceived social competence and the corresponding bias-corrected bootstrap confidence intervals for peer variables as predictors of loneliness. Independent variable Estimate SE 99% BC CIa Rmed2 b Friendship quality − .21⁎⁎ .03 (− .28, − .14) .13 Social preference − .17⁎⁎ .03 (− .24, − .10) .13 Proximity prestige − .13⁎⁎ .04 (− .23, − .04) .09 Note. Data set contains 4 cases with missing on all three variables (friendship quality, self-perceived social competence and loneliness). These cases were not included in the analysis of the mediation model with friendship quality (N = 505). All cases were included in the analyses of the other two mediation models (N = 509). Data in the three single mediation models were also analyzed using MLR in Mplus to correct for non-normality and the results did not change. a BC CI represents bias-corrected confidence intervals. b Rmed2 represents one effect size measure for mediated effect. Rmed2 is computed by using squared bivariate correlations and the overall model R2 from a model where Y is predicted from both X and M, as following ( Fairchild & McQuillin, 2010): Rmed2 = rMY2 − (RY, MX2 − rXY2). rMY2 is the squared correlation between the outcome (Y) and the mediator (M), RY, MX2 is the overall model R2 from the regression equation where Y is predicted from X and M. rXY2 is the squared correlation between the outcome and the independent. ⁎⁎ p < .01 (significant mediation effect). Table options 3.4. Path model linking peer relations with loneliness The hypothesized integrative model was a just-identified model, the information for model fit as following: df = 0, CFI = 1, TLI = 1.000, RMSEA = 0, and SRMR = 0. There was no significant pathway from proximity prestige to self-perceived social competence (.06, p = .24 > .01). The 99% bias-corrected bootstrap confidence intervals (99% BC CI) for the indirect effect of friendship quality or social preference did not contain zero (− .240, − .083; − .084, − .009, respectively). However, there was zero in the 99% BC CI for the indirect effect of proximity prestige (− .949, .376). The indirect effect of proximity prestige was not significant. The insignificant pathway was then excluded and the modified model was set. The result showed that the modified model was acceptable and a good fit to the data: χ2 (1) = 1.39 (p = .24), CFI = 1.00, TLI = .99 and RMSEA = .03 (90% confidence interval values ranging from .00 to .13). The Monte Carlo simulations also gave support for the modified model over the hypothesized integrative model. The result for the hypothesized integrative model from the Monte Carlo simulation showed that: for the information criteria (AIC, BIC, ABIC), the values observed in the Monte Carlo replications were close to the theoretical values; for χ2, RMSEA, and SRMR the observed values were not close to the theoretical values. Meanwhile, for the pathway from proximity to self-perceived social competence, there was only 77.8% of 10 000 replications for which the parameter estimate was significantly different from zero at α = .05, and it was smaller than the cut-off point value of .80 often used ( Cohen, 1988). In regard to the modified model, the observed values were all close to the theoretical values for the model fit statistics and indices from the Monte Carlo simulation, and the lowest percentage of 10 000 replications for which the parameter estimate was significantly different from zero at α = .05 was 92.4% (larger than the cut-off point value of .80). The Monte Carlo simulation for the modified model in present study provided good results, and demonstrated that the modifications made via modification indices would generalize. Fig. 3 presents the final integrative model of the SEM analyses. Path model showing standardized path coefficients for associations among peer ... Fig. 3. Path model showing standardized path coefficients for associations among peer variables, self-perceived social competence and children's loneliness. Data were also analyzed using MLR in Mplus to correct for non-normality and the results did not change. N = 509. Figure options The hypotheses were partially supported. In the modified model, there was a direct link from proximity prestige to loneliness, and a lack of a direct link between proximity prestige and self-perceived social competence. This means that self-perceived social competence did not act as a mediator between proximity prestige and loneliness when the influences of social preference and friendship quality were taken into account. There were direct links between friendship quality and loneliness, and between social preference and loneliness. Meanwhile, the effects of friendship quality on loneliness and of social preference on loneliness were transmitted through the intermediary predictor variable of self-perceived social competence, with the mediated (indirect) pathways having effects of − .16 and − .12 (p < .01), respectively. The 99% BC CI for the mediation (indirect) effect with friendship quality was (− .225, − .088), and that for social preference was (− 0.183, − .049). These mediating effects accounted for 55% of the total effects of friendship quality on loneliness (standardized path coefficient of − .29), and 38% of the total effects of social preference on loneliness (− .29). The peer relationship variables (friendship quality, social preference, and proximity prestige) and self-perceived social competence together accounted for 56% of the variance in loneliness.