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Who took the “×” out of expectancy-value theory? 1

This MS is the final prepublication (open access) version of the published article:

Nagengast, B., Marsh, H. W., Scalas, L. F., Xu, M. K., Hau, K.-T., & Trautwein, U. (2011). Who took the “×” out of expectancy-value theory? A psychological mystery, a substantive-methodological synergy, and a cross-national generalization. Psychological Science, 22(8), 1058-1066. doi:10.1177/0956797611415540

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Running Head: Who took the “×” out of expectancy-value theory?

Who Took The “×” Out Of Expectancy-Value Theory? A Psychological Mystery,

A Substantive-Methodological Synergy, And A Cross-National Generalization

Benjamin Nagengast, University of Oxford, UK

Herbert W. Marsh, University of Oxford, UK, and University of Western Sydney

L. Francesca Scalas, University of Oxford, UK, and University of Cagliari, Italy

Man Xu, University of Oxford, UK, and University of Cambridge, UK

Kit-Tai Hau, The Chinese University of Hong Kong, Hong Kong

Ulrich Trautwein, University of Tübingen, Germany

26 November 2010

Revision: 23 March 2011

Author note


This research was supported in part by a grant to the second author from the UK Economic

and Social Research Council. Requests for further information about this investigation should

be sent to Benjamin Nagengast, Department of Education, University of Oxford, 15 Norham

Gardens, Oxford OX2 6PY UK; E-mail: [email protected]


Abstract

Expectancy-value theory (EVT) is a dominant theory of human motivation. Historically, the

expectancy-by-value (E×V) interaction was central to EVT; motivation is high only if both

expectancy and value are high. However, the E×V interaction mysteriously disappeared more

than 25 years ago. Apparently solving the mystery, we tested E×V interactions using

evolving latent-variable models with interactions, based on large representative samples of

15-year-olds (N = 398,750) from 57 diverse countries. The effects of expectancy (science

self-concept), value (enjoyment), and the E×V interaction were all statistically significant and

positive for both engagement in science activities and intentions of pursuing scientific careers

for the total sample and for nearly all of the 57 countries considered separately. This,

apparently the strongest cross-national test of EVT ever undertaken, supports the

generalizability of EVT predictions – including the “lost” E×V interaction. The culprit (weak

statistical methodology) was identified, re-establishing the E×V interaction to its rightful role

in EVT.


We are about to embark on a psychological mystery– a real-life academic thriller. In

the not so ancient history of psychology, the dominant theory of human motivation was

expectancy-value theory (EVT). EVT had a proud tradition based on strong theory and

empirical support from both early animal research (e.g., Tolman, 1938; 1955) and subsequent

human research (Atkinson, 1957; see review by Feather, 1959). The cornerstone of EVT was

the critical role of the expectancy-by-value interaction (E×V). As depicted in the right panel

of Figure 1, EVT predicted a multiplicative pattern of relations between expectancy, value,

and resulting motivation. To achieve a high level of motivation, both expectancy and value

had to be high. If either expectancy or value were low, high values on the other dimension

were of little or no consequence for motivation and behavior. The centrality of the E×V

interaction to EVT was evident into the early 1980s; Feather’s (1982) comprehensive review

concluded that EVT models “assume that expectations and subjective values combine

multiplicatively to determine force” (p. 414).

Then, apparently without warning, the “×” in EVT mysteriously disappeared.

Beginning in the mid-1980s, the E×V term stopped being reported in tests of modern versions

of EVT of achievement motivation and the focus of the theoretical models was shifted away

from a multiplicative relation of expectancy and value to purely additive models (see the left

panel of Figure 1 compared to the right panel). A striking example is the influential socio-

cultural EVT-model of achievement motivation in educational psychology (Eccles (Parsons),

1983; Wigfield & Eccles, 2000). While it clearly extends earlier EVT-models, e.g. by

differentiating the value domain into four dimensions (interest-enjoyment value, attainment

value, utility value and cost) and by identifying pre-cursors of expectancy and value

dimensions, its core component is the joint prediction of achievement-related motivation and

choices and ultimately performance by expectancy of success and value of the outcome.


However, the initial presentation (Eccles (Parsons), 1983), but also more recent reviews (e.g.,

Wigfield & Eccles, 2000; Wigfield, Tonks, & Eccles, 2004) of this model only allude to the

positive correlation between expectancy and value. There is no explicit mention of an E×V

interaction that had been the main focus in Atkinson’s (1957) original model.

Does the omission of the multiplicative relation between expectancies and value

matter? Figure 1 illustrates that an additive model (left panel) as tacitly implied by current

EVT-models and a multiplicative relation implied by the original EVT-models (right panel)

lead to fundamentally different predictions. In the additive model (left panel), the effects are

compensatory; low scores in one domain (e.g., a small expectancy of success) can be offset

by corresponding larger scores in the other (e.g., a high value attributed to the outcome).

However, in the multiplicative model (right panel), the effects of expectancy and value are

non-compensatory; both expectancy and value have to be high in order to instigate motivated

behavior. This fundamental difference between the two models has important implications for

the application of EVT in policy and practice.

Mysteriously, the multiplicative relation between expectancy and value seems to have

vanished from EVT of achievement motivation without a trace. There seems to have been no

explicit theoretical argument for dropping it and hardly any reported empirical tests of it.

How could such a prominent theoretical feature of EVT have been so central in 1982 (Feather,

1982), and then be simply ignored for the next quarter century and more? Surely someone

must have noticed that it went missing! Surely there must have been a good reason for

discarding the cornerstone of EVT! Who was the culprit?

The Culprit: Weak statistical models for testing interaction effects

Our main suspect – the culprit who is responsible for the missing E×V interaction – is

weak statistical models and methodology. Appropriate tests of multiplicative relations

between variables require the test of interaction effects using a statistical model that includes


both the corresponding main or first-order effects of the predictors as well as the effect of

their product (Arnold & Evans, 1979; Blanton & Jaccard, 2006; Busemeyer & Jones, 1983;

Cohen, 1978). Especially in psychology with its emphasis on individual differences, many

theoretical models – like EVT – explicitly posit interaction effects; examples include

aptitude-treatment interactions (Cronbach & Snow, 1977), importance-weighted average

models (Marsh, 1993, 2008), theoretical accounts of attitude behavior-relations (Ajzen, 1987),

and models of person-environment fit. Even though they are a critical part of many

psychological theories, interaction effects, particularly in observational studies, are typically

small, non-significant, or not easily replicated – and hence easily overlooked and forgotten.

Part of the problem is that multiple regression, even if it uses a correctly specified model,

falls short when predictor variables are measured with error: The measurement errors in the

original variables combine multiplicatively in the product variable used for testing the

interaction effect. This leads to an underestimation of the size of the interaction effect that is

even stronger than the corresponding attenuation of the main effects (Busemeyer & Jones,

1983).1 Alternative approaches based on structural equation models (SEMs) have been

evolving during the last decade, but have often not been followed in applications (Marsh,

Wen, Nagengast, & Hau, in press-b).

The E×V interaction disappeared partly as a consequence of the shift from a focus on

behavioral choice in laboratory studies – prevalent in the historical approaches to tests of

EVT – to the assessment of expectations of success and value with surveys and

questionnaires in real-world contexts (see Busemeyer & Jones, 1983; Mitchell, 1974). In

laboratory studies, EVT constructs were operationally defined and experimentally

manipulated; this resulted in a small number of levels for each dimension that were free of

measurement error and were independent of each other. Using within-person designs, the

interaction hypothesis could be tested with analysis of variance. The analysis of surveys and


questionnaires shifted the focus from within-person differences in motivation to engage in

different tasks to between-person (interindividual) comparisons of motivation to engage in

identical subjects or tasks. Multiple regression and path analysis with scale scores (e.g.,

Eccles (Parsons), 1983) and linear SEMs (e.g., Meece, Wigfield, & Eccles, 1990) became the

analytical techniques of choice. Although in principle appropriate for the analysis of

interaction effects, both methods pose obstacles for detecting interaction effects: multiple

regression with scale scores will underestimate interaction effects due to measurement error

in the variables (Busemeyer & Jones, 1983); linear SEM approaches that control

measurement error could traditionally only model additive relations between latent variables

(see Marsh et al., in press-b).

Consistent with this perspective, Eccles (9 March, 2011, personal communication)

noted that she and colleagues sought evidence for E×V interactions, but did not report them

because they were consistently non-significant.

“So we replaced the × symbol with a dash in order to shift the focus away from the

multiplicative function. Note that we did not replace the × with a + sign but we did

stop testing for a significant interaction because we so rarely found it to be significant

in our studies. So one could certainly infer from our regression models that we were

focused on an additive model.”

Hence, although Eccles’ published work has emphasized the additive component of EVT that

is the focus of modern EVT, her personal account provides a richer picture.

In summary, our search seems to have narrowed down the list of likely suspects to

weak statistical models as the main culprit. E×V interactions in traditional multiple regression

were plagued by measurement error. Linear SEMs could not be used to test interaction effects.

Latent variable models with interactions became available in principle in the 1980s (e.g.,

Kenny & Judd, 1984). Despite the widespread use of SEMs and the importance of interaction


effects, practically no substantive researcher used them. The paucity of such applications is

not due to a lack of relevant substantive applications that require tests of interaction terms.

Rather, as noted by Rigdon, Schumacker and Wothke (1998), difficulties in specifying

complicated constraints necessary for SEMs with latent interactions have led researchers to

pursue other approaches. However, these models have recently become more easily

accessible to applied researchers (Klein & Moosbrugger, 2000; Marsh, et al., in press-b;

Marsh, Wen, & Hau, 2004). Here, we apply them to large nationally representative databases

to test for E×V interactions.

The Present Investigation

Despite wide acknowledgment that science skills are fundamental to socio-economic

development in technology-based societies, the number of students pursuing careers in

science is declining worldwide (OECD, 2007). Schools fail at their core academic business of

fostering engagement and educational aspirations in this field. While schools focus primarily

on academic achievement, the fundamental problem is one of motivation to engage in science

activities and to pursue a science-related career path. The first step in reversing this trend is a

better understanding of the key motivational determinants of career choices and their

interplay, as well as the underlying motives to students’ growing disengagement in science

subjects. What is needed to tackle this problem is a synergy between good substantive theory

(EVT) and strong statistical methodology (latent-variable models of interaction effects).

Understanding the interplay of expectancy and value in bringing about engagement and

career motivation in science is critical for the development of strategies to address this

shortage.

We analyze data from the OECD Program for International Student Assessment

(PISA) 2006 that focused on achievement and motivational outcomes in science. In line with

the socio-cultural EVT-model of achievement motivation (Eccles (Parsons), 1983; Wigfield


& Eccles, 2000), we use academic self-concept in science (Marsh, 2006; representing

expectancy of success, see Eccles & Wigfield, 1995) and enjoyment of science (representing

intrinsic value) to predict extra-curricular engagement in science (a behavioral measure of

motivation) and career aspirations in science (a proxy measure for choice) using a SEM with

latent interactions. The international comparability and the use of nationally-representative

samples in each of the participating countries make PISA the strongest available resource for

testing the cross-national generalizability of EVT, including the apparently lost E×V

interaction.

The large and diverse sample of 57 countries that participated in PISA allows

demanding tests of the cross-national generalizability of the findings. Although the EVT-

model explicitly includes cultural background variables and hypothesizes that they influence

academic motivation and choice (Wigfield & Eccles, 2000), most tests of the model have

been conducted in Western cultures (Wigfield, et al., 2004). The few international

comparisons to-date have addressed differences and similarities in the factor structure of

expectancy constructs and compared the absolute levels of components of expectancy and

value, showing e.g., that competence perceptions are higher in Western cultures, but

providing only limited results for value components of EVT (see the review in Wigfield et al.,

2004). However, these studies have not reported E×V interactions and were typically based

on convenience samples instead of nationally-representative data available in the PISA 2006

database.

Methods

Participants

We used data collected as part of PISA 2006. The sample consisted of 15-year-old

students (N = 398,750, 50.5% female) from 57 countries (OECD, 2007, 2009). A complex


two-stage sampling process was used to guarantee the representativeness of the samples for

the national populations of 15-year-old students (for details see OECD, 2009).

Measures

The measures of the expectancy-value constructs and the motivational outcomes were

selected from the student background questionnaire. All scales were based on positively

worded items with a 4-point Likert answering scale ranging from “Agree completely” to

“Disagree completely”, if not otherwise mentioned. All answers were recoded so that higher

scores indicated a higher value on the underlying construct. All scales have been validated

extensively and possess good measurement properties (OECD, 2009; also see the

supplemental appendix). Complete descriptions of the scales, items and their properties are

given in OECD (2009).

Science self-concept was used to operationalize students’ expectancy of success. The

scale consisted of six items assessing students’ perception of their competencies in science

(e.g. “<School science> topics are easy for me.”).

Enjoyment of science was used to represent one of the subdomains of value (interest-

enjoyment value). The scale consisted of five items that assessed the enjoyment students

experienced when engaging in science-related activities (e.g., “I generally have fun when I

am learning <broad science> topics.”).

Science-related extra-curricular activities were used as a behavioural outcome. The

scale consisted of six items assessing the frequency of participation in non-compulsory and

after-school science activities, e.g., watching TV programs, reading magazines or attending

science clubs. The labels of the answering categories ranged from “never or hardly ever” to

“very often”.

Career aspiration in science was used as a proxy measure for long-term academic

choice. The scale consisted of four items that assessed the students’ intentions of studying


science after school and taking up a science-related career (e.g., “I would like to work in a

career involving <broad science> after <secondary school>.”).

Data Analysis

We tested the joint influence of science self-concept, science enjoyment and their

latent interaction on science career aspirations and extracurricular activities (see the path

diagram Figure 2) implementing the unconstrained approach (see the supplemental technical

appendix) to latent interactions in Mplus 5.21 (Muthén & Muthén, 1998-2009) using robust

standard errors and test statistics to account for non-normality of the indicators. It is

important to emphasize that appropriately specified latent interaction models include both

main effect variables as well as the product term (Cohen, 1978, Cronbach, 1987; Marsh, et al.,

in press-a; see also footnote 1 and supplemental materials). All analyses used the

standardized student weight to obtain unbiased estimates of population parameters and the

complex design correction in Mplus to control standard errors for the nesting of students

within schools (see supplemental materials, for more details on the model and its

implementation including the standardization of variables, treatment of missing data and

criteria for the assessment of model fit).

Two sets of analyses were conducted: (1) In the total group analysis, the latent

interaction model was fitted to the full international sample (see also the Mplus syntax in the

supplemental material); (2) Using the 57 countries as separate groups, we then tested a

sequence of increasingly restrictive multigroup models to test the invariance of the factor

loadings for the first-order constructs across the 57 countries (Meredith, 1993). After that the

multigroup SEM with latent interactions was fitted simultaneously in each of the 57

countries. In order to test whether the structural effects varied across the countries, the path

coefficients in the structural model were fixed to equality across countries and the fit of this

model was compared to the model in which these effects were freely estimated.


Results

Total group analysis

The SEM with science self-concept, science enjoyment, their latent interaction as

predictors for career aspirations and extra-curricular activities fit the data in the total sample

well (χ2(289) =51384.01, CFI = 0.975, TLI = 0.971, RMSEA = 0.021). The path coefficients

of the completely standardized solution and correlations between the latent variables are

shown in Figure 2.

For both dependent measures, significant main effects of the latent predictor variables

and their latent interaction emerged (see Table 1). A higher science self-concept led – on

average – to more engagement in science-related activities and to higher career aspirations in

science. Similarly, a higher enjoyment of science led – on average – to more engagement in

science-related activities and to higher career aspirations in science. In line with previous

research on the joint effects of expectancy and value (see Wigfield & Eccles, 2000),

enjoyment proved to be a relatively more important predictor for engagement and choice; the

path coefficients for enjoyment were larger than those for self-concept.

Of central importance, the critical E×V interaction was statistically significant for

both the pursuit of science-related activities and science career aspirations. The effect of self-

concept varied with the level of enjoyment and the effect of enjoyment varied with the level

of self-concept. The effect of the latent interaction variable on science-related behaviours was

positive and highly significant (b = 0.073, s.e. = 0.003, p < 0.001) as was the effect on career

aspirations in science (b = 0.053, s.e. = 0.002, p < 0.001). Simple slopes plots (Bauer &

Curran, 2005; Preacher, Curran, & Bauer, 2006; see Figure 3) show the effect of science self-

concept on the respective outcome variables at the mean of enjoyment (the main effect) and

at one and two SD below and above the mean. The non-linear, multiplicative relation is

clearly evident from the graphs; the effect of self-concept on outcomes becomes more


positive when enjoyment is high and the effect of enjoyment becomes more positive when

self-concept is high. This interaction is particularly evident for engagement in science-related

activities such that at low levels of enjoyment, self-concept has no effect whereas at high

levels of enjoyment, an increase of one SD in self-concept is associated with about .4SD

change in science-related activities. In supplemental analyses (see supplemental materials)

we tested the generalizability of the results over gender. The results showed that these main

and interaction effects of self-concept and enjoyment were similar for boys and girls.

Multicountry Analysis

We now turn to the evaluation of the generalizability of EVT effects across the 57

countries represented in PISA 2006. First, we established the invariance of factor loadings

(Meredith, 1993) across the 57 countries. This was accomplished by comparing a multigroup

confirmatory factor analyses model in which all factor loadings of the indicators were fixed

(factorial invariance) to a model in which the factor loadings could vary between countries

(configural invariance). The unrestricted model showed a good fit to the data (χ2(10431) =

101151.241, CFI = 0.967, TLI = 0.962, RMSEA = 0.035). Restricting the factor loadings to

be equal across countries did not reduce the fit substantially (χ2(11383) = 130653.650, CFI =

0.957, TLI = 0.954, RMSEA = 0.039), supporting the invariance of the measurement model

across countries (e.g., Chen, 2007; Cheung & Rensvold, 2002) and fulfilling a sufficient

condition for comparability of path coefficients of the latent interaction model across

countries.

Next we tested a multigroup SEM that included the main effects of science self-

concept and enjoyment on career-aspirations and extracurricular activities as well as their

latent interaction (E×V). All factor loadings for the latent predictor and outcome variables

and the latent interaction were held invariant across countries. The model demonstrated good

fit to the data (χ2 (17649) = 144031.107, CFI = 0.955, TLI = 0.953, RMSEA = 0.032).


The main effects of science self-concept were statistically significant for all 57 countries on

career aspirations (M = 0.263, SD = 0.075), and for all but one country on extra-curricular

activities (M = 0.263, SD = 0.075; the exception being Liechtenstein with a sample size of

339). The main effects of enjoyment on extra-curricular activities (M = 0.503, SD = 0.108)

and on career aspirations (M = 0.481, SD = 0.063) were statistically significant in all 57

countries. For the latent interaction between science self-concept and enjoyment (E×V) the

following pattern emerged: The latent interaction effect on science-related activities (M =

0.067, SD = 0.017) was positive and statistically significant in all countries. The latent

interaction effect on career aspirations was statistically significant (M = 0.055, SD = 0.023)

in 49 countries; it was non-significant in 8 counties (see individual country parameters in

Table S1, supplemental materials)

Finally, we explored the cross-cultural generalizability of our results by constraining

the structural coefficients—path coefficients leading from EVT constructs to outcomes—to

be invariant across countries. The fit indices for the restricted model did not change

substantively (χ2 (17985) = 150537.360, CFI = 0.953, TLI = 0.952, RMSEA = 0.032); in fact,

changes in these fit indices were less than cut-off criteria typically used (∆CFI = 0.002, ∆TLI

< 0.001, ∆RMSEA < 0.001) to support the invariance of parameters over multiple groups (see

Chen, 2007; Cheung & Rensvold, 2002). These results suggest that the restricted model fit

the data nearly as well as the unrestricted model, and support the generalizability of the first-

order effects as well as the critical E×V interaction effect across the diverse set of 57

countries participating in PISA 2006.

Discussion

Consistent with most good psychological mysteries, we began with an apparent crime

(the missing E×V interaction), identified the culprit (weak statistical methodology), gathered

evidence to support our speculations, and returned the missing E×V interaction to its rightful


place at the heart of EVT. In accomplishing this task, we demonstrated apparently the

strongest support for the cross-national generalizability of EVT predictions ever undertaken.

Based on nationally representative samples of 15-year-olds from 57 diverse countries, there

was good support for the generalizability of the effects of self-concept, enjoyment, and their

interaction for both engagement in science activities and plans to pursue science careers.

Although it might be premature to claim the universality of EVT predictions – including the

“lost” E×V interaction – the support is very strong.

The presented research is a prime example for a substantive-methodological synergy

(Marsh & Hau, 2007): The application of new and emerging quantitative research

methodology – SEMs with latent interactions – to a long-standing and unresolved substantive

issue – the missing E×V interaction. Not only do our findings have important theoretical and

practical implications for applied motivational researchers, they also demonstrate evolving

methodology to test interaction effects in observational, non-experimental data (McClelland

& Judd, 1993). SEMs with latent interactions based on large Ns allow strong tests of

hypothesized multiplicative relations with models that are correctly specified to include both

main effects and the interaction effect (Blanton & Jaccard, 2006; Cohen, 1978; Marsh et al.,

in press-a; also see footnote 1). In supplemental power analyses, we found that sample sizes

of at least N=1000 that are typical in some large-scale studies are needed to consistently

identify latent E×V interactions similar to those in the international sample. However, with

sample sizes of less than 500 that are typical in many applied studies, power was insufficient

– less than a 50% chance of identifying such E×V interactions (see Technical Appendix for

further discussion).

Our findings also have direct implications for policy-practice and interventions that

seek to increase long-term engagement and pursuit of careers in science. The essence of the

non-compensatory, multiplicative relation between expectancy and value is that both have to


be high. It is not sufficient to either enhance academic self-concept or to enhance value;

teachers – supported by appropriate policy-practice – must be sufficiently skilled to

simultaneously enhance both constructs. If teachers focus on one to the exclusion of the

other, then the influence of each is undermined.

We have been successful in re-establishing the E×V interaction to its rightful position

at the centre of EVT, but much work is still needed. Within modern approaches to EVT of

achievement motivation (e.g., Wigfield & Eccles, 2000), the value component is

multifaceted, consisting of extrinsic value and cost in addition to intrinsic value represented

by enjoyment in our study. Hence further research is needed in order to disentangle the

interplay among these value components in motivating achievement-related behavior and

choices. Our supplemental analyses show that support for the E×V interaction generalizes

over gender for science, but further research is warranted to evaluate its cross-cultural

generalizability in other disciplines. More generally, evaluations of EVT require longitudinal

data to provide stronger tests of the implicit assumptions of the causal ordering of the EVT

constructs in relation to a wider variety of short- and long-term outcomes. However, based on

the present investigation, we posit that E×V interactions will continue to play an important

role in such extensions of EVT.


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Footnotes

1 Another common mistake in specifying models to test multiplicative effects is the

omission of main effects (see e.g., Blanton & Jaccard, 2006; Cohen, 1978) and exclusive tests

of the product variable. The effect of the product variable alone is a mixture of main and

interaction effects (Cohen, 1978) and can incorrectly lend support to a multiplicative model

even though an additive model holds (Blanton & Jaccard, 2006; Cohen, 1978; Cronbach,

1987). Appropriate (latent and manifest) models for tests of interactions must always include

both main effects and the product variable (Cohen, 1978, see also Marsh, Hau, Wen,

Nagengast, & Morin in press-a).


Table 1

Path coefficients of the latent interaction models for the total international sample and

average path coefficients of the multigroup analysis

Total sample Multigroup analysis

Predictor B s.e. p M SD

Extra-curricular activities

Self-concept 0.165 0.003 < 0.001 0.171 0.073

Enjoyment 0.528 0.003 < 0.001 0.503 0.108

SCxEnj 0.073 0.003 < 0.001 0.067 0.017

Career aspirations

Self-concept 0.251 0.003 < 0.001 0.263 0.075

Enjoyment 0.488 0.003 < 0.001 0.481 0.063

SCxEnj 0.053 0.002 < 0.001 0.055 0.023

Note: All parameter estimates were obtained from the fully standardized solution in Mplus 5.21, interaction effects have been restandardized according to Wen, Marsh and Hau (2010). The reported values for the multigroup analysis are means and standard deviations of the standardized parameter estimates over 57 countries (using the variance from the total international sample), so no meaningful p-values can be reported. SCxEnj = interaction of self-concept and enjoyment, b = standardized parameter estimate, s.e. = standard error, p = p-value, M = mean of standardized parameter estimate over 57 countries; SD = standard deviation of parameter estimate over 57 countries.


Figure 1. Hypothetical relations between expectancy and value in predicting motivation. The

left panel shows the implication of an additive relation, the right panel shows the implications

of a multiplicative relation.

Additive: Expectancy + Value

Expectancy

Mot

ivat

ion

Multiplicative: Expectancy x Value

Expectancy

Mot

ivat

ion

Theoretical relations between expectancy, value and motivation

Value = ++Value = +Value = 0Value = −Value = −−


Figure 2. Path diagram and parameter estimates of the structural equation model with a latent

interaction in the total sample of PISA 2006.

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Figure 3. Simple slope plots of the relation between self-concept and enjoyment jointly

predicting extra-curricular activities in science (left plot) and science career aspirations (right

plot) in the total sample of PISA 2006.

Extra−Curricular Activities

Self−Concept

−2 SD −1 SD 0 1 SD 2 SD

−1 S

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Career Aspirations

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Enjoyment = 2 SDEnjoyment = 1 SDEnjoyment = 0Enjoyment = −1 SDEnjoyment = −2 SD


Technical Appendix

Description of PISA database and Statistical Analyses

In this technical appendix, we give more information on the PISA 2006 sample and the

items that were used to measure the latent variables used in our research. We then describe

some details of the implementation of structural equation models with latent interactions that

we used to analyze the expectancy-value interaction.

PISA 2006

The Program for International Student Assessment (PISA) is an ongoing project

commissioned by the Organization for Economic Co-Operation and Development (OECD).

Every three years (starting in 2000) PISA collects cross-sectional data on achievement,

motivation and policy-relevant background data from nationally representative samples of

15-year-olds – an age when students in most countries approach the end of their compulsory

schooling. PISA aims to assess not only knowledge on a specific academic subject, but also

the extent to which students can use their knowledge to solve problems encountered in real

life. For PISA 2006, the data collection focuses on achievement and background data for

science based on responses of 398,750 students from 57 countries (30 OECD countries and

27 partner countries). PISA used a two-stage sampling model in each country; a nationally

representative sample of at least 150 schools enrolling 15 year-olds (sampling probabilities

were proportional to school size) followed by a random selection of about 35 15-year-old

students from each school. PISA measures were compiled by international experts in the

respective fields working in close collaboration with the participating countries. The cross-

cultural validity of the assessment material and the background measures are further ensured

by employing state-of-the-art translation and back-translation protocols, and by extensive

pre-testing in field trials before the main data collection (for further discussions see Marsh,

Hau, Artelt, Baumert, & Peschar, 2006; OECD, 2007; OECD, 2009)


Measures

In our analysis of the EVT-model, we used the following scales. All scales, unless

otherwise indicated, were measured with positively worded items using 4-point Likert scales

ranging from “strongly agree” to “strongly disagree”. For the analysis, the variables were

recoded so that higher numeric values indicated higher values on the corresponding construct.

Expectancy. The scale Science Self-Concept was used to operationalize students’

expectancy of success. The scale consisted of six items assessing students’ perception of their

competencies in science (e.g. “<School science> topics are easy for me.”). The median

reliability in the OECD countries was α = 0.92 and in the non-OECD-countries α�= 0.87

(OECD, 2009).

Value. The scale Enjoyment of Science was used to represent intrinsic value of science.

The scale consisted of five items that assessed the enjoyment students experienced when

engaging in science-related activities (e.g., “I generally have fun when I am learning <broad

science> topics.”). The median reliability in the OECD countries was α�= 0.92 and in the

non-OECD-countries �α = 0.87 (OECD, 2009).

Behavioral Outcome. The scale Science-Related Extra-Curricular Activities was used as

a behavioural outcome measure. The scale consisted of six items assessing the frequency of

participation in non-compulsory and after-school science activities, e.g., watching TV

programmes, reading magazines or attending science clubs. The labels of the answering

categories ranged from “never or hardly ever” to “very often”. The median reliability in the

OECD countries was α�= 0.78 and in the non-OECD-countries α = 0.76 (OECD, 2009).

Career Aspirations. The scale Future-Oriented Motivation in Science that measured

career aspiration in science-related fields was used as a proxy measure for long-term

academic choice. The scale consisted of four items that assessed the students’ intentions of


studying science after school and taking up a science-related career (e.g., “I would like to

work in a career involving <broad science> after <secondary school>.”). The median

reliability in the OECD countries was α = 0.92 and in the non-OECD-countries α�= 0.90

(OECD, 2009).

Structural equation models with latent interactions (SEM-LIs).

SEM-LIs have only recently become available to the applied social science research

community, e.g. in form of distribution-analytic approaches such as the Latent Moderated

Structural Equations model (LMS, Klein & Moosbrugger, 2000) or Quasi-Maximum

Likelihood (QML, Klein & Muthen, 2007) and product-indicator approaches such as the

unconstrained approach (Marsh, Wen, & Hau, 2006; Marsh, Wen, & Hau, 2004). For the

analyses in this paper, we adapted the unconstrained approach to latent interactions (Marsh,

et al., 2004) for cross-cultural studies.

SEM-LIs seek to estimate the regression of the latent outcome variable

η��(achievement motivation in our case) on the latent predictor variables ξ��(expectancy

in our case) and ξ2 (value in this case) and their cross-product ξ1ξ2, representing the

interaction effect. The inclusion of the cross-product distinguishes these models from

conventional structural equation models (e.g., Jöreskog, 1970; McDonald, 1978) and the best

way to estimate this effect has been an issue of longstanding debate (e.g., Marsh, Wen, et al.,

2006; Marsh, et al., 2004; Schumacker & Marcoulides, 1998). The SEM-LI with two latent

predictors is typically specified as:

η = γ1ξ1 + γ2ξ2 + γ3ξ1ξ2 + ζ, (1)

where γ1�, γ2 and γ3 are the partial regression coefficients of the latent predictor variables and

their cross-product and ζ� is the structural model residual. The latent predictors ξ1� and ξ2 as

well as the latent outcome variable η� are each inferred from at least two indicators as

specified in the corresponding measurement models


ijixiijx δξλ += kykky εηλ += , (2)

where ijx is the jth indicator of the ith latent predictor variable iξ , xiλ is the corresponding

factor loading and ijδ is the corresponding residual, ky is the kth indicator of the latent

outcome variable η , ykλ is the corresponding factor loading, and kε is the corresponding

residual.

Product-indicator approaches such as the unconstrained approach identify the latent

cross-product ξ1ξ2 by products of indicators of the latent predictor variables ξ1 and ξ2 (Marsh,

Wen, et al., 2006; Marsh, et al., 2004) according to the following measurement model

lilili xx 21212121 δξξλ += , (3)

where ix1 is the ith indicator of ξ1 and lx2 is the lth indicator of ξ2�, li 21λ is the corresponding

factor loading on the latent product variable and li 21δ is the corresponding residual (for

details on selecting the product indicators see, Marsh, Wen, et al., 2006; Marsh, et al., 2004).

In order to minimize the number of necessary constraints in the model, all indicators

of the latent variables are standardized before the product-indicators are computed (Marsh, et

al., 2004; Marsh, et al., 2007). Marsh et al. (2004) showed that it is only necessary to

constrain the mean of the latent product variable to be equal to the covariance of the latent

predictor variables, i.e.,

)( 21ξξE = cov(ξ1,ξ2), (4)

in order to identify the model. Estimation is carried out with a conventional statistical

package for structural equation modelling (e.g., Mplus, L. K. Muthén & Muthén, 1998-

2009). Model estimation in Mplus was based on the robust maximum likelihood estimator

(Yuan & Bentler, 2000) that provides standard errors and fit statistics that correct for non-

normality of the product-indicators. This estimator has been shown to provide good results


when indicators are based on Likert-scales with at least four answer categories (Beauducel &

Herzberg, 2006; DiStefano, 2002; Dolan, 1994; Muthén & Kaplan, 1985).

In order to test the cross-cultural generalizability of the EVT interaction, we extended

the unconstrained approach to multiple groups (defined by the individual countries in the

PISA 2006 data). Multiple group structural equation models specify separate measurement

and structural models (i.e., equations (1) – (4)), for each country g, exemplified here with the

structural model including the interaction effect

)()(2

)(1

)(3

)(2

)(2

)(1

)(1

)( ggggggggg ςξξγξγξγη +++= (5)

and testing parameter invariance constraints across countries (see Marsh, et al., 2009, for a

full taxonomy of invariance models and tests).

Appropriate Model Specification for Tests of Interaction Effects

A common mistake in specifying models to test multiplicative effects is the omission

of main effects (see e.g., Blanton & Jaccard, 2006; Cohen, 1978) and exclusive tests of the

product variable. The effects of the product variable alone are a mixture of main effects and

interaction effects (Cohen, 1978) and can incorrectly lend support to a multiplicative model

even though an additive model holds (Blanton & Jaccard, 2006; Cohen, 1978; Cronbach,

1987). Appropriate models for tests of interaction effects must always include both main

effects and the product variable (Cohen, 1978, see also Marsh, Hau, Wen, Nagengast, &

Morin in press). This appropriate model specification applies to latent interaction models like

those presented here as well as multiple regression models based on manifest effects like

those described by Cohen (1978) and Cronbach (1987).

Standardization

As recommended by Marsh et al. (2004), all manifest variables were standardized for

the total group before the analysis. Product indicators for the latent interactions were formed

on the basis of matched pair-strategy and not re-standardized in line with the


recommendations given in Marsh et al. (2004). Because of the different number of indicators

for the two latent predictors (Self-concept: six; Enjoyment: five), two self-concept items were

averaged for the construction of the product indicators. For the multiple group analysis, the

standardized manifest indicators were centred (but not re-standardized) around their country-

specific mean before computing the product-indicators for the latent interaction variable.

Assessment of model fit

Following Marsh, Balla, and Hau (1996; also see Marsh, Balla, & McDonald, 1988;

Marsh, Hau, & Wen, 2004) we considered the Tucker-Lewis index (TLI), the comparative fit

index (CFI), the root mean square error of approximation (RMSEA), and the Standardized

Root Mean Square Residual (SRMR) to evaluate goodness of fit, as well as the χ2-test

statistic and an evaluation of parameter estimates. The TLI and CFI vary along a 0-to-1

continuum in which values greater than .90 and .95 are typically taken to reflect acceptable

and excellent fits to the data, respectively. SRMR values below .09 indicate good fit. RMSEA

values of less than .06 are taken to reflect a reasonable fit. Whereas, RMSEA values greater

than .10 are unacceptable, although no golden rule exists (Chen, Curran, Bollen, Kirby, &

Paxton, 2008; Hu & Bentler, 1999; Marsh, Hau & Wen, 2004). The CFI contains no penalty

for a lack of parsimony, so that improved fit due to the introduction of additional parameters

may reflect capitalization on chance, whereas the TLI and RMSEA contain penalties for a

lack of parsimony (for further discussion see Cheung & Rensvold, 2002; Hu & Bentler, 1999;

Marsh et al., 2004). For comparison of nested models by differences in fit indices, we

followed the recommendations given by Chen (2007) and Cheung and Rensvold (2002) who

suggested that a decrease in fit for the more parsimonious model of less than .01 for

incremental fit indices like the CFI, should be treated as support for this model. Chen (2007)

suggested that when the RMSEA increases by less than .015 there is support for the more

constrained model.


Invariance tests

For the present analysis two sets of invariance constraints across the 57 countries

were most crucial: Invariant factor loadings (factorial invariance, Marsh, et al., 2009;

Meredith, 1993) and invariance of the path coefficients in the structural model (equation (5)).

Invariance of factor loadings in the measurement model guarantees that the latent variables

have identical meanings in each of the 57 countries and that relations between them can be

meaningfully compared. In order to test factorial invariance, we compared two multigroup

confirmatory factor analysis models: In the first model, the configural invariance model, a

four-factor model for self-concept, enjoyment, extra-curricular activities and career aspiration

was assumed to hold in each of the countries, but no further constraints were put on the

parameters of this model. In the next step, factor loadings were restricted to be invariant

across the countries and the fit of these two models was compared. The second set of

invariance constraints, invariance of path coefficients in the structural model was tested

within the multigroup latent interaction model specified with the unconstrained approach:

Here, we first specified a model in which factor loadings were constrained to be equal across

countries, but parameters in the structural model could vary freely. We compared the fit of

this model to a model where additionally all parameters of the structural model were held

constant across the 57 countries. A small reduction in fit of this second model would be a

strong indication of the cross-cultural generalizability of the main tenets of expectancy-value

theory.

Weighting

Consistent with its two-stage stratified sampling design, PISA provides both a student

and a school weight. We used the survey weighting facilities implemented in Mplus (see

Asparouhov, 2006, 2008; Carle, 2009; Stapleton, 2006) to obtain results that are


representative for the respective national populations of 15-year-old students using the final

student weight provided in the international PISA database.

Missing data

Missing data in the responses to the questionnaire items was handled with multiple

imputation (Graham, 2009; Schafer, 1997) including achievement and a variety of variables

from the PISA background questionnaires. Ten imputed datasets were created using the

multiple imputation procedure implemented in SPSS 17 and all analyses were repeated with

all imputed datasets. The final parameter estimates and fit statistics of the structural equation

models with latent interactions were obtained with the automatic aggregation procedure

implemented in Mplus 5.21(L. K. Muthén & Muthén, 1998-2009).

Supplemental Tests of Gender Effects

Given gender differences in motivation have been widely studied in EVT (e.g.,

Eccles, 1987; Eccles, Barber & Jozefowicz, 1998) and motivation research more generally

(e.g. Marsh, 1989; Marsh, Martin & Cheng, 2008; Marsh & Yeung, 1998 ), a relevant

question is whether our results – and particularly support for the E×V interaction – generalize

over gender. Because of space limitations, we pursued this issue as a supplement analysis

reported here. In order to address this issue, we tested the invariance of the findings over

gender in the full international sample with two multigroup models. First, we tested

invariance of the factor loadings and (in order to compare mean gender differences) item

intercepts over gender groups by comparing increasingly restrictive multigroup confirmatory

factor analyses models in which the factor loadings and item intercepts were estimated freely

or fixed to be equal. Next, we tested invariance of the structural coefficients with a

multigroup latent interaction model specified with the unconstrained approach. We first

tested a model in which the structural coefficients (the effects of science self-concept,

enjoyment of science, and the interaction between science self-concept and enjoyment of


science) were free to vary between gender groups. We then compared the fit of this model to

a model in which the structural coefficients were restricted to be equal for female and male

students. Both models assumed that the factor loadings of all constructs were invariant across

gender.

The multigroup confirmatory factor analyses confirmed invariance of the factor

loadings over gender: The model in which the factor loadings of the four constructs were

restricted to be equal for females and males did not fit worse (χ2(383) = 49869.447, CFI =

0.974, TLI = 0.971, RMSEA = 0.025, SRMR = 0.027) than the configural invariance model

in which the factor loadings could vary between the gender groups (χ2(366) = 48713.735,

CFI = 0.975, TLI = 0.971, RMSEA = 0.026, SRMR = 0.026). Hence, comparisons of the

structural coefficients in a multigroup model with latent interactions were justified. Further

restricting item intercepts to be equal across gender groups did not reduce the model fit

substantially (χ2(400) = 54634.727, CFI = 0.971, TLI = 0.970, RMSEA = 0.026, SRMR =

0.028), so gender differences in the four constructs could be compared. Male students

reported significantly higher values on all four scales, although the differences were small in

absolute size (mean differences for standardized latent variables: academic self-concept: M =

0.229, s.e. = 0.005, p < 0.001; enjoyment of science: M = 0.033, s.e. = 0.005, p < 0.001;

extra-curricular activities: M = 0.140, s.e. = 0.005, p < 0.001; career aspirations: M = 0.098,

s.e. = 0.005, p < 0.001).

The multigroup latent interaction model with separate groups for female and male

students showed that the findings in the total sample generalized over gender in spite of the

mean differences. The model in which the structural coefficients (i.e. main and interaction

effects of self-concept and enjoyment on career aspirations and extra-curricular activities)

were restricted to be equal across males and females did not fit markedly worse (χ2(605) =

58902.115, CFI = 0.972, TLI = 0.969, RMSEA = 0.022, SRMR = 0.026) compared to the


model in which the structural coefficients were free to vary across gender groups (χ2(599) =

51384.01, CFI = 0.972, TLI = 0.970, RMSEA = 0.022, SRMR = 0.025). The effects of

science self-concept, enjoyment of science, and the interaction between science self-concept

and enjoyment of science were all highly significant and of a similar size for both males and

females (see Table S2 for the structural coefficients of the two models). In summary, these

supplemental analyses provide good support of the generalizability of these results over

gender.

Power Analysis

In order to test the power of the latent interaction model to detect latent interaction

effects similar to the ones in the total sample, we conducted a small simulation study. We

used the parameter estimates from the international total sample results to create artificial

datasets with sample sizes of N=500, 1000 and 5000. For each sample size, we simulated

1000 datasets and tested how often the two latent interaction effects were statistically

significant at p < 0.05. The results showed that the power to detect such effects was

comparatively small for N = 500 (extra-curricular activities: 39.2 % significant tests; career

aspirations: 48.9%), reasonably large for N = 1000 (extra-curricular activities: 68.4% ; career

aspirations: 80.3%) and essentially a 100% for N = 5000 (extra-curricular activities: 99.9 %

significant tests; career aspirations: 100%). These results indicate that latent interaction

models might not solve the problem of small power to detect E×V interactions for typical

sample sizes in educational psychology which will usually be smaller than 500.


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Table S1 Country-specific parameter estimates for the multigroup structural equation model with latent interaction

Extra-Curricular Activities Career Aspirations Self-Concept . Enjoyment Self-Concept x Enjoyment Self-Concept Enjoyment Self-Concept x Enjoyment Country Unst. Stand. s.e Unst. Stand. s.e Unst. Stand. s.e Unst. Stand. s.e Unst. Stand. s.e Unst. Stand. s.e Azerbaijan 0.314 (0.337) 0.030 0.387 (0.480) 0.033 0.083 (0.062) 0.019 0.458 (0.374) 0.031 0.424 (0.400) 0.030 0.089 (0.059) 0.025 Argentina 0.190 (0.204) 0.028 0.493 (0.612) 0.018 0.127 (0.095) 0.018 0.322 (0.263) 0.028 0.508 (0.479) 0.026 0.084 (0.055) 0.024 Australia 0.115 (0.124) 0.011 0.300 (0.372) 0.009 0.101 (0.076) 0.006 0.306 (0.250) 0.018 0.555 (0.523) 0.014 0.128 (0.084) 0.007 Austria 0.066 (0.071) 0.014 0.394 (0.489) 0.012 0.053 (0.040) 0.010 0.140 (0.114) 0.022 0.508 (0.479) 0.016 0.113 (0.074) 0.014 Belgium 0.126 (0.135) 0.015 0.360 (0.447) 0.014 0.098 (0.073) 0.008 0.381 (0.311) 0.020 0.486 (0.458) 0.018 0.151 (0.099) 0.011 Brazil 0.220 (0.236) 0.025 0.515 (0.639) 0.023 0.090 (0.067) 0.018 0.346 (0.282) 0.030 0.545 (0.514) 0.028 0.058 (0.038) 0.021 Bulgaria 0.255 (0.274) 0.028 0.442 (0.549) 0.027 0.071 (0.053) 0.022 0.361 (0.295) 0.033 0.465 (0.438) 0.032 0.046 (0.030) 0.033 Canada 0.082 (0.088) 0.011 0.351 (0.436) 0.011 0.095 (0.071) 0.006 0.253 (0.207) 0.014 0.644 (0.607) 0.012 0.098 (0.064) 0.008 Chile 0.260 (0.279) 0.027 0.463 (0.575) 0.020 0.103 (0.077) 0.015 0.390 (0.318) 0.027 0.543 (0.512) 0.022 0.092 (0.061) 0.022 Chinese Taipei 0.129 (0.139) 0.014 0.424 (0.526) 0.014 0.085 (0.064) 0.010 0.340 (0.278) 0.019 0.505 (0.476) 0.018 0.065 (0.043) 0.011 Colombia 0.251 (0.270) 0.037 0.521 (0.647) 0.037 0.136 (0.102) 0.027 0.285 (0.233) 0.039 0.626 (0.590) 0.032 0.043 (0.028) 0.036 Croatia 0.134 (0.144) 0.016 0.482 (0.598) 0.014 0.108 (0.081) 0.016 0.205 (0.167) 0.023 0.536 (0.505) 0.020 0.084 (0.055) 0.018 Czech Republic 0.086 (0.092) 0.020 0.456 (0.566) 0.015 0.077 (0.058) 0.015 0.181 (0.148) 0.028 0.553 (0.521) 0.023 0.072 (0.047) 0.021 Denmark 0.102 (0.110) 0.016 0.362 (0.449) 0.013 0.088 (0.066) 0.009 0.220 (0.180) 0.022 0.541 (0.510) 0.018 0.108 (0.071) 0.014 Estonia 0.103 (0.111) 0.019 0.426 (0.529) 0.017 0.050 (0.037) 0.017 0.238 (0.194) 0.035 0.477 (0.450) 0.029 0.099 (0.065) 0.018 Finland 0.124 (0.133) 0.015 0.297 (0.369) 0.014 0.108 (0.081) 0.012 0.273 (0.223) 0.022 0.483 (0.455) 0.016 0.112 (0.074) 0.014 France 0.089 (0.096) 0.019 0.366 (0.454) 0.018 0.100 (0.075) 0.009 0.383 (0.313) 0.028 0.554 (0.522) 0.027 0.138 (0.091) 0.014 Germany 0.122 (0.131) 0.016 0.363 (0.451) 0.012 0.073 (0.055) 0.009 0.340 (0.278) 0.022 0.431 (0.406) 0.017 0.115 (0.076) 0.013 Greece 0.190 (0.204) 0.025 0.426 (0.529) 0.018 0.082 (0.061) 0.011 0.498 (0.406) 0.025 0.412 (0.389) 0.022 0.086 (0.057) 0.013 Hongkong 0.142 (0.153) 0.019 0.494 (0.613) 0.017 0.134 (0.100) 0.014 0.292 (0.238) 0.024 0.611 (0.576) 0.022 0.109 (0.072) 0.013 Hungary 0.163 (0.175) 0.018 0.463 (0.575) 0.015 0.072 (0.054) 0.014 0.375 (0.306) 0.026 0.412 (0.389) 0.025 0.084 (0.055) 0.021 Iceland 0.123 (0.132) 0.017 0.296 (0.367) 0.013 0.087 (0.065) 0.009 0.260 (0.212) 0.022 0.516 (0.487) 0.017 0.100 (0.066) 0.011 Indonesia 0.242 (0.260) 0.024 0.250 (0.310) 0.020 0.070 (0.052) 0.020 0.414 (0.338) 0.024 0.489 (0.461) 0.028 -0.022 (-0.014) 0.022 Ireland 0.100 (0.107) 0.017 0.297 (0.369) 0.017 0.090 (0.067) 0.009 0.278 (0.227) 0.024 0.563 (0.531) 0.019 0.103 (0.068) 0.011 Israel 0.227 (0.244) 0.026 0.374 (0.464) 0.019 0.083 (0.062) 0.012 0.434 (0.354) 0.030 0.378 (0.356) 0.025 0.027 (0.018) 0.014 Italy 0.171 (0.184) 0.015 0.444 (0.551) 0.013 0.114 (0.085) 0.012 0.355 (0.290) 0.016 0.528 (0.498) 0.016 0.085 (0.056) 0.013 Japan 0.062 (0.067) 0.014 0.264 (0.328) 0.010 0.092 (0.069) 0.010 0.218 (0.178) 0.021 0.484 (0.456) 0.019 0.059 (0.039) 0.012 Jordan 0.201 (0.216) 0.030 0.487 (0.604) 0.029 0.088 (0.066) 0.022 0.502 (0.410) 0.023 0.491 (0.463) 0.026 0.053 (0.035) 0.021 Korea 0.104 (0.112) 0.016 0.291 (0.361) 0.013 0.060 (0.045) 0.010 0.243 (0.198) 0.029 0.503 (0.474) 0.026 0.095 (0.063) 0.012 Kyrgyztan 0.337 (0.362) 0.030 0.543 (0.674) 0.027 0.083 (0.062) 0.028 0.440 (0.359) 0.025 0.533 (0.503) 0.027 0.037 (0.024) 0.028 Latvia 0.147 (0.158) 0.026 0.507 (0.629) 0.021 0.127 (0.095) 0.026 0.236 (0.193) 0.030 0.559 (0.527) 0.026 0.079 (0.052) 0.027 Liechtenstein 0.083 (0.089) 0.045 0.301 (0.374) 0.031 0.077 (0.058) 0.036 0.183 (0.149) 0.056 0.449 (0.423) 0.061 0.030 (0.020) 0.059 Lithuania 0.175 (0.188) 0.023 0.379 (0.470) 0.017 0.101 (0.076) 0.023 0.262 (0.214) 0.027 0.492 (0.464) 0.021 0.051 (0.034) 0.026 Luxembourg 0.115 (0.124) 0.018 0.365 (0.453) 0.012 0.078 (0.058) 0.009 0.305 (0.249) 0.022 0.462 (0.436) 0.022 0.098 (0.064) 0.012 Macao-China 0.163 (0.175) 0.022 0.489 (0.607) 0.016 0.127 (0.095) 0.014 0.280 (0.229) 0.026 0.441 (0.416) 0.030 0.063 (0.041) 0.030 Mexico 0.239 (0.257) 0.022 0.588 (0.730) 0.019 0.103 (0.077) 0.023 0.255 (0.208) 0.025 0.604 (0.570) 0.027 0.097 (0.064) 0.020 Montenegro 0.264 (0.284) 0.022 0.405 (0.503) 0.019 0.100 (0.075) 0.015 0.399 (0.326) 0.028 0.539 (0.508) 0.029 0.110 (0.072) 0.024 Netherlands 0.092 (0.099) 0.018 0.371 (0.460) 0.014 0.093 (0.070) 0.010 0.342 (0.279) 0.023 0.482 (0.455) 0.020 0.136 (0.089) 0.013 New Zealand 0.090 (0.097) 0.019 0.348 (0.432) 0.014 0.113 (0.085) 0.009 0.250 (0.204) 0.027 0.623 (0.587) 0.021 0.136 (0.089) 0.011


Norway 0.180 (0.193) 0.021 0.296 (0.367) 0.017 0.092 (0.069) 0.011 0.309 (0.252) 0.026 0.370 (0.349) 0.020 0.082 (0.054) 0.013 Poland 0.144 (0.155) 0.025 0.301 (0.374) 0.017 0.034 (0.025) 0.014 0.185 (0.151) 0.031 0.447 (0.422) 0.023 0.069 (0.045) 0.017 Portugal 0.154 (0.165) 0.030 0.572 (0.710) 0.031 0.103 (0.077) 0.022 0.375 (0.306) 0.031 0.571 (0.538) 0.031 0.102 (0.067) 0.022 Qatar 0.235 (0.252) 0.025 0.393 (0.488) 0.018 0.051 (0.038) 0.012 0.523 (0.427) 0.027 0.403 (0.380) 0.026 0.054 (0.036) 0.012 Romania 0.259 (0.278) 0.037 0.422 (0.524) 0.035 0.069 (0.052) 0.031 0.557 (0.455) 0.036 0.353 (0.333) 0.031 0.040 (0.026) 0.027 Russian Federation

0.305 (0.328) 0.030 0.490 (0.608) 0.019 0.056 (0.042) 0.023 0.300 (0.245) 0.031 0.483 (0.455) 0.023 0.046 (0.030) 0.022

Serbia 0.156 (0.168) 0.022 0.400 (0.496) 0.020 0.111 (0.083) 0.015 0.362 (0.295) 0.025 0.519 (0.489) 0.023 0.094 (0.062) 0.016 Slovak Republic

0.096 (0.103) 0.019 0.480 (0.596) 0.019 0.094 (0.070) 0.016 0.181 (0.148) 0.026 0.594 (0.560) 0.027 0.064 (0.042) 0.024

Slovenia 0.097 (0.104) 0.017 0.441 (0.547) 0.012 0.068 (0.051) 0.012 0.292 (0.238) 0.029 0.495 (0.467) 0.020 0.091 (0.060) 0.014 Spain 0.113 (0.121) 0.013 0.347 (0.431) 0.012 0.083 (0.062) 0.008 0.409 (0.334) 0.019 0.523 (0.493) 0.019 0.110 (0.072) 0.011 Sweden 0.073 (0.078) 0.014 0.249 (0.309) 0.016 0.066 (0.049) 0.009 0.250 (0.204) 0.020 0.486 (0.458) 0.025 0.156 (0.103) 0.019 Switzerland 0.092 (0.099) 0.019 0.373 (0.463) 0.012 0.067 (0.050) 0.009 0.239 (0.195) 0.021 0.507 (0.478) 0.020 0.124 (0.082) 0.012 Thailand 0.163 (0.175) 0.022 0.600 (0.745) 0.021 0.095 (0.071) 0.025 0.372 (0.304) 0.026 0.525 (0.495) 0.026 0.002 (0.001) 0.033 Tunisia 0.215 (0.231) 0.032 0.373 (0.463) 0.034 0.107 (0.080) 0.029 0.400 (0.326) 0.033 0.617 (0.582) 0.046 0.071 (0.047) 0.033 Turkey 0.183 (0.197) 0.026 0.515 (0.639) 0.022 0.099 (0.074) 0.012 0.328 (0.268) 0.024 0.651 (0.614) 0.025 0.085 (0.056) 0.014 United Kingdom

0.119 (0.128) 0.015 0.330 (0.410) 0.013 0.125 (0.094) 0.008 0.353 (0.288) 0.024 0.522 (0.492) 0.021 0.139 (0.091) 0.014

United States 0.105 (0.113) 0.017 0.355 (0.441) 0.014 0.078 (0.058) 0.011 0.297 (0.242) 0.021 0.501 (0.472) 0.022 0.066 (0.043) 0.013 Uruquay 0.170 (0.183) 0.024 0.399 (0.495) 0.017 0.104 (0.078) 0.020 0.388 (0.317) 0.034 0.553 (0.521) 0.027 0.092 (0.061) 0.023 Average 0.159 (0.171) 0.406 (0.503) 0.090 (0.067) 0.322 (0.263) 0.510 (0.481) 0.084 (0.055) Standard Deviation

0.068 (0.073) 0.087 (0.108) 0.022 (0.017) 0.092 (0.075) 0.067 (0.063) 0.036 (0.023)

Note: Unst. = unstandardized parameter estimate; Stand. = standardized parameter estimate using the variances in the total group; s.e. = standard error of the unstandardized parameter estimates. Bold fonted parameters indicate statistically significant parameters at p < 0.05. Tests were based on the unstandardized coefficients and their standard error (see Wen, Marsh & Hau, 2010). Parameters were significant at p < 0.05 when their absolute value exceeded 1.96 times the s.e.


Table S2

Path coefficients of the latent interaction models for gender groups in the total international samp

Model with free path coefficients Model with fixed path coefficients

Female Males Females and Males

Predictor b s.e. p b s.e. p b s.e. p

Extra-curricular activities

Self-concept 0.145 0.004 < 0.001 0.167 0.004 < 0.001 0.156 0.003 < 0.001

Enjoyment 0.543 0.004 < 0.001 0.516 0.004 < 0.001 0.528 0.003 < 0.001

SCxEnj 0.084 0.003 < 0.001 0.063 0.002 < 0.001 0.073 0.002 < 0.001

Career aspirations

Self-concept 0.225 0.005 < 0.001 0.271 0.005 < 0.001 0.249 0.004 < 0.001

Enjoyment 0.558 0.005 < 0.001 0.433 0.004 < 0.001 0.491 0.004 < 0.001

SCxEnj 0.065 0.003 < 0.001 0.049 0.003 < 0.001 0.052 0.002 < 0.001

Note: All parameter estimates interaction effects have been standardized according to Wen, Marsh and Hau (2010) using the variances from the total international sample. SCxEnj = interaction of self-concept and enjoyment, b = standardized parameter estimate, s.e. = standard error, p = p-value.


Supplemental Material

Mplus -Syntax for structural equation model with latent interaction estimated with the unconstrained approach

TITLE: Expectancy-value model with latent interactions Unconstrained approach after Marsh, Wen & Hau (2004) DATA:

FILE IS imputation_i.dat; type = imputation; VARIABLE: NAMES ARE studentid schoolid

! indicators for the latent main effects and outcome variables sc1-sc6 enj1-enj5 activ1-activ6 career1-career4

! product-indicators were built in SPSS ! matching order was based on preliminary CFA ! see Marsh et al. (2004)

sc3xen5 sc4xen4 sc6xen1 sc5xen2 sc12xen3 ! standardized final student weight

WSTUCNT; USEVARIABLES ARE enj1-enj5 sc1-sc6 activ1-activ6 career1-career4 sc3xen5 sc4xen4 sc6xen1 sc5xen2 sc12xen3 WSTUCNT contschl; ! Options for design-based correction of standard errors ! and survey weighting

CLUSTER = CONTSCHL; STRATIFICATION = COUNTRY; WEIGHT = WSTUCNT; DEFINE: CONTSCHL = (COUNTRY*10000) + SCHOOLID; ANALYSIS:

ESTIMATOR = MLR; PROCESSORS = 2; TYPE = COMPLEX;


MODEL: ! Measurement models for first-order effects

sc by sc1-sc6; enjoy by enj1-enj5 ;

! Measurement model for latent product-variable

scxenj by sc3xen5 sc4xen4 sc6xen1 sc5xen2 sc12xen3; ! Measurement model for latent outcomes

activ by activ1-activ6; career by career1-career4;

! Covariances between latent outcomes are ! estimated freely

activ with career;

sc with enjoy (1); sc with scxenj; enjoy with scxenj;

! Intercepts of indicators are estimated freely

[sc1-sc6]; [enj1-enj5]; [sc3xen5 sc4xen4 sc6xen1 sc5xen2 sc12xen3];

[activ1-activ6]; [career1-career4];

! Means of latent first-order predictors are fixed to zero

[sc@0]; [enjoy@0]; ! Mean of latent product-variable is fixed to covariance ! between the latent first-order variables [scxenj] (1);

! Structural model with latent interaction between ! self-concept and enjoyment

activ on sc enjoy scxenj; career on sc enjoy scxenj;

OUTPUT: stand tech1;

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