SelfDetermination Theory (SDT) provides a framework for the study of motivation, emotions and their development. SDT assumes the importance of internal sources for the development of personality and selfregulation of behavior (^{Ryan & Deci, 2000a}). SDT also seeks to identify and study the factors that facilitate or undermine a person's initiative and personal sense of volition (^{Deci & Ryan, 2000}, ^{2008}; ^{Ryan & Deci, 2000a}, ^{2000b}; ^{Ryan, Mims, & Koestner, 1983}). In this sense, it defines the sources of intrinsic and extrinsic motivation, characterizing its role in cognitive and social development and in individual differences. SDT considers subjects to be by nature "active and selfmotivated, curious and interested, vital and eager to succeed because success itself is personally satisfying and rewarding" (^{Deci & Ryan, 2008, p. 14}). In short, the theory recognizes differences between subjects in terms of these characteristics, which result from "the interaction between people's inherent active nature and the social environments that either support or thwart that nature" (^{Deci & Ryan, 2008, p. 14}).
To understand and explain motivated behaviors SDT maintains that there are basic and innate psychological needs which are essential to the psychological development and wellbeing of individuals. In this sense, SDT takes into consideration three basic psychological needs: autonomy, competence and relatedness (e.g. ^{Deci & Ryan, 1985}, ^{2000}; ^{Deci, Vallerand, Pelletier, & Ryan, 1991}; ^{Ryan & Deci, 2000b}). Autonomy refers to acting with volition, with a sense of choice. Competence refers to the perception that the individual has of his/her effectiveness in a given task because these feelings of competence allow for the satisfaction of the basic psychological need of competence (^{Ryan & Deci, 2000a}). Finally, relatedness is defined as an individual's feeling of support and affection that one might give and receive from interactions with others (^{Ryan & Deci, 2000b}).
According to SDT, environments and social context affect individual experience and the satisfaction of some basic psychological needs, promoting intrinsic motivation. Intrinsically motivated behaviours are more flexible, autonomous, persistent, creative and effective, and beyond that they are also more enjoyable, providing pleasure and satisfaction (^{Filak & Sheldon, 2003}).
Education is one of the domains where the SDT has been applied (^{Deci et al., 1991}; ^{Filak & Sheldon, 2003}; ^{Ryan & Niemiec, 2009}) because motivation is considered as a central factor in the performance and engagement of students. SDT provides a framework that allows for an understanding of the reasons why people in general and students in particular, engage in activities and how contextual factors and regulatory processes associated either with interactions or to the characteristics of tasks can interfere (^{Boiché, Sarrazin, Gouzet, Pelletier, & Chanal, 2008}; ^{Deci & Ryan, 2008}; ^{Filak & Sheldon, 2003}; ^{Ryan & Niemiec, 2009}). According to ^{Filak and Sheldon (2003)} allowing students to learn in their own way (autonomy) by providing them with opportunities to succeed (competence) and by "defusing or removing authoritarian barriers (relatedness), instructors can give their students an interesting, challenging, and intrinsically motivating educational experience" (^{Filak & Sheldon, 2003, p. 245}).
The SDT framework has been the subject of a great deal of research taking place over several years at the University of Rochester where a group of researchers have been developing a multidimensional measure of the motivational characteristics of participants related to a target activity (^{Grolnick & Ryan, 1987}; ^{Ryan & Connell, 1989}; ^{Ryan et al., 1983}). The result of this ongoing research is an instrument called Intrinsic Motivation Inventory  IMI (SDT, n.d.). This instrument determines the levels of intrinsic motivation as the outcome of a set of subscales: Interest / Enjoyment, Perceived Competence, Effort, Value / Usefulness, Pressure / Tension, Relatedness and Perceived Choice.
The subscale Interest / Enjoyment is the most direct measure (selfreport) of intrinsic motivation. This subscale assesses the interest and inherent pleasure when doing a specific activity. Perceived Choice and Perceived Competence are theorized as positive predictors of intrinsic motivation and are related to the SDT innate psychological needs of autonomy and competence. Perceived Choice evaluates how individuals feel they engage in one activity because they choose to do it, and Perceived Competence measures how effective individuals feel when they are performing a task. Pressure/Tension, conceived as a negative predictor of intrinsic motivation, evaluates if participants feel pressure to succeed in an activity. Effort is a separate variable, which is important when taking into account motivation in specific issues and contexts. It assesses the person's investment of his/her capacities in what he/she is doing. The Value / Usefulness subscale embodies the idea that people internalize and develop more selfregulatory activities when experience is considered as valuable and useful for them. Finally Relatedness refers to the degree of a person's feelings connected to others and is used in studies where interpersonal interactions are relevant (SDT, n.d.).
The IMI has been used in research focused on intrinsic motivation and selfregulation in diverse fields such as sports activities (^{Fonseca & Brito, 2001}; ^{Gutiérrez, Ruiz, & López, 2010}; ^{McAuley, Duncan, & Tammen, 1989}; ^{Tsitskari & Kouli, 2010}), reading (^{Grolnick & Ryan, 1987}), computer activities (^{Deci, Eghrari, Patrick, & Leone, 1994}), performance of puzzles (^{Ryan et al., 1983}) and training / education (^{Filak & Sheldon, 2003}). In all these studies the IMI versions used varied in subscales and items depending on the characteristics of tasks and participants. As the authors of IMI (SDT, n.d.) point out, it is necessary to adjust the instrument according to specific tasks and fields and even to different populations. However IMI adaptations are diversified in consistency and methods used, and also in the conceptual framework presented in tested models. As ^{Markland and Hardy (1997)} argued, Ryan and collaborators (e.g. ^{Ryan & Connell, 1989}; ^{Ryan et al., 1983}) briefly described the use of a set of scales to assess levels of enjoyment, pressure, effort and perceived competence in respect to a task, without presenting data on their psychometric characteristics. Despite the above references concerning the use of IMI and its characteristics, one of the first reported analyses of IMI scale structure and psychometric properties was conducted by ^{McAuley et al. (1989)} in a competitive sport setting. The authors focused on four different dimensions: Enjoyment, Perceived Competence, Effort and Pressure. Internal consistency was adequate with alpha coefficients varying between .68 and .87 for specific dimensions and .85 for the overall scale. Confirmatory factor analyses were used to test and compare several alternative models: a four factor model (M4), a single factor model (M1) and a five factor hierarchical model (M5) with four firstorder factors and a global second order factor. M4 and M5 presented similar appropriate fit indicators. However the authors maintain that the M5 has greater adequacy as it allows for a perspective beyond the firstorder factors.
^{Goudas and Biddle (1994)} using this ^{McAuley et al. (1989)} IMI version conducted an exploratory principal components factor analysis with oblique rotation and identify four dimensions, but as the pressure/tension subscale alpha was low and considered unreliable it was excluded from further analyses. Therefore authors use a three dimensions instrument in their research with Enjoyment (alpha .82), Competence (alpha .83) and Effort (alpha .82). They also calculate a global IMI score (alpha .88) but this global measure wasn't tested when conducting factor analysis. Later, Goudas and collaborators (^{Goudas, Dermitzaki, & Bagiatis, 2000}) used IMI and tested the validity of the hierarchical structure. When conducting a confirmatory factor analysis with a nested factor model they concluded that the model fit was satisfactory for a general factor and three dimensions.
^{Fonseca and Brito (2001)} also analyze the psychometric properties of the Portuguese version of IMI adaptation from ^{McAuley et al. (1989)}. Confirmatory Factor Analysis (CFA) was used to examine the four factor and one factor model. Results showed the superiority of the four factor model to the one factor model. However, the fit indexes suggested that IMI can be used with confidence as a global measure for the evaluation of individual intrinsic motivation for sport and exercise activities, as well as to measure each one of its four facets.
As we can see IMI models tested and used in research are diverse. Some studies tested hierarchical models focusing on global and partial facets (e.g. ^{Goudas et al., 2000}; ^{McAuley et al., 1989}) whereas other authors focused on only one multidimensional level (e.g. ^{Fonseca & Brito, 2001}; ^{Goudas & Bidle, 1994}) and others considering intrinsic motivation as a unidimensional construct (^{Fonseca & Brito, 2001}; ^{McAuley et al., 1989}). Conclusions are also diversified because the unidimensional model shows good fit index in ^{Fonseca and Brito's (2001)} research but not in ^{McAuley et al. (1989)}. Therefore, the multifaceted IMI organization is conceptualized either as a hierarchical structure or as a group of correlated factors which can lead to different results and interpretations.
^{Chen, Hayes, Carver, Laurenceau and Zhang (2012)} state that when researchers work with multifaceted constructs they often have to choose between two approaches: a total score approach using a composite score obtained from individual facets or the facets score approach which analyses each facet of the construct separately. The authors maintain that both approaches miss some information, as a composite score highlights the shared effects but does not separate the unique effects from the shared variance. Analyzing the facets separately taps into their unique contributions, but the specific effects of the facets are often entangled with the effects of the shared general construct (^{Chen et al., 2012}). Along the same lines, ^{Reise, Moore and Haviland (2010)} assert that if a researcher intends to both recognize multidimensionality and simultaneously retain the idea of a single important target construct, the secondorder (hierarchical) or bifactor models are the only choices. ^{Chen et al. (2012)} and ^{Reise et al. (2010)} maintain that the bifactorial model can be an even better alternative than secondorder models. A secondorder model hypothesizes that there is a higher order factor that accounts for the communality shared by the facets, which consists of the lower order factors. However, the bifactor model can separate the specific factors from the general factor; the factor loadings in bifactor model, reflect the strength of the relation between the specific factors and their associated items. The bifactor model can also identify whether a facet still exists after partialling out the general factor (^{Chen et al., 2012}). Therefore, the bifactor model allows one to directly explore the extent to which items reflect a common target trait and the extent to which they reflect a primary or subtrait (^{Reise et al., 2010}). In particular, the bifactor structural model is more amenable to study (a) whether scale items measure a single common dimension, (b) how well the scale items measure a single common dimension, (c) the effect of multidimensionality on scale scores, and (d) the feasibility of applying a unidimensional model in the presence of multidimensional data (^{Reise et al., 2010}).
As we have seen above, researchers interested in assessing intrinsic motivation using the IMI reveal evidence of a general factor running through the items but also there is some evidence of multidimensionality in this instrument. These findings encouraged us to evaluate the Portuguese version of IMI (for Mathematics and for Portuguese Language), particularly whether this construct is unitary or multifaceted.Thus, the aim of our research is to analyze the characteristics of IMI in an adaptation that characterizes the learning motivation for Portuguese students of Mathematics and the Portuguese Language. Furthermore, our aim is to test the best model to describe the internal structure of IMI.
Method
Participants
A total of 3685 preadolescents and adolescents (M = 13.67, SD = 2.26) from schools in the metropolitan area of Lisbon participated in the study. The sample consisted of 49.6% (1829) girls and 50.4% (1856) boys. Overall 1250 (33.9%) were attending the second cycle of school (5^{th} and 6^{th} grades), 1691 (45.9%) the 3^{rd} cycle (7^{th} to 9^{th} grade), and 744 (20.2%) Secondary Education (10^{th} to 12^{th} grades). In relation to academic achievement 70.2% had never retaken a year whereas 29.8% had retaken at least once.
From the global sample, 1882 (50.1%) completed a version of the questionnaire on Mathematics whereas 1803 (48.9%) completed the questionnaire on Portuguese Language.
Instrument
The Intrinsic Motivation Inventory (IMI; SDT, n.d.), is an instrument that aims to assess motivation in a broad array of situations and contexts and is rooted in Self Determination Theory (^{Deci & Ryan, 1985}, ^{2000}). The IMI has a multidimensional structure organized in 7 subscales directed towards assessing motivation in specific tasks. Authors of IMI have suggested that, according to the situation, different options can be taken with regard to the subscales to be used depending on their relevance to the issues researchers are exploring (SDT, n.d.). Items can also be removed if they appear redundant or less adapted to the situations under analysis. Moreover, the items are also flexible in formulation, and some may be adjustable to the specific activity of the study (SDT, n.d.).
In these versions of IMI, directed towards motivation in Portuguese Language and in Mathematics, we used five subscales: Interest/Enjoyment, Perceived Competence, Pressure/Tension, Perceived Choice and Value/Utility. We have not included Effort and Relatedness subscales because, in previous studies using exploratory factor analyses, the Effort subscale failed to present consistent or reliable characteristics. Relatedness items did not fit to the tasks that we sought to assess (e.g. ^{Sousa, Monteiro, Mata, & Peixoto, 2010}).
Thus, both versions of the questionnaire (Math and Portuguese Language) comprised 21 items, distributed over the 5 referred subscales: Interest/Enjoyment  items 1, 6, 11, 15 and 19 (e.g. "I enjoy doing Math's/Portuguese Language assignments, very much"); Perceived Competence  items 2, 7, 12, 16 (e.g. "I think I am pretty good at Math's/Portuguese Language activities"); Pressure/Tension  items 3 and 8 (e.g. "I felt very tense while doing Math's/Portuguese Language tasks"); Value/Utility  items 4, 9, 13, 17, 20 (e.g. "Math's/Portuguese Language are valuable to me"); Perceived Choice  items 5, 10, 14, 18, 21 (e.g. "I only do the Math's/Portuguese Language tasks because teacher orders."). All items were scored on a 6point scale ranging from 1 ("Never") to 6 ("Always"). All items from the Perceived Choice subscale were reversed due to their negative formulation. Items were scored in a way that higher scores are related to intrinsic motivation characteristics except for Pressure/Tension, where higher scores mean greater feelings of pressure, which are negatively related to intrinsic motivation.
Data Collection Procedure
The data for this study were collected at schools after obtaining the consent from the school boards and a written consent from parents for their children to participate. Questionnaires were done in the classroom. The researcher carefully explained how to fill out the questionnaire and then assured participants that their responses would remain anonymous. Following these instructions, students completed the IMI individually.
Statistical Analysis
Confirmatory factor analysis (CFA) was used to assess the factorial structure of IMI. We started by testing four different models: unidimensional, multidimensional, second order and bifactor model (Figure 1). These analyses were carried out on the data from both the Portuguese Language Motivation and Math's Motivation Questionnaires. After choosing the model(s) that best fit the data we undertook separate analyses for Motivation in Portuguese Language and Mathematics. All the analyses were carried out using the Asymptotically Distribution Free estimator (ADF) with Amos 19.0 (^{Arbuckle, 2010}). The option to use ADF was due to the fact that multivariate normality was not guaranteed , despite the fact that all variables presented values for skewness and kurtosis below the thresholds usually considered (^{Byrne, 2010}; ^{Kline, 2011}). To assess global fit of the models we used the χ², the Adjusted Goodness of Fit Index (AGFI), the TuckerLewis Index (TLI), the Comparative Fit Index (CFI) and the Root Mean Square Error of Approximation (RMSEA). As the χ² test of models is highly sensitive to sample size (^{Bentler, 1990}; ^{Kline, 2011}; ^{Shumacker & Lomax, 2004}), and, as we were looking at a large sample, we based our decision of model fit on the others indices. For these, .90 can be considered an acceptable threshold for AGFI, TLI and CFI (^{Hair, William, Barry, Rolph, & Ronald, 2010}; ^{Kline, 2011}; ^{Loehlin, 2004}). ^{Shumacker and Lomax (2004)}, however, adopted a more conservative criterion, maintaining that only models presenting values higher than .95 for those indices can be accepted. For RMSEA acceptable values were those lower than .05 with a not excessively high confidence interval (^{Raykov & Marcoulides, 2006}; ^{Shumacker & Lomax, 2004}).
To compare the models we conducted chisquare difference tests. However, as this difference can be inflated due to the large sample used we also based our judgment on TLI difference, considering that it was significant if it exceeded .01 (^{Gignac, 2007}; ^{Murray & Johnson, 2013}).
Reliability was analyzed using Cronbach's alpha and composite reliability (^{Maroco, 2010}) where scores should be above .70 (^{Hair et al., 2010}). Composite reliability indices were computed following the approach of ^{Geldhof, Preacher and Zyphur (2014)}.
Results
We examined four models for IMI organization, where construction was based on the arguments previously presented concerning the different models that can be used to study the adequacy of this instrument (Figure 1). Thus we tested a model who presupposes that all the items load in a single factor (Figure 1  Unidimensional) and three different multidimensional structures: one with five interrelated latent factors (Figure 1  Multidimensional), another with a common high order latent factor that aggregates the five latent first order factors (Figure 1  Second order Factor), and a bifactor model in which each item loads simultaneously in a general factor and in one of the specific dimensions (Figure 1  Bifactor).
Results displayed in Table 1 show that all the models tested presented poor fit. To improve model fit we used the "modification indices" provided by AMOS, excluding: items that had crossloadings with factors other than those they were supposed to load; items in which error terms were correlated with error terms of items which belonged to a different factor. Following this strategy we excluded 2 items from the dimensions Value and Enjoyment and one item from the dimensions Perceived Competence and Perceived Choice. The items of Pressure/Tension factor were also excluded due to low correlations with the remaining factors and the inspection of the factor loadings of the items in the bifactor model showing low factor loadings in the general factor. In addition the fact that this dimension only has 2 items presented some model identification problems, namely in the bifactor model. The 12 items that remained were tested within the model framework previously tested.
χ^{2}  df  AGFI  TLI  CFI  RMSEA (90% CI)  

Null Model  4955.3  190  .67  .70  .76  .092 (.090.094) 
Unidimensional Model  3111.5  170  .74  .35  .42  .072 (.070.075) 
Multidimensional Model  1724.3  160  .84  .63  .69  .054 (.052.057) 
SecondOrder Factor Model  1974.7  165  .86  .59  .64  .058 (.055.060) 
BiFactor Model  1283.4  151  .89  .70  .76  .050 (.048.053) 
Note. df  Model degrees of freedom; AGFI  Adjusted GoodnessofFit Index; CFI  Comparative Fit Index; RMSEA  Root Mean Square Error of Approximation
The results presented in Table 2 show that, with the exception of the unidimensional model, all the models presented adequate goodnessoffit indices. The chisquare difference tests show that the bifactor model outperformed both the secondorder factor model, ∆ χ² (7) = 79, p < .005, and the multidimensional model, ∆ χ² (5) = 38.3, p < .005. The bifactor model also showed significant TLI difference from the secondorder factor (∆ TLI = .018) but not from the multidimensional model (∆ TLI = .006), according to the criteria proposed by ^{Gignac (2007)}. In relation to the specific subjects, Mathematics and Portuguese Language, there are slight differences. Thus, for Mathematics, the results are similar to those of the global sample [∆χ² (5) = 20.4, p < .01; ∆TLI = .004, for the comparison between the bifactor model and the multidimensional; ∆χ² (7) =37.9 = 20.4, p < .001; ∆TLI = .011, for the comparison between the secondorder factor and the bifactor model]. In Portuguese Language the comparison between the bifactor model and the multidimensional present significant differences in the TLI indices (∆TLI = .016) which does not happen either in the global sample or in Mathematics. The remaining comparisons are similar to the results of both the global sample and Mathematics with significant differences both in chisquare and in TLI [∆χ² (5) = 32.7, p < .001 for the comparison between the multidimensional and the bifactor model; ∆χ² (7) = 53.4, p < .001; ∆TLI = .027 for the comparison between the secondorder and the bifactor model].
χ^{2}  df  AGFI  TLI  CFI  RMSEA (90% CI)  

Global Sample  
Null Model  3712.5  66  .61  .00  .00  .129 (.126.133) 
Unidimensional Model  2024.1  55  .75  .35  .46  .104 (.100.108) 
Multidimensional Model  213.6  48  .97  .94  .96  .032 (.028.037) 
SecondOrder Model  254.3  50  .97  .93  .94  .035 (.031.039) 
BiFactor Model  175.3  43  .97  .94  .96  .030 (.026.035) 
Mathematics  
Null Model  2149.5  66  .60  .00  .00  .137 (.132.142) 
Unidimensional Model  1109.1  55  .75  .39  .49  .107 (.101.112) 
Multidimensional Model  145.2  48  .96  .94  .95  .035 (.028.041) 
SecondOrder Model  162.7  50  .96  .93  .95  .037 (.030.043) 
Bifactor Model  124.8  43  .96  .94  .96  .034 (.027.041) 
Portuguese Language  
Null Model  1767.9  66  .61  .00  .00  .126 (.121.131) 
Unidimensional Model  967.1  55  .74  .36  .46  .101 (.095.106) 
Multidimensional Model  145.8  48  .96  .92  .94  .035 (.029.042) 
SecondOrder Model  166.5  50  .95  .91  .93  .038 (.031.044) 
BiFactor Model  113.1  43  .96  .94  .96  .032 (.025.039) 
Note. df  Model degrees of freedom; AGFI  Adjusted GoodnessofFit Index; CFI  Comparative Fit Index; RMSEA  Root Mean Square Error of Approximation.
Table 3 shows the factor loadings for the bifactor model, for the global sample and for Mathematics and Portuguese Language. The general pattern is similar both in the global sample and in the specific subjects in which the items of the Enjoyment factor show substantial lower loadings in the specific facet than in the general factor (Motivation) and the others showing more or less similar factor loadings in both specific and general factors. Reliability results are also presented in Table 3. Cronbach's Alpha values are quite acceptable, both for the specific facets of IMI and for the global score in all situations considered (Global Sample, Mathematics and Portuguese Language).
Motivation  Value  Enjoyment  Perceived Competence 
Perceived Choice 


Item 13 
.62 (.60/.64) 
.57 (.64/.51) 

Item 17 
.65 (.64/.68) 
.58 (.61/.55) 

Item 20 
.69 (.66/.72) 
.58 (.64/.52) 

Item 6 
.77 (.76/.77) 
.26 (.29/.22) 

Item 15 
.83 (.84/.82) 
.47 (.45/.48) 

Item 19 
.73 (.75/.73) 
.13 (.11/.13) 

Item 2 
.56 (.60/.51) 
.71 (.68/.74) 

Item 7 
.48 (.48/.46) 
.48 (.54/.42) 

Item 12 
.43 (.44/.39) 
.46 (.46/.47) 

Item 5 
.58 (.61/.57) 
.48 (.58/.59) 

Item 10 
.57 (.63/.53) 
.46 (.62/.62) 

Item 18 
.52 (.59/.47) 
.58 (.62/.64) 

Cronbach Alpha 
.89 (.90/.88) 
.91 (.92/.89) 
.86 (.86/.85) 
.82 (.78/.73) 
.86 (.88/.84) 
Composite Reliability 
.92 (.93/.92) 
.70 (.76/.63) 
.25 (.25/.24) 
.67 (.68/.66) 
.75 (.74/.75) 
The values for Composite Reliability were computed for the bifactor model and show that the global score is a reliable measure and also that composite reliability is acceptable for Value, Perceived Competence and Perceived Choice but not for Enjoyment (Table 3).
Discussion
The central aim of this research was to analyze the characteristics of IMI, in a version adapted for the evaluation of the motivation for Portuguese Language and Mathematics. The intention was, therefore, to analyse the psychometric properties, including internal validity and reliability, and to test the model that best describes the structure of IMI. The results obtained in the confirmatory factor analysis highlight a multidimensional structure of IMI since among the four models tested only the Unidimensional model showed inadequate goodnessoffit indices. These indices were poor not only when considering data of Math and Portuguese Language separately but also when taken together. The inadequacy of the unidimensional model is consistent with the results of ^{McAuley et al. (1989)} in a competitive sport setting. These data confirm the theoretical framework which underlies the construction of the original IMI (e.g. ^{Ryan & Connell, 1989}; SDT, n.d.). Therefore we may conclude that regardless of the specificity of contexts and tasks, this instrument supports the multidimensionality of motivation and is consistent with the model that takes into consideration the existence of different indicators to understand intrinsic motivation (^{Deci & Ryan, 1985}; ^{Grolnick & Ryan, 1987}; ^{Ryan & Deci, 2000b}; ^{Ryan et al., 1983}).
Although the three multidimensional models present goodfit indices, if we take each one separately, we must decide which one is the most welladjusted to the theoretical construct underlying IMI. In order to take into account the multidimensionality while simultaneously retaining the idea of a general construct, the secondorder and the bifactor models are the only options (^{Murray & Johnson, 2013}; ^{Reise et al., 2010}). Looking at the results from these two multidimensional models, the bifactor model significantly outperforms the hierarchical model (secondorder). Also, if we consider the possibility that the model global fit indices could be biased in favour of the bifactor model (^{Murray & Jonhson, 2013}), in this case, then the bifactor model can be a good description of IMI structure in the sense that it can separate the common variance shared for all items from the variance associated with each factor (^{Chen et al., 2012}; ^{Reise et al., 2010}). Therefore it is possible to understand to what extent the items reflect the general construct without the impact of multidimensionality and how they reflect a specific construct controlling the general (^{Reise et al., 2010}). As IMI intends to tap into a general construct (Intrinsic Motivation) as well as some specific dimensions which are considered predictors (e.g. Choice) or indicators of intrinsic motivation (Enjoyment), the use of the bifactor model can help to understand the best way to interpret IMI responses. In this sense results of the analyses of the bifactor model point to several important aspects related to the general trait (Intrinsic Motivation) and the four specific scales: (a) a general factor clearly emerges; (b) the multidimensionality of IMI is confirmed; (c) the specific scales don't have all the same organizational pattern; (d) the IMI structure is similar considering different specific school subjects. The item loading scores in the general factor and on the specific facets, allows us to evaluate the viability of considering only general or specific dimensions (^{Reise, 2012}; ^{Reise et al., 2010}). As we have seen, in IMI, factor loadings of almost all items were very similar in the general factor and in the specific. Only Enjoyment items show a different pattern in that the lowest loading scores are in the specific and the highest loading scores are in the general factor. These results are consistent with SDT theory which maintains the enjoyment/pleasure as a direct index of intrinsic motivation and in IMI specifically is considered as a selfreport measure of intrinsic motivation (SDT, n.d.). Perceived Competence, Value and Perceived Choice are considered only as predictors of intrinsic motivation, as they are related to the basic needs which are central to promoting it. The experiences of competence and autonomy are essential to intrinsic motivation, but they are not intrinsic motivation (^{Grolnick & Ryan, 1987}; ^{Ryan & Connell, 1989}; ^{Ryan et al., 1983}). Therefore, equivalent loading scores in both general and group factors can be explained by the type of contribution of these domains to intrinsic motivation.
In addition to the contribution of the bifactor model for the conceptualization of intrinsic motivation, it can also be useful to decide on which measures to compute. According to ^{Chen et al. (2012)} and ^{Reise et al. (2010)} positions on bifactor models when items have low loadings in the specific factor primarily reflect the general, so it does not make much sense to compute scores for the specific facet. Nevertheless when factor loadings are substantial and more or less identical in the general and specific factors, then we can take into consideration to compute scores for both the aggregate measure and the specific dimension (^{Reise et al., 2010}). Thus, the results presented suggest that we can compute measures for Perceived Competence, Value and Perceived Choice and also for Intrinsic Motivation global score. In regard to the reliability of the instrument, results also point to the adequacy of IMI. All the specific facets present satisfactory Cronbach's alpha scores, either when the data is analyzed together (Portuguese Language and Mathematics) or in separate analyses of the scales. In the case of Cronbach's alpha for the global analysis, scores ranged from .82 to .91. Note that these scores, although similar, are higher in all dimensions than those found in other research which adapted IMI for sport activities (^{Gutiérrez et al., 2010}; ^{McAuley et al., 1989}; ^{Tsitskari & Kouli, 2010}). In addition to Cronbach alpha scores, composite reliability was computed, which can be considered a better estimator of reliability as it acknowledges the heterogeneous itemconstruct relations taking into account the factor loadings of each factor (^{Geldhof et al., 2014}). Composite reliability scores show that not only the global measure but also some specific dimensions scores (Value, Perceived Competence, and Perceived Choice) are reliable measures. The low composite reliability value for Enjoyment is consistent with the weak loadings on the specific factor referred to above and also points to the inadequacy of computing a measure based only on the items of this dimension.
Some limitations should be highlighted when considering the present findings and which also have the benefits of pointing to new directions for future research. Firstly the data in our research is all selfreported. Future studies are needed in order to provide additional support for the validation of the IMI dimensions in these specific tasks, through correlations with other assessments of motivation indicators such as behavioral indices or through the observations of others who can corroborate the reports made independently by students (e.g teachers). Another concern focuses on something already mentioned by authors of the original IMI (SDT, n.d.) and refers to the redundancy of items within the subscales which can sometimes overlap with each other. However, we have tried reduce this so as not to include as many items in each dimension and randomizing their presentation in the scale and mixing the items of the five subscales. We think that this option was a positive step as the result was a relatively short instrument, allowing for a tool that can be utilized easily and usefully in research and intervention in school and educational settings.
To summarize, the Portuguese version of IMI showed adequate validity and reliability for the assessment of intrinsic motivation in Portuguese Language and Mathematics. The bifactor model was shown to be the best model, which enabled us to highlight the contribution of the different dimensions for the general factor of intrinsic motivation. Therefore the Portuguese version of IMI not only represents a crosscultural and specific domain adaptation of the initial IMI but it is also a reliable and valid instrument in its own right. Its ability to assess three dimensions of motivation and a global measure is of central importance as it allows for a diverse understanding of student engagement with academic tasks. This may also help technicians and teachers to understand levels and types of student motivation in specific curricular subjects. Such information may be fundamental to the definition of new teaching strategies, aims and tasks in order to promote intrinsic motivation and a deeper engagement of students with school subjects and knowledge.