Comparing effectiveness of additive, interactive and ...

Learning and Individual Differences 50 (2016) 203–209

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Comparing effectiveness of additive, interactive and quadratic models in detecting combined effects of achievement goals on academic attainment Nikos L.D. Chatzisarantis a,⁎, Qi Bing b, Cui Xin d, Masato Kawabata c, Severine Koch a, Rosanna Rooney a, Martin S. Hagger a a

School of Psychology and Speech Pathology, Curtin University, Australia School of Education, Hebei University, China National Institute of Education, Singapore d Department of Foreign Language Teaching and Research, Hebei University, China b c

a r t i c l e

i n f o

Article history: Received 14 August 2015 Received in revised form 6 July 2016 Accepted 11 August 2016 Available online xxxx Keywords: Achievement goals Combined effects Quadratic models Academic achievement

a b s t r a c t This study compared effectiveness of additive, interactive, and quadratic statistical models in detecting the combined effects of achievement goals on academic achievement. In a prospective study that aimed to predict college students' grades in an English course, we found that the quadratic model was more effective in detecting the combined effects of achievement goals on course grades than the additive and interactive models. In addition, a response surface analysis showed that the combined effects of achievement goals on course grades corresponded to a goal profile that involved tendencies to endorse mastery goals at high levels and performance goals at moderate levels. Findings suggest that the quadratic model is a viable data analytic technique that assists researchers in detecting combined effects of achievement goals on academic achievement. Crown Copyright © 2016 Published by Elsevier Inc. All rights reserved.

1. Introduction The study of achievement motivation has long been concerned with the question of which types of goals are most strongly associated with desirable outcomes such as high levels of self-esteem, intrinsic motivation, productivity in the workplace, and academic performance. Building upon Nicholls' (1989) or Dweck's (1986) achievement goal theories, early research distinguished between two major classes of achievement goals: mastery goals that focus on developing competence through task mastery and learning, and performance goals that focus on demonstrating competence by outperforming others (Duda, 1989). This dichotomous conceptualisation of achievement goals has since been extended to a 2 × 2 hierarchical model that differentiated achievement goals into mastery-approach goals (i.e., understand and master a task), mastery-avoidance goals (i.e., avoid misunderstanding or making mistakes), performance-approach goals (i.e., try to do better than others) and performance-avoidance goals (i.e., avoid doing poorly relative to others) (Elliot, 1999; Elliot & Church, 1997; Elliot & McGregor, 2001). ⁎ Corresponding author at: Laboratory of Self-Regulation, Health Psychology and Behavioural Medicine Research Group, School of Psychology and Speech Pathology, Curtin University, Australia. E-mail address: [email protected] (N.L.D. Chatzisarantis).

http://dx.doi.org/10.1016/j.lindif.2016.08.015 1041-6080/Crown Copyright © 2016 Published by Elsevier Inc. All rights reserved.

To date, research has generally shown that avoidance goals (mastery-avoidance or performance-avoidance goals) are almost uniformly associated with maladaptive outcomes such as high anxiety, disorganised study habits, fear or failure, self-handicapping, and low achievement or task interest (Senko, Huleman, & Harackiewitz, 2011). Mastery-approach and performance-approach goals have been associated with adaptive outcomes such as elevated task (mental) focus (Lee, Sheldon, & Turban, 2003), task absorption (Barron & Harackiewicz, 2001) or positive peer relationships and classroom belongingness (Senko et al., 2011). However, in comparison to performance-approach goals, mastery-approach goals are more strongly associated with high intrinsic motivation, high task-interest, and use of deep learning strategies (Harackiewicz, Barron, Carter, Lehto, & Elliot, 1997; Harackiewicz, Barron, Tauer, Carter, & Elliot, 2000; Harackiewicz, Durik, Barron, Linnenbrink, & Tauer, 2008). Interestingly, performance-approach goals exhibit a stronger relationship with academic achievement than mastery-approach goals (Senko et al., 2011; Van Yperen, Blaga, & Potmes, 2014). Nevertheless, a number of studies clarified that the positive effects of performance-approach goals on educational outcomes are specific, and they are more likely to be observed among boys than girls, among older students than younger students, in competitive learning environments and if mastery goals are also endorsed at high levels (Midgley, Kaplan, & Middleton, 2001; Richardson & Remedios, 2014).

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The adaptive effects of performance-approach and mastery-approach goals on the same or distinct outcomes have compelled researchers to examine the extent to which the two types of approach goals combine to optimise performance, and the processes involved (Harackiewicz, Barron, & Elliot, 1998). According to this multiple goal perspective, students who adopt both mastery-approach and performance-approach goals may experience more positive outcomes than students who adopt only one type of goal. The reason for this is that students who endorse both types of approach goals may reap the benefits associated with each goal by pursuing both goals simultaneously (Senko et al., 2011). Accordingly, Barron and Harackiewicz (2001) advised researchers to examine combined effects of achievement goals on outcomes by testing additive and interactive statistical models. However, to our knowledge, only six studies have supported combined effects of performance-approach goals and mastery-approach goals on academic achievement using additive or interactive models (see Table 1). One reason why previous research has been inconsistent in observing combined effects of achievement goals on academic achievement is that quadratic terms, which test for non-linear functional relationships, are not included in the additive and interactive models (Aiken & West, 1991, p. 62; Cortina, 1993; Krantz & Tversky, 1971; Lubinski & Humphreys, 1990). This practice can mislead researchers to reject a combined effect when, in fact, there is an alternative model that supports combined effects of achievement goals on academic achievement (Ganzach, 1997). Given this, the purpose of the present article was to compare utility of the additive, interactive, and quadratic regression models in detecting combined effects of achievement goals on academic performance. 2. Differences between additive, interactive, and quadratic models According to the multiple goal approach, the additive or interactive models can be tested by examining whether the following regression equation explains observations (Barron & Harackiewicz, 2001): TP ¼ b0 þ b1 M þ b2 P þ b4 M  P þ e1

ð1Þ

In Eq. (1), TP represents students' performance on educational tasks as reflected, for example, on grades achieved in an exam. The terms M and P are individuals' responses to instruments measuring mastery goals and performance goals. The product term M × P represents the interaction between performance goals and mastery goals. The coefficient b0 is the intercept of the regression equation. The term e1 indicates residual variance that is not explained by the regression equation. The coefficients b1, b2 and b4 are unstandardised regression coefficients indicating the main and interactive effects of mastery goals or performance goals on task performance. Eq. (1) supports the additive model if the main effects of mastery goals and performance goals on task performance are positive and statistically significant (Senko et al., 2011). In this case, the additive model supports the notion that task performance is maximised, in the

sense that it reaches the highest possible level, when mastery goals and performance goals are also endorsed at the highest possible levels (Barron & Harackiewicz, 2001). A statistically significant value for b4 indicates presence of an interaction effect whereas a statistically nonsignificant b4 does not support an interaction (Aiken & West, 1991). A positive b4 implies a type of interactive effect, termed synergistic, when the main effects of achievement goals on course grades are zero or positive. In these cases, the interactive effect indicates that, among students who endorse mastery goals at high levels, those students who also endorse performance goals at high levels achieve higher performance levels than all other students. The quadratic model is estimated by introducing quadratic terms into Eq. (2) (Edwards, 1994): TP ¼ b0 þ b1 M þ b2 P þ b3 M2 þ b4 M  P þ b5 P2 þ e1

ð2Þ

In this equation, M2 and P2 are quadratic terms that represent nonlinear relationships between achievement goals and task performance. The coefficients b3 and b5 are unstandardised regression coefficients that capture effects associated with the quadratic terms. In Eq. (2), negative values of b3 or b4 imply a concave-shaped relationship between achievement goals and task performance. A relationship is concave in shape when performance levels increase as achievement goals increase but only up to a given point beyond which any further increases in achievement goals will yield lower (or the same) performance levels. Hence, a concave function indicates that achievement goals yield higher performance levels when they are endorsed at a moderate level (Edwards & Parry, 1993). In contrast, positive values of b3 or b4 imply a convex function and that performance levels decrease (or remain constant) as achievement goals increase but up to a point beyond which further increases in achievement goals increase task performance. Accordingly, a convex function yields low performance levels when achievement goals are endorsed at moderate levels (Edwards & Parry, 1993). An important difference between the additive, interactive, and quadratic models concerns the types of combined effects that these models enable researchers to test during the analysis. The additive and interactive models enable researchers to test the hypothesis that people who endorse both mastery goals and performance goals at the highest possible levels perform best in achievement contexts. These models cannot test the hypothesis that individuals who endorse one goal at the highest possible level and the other goal at a marginally lower level are the best performers (i.e., individuals who adopt a high-mastery/moderate-performance goal profile). The reason for this is that the additive and interactive models assume that the effects of a goal (i.e., performance goal) on course grades linearly increase within high or low levels of endorsement of the other goal, i.e. the mastery goal (Edwards, 1994, 2001). As a consequence, when these models support combined effects, they always “force” researchers to conclude that a high-mastery/high-performance goal profile is the most optimal goal profile (see Appendix). However, conclusions based on additive or interactive models can be

Table 1 Characteristics of studies that detected combined effects of achievement goals on academic achievement. Effects Study

Performance outcome

Mastery goal

Performance goal

Mastery × Performance interaction

Bodmann, Hulleman, and Harackiewicz (2008) Church, Elliot, and Gable (2001) Finney, Pieper, and Barron (2004) Pekrun, Elliot, and Maehr (2009) Senko and Harackiewicz (2005) Senko et al. (2013)

Final grade Final grade Semester GPA Exam grade Exam grade Exam grade

0.19⁎ 0.20⁎ 0.09⁎ 0.11⁎ 0.16⁎ 0.21⁎

0.24⁎ 0.14⁎ 0.04⁎ 0.38⁎ 0.28⁎ 0.18⁎

−0.08 ns ns ns ns ns

Note. Parameters with an asterisk are statistically significant at p b 0.05 level. An additive model is supported when main effects of mastery goals and performance goals are statistically significant. A synergistic effect is supported when the interaction between mastery goals and performance goal is statistically significant. The term ns denotes nonsignificant finding from studies that did not actually report regression coefficients.

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misleading if performance goals and mastery goals relate to academic achievement by concave and linear functions respectively. The reason for this is that the combination of linear and concave relationships implies that a high-mastery/moderate-performance goal profile is the most optimal (see Appendix). In the context of education, there is some evidence to suggest that performance class climates or performance goals do indeed relate to academic achievement by non-linear (concave) functions (Sideridis, Antoniou, & Simos, 2013; Sideridis & Stamovlasis, 2015; Stamovlasis & Sideridis, 2014). Using person-centred profile analysis (i.e., cluster analysis), Conley (2012) demonstrated that students who endorsed mastery goals at high levels and performance goals at moderate levels achieved the highest grades in math. However, other longitudinal studies that employed person-centred analysis did not observe similar findings (Meece & Holt, 1993; Shim, Ryan, & Anderson, 2008). Nevertheless, using a catastrophe model of achievement goals (i.e., cusp model), Sideridis, Stamovlasis, and Antoniou (2015) showed that a mild performance class-climate, in which teachers did not actively dissuade adoption of performance goals, increased reading performance among students with learning difficulties. A possible reason behind these positive effects of performance goals on academic achievement is that performance-approach goals motivate students to prepare for exams and actively attend to teachers' instructions, guides and hints (Senko, Hama, & Belmonte, 2013). However, in support of the hypothesis that the relationship between performance-approach goals and academic achievement is concave, empirical evidence shows that increases in performance class-climate or performance-approach goals beyond a moderate level increase worry and anxiety levels that in turn cause abrupt or discontinuous changes in academic performance (Eisenbarth & Petlichkoff, 2012; Sideridis, 2007). Interestingly, Sideridis et al. (2015) also observed a linear relationship between reading performance and a mastery class-climates in which teachers strongly recommended adoption of mastery goals. This combination of linear and concave relationships, in turn, implies that a high-mastery/ moderate-performance class-climate yields high levels of reading performance.

2.1. Hypotheses In the present article, we compared the utility of the additive, interactive, and quadratic models in detecting combined effects of achievement goals on academic achievement (i.e., course grades). Based on previous studies (i.e., Elliot & McGregor, 2001), we expected that there would be no additive or synergistic effects of achievement goals on academic achievement. However, we expected the quadratic regression model to support a combined effect that corresponded to a highmastery/moderate-performance goal profile (Sideridis et al., 2015). We tested this hypothesis in the context of a prospective study that aimed to predict grades achieved by university students in an English course. In the present study, we also statistically controlled for the effects that perceptions of competence and academic ability exert on academic achievement. It was important to control for these variables because they have been shown to be positively associated with achievement goals (Brophy, 2005). Hence, the relationship between performance goals and academic achievement may be spurious reflecting a confound between ability and competence. However, in the present study we did not expect perceived competence or ability to completely attenuate the relationship between achievement goals and grades because their effects tend to be independent of the corresponding effects of achievement goals on grades (Senko et al., 2011). Despite this, we controlled for the effects of these variables to provide a more precise estimate of effects of achievement goals on academic achievement.

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3. Method 3.1. Participants Participants were second-year university students who attended an advanced English course in a public university (N = 200, M age = 20.82, SD = 0.83, Male = 86, Female = 114). The English course was delivered in small groups (7 groups) of no more than 30 students. Students' consent and permission from university's ethics committee were obtained. Fifteen students did not take the final exam leaving a final sample of 185 (M age = 20.84, SD = 0.84, Male = 78, Female = 107). This sample size permits detection of a small to medium squared multiple correlation (R2 = 0.076) at 0.05 alpha level, 80% of the time (Cohen, 1988, p. 410–414). 3.2. Procedure and design We employed a prospective design that measured psychological variables at the beginning of the semester and final grades at the end of the semester. Students completed measures of achievement goals and perceptions of competence at the beginning of the semester. The university provided us with records indicating students' academic ability. At the end of the semester, students' grades were provided by the teacher. 3.3. Measures 3.3.1. Achievement goals We used Duda's (1989) task- and ego-orientation questionnaire to measure achievement goals because: (i) this questionnaire captures the two approach goals (i.e., mastery approach and performance approach goals) and (ii) according to Barron and Harackiewicz (2001) the combined effects of achievement goals on academic attainment are function of approach goals and not of avoidance goals. It comprises 13 items tapping achievement goals on 5-point scales ranging from strongly disagree (1) to strongly agree (5). The instrument was modified to reflect perceptions related to the English course. An example item for performance goals was: “I feel most successful on the English course when my final grade is greater than the final grade achieved by others”. An example item for mastery goals was: “I feel most successful on the English course when I do my best”. In the current study, the alpha reliability for the mastery (α = 0.74) and performance scales (α = 0.79) were satisfactory. 3.3.2. Perceived competence We used five items from McAuley, Duncan, and Tammen's (1989) intrinsic motivation scale to measure perceived competence. An example item was: “I feel pretty competent on the English course”. All items were measured on 7-point scales ranging from strongly disagree (1) to strongly agree (7). In the present study, the alpha reliability for this scale was satisfactory (α = 0.92). 3.3.3. Academic achievement (course grades) Academic achievement was measured through students' final grade on the English course. This grade reflected students overall performance on the English course and it was function of coursework and a grade achieved on a final exam taken at the end of the semester. Grades could range from 0% to 100%. 3.3.4. Academic ability Upon entering the university students take an exam that indicates students' ability on core subjects such as Math and English. On the basis of this exam, the university identifies students as “higher” and “lower” ability. Accordingly, in our statistical analysis, we assigned the value of (+ 1) to students who were identified, by the university, as

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“higher” ability students and (−1) to students who were identified as “lower” ability students. 3.3.5. Analysis We initially calculated descriptive statistics and Pearson's correlations for all psychological variables. For the main analysis, we conducted two separate regression analyses. First, Following Aiken and West's (1991) recommendations, we carried out a hierarchical regression analysis to examine whether the additive and interactive models supported the additive or synergistic effects of achievement goals on grades. In this analysis, we did not estimate quadratic effects. Second, we examined the combined effects of achievement goals on grades by conducting a separate regression analysis in which we tested whether the following quadratic equation explained the observed data (Edwards, 1994): G ¼ b0 þ b1 M þ b2 P þ b3 M2 þ b4 M  P þ b5 P2 þ b6 C þ b7 A þ e1

ð3Þ

In Eq. (3), G represents students' course grades and the terms A and C represent ratings of academic ability (A) and perceptions of competence (C). The regression coefficients b6 and b7 represent main effects of academic ability and perceptions of competence on academic achievement. The unstandardised coefficients from Eq. (3) were used to plot and analyse a three-dimensional response surface in which mastery goals and performance goals were represented on the perpendicular axes (x and y) and course grades were represented on the vertical axis (z) (Edwards & Parry, 1993). Fig. 1 presents a hypothetical response surface that is consistent with our hypothesis that course grades are maximised when mastery goals and performance goals are endorsed at high and moderate levels respectively. On the bottom of Fig. 1 there is the first principal axis of the response surface. For concave surfaces, this axis captures goal profiles that are likely to yield maximum grades (Edwards & Parry, 1993). The slope and the intercept of the first principal axis indicate which particular goal profiles run along the first principal axis. In Fig. 1, responses on the first principal axis capture three distinct goal profiles: (i) a highmastery/moderate-performance goal profile (ii) a moderate-mastery/ moderate-performance goal profile and (iii) a low-mastery/moderateperformance goal profile. The reason for this is that the slope and the intercept of the first principal axis are zero (Edwards & Parry, 1993). Another feature of the first principal axis refers to the slope of the surface that corresponds to the first principal axis. This feature indicates which of the three goal profiles that run along the first principal axis exhibits maximum grades (Edwards & Parry, 1993). In the present study, we expected the slope of the surface that corresponded to the first principal axis to be positive – a finding that will be consistent with our

Note: The dashed line represents the first principal axis. Fig. 1. A hypothetical response-surface supporting combined effects of achievement goals on course grades. Note: The dashed line represents the first principal axis.

hypothesis that the high mastery/moderate performance goal profile is the most optimal. Finally, prior to this analysis the measures of achievement goals were scale-centred by subtracting the midpoint of the scale (Edwards, 1994). In addition, we estimated Cook's D and leverage values to identify potential outliers. However, no individual response exceeded the high cut-off value suggested by Bollen and Jackman (1990, p. 261-267). 4. Results 4.1. Preliminary analysis Table 2 presents descriptive statistics and correlations between psychological variables. Measures of achievement goals were normally distributed because the standard skewness and kurtosis estimates for these variables were close to zero. Correlations supported statistically significant and positive relationships between course grades and mastery goals. The correlation between performance goals and course grades was not statistically significant. In addition, perceptions of competence or measures of ability were positively associated with course grades. 4.2. Main analysis Table 3 presents results from the hierarchical regression analysis that examined main and interactive effects of achievement goals on course grades. In accordance with our expectations, results showed that the additive model did not support additive effects of achievement goals on course grades. The regression analysis did not support additive effects because although mastery goals predicted course grades, the effect of performance goals on course grades was not statistically significant in the first step of the analysis. Likewise, the interactive model did not detect synergistic effects because the interaction between mastery goals and performance goals was not statistically significant in the second step of the analysis. However, in the third step of the analysis, measures of academic ability predicted course grades whereas the effects of perceptions competence on course grades were not statistically significant. Overall, the additive and interactive models explained 7% of the variance in course grades. Table 4 and Fig. 2 present parameters of the quadratic regression equation and the corresponding response surface respectively. An incremental F-test indicated that the quadratic model improved predictive validity of the additive and interactive models by 2% (ΔF = 5.91, p b 0.02) (Edwards & Parry, 1993). Most critical, the quadratic effect of performance goals on course grades was negative and statistically significant, implying a concave relationship between performance goals and course grades. In contrast, although the quadratic effect of mastery goals on course grades was not statistically significant, it was, nevertheless, positive implying a weak convex relationship between mastery goals and course grades. Further, consistent with the hierarchical regression analysis, the quadratic regression analysis supported a positive and statistically significant main effect of mastery goals on course grades. The main and interactive effects of performance goals on course grades were not statistically significant.1 Fig. 2 presents the response surface that corresponded to the quadratic regression model. As shown, the first principal axis of the response surface run almost parallel to the mastery axis. This is supported by the fact that the slope and the intercept of first principal axis were not statistically significantly different from zero (see Table 4). Most critical, in accordance with our hypothesis, the slope of the surface that corresponded to the first principal axis was positive. This 1 In accordance with Edwards' (1994) recommendations we also tested a higher-order model that assumed a cubic relationship between performance-approach goals and course grades. However, an incremental F-test indicated that the cubic model did not explain more variance on academic achievement than the quadratic model (ΔF = 1.46, p N 0.10).

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Table 2 Descriptive statistics and correlations among study variables.

1. Academic ability 2. Mastery goals 3 Performance goals 4. Competence 5. Course grades

M

SD

– 4.04 3.23 3.54 66.81

– 0.49 0.68 1.17 10.77

Standard skewness 0.12 −1.76 −1.8 −2.33

Standard Kurtosis

1

2

3

4

5

0.47 0.88 0.38 0.83

– 0.13 0.10 0.34⁎ 0.48⁎

– 0.19⁎ 0.22⁎ 0.28⁎

– 0.26⁎ 0.12

– 0.26⁎

–

Note. Correlations with an asterisk are statistically significant at p b 0.05 level. Significance tests are two-tailed.

finding supports our hypothesis that the high-mastery/moderate-performance goal profile yielded the highest grades. As an example, a student endorsing mastery goals at a high level and performance goals at a moderate level is predicted to attain a grade of 85.2%. In contrast, a student who endorsed both achievement goals at high levels is expected to attain a grade of 70.8%. Hence, the predicted decline in grades when increasing performance goals from a moderate to high level was 14.4 percentage points. Likewise, a student who endorsed mastery goals at a high level and performance goals at a low level is expected to achieve a grade of 77.6, a predicted decline in academic achievement of 7.6 percentage points. 5. Discussion The purpose of the present study was to compare the utility of the additive, interactive, and quadratic models in detecting combined effects of achievement goals on course grades. In accordance with our initial expectations and previous research, the hierarchical regression analysis did not support additive and interactive effects of achievement goals on course grades. Rather, results from the hierarchical regression analysis supported a positive relationship only between mastery goals and course grades and not between performance goals and course grades. At that juncture, therefore, the additive and interactive models might have led us to reject combined effects altogether and conclude that students who pursued mastery goals were the best-performing students. Contrary to these conclusions, the quadratic model explained more variance on course grades than the additive or interactive models. Most critical, the response surface analysis supported a concave relationship between performance goals and course grades even after we statistically controlled for effects of academic ability and perceptions of competence. In addition, the surface analysis pointed out a combined effect that corresponded to a high-mastery/moderate-performance goal profile. Given these findings, the current study suggests that the additive and interactive models can mislead researchers to reject a combined effect when, in fact, there is an alternative model that supports a combined effect that is consistent with the multiple goal approach. This can happen when researchers do not include quadratic terms alongside main and interactive terms in the regression analysis (see also Cortina, 1993; Lubinski & Humphreys, 1990; Ganzach, 1997). Therefore, researchers who test additive and synergistic effects, through Table 3 Results of a hierarchical regression analysis predicting course grades from mastery and performance achievement goals. Variables

Step 1

Step 2

Step 3

Mastery goals Performance goals Mastery × Performance goals Academic ability Competence Fchange R2adj

5.70⁎ 1.06

5.61⁎ 0.99 1.40

4.36⁎ 0.39 0.15 5.03⁎

7.69⁎ 0.07

0.42 0.07

Note. Parameters are unstandardised regression coefficients. Coefficients with an asterisk are statistically significant at p b 0.05. Significance tests are two-tailed.

0.56 24.31 0.25

additive and interactive models, are advised to also include quadratic terms into regression equations. Apart from demonstrating utility of the quadratic model in detecting combined effects of achievement goals on academic achievement, the present study makes several contributions to the literature. At an empirical level, the current study supports Harackiewicz et al.'s (1998) multiple goal approach because it shows that a goal profile involving tendencies to adopt both mastery goals and performance goals simultaneously was most optimal in terms of yielding the highest levels of academic performance (Barron & Harackiewicz, 2001; Pintrich, 2000). However, the current study adds to achievement goal literature because it identified a unique combined effect that corresponded to a high-mastery/moderate performance goal profile. This goal profile, and the corresponding combined effect, are unique because previous studies could not identify it using the additive and interactive models. Most critical, the results of the current study compare favourably to previous research that observed a non-linear relationship between performance class-climate and academic achievement (Sideridis et al., 2015) or between performance goals and help-seeking behaviour (Sideridis & Stamovlasis, 2015). The present study adds to this growing body of literature by demonstrating that performance goals predict academic achievement with a concave function, similar to the findings for performance class-climate. These findings are essential for theoretical progress in achievement goal research because functional relationships determine which particular goal profile is identified as the most optimal goal profile. For example, in the current study, the response-surface analysis showed that the high-mastery/moderate performance goals profile was the most optimal because observations supported linear and concave functional relationships between achievement goals and grades. If both achievement goals related to academic achievement by concave functions then the surface analysis might have shown that a moderate-mastery/moderate-performance goal profile is the most optimal goal profile (Edwards, 1994). The functional relationships that the different statistical models and techniques impose upon observations is an issue that warrants careful attention in achievement goal research because functional relationships determine recommendations for practice. In the present study, the additive and interactive models would have led us to conclude that the best recommendation to increase academic achievement involves creating environments that facilitate mastery goals. Importantly, we would not make any recommendation for promoting performance goals because the additive and interactive models did not identify a statistically significant effect of performance goals on academic achievement. However, teachers and administrators need to know how to best manage performance goals because competition naturally emerges in achievement contexts (Fryer & Elliot, 2007; Van Yperen et al., 2014). The quadratic model provides evidence for practical recommendations related to performance goals. The statistically significant concave and linear functions observed for performance and mastery goals imply that teachers and administrators should encourage students to assign a marginally lower importance on performance goals than mastery goals. This balance between achievement goals can be facilitated by presenting learning goals as an important means to attain performance goals or by emphasising mastery goals without actively discouraging adoption of performance goals. As Fryer and Elliot (2007) pointed

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Table 4 Results of the first principal axis and quadratic regression analysis predicting student English grades from mastery and performance achievement goals. Quadratic equation b1M 4.44⁎

First principal axis b2P −0.09

b3M2 1.72

b4M × P −0.80

b5P2 −2.79⁎

b6C 0.66

b7A 5.21⁎

b0 69.6

R2adj 0.27⁎

CI95

Slope −0.08 [−0.51, 0.46]

Intercept 0.07 [−0.55, 0.84]

Slope of surface 4.1 [1.3, 8.2]

Note. Regression coefficients are unstandardised regression coefficients. Regression coefficients were estimated using 10,000 bootstrap samples because estimates of slopes of the first principle axis involve non-linear combinations of regression coefficients. The term CI95 indicates the 95% biased corrected confidence intervals for the slope, intercept and the slope of the surface of the first principal axis. Parameters with an asterisk are statistically significant at p b 0.05. Significance tests are two-tailed.

out, by not actively dissuading low-to-moderate levels of performance goals, students may endorse them at moderate levels as a function of the achievement context in which performance goals and competition are emerging as an integral part of the self-regulatory process. Finally, it will be remiss to not mention limitations of the current study that can provide directions for future research. Current findings may not generalise to younger students or schools because the sample of the present study comprised university students. In addition, some of the items in McAuley et al.'s (1989) measure of competence describe moderate levels of competence (i.e., I feel pretty competent in the English course). Hence, it is unclear whether a high score on the competence scale reflects a high or a moderate level of perceived competence. Further, Duda's (1989) measure of goal orientations captures the standards that people are inclined to adopt when they evaluate personal competence. As a consequence, results of the present study may not generalise to Elliot and McGregor's (2001) 2 × 2 model that captures goal adoption (Hulleman, Schrager, Bodman, & Harackiewicz, 2010). In addition, results of the present study do not indicate whether the concave relationship between performance goals and academic achievement is due to performance-approach or performance-avoidance goals. However, the positive correlations between performance goals with perceptions of competence and academic achievement provide some preliminary support to the conclusion that effects observed for performance goals are due to performance-approach rather than performance avoidance goals (see Table 2). This is because approach goals tend to be positively associated with adaptive outcomes such as competence and academic achievement (Elliot & Church, 1997). Despite this, we think that it is important to examine whether the quadratic model detects combined effects when achievement goals and perceptions of competence are measured using instruments that capture approach and avoidance reactions. In the current study we also assume that there is a true midpoint in the measurement scale of performance goals that represents a true psychological state that is function of tendencies to manage anxiety that may arise from being too much

concerned with performance goals, exams and competition. However, we did not examine whether measures of anxiety or students' concerns about exams predicted performance goals. It is also important to keep in mind that an assumption of the multiple goal approach is that the process of goal endorsement is dynamic in that individuals regulate goal endorsement according to task and environmental demands (Barron & Harackiewicz, 2001; Harackiewicz et al., 1998; Senko et al., 2011). Hence, in the present study, we might have found a high-mastery/moderate-performance goal profile to be the most optimal goal profile because we measured achievement goals at the beginning of the semester during which students devote a considerable time on mastering and learning course materials. However, the same students may choose to endorse performance goals at a slightly higher level than mastery goals at the end of the semester when they study for exams or prepare assignments (Fryer & Elliot, 2007; Pintrich, 2000; Senko et al., 2011). As a consequence, it is possible the quadratic model to show that a high-performance/moderate-mastery goal profile is the most optimal goal profile when achievement goals are measured at the end of the semester. Such speculation would also be consistent with Barron and Harackiewicz (2001) selective goal hypothesis that predicts students to focus on different goals at different times within a semester or across different situations. Hence, results of the present study should not be interpreted as evidence that a high-mastery/moderate-performance will always appear to be the most optimal goal profile but that the quadratic model is a viable data analytic technique that assists researchers in detecting combined effects of achievement goals on course grades. In conclusion, the present study demonstrated that the additive and interactive models can lead researchers to reject combined effects of achievement goals on academic performance when, in fact, there is an alternative model that supports combined effects that are consistent with the multiple goal approach. Most critical, we showed that the quadratic model detected combined effects that corresponded to tendencies to endorse mastery goals at high levels and performance goals at moderate levels. Therefore, by applying the quadratic model future research may avoid the problems associated with the interactive and additive models and provide a more complete description of the combined effects that achievement goals exert on academic achievement. Supplementary data Supplementary data to this article can be found online at http://dx. doi.org/10.1016/j.lindif.2016.08.015. References

Note: The dashed line represents Fig. 2. A response-surface capturing effects of achievement goals on course grades. Note: The dashed line represents the first principal axis.

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