Criterion validity

& Gerbing, 1988). In the first step, I estimated the measurement model using all of the constructs from the research framework in Figure 59. For this model, I kept the correlated error terms from the potential quasi-transformational instructor-leadership dimensions (the terms ‘potential’ and ‘quasi’ were used because respectively the item wordings did not always reflect personal leadership, and two of the three dimensions were not representative of transformational leadership as typically proposed in the organizational behaviour literature). The measurement model was estimated for the total sample using the maximum likelihood bootstrapping procedure because this procedure assessed the stability of the parameter estimates (Byrne, 2001). The number of bootstraps was set to 500 and the bias-corrected (BC) confidence interval was set to 95%. With the number of observed variables exceeding 30 and the sample size exceeding 250, Hair et al. (2009) suggested that significant p-values were to be expected; CFI should be above 0.90; and RMSEA should be less than .07. Based on these thresholds, all fit indices indicated good model fit (𝜒2_{⁄ = 3.42, RMR = 0.05, CFI = .94, RMSEA = 0.035, GFI = .95, AGFI = 𝑑𝑓}

.94). ). 𝜒2 _{was 1,533.58 (df = 449) and significant at p < 0.001.}

9 _{For the single-item measures of age, gender, and student achievement, their loading and error terms were specified}

(Hair, Black, Babin, & Anderson, 2009). No measurement error was assumed, and the loadings and error terms were set to one and zero respectively. This approach was similar to multiple regression because it assumed no

For the second step, I transformed the measurement model into a structural model to test H2 to H9 (results shown in Figure 6). Like the measurement model, 500 bootstraps were used at a 95% BC confidence interval. The structural model showed very good fit (𝜒2_{⁄ = 3.4, RMR = 𝑑𝑓}

0.05, CFI = 0.94, RMSEA = 0.035, GFI = 0.95, AGFI = 0.94). 𝜒2 _{was 1,591.27 (df = 468) and}

Figure 6. Structural model of the relationships between potential quasi-transformational instructor-leadership dimensions, facets of student engagement, and student achievement. Standardized maximum likelihood parameter estimates. Indicator variables and error variances excluded for ease in readability. Grey arrows used to indicate relationships to control variables.

Table 9

Parameter Estimates for the Structural Model of Potential Quasi-Transformational Instructor- Leadership Dimensions, Facets of Student Engagement, and Student Achievement (Study 1) Structural

relationship

Unstandardized parameter estimate

Standard error Critical ratio Standardized parameter estimate H2: IS  IE 0.61*** 0.06 9.60 0.41 H3: IS  DAe 0.29*** 0.03 11.33 0.41 H4: Co  IE 0.35*** 0.07 4.76 0.20 H5: Co  SCS 0.44*** 0.03 12.70 0.45 H6: DC  IE 0.35*** 0.07 5.09 0.20 H7: IE  SA 0.51*** 0.04 13.27 0.36 H8: SCS  SA -0.04 0.06 -0.70 -0.02 H9: DAe  SA 0.54*** 0.09 6.11 0.18 DAl  DAe 0.37*** 0.03 13.14 0.54 GN  SA -0.21*** 0.06 -3.65 -0.07 AG  SA -0.00 0.01 -0.29 -0.01

Note. IS = intellectual stimulation; DC = direction and congruence; Co = consideration; IE = interest and enjoyment; SCS = student collegial support; DAe = deep approach in module; SA = student achievement; DAl = deep approach orientation; AG = age; GN = gender.

*** p < .001.

Table 9 shows the standardized and unstandardized structural path estimates for each hypothesis and control variable. H8 was the only unsupported hypothesis, i.e., the relationship between student collegial support and student achievement was not significant. In addition, the relationship between gender and student achievement was negative. Even though this

relationship was not expected, women have been shown to underperform in some universities in England and Wales (McNabb et al., 2002). McNabb et al. (2002) ruled out differences in

academic ability, male prejudice, subject specific effects, and institution-specific factors as potential causes of female academic underperformance. They suggested that the reason for women underperformance might be more biological or psychological, whereby men tend to take greater risks compared to women who preferred to take a safer route. Alternatively, the gender

difference in self-ratings of academic performance could be due to differences in modesty between genders. In a review of self-other rating agreement in leadership, Fleenor, Smither, Atwater, Braddy, and Sturm (2010) explain that, in general, men overestimate their abilities and effectiveness more than women. Therefore, men tend to inflate their self-ratings of performance, while women tend to be more accurate in their estimation of performance because they give self- ratings that are more aligned with other-ratings (Fleenor et al., 2010).

To check the mediating effects in the structural model, the first step was to establish significant relationships between the constructs. This analysis was conducted by checking (a) the direct unmediated relationship, (b) the relationship between the mediator and the ‘input’

construct, and (c) the relationship between the mediator and the outcome construct (see Table 10) (Hair et al., 2009). The findings in Table 10 showed that all of the relationships were statistically significant.

Table 10

Assessment of the Level of Mediation Through the Mediated Model (Study 1) (a) Direct unmediated relationship Cov(X,Y) (b) Relationship between mediator and “input” Cov(X,Y) (c) Relationship between mediator and outcome Cov(X,Y) IS ↔ SA 0.34*** IS ↔ IE IS ↔ DAe 0.48*** 0.21*** IE ↔ SA DAe ↔ SA 0.63*** 0.24*** C ↔ SA 0.24*** C ↔ IE C ↔ SCS 0.37*** 0.15*** IE ↔ SA SCS ↔ SA 0.63*** 0.10*** DC ↔ SA -0.26*** DC ↔ IE 0.35*** IE ↔ SA 0.24***

Note. IS = intellectual stimulation; DC = direction and congruence; Co = consideration; IE = interest and enjoyment; SCS = student collegial support; DAe = deep approach in module; SA = student achievement.

Table 11

Model Comparisons of the Effect of Adding Direct Unmediated Relationships (Study 1) Direct

unmediated relationship

χ2 _{df CFI ∆χ}2_{/∆df Bootstrap two-}

tailed significance estimate (BC) Baseline model 1,591.27 468 .94 - - IS  SA 1,591.23 467 .94 0.04/1 -0.02 Co  SA 1,590.87 467 .94 0.40/1 -0.06 DC  SA 1,588.37 467 .94 2.9/1 0.14

Note. IS = intellectual stimulation; Co = consideration; DC = direction and congruence; SA = student achievement.

Having established the relationships between the constructs, the next step was to assess the level of mediation through the mediated model. To do so, each of the direct unmediated relationships in the structural model was added separately in a series of models (see Table 11). Both the ∆χ2 _{test and the bootstrap two-tailed significance (bias-corrected) indicated that none of}

the direct unmediated relationships significantly improved the model. This meant that the

engagement facets in the model were full mediators between the potential quasi-transformational instructor-leadership dimensions and student achievement.

The magnitude of the mediating effects between the potential quasi-transformational instructor-leadership dimensions, facets of student engagement, and student achievement were decomposed into direct and indirect effects as shown in Table 12. As expected, each student engagement facet was directly associated with student achievement (although the strength of the association between student collegial support and student achievement was extremely weak). Also, the three potential quasi-transformational leadership dimensions were all indirectly associated with student achievement, indicating full mediation as mentioned before.

Table 12

Mediation Between Potential Quasi-Transformational Instructor-Leadership Dimensions; Facets of Student Engagement; And Student Achievement Decomposed Into Direct and Indirect Effects (Study 1)

Constructs in relation to SA Direct effect Indirect effect Total effect

Intellectual stimulation 0.00 0.23 0.23

Consideration 0.00 0.07 0.07

Direction and congruence 0.00 0.07 0.07

Interest and enjoyment 0.36 0.00 0.36

Deep approach 0.18 0.00 0.18

Student collegial support -0.02 0.00 -0.02

Note. SA = student achievement. Values in the table represent standardized effects.

most strongly related to student achievement via interest and enjoyment and deep approach to learning. This may be an indication that intellectual stimulation is a relatively more impactful leader behaviour in the HEI module context.