Hierarchical Regression Analysis: Statistical Analysis

Hierarchical multiple regression was used to analyse the relationship with

expertise, culture and gaze as predictors and interpersonal style, agency or communion as outcome variables. Expertise and culture were included in the regression analysis because they are central to this thesis. As such, they were generally expected to play a role in how teachers gaze and, in turn, how gaze relates to interpersonal style. Hierarchical multiple regression was selected rather than multiple regression because notable changes were seen when expertise and culture were not controlled for via hierarchical multiple regression— that is, when gaze variables predicted teacher interpersonal style on their own. The hierarchical multiple regression model was therefore run three times: once with

interpersonal style as the outcome variable, then agency, finally with communion as the outcome variable.

Student age was run as a covariate because the two cultural groups differed notably in age. Any ‘cultural’ effect may therefore be conflated with the age differences between the two cultural samples. Specifically, the Hong Kong sample consisted of students aged M=13.65 (12-16) years whereas the UK students were M=12.00 (11-14) years old, which was a significant age difference between the two cultural groups, F(1,39) = 14.59, p < .001, 𝜂_𝑝2_{=.27. UK teachers may thus have been rated more highly than Hong Kong} teachers because UK students were younger than the Hong Kong students. Indeed,

although the variance accounted for by student characteristics is typically smaller (Levy & Wubbels, 1992), students’ ratings of teacher interpersonal style—that is each, agency and communion—do change with student age. In view of this, student age was explored as a covariate (cf. Den Brok, Levy, Wubbels & Rodriguez, 2003; Den Brok, van Tartwijk,

Wubbels & Veldman, 2010), to ensure that any cultural effect would be genuine effect on each, attentional and communicative TIS, agency and communion, rather than being conflated with the sample differences in age. To do this, the same hierarchical regression models run with student age as a covariate, which is accomplished by adding student age as a factor before all predictors of interest.

In all, my hierarchical regression model always consisted of four stages. Stage 1 involved only student age, the present covariate. Stage 2 analysis involved only expertise as the predictor; in Stage 3, culture was added; Stage 4, the expertise × culture interaction term was added; in Stage 5, all the gaze variables were added that could be used from the present thesis. Specifically, measures from the scanpath analysis could not be included in the interpersonal gaze analysis, since the similarity scores can only relate gaze strings to each other and are inapplicable in relation to anything else. Stage 5 was then refined until the best-fitting model was identified, in which only the relevant gaze predictors are

included in the regression analysis. For this model refinement process, the 𝑅̅2 _{value was} examined until it ceased to improved (i.e., stayed the same) or started declining. For model refinement, the (standardised) 𝛽 coefficients of the gaze variables were also examined: the strongest were identified, the moderately strong listed, and the weakest were identified and dispensed from the subsequent model. When the 𝑅̅2 _stopped

improving, the most recent decision to dispense gaze predictors was retracted so that the preceding model is chosen. In this way, Stage 5 was the key part of the hierarchical multiple regression model by which the relevant gaze predictors from this whole thesis were identified. Outcomes reported below are derived from the best-fitting (i.e., optimal) model.

The advantages to reporting standardised regression outcomes (i.e., standardised beta coefficients) are well established (e.g., Pianta et al., 2014; Reeve & Lee, 2014). Yet

SPSS, the package used in the present analysis, does not make available standard error values that are associated with standardised beta coefficients, only the unstandardised B. To obtain standard errors that were associated with the standardised beta coefficient, I obtained a z-score for every variable so that the ‘unstandardised’ beta coefficients and associated standard error values could now be read as standardised values instead: these are presently reported as standardised beta coefficients and associated standard errors.

In summary, all the variables that will be analysed and reported in the Results section are in Table 7.10 below.

Table 7.10

List of all the variables that underwent statistical analysis

Analysis Measure Type Variable Description

Frequency

Proportion 1 Student gaze Fixations on students (i.e., ≥ 4 key frames) 2 Student material Gaze towards student learning materials 3 Teacher material Gaze towards teacher materials

4 Other Non-student and non-instructional gaze targets Temporal

Duration per visit 5 Student gaze Student fixations and scans

6 Non-student gaze Student materials, teacher materials, other

Attractor 7 Rate of efficient gaze How much an efficient gaze is used; attractor quantity 8 Strength of efficient

gaze

How strong the efficient gaze is; attractor strength Transition entropy 9 Gaze flexibility Rate of gaze shift between students and non-students

Dispersion 10 Strategic

(in)consistency

How little gaze moves across the whole ‘state space’ Scanpath

Single-IV SED 11 Within-expertise Comparisons without culture controlled for 12 Within-culture Comparisons without expertise controlled for 13 Across-expertise Comparisons without culture controlled for 14 Across-culture Comparisons without expertise controlled for Dual-IV SED 15 Within-expertise Comparisons with culture controlled for

16 Within-culture Comparisons with expertise controlled for 17 Across-expertise Comparisons with culture controlled for 18 Across-culture Comparisons with expertise controlled for

19 Within sub-group Comparisons within expertise + culture groupings 20 Across sub-group Comparisons across expertise + culture groupings HMR

Stage 1 21 Class size Number of students taught by each teacher

Stage 2 22 Expertise Teacher expertise category: expert or novice

Stage 3 23 Culture Teacher culture category: Hong Kong or UK

Stage 4 24 Expertise × Culture Interaction term between expertise and culture Stage 5 - (Gaze variables) Variables from frequency and temporal analyses

Note. All these variables were analysed for attentional gaze and for communicative gaze. This means that, 24 variables for each gaze type: in total, there were 48 variables. HMR = Hierarchical multiple regression.