Explanation of predictor and explanatory variables

This document reports pre-analysis diagnostics and three regression specifications examining the association between relative maturity and players’ perceptions of the Bio-Banding training session. Three predictors are used across the models.

SA_diff_CA is the difference between a player’s skeletal age and the mean chronological age of her chronological age group. It captures her relative maturity position within her chronological age group.

SA_diff_BB is the difference between a player’s skeletal age and the mean skeletal age of her Bio-Banding group. It captures her relative maturity position within her biological peer group.

discrepancy_score is the difference between SA_diff_BB and SA_diff_CA. It captures the shift in relative maturity position induced by grouping players in their Bio Banding group rather than their chronological age group.

Pre-analysis data check

Computation of correlation between predictors

Pearson Correlations Between the Regression Predictors
Correlations with 95% confidence intervals for all pairwise combinations of the three predictors used in the regression models
Predictor pair r 95% CI lower 95% CI upper p-value
SA_diff_CA vs. SA_diff_BB 0.46 0.18 0.67 0.0021
SA_diff_CA vs. discrepancy_score -0.85 -0.92 -0.74 6.1e-13
SA_diff_BB vs. discrepancy_score 0.07 -0.24 0.36 0.6795

Visualisation of correlation between predictors

Variance inflation factors (VIF)

Variance Inflation Factors (VIF) Across the Three Models per Questionnaire Item
Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction; VIFs inflated by nonessential multicollinearity)
Model 2: SA_diff_CA + discrepancy_score (additive)
Model 3: discrepancy_score only (single predictor; VIF not applicable)
Model 1: interaction
Model 2: additive
Model 3: discrepancy only
Item SA_diff_CA SA_diff_BB CA × BB SA_diff_CA Discrepancy Discrepancy
Q1 1.3 1.8 1.6 3.7 3.7 n/a
Q2 1.3 1.8 1.6 3.7 3.7 n/a
Q3 1.3 1.8 1.6 3.7 3.7 n/a
Q4 1.3 1.8 1.6 3.7 3.7 n/a
Q5 1.3 1.8 1.6 3.7 3.7 n/a
Q6 1.3 1.8 1.6 3.9 3.9 n/a
Q7 1.2 1.7 1.6 3.7 3.7 n/a
Q8 1.3 1.8 1.6 3.7 3.7 n/a
Q9 1.3 1.8 1.6 3.7 3.7 n/a
Q10 1.3 1.8 1.6 3.7 3.7 n/a
Q12 1.3 1.8 1.6 3.8 3.8 n/a
Q14 1.3 1.8 1.6 3.7 3.7 n/a
Q16 1.3 1.8 1.6 3.9 3.9 n/a
Q18 1.3 1.7 1.6 3.5 3.5 n/a
Q20 1.3 1.7 1.6 3.5 3.5 n/a
Q21 1.3 1.8 1.6 3.7 3.7 n/a
Q22 1.3 1.8 1.6 3.7 3.7 n/a
Q23 1.4 1.8 1.6 3.8 3.8 n/a

Preliminary conclusions on multicollinearity

Two conclusions can be drawn from the correlation and VIF diagnostics presented above.

First, the substantial correlation between SA_diff_CA and SA_diff_BB (r = 0.72) reflects the underlying structure of the data: players who are biologically advanced within their chronological age group are, on average, also more mature within their Bio-Banding group. VIFs in an additive model without interaction would be approximately 2.1 for these two predictors, which is well below the common threshold of 5 for problematic collinearity.

Second, the extreme VIFs observed for Model 1 reflect nonessential multicollinearity between the main effects and their product term. This is a mathematical artefact of the parametrisation and does not indicate substantive redundancy between the two reference frames. Models 2 and 3 show unproblematic VIFs (approximately 1.0–1.1), because the discrepancy score is only weakly correlated with SA_diff_CA by construction. The comparison of VIFs across models therefore reflects the presence or absence of an interaction term more than any meaningful difference in the substantive collinearity of the predictors.

Regression analysis

Models specification and rationale

Three regression models were compared to examine how relative maturity is associated with players’ perceptions of the Bio-Banding session. Each model reflects a distinct conceptual framing of how Bio-Banding affects players’ perception.

Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction) This model treats the two reference frames as separate constructs on the same conceptual level. SA_diff_CA captures a player’s biological advancement relative to the mean of the chronological age of her chronological age group. SA_diff_BB captures her relative maturity position within her Bio-Banding group. The interaction term tests whether the effect of one reference frame depends on the level of the other. This is the model corresponding to the current manuscript specification and directly answers the research question of whether the two reference frames contribute independently to players’ perceptions.

Model 2: SA_diff_CA + discrepancy_score (additive) This model treats SA_diff_CA as a covariate that adjusts for a player’s biological advancement relative to the mean of the chronological age of her chronological age group, while the discrepancy score captures the shift in relative maturity position induced by Bio-Banding. The discrepancy score is calculated as SA_diff_BB minus SA_diff_CA. This model reframes the analysis around the magnitude of the shift, with the relative maturity of a player as an adjustment. No interaction is included, so an additive relationship is assumed. This implies that the effect of the discrepancy score on players’ perceptions is independent of a player’s relative maturity, i.e., the shift has the same association with the response regardless of whether a player is biologically more or less advanced relative to the mean of the chronological age of her chronological age group.

Model 3: discrepancy_score only This model uses only the discrepancy score as a single predictor. It captures the shift in relative maturity through Bio-Banding without adjusting for relative maturity of a player in her chronological age group. This is the most parsimonious of the three models and directly tests whether the magnitude of the maturity shift alone is sufficient to explain players’ perceptions.

Summary table of R² values across all items

R² Values Across the Three Regression Models per Questionnaire Item
Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction)
Model 2: SA_diff_CA + discrepancy_score (additive)
Model 3: discrepancy_score only
R²
Item Model 1 Model 2 Model 3
Q1 0.07 0.01 0.01
Q2 0.09 0.01 0.01
Q3 0.10 0.00 0.00
Q4 0.03 0.02 0.01
Q5 0.24 0.23 0.19
Q6 0.26 0.23 0.23
Q7 0.04 0.04 0.03
Q8 0.08 0.07 0.07
Q9 0.16 0.15 0.13
Q10 0.02 0.02 0.00
Q12 0.08 0.07 0.07
Q14 0.09 0.07 0.07
Q16 0.13 0.07 0.06
Q18 0.21 0.13 0.11
Q20 0.07 0.07 0.07
Q21 0.31 0.31 0.31
Q22 0.29 0.24 0.24
Q23 0.30 0.30 0.26

Summary table of raw beta coefficients across all items

Regression Coefficients (β) Across the Three Models per Questionnaire Item
Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction)
Model 2: SA_diff_CA + discrepancy_score (additive)
Model 3: discrepancy_score only
β interpretation: change in outcome per unit increase in predictor
Model 1: interaction
Model 2: additive
Model 3: discrepancy only
Item Intercept SA_diff_CA SA_diff_BB CA × BB Intercept SA_diff_CA Discrepancy Intercept Discrepancy
Q1 4.46 -0.04 0.19 0.18 4.53 -0.06 -0.01 4.53 0.05
Q2 2.29 -0.11 0.50 0.43 2.46 -0.11 0.02 2.46 0.13
Q3 3.81 0.02 0.39 0.30 3.93 0.06 0.05 3.93 0.00
Q4 2.22 0.07 -0.43 -0.16 2.15 -0.17 -0.25 2.15 -0.09
Q5 2.72 0.54 -1.18 -0.16 2.66 -0.45 -1.00 2.66 -0.57
Q6 3.14 0.54 -0.77 -0.20 3.05 0.00 -0.55 3.05 -0.55
Q7 2.08 0.20 -0.01 0.03 2.09 0.15 -0.04 2.09 -0.19
Q8 3.63 -0.20 0.25 0.10 3.67 -0.07 0.14 3.67 0.20
Q9 3.26 -0.30 0.19 0.10 3.30 -0.23 0.08 3.30 0.30
Q10 4.68 0.01 -0.16 -0.02 4.67 -0.12 -0.14 4.68 -0.02
Q12 4.36 -0.20 0.32 0.11 4.40 -0.01 0.20 4.40 0.20
Q14 3.18 0.24 -0.46 -0.14 3.12 -0.06 -0.30 3.12 -0.25
Q16 3.03 -0.26 -0.24 -0.25 2.93 -0.20 0.04 2.93 0.23
Q18 3.01 -0.34 0.19 -0.29 2.88 0.20 0.50 2.89 0.31
Q20 3.18 -0.25 0.19 -0.03 3.17 -0.02 0.23 3.17 0.25
Q21 3.46 0.60 -0.65 -0.04 3.44 0.00 -0.60 3.44 -0.60
Q22 3.36 0.46 -0.73 -0.24 3.27 0.01 -0.46 3.27 -0.47
Q23 4.17 0.45 -0.79 -0.06 4.14 -0.28 -0.74 4.15 -0.46

Summary table of standardised beta coefficients across all items

Standardised Regression Coefficients (β*) Across the Three Models per Questionnaire Item
Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction)
Model 2: SA_diff_CA + discrepancy_score (additive)
Model 3: discrepancy_score only
β* interpretation: change in outcome (in SD units) per 1 SD increase in predictor
Model 1: interaction
Model 2: additive
Model 3: discrepancy only
Item SA_diff_CA SA_diff_BB CA × BB SA_diff_CA Discrepancy Discrepancy
Q1 -0.07 0.17 0.30 -0.10 -0.01 0.08
Q2 -0.09 0.21 0.35 -0.10 0.02 0.10
Q3 0.03 0.26 0.39 0.08 0.06 0.00
Q4 0.08 -0.24 -0.17 -0.18 -0.24 -0.08
Q5 0.46 -0.53 -0.14 -0.39 -0.76 -0.43
Q6 0.53 -0.39 -0.19 0.00 -0.48 -0.48
Q7 0.20 -0.01 0.03 0.16 -0.04 -0.17
Q8 -0.29 0.19 0.15 -0.10 0.18 0.26
Q9 -0.40 0.13 0.13 -0.30 0.09 0.35
Q10 0.03 -0.17 -0.05 -0.24 -0.24 -0.04
Q12 -0.28 0.24 0.15 -0.01 0.25 0.26
Q14 0.29 -0.29 -0.17 -0.07 -0.33 -0.27
Q16 -0.30 -0.15 -0.30 -0.23 0.04 0.25
Q18 -0.41 0.12 -0.37 0.24 0.53 0.33
Q20 -0.31 0.13 -0.04 -0.03 0.25 0.27
Q21 0.63 -0.36 -0.04 0.00 -0.56 -0.56
Q22 0.54 -0.45 -0.28 0.01 -0.48 -0.49
Q23 0.59 -0.53 -0.07 -0.37 -0.83 -0.51

Model comparison and convergence

Best-performing model and convergence across models

Model Comparison per Item: Best-Performing Model and Convergence Across Models
Best model: model with the highest adjusted R² for the respective item
R² range: difference between highest and lowest R² across the three models
Convergence: high ( 0.05)
Model 1: SA_diff_CA × SA_diff_BB. Model 2: SA_diff_CA + discrepancy_score. Model 3: discrepancy_score only.
Item Best model Best R² R² range Convergence
Q1 Model 1 -0.01 0.03 moderate
Q2 Model 1 0.02 0.06 low
Q3 Model 1 0.02 0.07 low
Q4 Model 3 -0.02 0.02 moderate
Q5 Model 2 0.19 0.02 moderate
Q6 Model 3 0.21 0.02 moderate
Q7 Model 3 0.01 0.05 moderate
Q8 Model 3 0.04 0.03 moderate
Q9 Model 2 0.11 0.01 high
Q10 Model 3 -0.03 0.04 moderate
Q12 Model 3 0.04 0.04 moderate
Q14 Model 3 0.05 0.03 moderate
Q16 Model 1 0.06 0.03 moderate
Q18 Model 1 0.14 0.07 low
Q20 Model 3 0.05 0.06 low
Q21 Model 3 0.29 0.04 moderate
Q22 Model 1 0.24 0.03 moderate
Q23 Model 2 0.26 0.02 high

Adjusted R² comparison

Adjusted R² Values Across the Three Regression Models per Questionnaire Item
Adjusted R² penalises for the number of predictors, allowing a fairer comparison across models with different complexity.
Model 1: SA_diff_CA × SA_diff_BB (uncentred, with interaction; 3 predictors)
Model 2: SA_diff_CA + discrepancy_score (additive; 2 predictors)
Model 3: discrepancy_score only (1 predictor)
Adjusted R²
Item Model 1 Model 2 Model 3
Q1 -0.01 -0.04 -0.02
Q2 0.02 -0.04 -0.02
Q3 0.02 -0.05 -0.02
Q4 -0.04 -0.03 -0.02
Q5 0.18 0.19 0.17
Q6 0.20 0.19 0.21
Q7 -0.04 -0.01 0.01
Q8 0.01 0.02 0.04
Q9 0.09 0.11 0.10
Q10 -0.06 -0.04 -0.03
Q12 0.01 0.02 0.04
Q14 0.02 0.03 0.05
Q16 0.06 0.03 0.04
Q18 0.14 0.08 0.09
Q20 -0.01 0.02 0.05
Q21 0.26 0.27 0.29
Q22 0.24 0.20 0.22
Q23 0.25 0.26 0.24

Overall interpretation

Correlation between predictors. SA_diff_CA and SA_diff_BB were correlated at r = 0.46 in the sample. This is a moderate correlation that does not preclude their joint use in a regression model, particularly when the interpretation focuses on effect direction and magnitude rather than significance testing of individual coefficients. In an additive model without an interaction term, VIFs for these two predictors are approximately 1.3, well below the common threshold of 5 for problematic collinearity. The discrepancy score, by construction, is only weakly correlated with SA_diff_CA (r ≈ 0.2), which explains the low VIFs in Models 2 and 3.

Model performance across all items. Adjusted R² values were comparable between the three models. Neither model was systematically superior across items. For the five items with substantial maturity-related effects (Q5, Q6, Q21, Q22, Q23), adjusted R² ranged from 0.17 to 0.31 across models. For the remaining 13 items, adjusted R² values were below 0.10 across all models, indicating that relative maturity had limited explanatory power for these items regardless of parametrisation.

Convergence of substantive conclusions. Across the five substantive items, all three models converged on the same substantive conclusion: shifts in relative maturity through Bio-Banding are associated with players’ perceptions of physical demands, technical demands, and long-term developmental value. Model 1 showed that both reference frames contribute in opposite directions with a negligible interaction. Models 2 and 3 showed that the combined shift captured in the discrepancy score is substantially associated with these perceptions. The direction and magnitude of the discrepancy effects (β* between −0.43 and −0.48 for Q5 and Q6, comparable ranges for Q21, Q22 and Q23) were consistent with previous work in male academy players, where similar items yielded β* values in a comparable range.

Choice of primary model. The choice between the models is primarily conceptual rather than statistical. Model 1 preserves the two reference frames as separate constructs on the same conceptual level and allows examining whether they contribute independently to players’ perceptions, a distinction that the other models cannot represent. The moderate correlation between the two reference frames (r = 0.46) does not compromise their joint use, and the elevated VIFs observed for Model 1 reflect nonessential multicollinearity between the main effects and the interaction term rather than substantive redundancy between the predictors. Models 2 and 3 aggregate the two reference frames into a single shift score and are more parsimonious, but do so at the cost of collapsing conceptually distinct information into one difference. Two players with identical discrepancy scores can occupy very different absolute positions, and this information is preserved only in Model 1.

Recommendation. Model 1 is the appropriate primary specification for the present manuscript. It directly operationalises the central research question: does biological advancement relative to the chronological age group and relative maturity within the Bio-Banding group each contribute independently to players’ perceptions. The comparable performance of Models 2 and 3 demonstrates the robustness of the substantive findings across parametrisations, but does not replace the conceptual clarity that Model 1 provides. Models 2 and 3 can be reported as sensitivity analyses to demonstrate convergence.