Table 1. Result from fitted Y^* response function for TEST 1 analysis (first row), and from the different eye movement parameters (LMM analysis), for TEST 2.

Parameters	Model	T		Z		T*Z		AIC		R2		Random Slope T factor		Random Slope Z factor
		βT	PT †	βZ	PZ †	γZ	Pγ †	AIClmm	AIC lm	Marg.	Cond.	PS1	corr	PS2	corr
Y*	§	5.54	***	0.02	0.68			632.9	765.9	0.06	0.49
N° Saccades	†	3.02	***	0.34	***			1170.8	1283.9	0.23	0.28
Δφ [°]	†††	8.69	***	1.71	***			1833.8	2024.7	0.19	0.83	***	-0.98
LTs [s]	††	-0.03	0.12	-0.02	**	0.04	**	-3080.4	-2769.5	0.02	0.46
CUSPn	†	0.37	***	0.07	***			-737.6	-667.5	0.14	0.28
SPn	††††	0.50	***	-0.06	***			-1021.7	-866.8	0.28	0.40			***	-0.75
NTP [s]	‡	-0.17	**	-0.06	*	0.09	*	-1902.1	-1902.8	0.04
Gain	†††	0.32	**	0.01	*			-832.79	-504.7	0.02	0.80	*	-0.95
TAU [s]	†††	-0.01	0.41	0.00	0.76			-2841.8	-2754.9	0.00	0.66	**	-0.92
ΔHn θ	†	0.13	0.62	0.1	0.01			1279.8	1287.8	0.004	0.1
ΔHn φ	†††	0.61	0.012	0.12	***			-238.37	-3.98	0.05	0.78	***	-0.92

The second column reports the model selected for the specific parameter under analysis (see below for the symbol legend). The subsequent columns report the regression coefficients (β) and p-values (p) of the fixed factors (flight time T, ball arrival height Z) and their interaction term (when significant). The sixth column reports the AIC values computed including the random factor (AIC_lmm) and without it (AIC_lm). If AIC_lmm < AIC_lm the inclusion of the random factor (i.e. subject) is justified, indicating that the particular eye movement parameter varies across subjects. The seventh column reports the marginal and the condition R² coefficient of the regression. Finally the two rightmost columns show the significance of the by-subject adjustment of the slope relatively to both the T factor (p_s1), and the Z factor (p_s2), and the correlation between the two random parameters (intercept and slope for each factor).

*: p_value <0.05;

**: p_value<0.01;

***: p_value<0.001.

^§ (GLMM)Model type Y_ij* = β₀ + β_Tt_j + β_zz_j + ε_ij + μ_i;

^† (LMM) Model type (LMM) v_ij = β₀ + β_Tt_j + β_zz_j + S_0i +ε_ij;

^†† (LMM) Model type: v_ij = β₀ + β_Tt_j + β_zz_j + γt_jz_j + S_0i +ε_ij;

^††† (LMM) Model type: v_ij = β₀ + (β_{T +} S_li)t_{j +} β_zz_j + S_0i + ε_ij;

^†††† (LMM) Model type: v_ij = β₀ + β_Tt_j + (β_{z +} S_2i)z_j + S_0i +ε_ij;

^‡ (LM) Model type: v_ij = β₀ + β_Tt_j + β_zz_j + γt_jz_j + ε_ij