## 3 Experimental Results
### Statistical Results  
A 3-way ANOVA over the SNR values with the independent factors being Azimuth (\(0^{\circ}\), \(10^{\circ}\), \(20^{\circ}\), \(30^{\circ}\)), Cue (TO, LO, TL) and shift Direction (inward, outward) confirmed the statistical significance of Azimuth [F(3,57) = 254.52, \(p<0.000\), \(\eta^{2}_{partial}=0.931\)], Cue [F(2,38) = 2282.04, \(p<0.000\), \(\eta^{2}_{partial}=0.992\)] and Direction [F(1,19) = 1945.69, \(p<0.000\), \(\eta^{2}_{partial}=0.990\)]. The values confirmed that each of the factors individually affect the performance of detecting shift direction. Interactions among all pairs of factors were also significant as follows. Azimuth x Cue: [F(6,114) = 365.42, \(p<0.000\), \(\eta^{2}_{partial}=0.951\)], Cue x Direction: [F(2,38) = 13.48, \(p<0.001\), \(\eta^{2}_{partial}=0.415\)] and Azimuth x Direction: [F(3,57) = 1011.58, \(p<0.000\), \(\eta^{2}_{partial}=0.982\)]. The differences in all of the tests are confirmed to be statistically significant. The Azimuth x Cue x Direction interaction is also found to be significant [F(6,114) = 5.66, \(p=0.002\), \(\eta^{2}_{partial}=0.230\)]. This shows that each of the pairwise interactions may change significantly depending on the levels of the remaining factors.  
The stimuli used in the study have three types: TO, LO and TL. The sum of the SNR values of the TO and LO conditions is denoted by \(SNR_{T+L}\) and compared to the \(SNR_{TL}\) to see if the cue combination is additive (\(SNR_{TL}=SNR_{T+L}\)), super-additive (\(SNR_{TL}>SNR_{T+L}\)) or sub-additive (\(SNR_{TL}<SNR_{T+L}\)).  
In order to test if \(SNR_{TL}\) and \(SNR_{T+L}\) are significantly different from each other, the error rate data of the five participants are pooled and 20 re-sampled (bootstrapped) data of size 100 samples were drawn. The median, quartile, minimum, and maximum SNR values computed from these error rates for the mentioned combinations are given as boxplots in Figure 3. The SNR values of T+L are calculated (\(SNR_{T+L}=SNR_{TL}+SNR_{L}\)). Then the SNR data used for testing the additivity, super- and sub-additivity of the cues in different azimuths. Based on the data a super-additivity is observed for \(0^{\circ}\) and \(10^{\circ}\). Increasing the azimuth leads to additivity and sub-additivity in rather peripheral azimuths of \(20^{\circ}\) and \(30^{\circ}\). To test the statistical significance of the differences between combinations, a 2-way ANOVA with the factors Combination (T+L and TL) and Azimuth (0, 10, 20, and 30) is applied. Significant effects are noted for Combination [F(1,19) = 5.99, \(p=0.024\), \(\eta^{2}_{partial}=0.240\)] and for Azimuth [F(3,57) = 9.41, \(p<0.000\), \(\eta^{2}_{partial}=0.331\)]. Interaction of Azimuth with Combination also proves to be significant [F(3,57) = 30.56, \(p<0.000\), \(\eta^{2}_{partial}=0.617\)], which indicates that the effect of Combination strongly depends on Azimuth. To study this dependency \(post-hoc\) t-tests are also applied for testing the significance of the differences between \(SNR_{TL}\) and \(SNR_{T+L}\) in the four azimuthal conditions. The differences are found significant, except for \(20^{\circ}\) where T+L and TL profile lines intersect each other, indicating a transition in the type of cue-combination from super-additivity to sub-additivity.