Serial versus parallel search: A model comparison approach based on reaction time distributions

Figures & data

Figure 1. Schematic representation of the tasks. From left to right: feature search, conjunction search, spatial configuration search.

Figure 2. Search functions of the three tasks (feature search, conjunction search, spatial configuration search) used in this study. The error bars indicate the standard error.

Figure 3. The graph at the top shows SAIM-WTA’s architecture. All nodes compete via global inhibitory neuron. The time course at the bottom shows an example of SAIM-WTA’s output activation. The line colours correspond to the network’s nodes. The dotted line indicates the threshold for this particular simulation. The simulation result came from the spatial configuration search with six search items and the parameters from the 8th participant (see Appendix I).

Figure 4. An illustrative example of how oKDE decomposes a distribution in several kernels with varying variance. The example also shows the location of the sigma points in this KDE which were relevant for the evaluation of the KDE (see Appendix III for details).

Figure 5. Results of fitting SAIM-WTA. The top-left graph shows the mean log-likelihood ratios (quality of fit) for the different tasks. The remaining graphs show the mean parameters. The error bars indicate the standard error.

Figure 6. SAIM-WTA. KDE-based illustration of RT distributions and response errors. Note that these graphs show the RT distributions for three participants.

Figure 7. Results of fitting CGS. The top-left graph shows the mean log-likelihood ratios (quality of fit) for the different tasks. The remaining graphs show the means of CGS’s seven parameters for the three tasks (Feature Search = FS; Conjunction Search = CS; Spatial configuration search = SC). Note, for the purpose of a better illustration the target saliency parameter was scaled logarithmically. The results replicate Moran et al.’s (2013) findings. The error bars indicate the standard error.

Figure 8. CGS. KDE-based illustration of RT distributions and response errors. Note that these graphs show the RT distributions for three participants.

Figure 9. Comparison of mean log likelihood ratios from SAIM-WTA and CGS for the three tasks (feature search, conjunction search, spatial configuration). The graphs indicate the contributions from the different display size to the overall log likelihood ratios. The error bars indicate the standard error without within-participant variance (Cousineau, 2005). The graphs demonstrate that CGS was better at explaining the data than SAIM-WTA. However, they also show that the performance of both models is best at display size 6 and worse at all other display sizes (see main text for more discussion).

Figure 10. Comparison of BIC scores and AIC scores from SAIM-WTA and CGS for the three tasks (feature search, conjunction search, spatial configuration search). The error bars indicate the standard error without within-participant variance (Cousineau, 2005). BIC and AIC penalize the quality with the model complexity as measured with the number of parameters. The graphs indicate that CGS performed better than SAIM-WTA despite SAIM-WTA being the simpler model.

Table 1. Comparison of BIC scores and AIC scores from SAIM-WTA and CGS for the three tasks (feature search, conjunction search, spatial configuration search) using Wilcoxon sign-rank test.

z-value, p-value	AIC	BIC
Feature	3.62, <0.001	2.29, 0.022
Conjunction	3.82, <0.001	3.58, <0.001
Spatial	3.55, <0.001	2.95, 0.003

Note: All comparisons were significant. For all comparisons CGS showed better results than SAIM-WTA.

Table

Constant parameters	Noise	Sigmoid slope	Sigmoid shift	Inhibitory weight
	0.1	50	0.5	−1

Table

Feature Search Participant	Decision Boundary	Accumulation	Distractor Saliency	Outlier
1	0.5128	0.0019	0.6642	No
2	0.5214	0.0018	0.6537	No
3	0.5020	0.0017	0.6196	No
4	0.5067	0.0018	0.6381	No
5	0.5276	0.0016	0.0635	Yes
6	0.5011	0.0018	0.6672	No
7	0.5035	0.0019	0.6849	No
8	0.5079	0.0019	0.7145	No
9	0.5190	0.0020	0.6636	No
10	0.5297	0.0020	0.6413	No
11	0.5249	0.0016	0.5686	No
12	0.5221	0.0018	0.6756	No
13	0.5068	0.0020	0.6695	No
14	0.5184	0.0019	0.2815	Yes
15	0.5503	0.0018	0.6467	No
16	0.5047	0.0020	0.6612	No
17	0.5412	0.0017	0.6884	No
18	0.5058	0.0021	0.5642	No
19	0.5075	0.0019	0.6968	No

Table

Conjunction Search Participant	Decision Boundary	Accumulation rate	Distractor Saliency	Outlier
1	0.5762	0.0015	0.7192	No
2	0.5709	0.0017	0.7067	No
3	0.5056	0.0015	0.6181	No
4	0.5503	0.0014	0.7780	No
5	0.7373	0.0012	0.6394	Yes
6	0.5228	0.0013	0.6968	No
7	0.5226	0.0014	0.8034	No
8	0.5927	0.0013	0.7208	No
9	0.5177	0.0015	0.6557	No
10	0.8288	0.0043	0.8395	Yes
11	0.5326	0.0013	0.6682	No
12	0.5696	0.0014	0.6703	No
13	0.5260	0.0015	0.6937	No
14	0.5445	0.0016	0.5276	Yes
15	0.6008	0.0014	0.6723	No
16	0.5666	0.0014	0.6761	No
17	0.5499	0.0012	0.6795	No
18	0.5120	0.0015	0.6671	No
19	0.6416	0.0014	0.7351	No

Table

Spatial Configuration Search Participant	Decision Boundary	Accumulation rate	Distractor Saliency	Outlier
1	0.7613	0.0013	0.7352	No
2	0.7843	0.0014	0.7419	No
3	0.6618	0.0013	0.7079	No
4	0.7788	0.0012	0.7797	No
5	0.5340	0.0012	0.7579	No
6	0.7609	0.0011	0.7613	No
7	0.7340	0.0012	0.7292	No
8	0.7389	0.0013	0.7465	No
9	0.7639	0.0011	0.7918	No
10	0.7017	0.0012	0.7877	No
11	0.6516	0.0014	0.7788	No
12	0.7391	0.0010	0.7551	No
13	0.7196	0.0011	0.7632	No
14	0.7221	0.0011	0.7393	No
15	0.6336	0.0011	0.8028	No
16	0.6639	0.0012	0.7321	No
17	0.6368	0.0013	0.7043	No
18	0.7149	0.0014	0.7519	No
19	0.6088	0.0012	0.8059	No
20	0.7118	0.0014	0.7651	No

Table

Feature Search Participant	$w_{\mathrm{target}}$	$\upsilon$	$\theta$	$\mathrm{\Delta }{w}_{\mathrm{quit}}$	$T_{\min }$	$\gamma$	$m$	Outlier
1	509.73	0.9012	0.1936	87.46	0.1750	59.309	0.0679	No
2	601.31	0.9948	0.2353	92.20	0.1632	53.809	0.0671	No
3	427.05	1.3736	0.2190	138.58	0.2381	45.042	0.0586	No
4	414.69	0.8981	0.1497	51.52	0.2202	41.927	0.0627	No
5	703.09	0.9442	0.2183	10.79	0.2367	66.394	0.0030	No
6	519.11	0.8869	0.1715	83.47	0.1964	64.822	0.0687	No
7	470.43	1.0780	0.1787	57.55	0.2049	38.983	0.1109	No
8	603.56	0.7128	0.1240	77.35	0.2003	53.991	0.0978	No
9	541.55	0.9225	0.1748	78.00	0.1883	76.430	0.0726	No
10	522.10	0.7007	0.1310	123.69	0.1514	28.384	0.0871	No
11	374.24	1.1795	0.2349	75.31	0.2422	38.389	0.0187	No
12	313.05	0.7203	0.1452	53.64	0.1854	24.450	0.0690	No
13	566.21	0.6860	0.0867	95.44	0.2251	49.145	0.0855	No
14	486.48	0.7606	0.1221	81.15	0.2149	34.367	0.0306	No
15	383.20	0.6314	0.0997	26.61	0.2311	42.630	0.0433	No
16	539.41	0.8274	0.0964	116.18	0.2310	63.428	0.0852	No
17	323.05	0.7722	0.2064	34.15	0.1694	47.416	0.0596	No
18	651.63	0.7536	0.0746	1.12	0.2386	46.928	0.0399	No
19	482.04	1.0000	0.2288	101.74	0.1539	60.497	0.0657	No
Mean	496.42	0.8813	0.1627	72.93	0.2035	49.281	0.0628

Table

Conjunction Search Participant	$w_{\mathrm{target}}$	$\delta$	$\theta$	$\mathrm{\Delta }{w}_{\mathrm{quit}}$	$T_{min}$	$\gamma$	$m$	Outlier
1	2.6567	0.7215	0.0385	0.0296	0.3235	22.987	0.0670	No
2	3.9355	0.7221	0.0385	0.0062	0.3076	21.287	0.1006	No
3	5.3075	0.4667	0.0243	0.0236	0.3804	38.041	0.0606	No
4	3.8513	0.8845	0.0538	0.0886	0.3268	8.6027	0.2885	Yes
5	1.4917	0.5548	0.0287	0.0047	0.4121	7.3934	0.0148	No
6	5.6134	0.6043	0.0414	0.0093	0.3641	15.702	0.0720	No
7	6.2059	0.8420	0.0540	0.0387	0.3214	13.922	0.2933	Yes
8	3.3363	0.6934	0.0436	0.0094	0.3609	10.769	0.1021	No
9	3.3293	0.6954	0.0367	0.0323	0.3248	19.404	0.0334	No
10	0.9580	0.8742	0.0439	0.0590	0.2298	12.375	0.1097	No
11	4.0060	0.5743	0.0337	0.0028	0.3974	11.466	0.0452	No
12	4.9320	0.6380	0.0422	0.0078	0.3572	14.878	0.0547	No
13	4.4002	0.6305	0.0319	0.0242	0.3430	21.556	0.1323	No
14	3.2589	0.6728	0.0362	0.0048	0.3081	24.146	0.0208	No
15	3.9908	0.5219	0.0323	0.0011	0.3907	20.714	0.0419	No
16	4.9043	0.6137	0.0369	0.0024	0.3598	17.587	0.0550	No
17	4.5750	0.6768	0.0459	0.0222	0.3855	12.596	0.0588	No
18	4.3901	0.7781	0.0400	0.0236	0.3163	18.593	0.0940	No
19	2.6121	0.8141	0.0519	0.0128	0.3034	7.3051	0.1020	No
Mean	3.8818	0.6831	0.0397	0.0212	0.3428	16.807	0.0919

Table

Spatial Configuration Search Participant	$w_{\mathrm{target}}$	$\upsilon$	$\theta$	$\mathrm{\Delta }{w}_{\mathrm{quit}}$	$T_{\min }$	$\gamma$	$m$	Outlier
1	0.7808	0.6625	0.0535	0.0002	0.4152	13.623	0.0143	No
2	0.8427	0.4798	0.0383	0.0016	0.3729	13.779	0.0567	No
3	2.4182	0.5085	0.0385	0.0119	0.4176	17.661	0.0993	No
4	0.5615	0.6705	0.0627	0.0021	0.4086	15.636	0.0821	No
5	2.9063	0.4395	0.0232	0.0132	0.4108	13.377	0.1779	No
6	1.3379	0.5003	0.0644	0.0122	0.4648	17.407	0.0370	No
7	0.8685	0.5045	0.0382	0.0002	0.4322	31.391	0.0369	Yes
8	1.9042	0.6199	0.0474	0.0120	0.3735	7.285	0.1433	No
9	1.1208	0.6009	0.0767	0.0279	0.4161	9.546	0.1216	No
10	2.0736	0.6429	0.0778	0.0436	0.4443	17.560	0.1167	No
11	2.6494	0.6500	0.0492	0.0319	0.3630	9.308	0.1728	No
12	1.5317	0.4810	0.0501	0.0211	0.4816	3.984	0.1087	No
13	1.8925	0.6139	0.0776	0.0288	0.4324	7.596	0.0944	No
14	1.3086	0.5378	0.0489	0.0109	0.4229	9.263	0.0551	No
15	3.2230	0.6341	0.0829	0.0324	0.3912	15.537	0.1257	No
16	2.2699	0.4245	0.0418	0.0304	0.4431	14.322	0.0563	No
17	2.3946	0.6768	0.0406	0.0047	0.3850	14.435	0.0559	No
18	0.8305	0.6198	0.0395	0.0040	0.4052	26.181	0.0837	No
19	4.0731	0.7604	0.0932	0.0554	0.3875	18.036	0.3250	Yes
20	2.2644	0.5925	0.0691	0.0225	0.3915	27.515	0.0630	No
Mean	1.8438	0.5659	0.0537	0.0176	0.4142	13.932	0.0917

Table II.1 Comparison between Moran et al.’s (2013) parameters and our parameters using Wilcoxon sum-rank test.

z-value, p-value	${\mathit{w}}_{\mathrm{target}}$	$\mathit{\upsilon }$	$\mathit{\theta }$	$\mathrm{\Delta }{\mathit{w}}_{\mathrm{quit}}$	${\mathit{T}}_{\min }$	$\mathit{\gamma }$	$\mathit{m}$
Feature	−0.84, 0.403	3.15, 0.002	3.00, 0.003	−3.84, <0.001	−1.13, 0.258	−4.18, <0.001	4.18, <0.001
Conjunction	−1.35, 0.176	3.74, <0.001	3.38, <0.001	−2.59, 0.010	−0.89, 0.371	0.85, 0.396	4.02, <0.001
Spatial	0.87, 0.383	4.22, <0.001	3.32, <0.001	−0.35, 0.724	0.92, 0.358	−4.22, <0.001	4.22, <0.001

Note: Results in bold font indicate a significant difference.

Figure III.1. The graph on the left shows the KDE of the original data with 70% confidence intervals (CI) at the sigma points. The graph on right shows the percentage of KDEs from different dataset sizes falling into the confidence intervals. The results indicate that for a 95% CI, data sets as small as 50 data points produce a 100% accurate representation of the original distribution.

Moran, R., Zehetleitner, M., Mueller, H. J., & Usher, M. (2013). Competitive guided search: Meeting the challenge of benchmark RT distributions. Journal of Vision, 13(8), 24–24. doi: 10.1167/13.8.24

 PubMed Web of Science ®Google Scholar

Cousineau, D. (2005). Confidence intervals in within-subject designs: A simpler solution to Loftus and Masson’s method. Tutorials in Quantitative Methods for Psychology, 1(1), 4–10. doi: 10.20982/tqmp.01.1.p004

 Google Scholar