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Rmd	bd4637a	knowlabUnimelb	2025-07-29	Update analysis code to use shared file repo
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Ruby Steinberg¹, Ami Eidels², and Daniel R. Little¹

¹ The University of Melbourne, ² The University of Newcastle

Method

Participants

Design

Data Cleaning

Subjects completed the experiment by clicking a link with the uniquely generated id code. Subjects were able to use the link multiple times; further, subjects were able to exit the experiment at any time. Consequently, the datafile contains partially completed data for some subjects which needed to be identified and removed.

Data Analysis

We first summarize performance by answering the following questions:

Task completions

How many tasks are completed on average?

RDK performance

We next analysed performance on the RDK discriminations. We then asked:

What was the average completion time and accuracy of the easy, medium, hard, and very hard tasks?

Version	Author	Date
2e6ecdf	knowlabUnimelb	2022-11-09

We further broke down RTs by condition, deadline, and difficulty.

Reward Rate

	name[none]	ss[none]	df[none]	ms[none]	F[none]	p[none]	partEta[none]
“Phase”	Phase	59.039	1	59.039	80.539	0.000	0.514
[“Phase”,“condition”]	Phase:condition	0.035	1	0.035	0.048	0.827	0.001
[“Phase”,“condition”,“.RES”]	Residual	55.712	76	0.733	NA	NA	NA
“Difficulty”	Difficulty	35.793	3	11.931	52.289	0.000	0.408
[“Difficulty”,“condition”]	Difficulty:condition	0.754	3	0.251	1.101	0.350	0.014
[“Difficulty”,“condition”,“.RES”]	Residual	52.023	228	0.228	NA	NA	NA
[“Phase”,“Difficulty”]	Phase:Difficulty	2.653	3	0.884	5.139	0.002	0.063
[“Phase”,“Difficulty”,“condition”]	Phase:Difficulty:condition	0.812	3	0.271	1.572	0.197	0.020
[“Phase”,“Difficulty”,“condition”,“.RES”]	Residual	39.238	228	0.172	NA	NA	NA

	name	ss	df	ms	F	p	partEta
“condition”	condition	0.196	1	0.196	0.229	0.633	0.003
“Residual”	Residual	65.024	76	0.856	NA	NA	NA

Optimality in each condition

What is the proportion of easy, medium, hard, and very hard tasks selected first, second, third or fourth?

Do the marginal distributions differ from uniformity?

We tested whether the marginal distributions were different from uniformally random selection using the fact that the mean rank is distributed according to a \(\chi^2\) distribution with the following test-statistic: \[\chi^2 = \frac{12N}{k(k+1)}\sum_{j=1}^k \left(m_j - \frac{k+1}{2} \right)^2\] see (Marden, 1995).

Chi2 test of uniformity
condition	phase	chi2	df	p
fixed_moving_shortdotlife	untimed	9.75	3	0.01
fixed_moving_shortdotlife	deadline	25.51	3	0.00
random_moving_shortdotlife	untimed	7.33	3	0.03
random_moving_shortdotlife	deadline	9.32	3	0.01

We compared the location conditions and phases using chi-2 analysis.

How optimal were responses?

Stability of selections

Selection Choice RTs

Mean RTs for each scheduling selection
condition	phase	mrt_sel1	mrt_sel2	mrt_sel3	mrt_sel4
fixed_moving_shortdotlife	deadline	690.29	719.05	696.63	570.98
fixed_moving_shortdotlife	untimed	1578.95	1285.66	1218.24	1236.19
random_moving_shortdotlife	deadline	790.40	825.50	736.52	559.88
random_moving_shortdotlife	untimed	1776.43	1438.74	1252.29	1276.20

REPEATED MEASURES ANOVA

Within Subjects Effects
─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Sum of Squares df Mean Square F p η²-p
─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Phase 7.429918e+7 1 7.429918e+7 231.0252920 < .0000001 0.7524634
Phase:condition 96868.91 1 96868.91 0.3012034 0.5847378 0.0039476
Residual 2.444208e+7 76 321606.28
Selection 8166318.31 3 2722106.10 30.8536450 < .0000001 0.2887468
Selection:condition 538616.71 3 179538.90 2.0349793 0.1097859 0.0260778
Residual 2.011562e+7 228 88226.40
Phase:Selection 3987223.62 3 1329074.54 17.0963411 < .0000001 0.1836414
Phase:Selection:condition 72319.05 3 24106.35 0.3100882 0.8180852 0.0040635
Residual 1.772479e+7 228 77740.29
─────────────────────────────────────────────────────────────────────────────────────────────────────────────── Note. Type 3 Sums of Squares

Between Subjects Effects
────────────────────────────────────────────────────────────────────────────────────────── Sum of Squares df Mean Square F p η²-p
────────────────────────────────────────────────────────────────────────────────────────── condition 1130408 1 1130408.5 2.113166 0.1501540 0.0270526
Residual 4.065514e+7 76 534936.0
────────────────────────────────────────────────────────────────────────────────────────── Note. Type 3 Sums of Squares

ASSUMPTIONS

Tests of Sphericity
─────────────────────────────────────────────────────────────────────────────────────────── Mauchly’s W p Greenhouse-Geisser ε Huynh-Feldt ε
─────────────────────────────────────────────────────────────────────────────────────────── Phase 1.0000000 NaN ᵃ 1.0000000 1.0000000
Selection 0.3617133 < .0000001 0.6039606 0.6178657
Phase:Selection 0.5244773 < .0000001 0.7022065 0.7227066
─────────────────────────────────────────────────────────────────────────────────────────── ᵃ The repeated measures has only two levels. The assumption of sphericity is always met when the repeated measures has only two levels.

Homogeneity of Variances Test (Levene’s)
───────────────────────────────────────────────────────── F df1 df2 p
───────────────────────────────────────────────────────── rt1_untimed 1.41914480 1 76 0.2372513
rt2_untimed 0.97399516 1 76 0.3268174
rt3_untimed 0.84632363 1 76 0.3605053
rt4_untimed 0.06239663 1 76 0.8034212
rt1_deadline 4.97280100 1 76 0.0287010
rt2_deadline 0.79821624 1 76 0.3744477
rt3_deadline 0.24128684 1 76 0.6246938
rt4_deadline 0.04943231 1 76 0.8246505
─────────────────────────────────────────────────────────

Selection model

We can treat each task selection as a probabilistic choice given by a Luce’s choice rule (Luce, 1959), where each task is represented by some strength, \(\nu\). The probability of selecting task \(i_j\) from set \(S = \{i_1, i_2, ..., i_J \}\), where J is the number of tasks, is:

\[p\left(i_j |S \right) = \frac{\nu_{i_j}}{\sum_{i \in S} \nu_{i}}. \]

Plackett (1975) generalised this model to explain the distribution over a sequence of choices (i.e., ranks). In this case, after each choice, the choice set is reduce by one (i.e., sampling without replacement). This probability of observing a specific selection order, \(i_1 \succ ... \succ i_J\) is:

\[p\left(i_j |A \right) = \prod_{j=1}^J \frac{\nu_{i_j}}{\sum_{i \in A_j} \nu_{i}}, \]

where \(A_j\) is the current choice set.

sessionInfo()

R version 4.3.1 (2023-06-16 ucrt)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 19045)

Matrix products: default


locale:
[1] LC_COLLATE=English_Australia.utf8  LC_CTYPE=English_Australia.utf8   
[3] LC_MONETARY=English_Australia.utf8 LC_NUMERIC=C                      
[5] LC_TIME=English_Australia.utf8    

time zone: Australia/Sydney
tzcode source: internal

attached base packages:
[1] stats4    grid      stats     graphics  grDevices utils     datasets 
[8] methods   base     

other attached packages:
 [1] statmod_1.5.0     betareg_3.2-0     jmv_2.4.9         pmr_1.2.5.1      
 [5] jpeg_0.1-10       rstatix_0.7.2     lmerTest_3.1-3    lme4_1.1-34      
 [9] Matrix_1.6-1.1    png_0.1-8         reshape2_1.4.4    knitr_1.44       
[13] english_1.2-6     gtools_3.9.4      DescTools_0.99.50 lubridate_1.9.3  
[17] forcats_1.0.0     stringr_1.5.0     dplyr_1.1.3       purrr_1.0.2      
[21] readr_2.1.4       tidyr_1.3.0       tibble_3.2.1      ggplot2_3.4.3    
[25] tidyverse_2.0.0   workflowr_1.7.1  

loaded via a namespace (and not attached):
 [1] gld_2.6.6           sandwich_3.0-2      readxl_1.4.3       
 [4] rlang_1.1.1         magrittr_2.0.3      multcomp_1.4-25    
 [7] git2r_0.32.0        e1071_1.7-13        compiler_4.3.1     
[10] flexmix_2.3-19      getPass_0.2-2       callr_3.7.3        
[13] vctrs_0.6.3         pkgconfig_2.0.3     fastmap_1.1.1      
[16] backports_1.4.1     labeling_0.4.3      utf8_1.2.3         
[19] promises_1.2.1      rmarkdown_2.25      tzdb_0.4.0         
[22] ps_1.7.5            nloptr_2.0.3        modeltools_0.2-23  
[25] xfun_0.40           cachem_1.0.8        jsonlite_1.8.7     
[28] later_1.3.1         afex_1.3-0          parallel_4.3.1     
[31] broom_1.0.5         R6_2.5.1            RColorBrewer_1.1-3 
[34] bslib_0.5.1         stringi_1.7.12      car_3.1-2          
[37] boot_1.3-28.1       estimability_1.4.1  lmtest_0.9-40      
[40] jquerylib_0.1.4     cellranger_1.1.0    numDeriv_2016.8-1.1
[43] Rcpp_1.0.11         zoo_1.8-12          base64enc_0.1-3    
[46] nnet_7.3-19         httpuv_1.6.11       splines_4.3.1      
[49] timechange_0.2.0    tidyselect_1.2.0    rstudioapi_0.15.0  
[52] abind_1.4-5         yaml_2.3.7          codetools_0.2-19   
[55] processx_3.8.2      lattice_0.21-8      plyr_1.8.9         
[58] withr_2.5.1         coda_0.19-4         evaluate_0.22      
[61] survival_3.5-5      proxy_0.4-27        pillar_1.9.0       
[64] carData_3.0-5       whisker_0.4.1       generics_0.1.3     
[67] rprojroot_2.0.3     hms_1.1.3           munsell_0.5.0      
[70] scales_1.2.1        rootSolve_1.8.2.4   minqa_1.2.6        
[73] xtable_1.8-4        jmvcore_2.4.7       class_7.3-22       
[76] glue_1.6.2          emmeans_1.8.8       lmom_3.0           
[79] tools_4.3.1         data.table_1.14.8   Exact_3.2          
[82] fs_1.6.3            mvtnorm_1.2-3       colorspace_2.1-0   
[85] nlme_3.1-162        Formula_1.2-5       cli_3.6.1          
[88] fansi_1.0.4         expm_0.999-7        gtable_0.3.4       
[91] sass_0.4.7          digest_0.6.33       TH.data_1.1-2      
[94] farver_2.1.1        htmltools_0.5.6     lifecycle_1.0.3    
[97] httr_1.4.7          MASS_7.3-60

Experiment 3c: Dynamic RDK Direction Judgement, Short Dot Life, 4 Tasks, 500 msec Error Penalty

knowlabUnimelb

2021-06-16