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File	Version	Author	Date	Message
Rmd	67e1aac	knowlabUnimelb	2022-11-09	Publish data and analysis files

Ruby Steinberg¹, Ami Eidels², and Daniel R. Little¹

¹ The University of Melbourne, ² The University of Newcastle

Method

Participants

We tested 97 participants (70 F, 23 M, 4 Undeclared). Participants were recruited through the Melbourne School of Psychological Sciences Research Experience Pool (Mean age = 19.17, range = 17 - 33). Participants were reimbursed with credit toward completion of a first-year psychology subject.

Fifty were assigned to the Fixed Difficulty condition. In this condition, the location of easy, medium, hard, and very hard random dot kinematograms (RDK’s) was held constant across trials.

Forty-seven were assigned to the Random Difficulty condition. In this condition, the location of easy, medium, hard, and very hard random dot kinematograms (RDK’s) were randomized from trial to trial.

The Fixed Difficulty experiment was completed before the Random Difficulty experiment. Participants only completed one of these.

Design

In each condition, participants completed multiple trials in which they selected and completed RDK tasks. On each trial, participants were shown a set of four RDKs that were dynamically moving on the selection screen at a coherence of 80%, 50%, 20%, or 0% for the Easy, Medium, Hard, and Very Hard locations, respectively. From the set of four RDKs, participants selected and completed one RDK at a time in any order. Once selected, the direction of the RDK was randomly resampled and the unselected tasks stopped moving. On successive selections only the subtasks remaining in the set continued to move. The goal of each trial was to complete as many as possible before a deadline. If an incorrect RDK response was made, that RDK was restarted at the same coherence but with a new randomly sampled direction after a 500 msec penalty, and the participant had to respond to the RDK again. A new task could not be selected until the RDK was completed successfully.

Participants first completed 10 trials with a long (30 sec) deadline to help participants learn the task, explore strategies, and allow for comparison to a short-deadline condition. We term this the no deadline condition since the provided time is well beyond what is necessary to complete all four RDK’s. Next, participants completed 30 trials with a 6 second deadline.

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.

A handful of subjects (N = 5) had less than chance accuracy on the easiest RDK indicating equipment problems or a misunderstanding of task directions. We removed these participants from further anlaysis leaving 46 and 46 in the fixed and random location conditions, respectively.

Data Analysis

We first summarize performance by answering the following questions:

Task completions

How many tasks are completed on average?

Across both conditions, participants completed 3.93 tasks during the phase and NaN tasks during the phase.

Average number of correctly completed tasks in each condition
condition	phase	mean
fixed	untimed	3.894855
fixed	deadline	3.177177
random	untimed	3.956332
random	deadline	3.201471

[ADD DESCRIPTION OF ANALYSIS OF TASK COMPLETIONS]

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?

[ADD DESCRIPTION OF PLOT]

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

Mean accuracy, RT, and RT summed across attempts for each difficulty and each phase
condition	phase	difficulty	n	Mean.Correct	SE.Correct	Mean.RT	SE.RT	Mean.crt	SE.crt
fixed	untimed	easy	802	0.85	0.01	727.31	24.93	855.17	30.01
fixed	untimed	medium	606	0.79	0.02	893.77	41.57	1134.00	54.22
fixed	untimed	hard	557	0.72	0.02	1174.31	59.25	1635.93	86.14
fixed	untimed	v. hard	515	0.54	0.02	1568.54	65.92	2885.25	144.05
fixed	deadline	easy	1548	0.84	0.01	439.70	8.12	512.29	11.73
fixed	deadline	medium	1452	0.80	0.01	503.17	9.51	613.88	13.86
fixed	deadline	hard	1425	0.71	0.01	543.58	10.52	719.48	16.60
fixed	deadline	v. hard	1369	0.59	0.01	580.98	11.11	834.28	21.23
random	untimed	easy	878	0.86	0.01	774.27	20.01	905.30	24.27
random	untimed	medium	589	0.85	0.01	912.73	33.67	1069.20	42.58
random	untimed	hard	533	0.77	0.02	1231.94	45.84	1598.27	76.57
random	untimed	v. hard	532	0.51	0.02	1718.35	64.96	3323.15	145.75
random	deadline	easy	1902	0.84	0.01	461.53	5.62	537.47	6.92
random	deadline	medium	1531	0.83	0.01	484.40	6.57	572.08	9.14
random	deadline	hard	1357	0.69	0.01	585.52	9.19	805.42	16.69
random	deadline	v. hard	1373	0.53	0.01	649.50	10.40	1090.33	22.82

Reward Rate

[ADD DESCRIPTION]

Optimality in each condition

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

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	untimed	20.13	3	0.00
fixed	deadline	988.81	3	0.00
random	untimed	1.85	3	0.22
random	deadline	80.09	3	0.00

[ADD DESCRIPTION]

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

Pearson’s chi-squared test
Comparison	chi2	df	p
Fixed: Untimed vs Deadline	270.17	15	0.00
Random: Untimed vs Deadline	30.90	15	0.01
Untimed: Fixed vs Random	86.08	15	0.00
Deadline: Fixed vs Random	442.80	15	0.00

How optimal were responses?

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 \(S = \left{i_1, i_2, ..., i_J \right}\), 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.

Sampling distribution anlaysis

In order to characterise performance, we examined three sampling distributions for comparison to our data. The first is the sampling distribution of edit distances from optimal assuming that orders are sampled uniformly at random. The second distribution assumes that the first choice was optimal but the remaining orders are sampled at random. Finally, the third distributions assumes that the first two choices are selected optimally but that the remaining are randomly selected. It is clear that the mode of the distribution moves from a distance of 3 to a distance of 0 as the sampling distribution summarises orders which better conform to optimality.

To characterise the optimality of each condition at each point in the experiment, we first computed the ks-test statistic between the data (the average partial distance data) and the random order distribution and the first-two optimal distribution. Since smaller ks-statistics indicate a closer match between the distributions, we then took the ratio of the ks-statistics (random over first two-optimal). Values less than one indicate that the data are more consistent with random than optimal responding. Values greater than one indicate that the data are more consistent with optimal rather than random responding.

This figure efficiently summarises the main result: responding is more optimal in the fixed deadline condition particularly during the last ten trials; in the random deadline conditions, responding was closer to a random sampling distribution than to an optimal sampling distribution.

Selection Choice RTs

Mean RTs for each scheduling selection
condition	phase	mrt_sel1	mrt_sel2	mrt_sel3	mrt_sel4
fixed	untimed	1580	1446	1355	1307
fixed	deadline	786	794	782	754
random	untimed	1442	1267	1269	1216
random	deadline	792	793	740	678

SD RTs for each scheduling selection
condition	phase	sd_sel1	sd_sel2	sd_sel3	sd_sel4
fixed	untimed	19.451724	14.869030	17.077954	15.709740
fixed	deadline	6.923495	6.428960	4.978323	4.450818
random	untimed	24.397078	12.561198	16.962563	15.429503
random	deadline	7.418030	5.950909	4.747539	3.511201

sessionInfo()

R version 4.1.3 (2022-03-10)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 10 x64 (build 19042)

Matrix products: default

locale:
[1] LC_COLLATE=English_Australia.1252  LC_CTYPE=English_Australia.1252   
[3] LC_MONETARY=English_Australia.1252 LC_NUMERIC=C                      
[5] LC_TIME=English_Australia.1252    

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

other attached packages:
 [1] pmr_1.2.5.1       jpeg_0.1-9        rstatix_0.7.0     lme4_1.1-29      
 [5] Matrix_1.4-0      png_0.1-7         reshape2_1.4.4    knitr_1.38       
 [9] hrbrthemes_0.8.0  english_1.2-6     gtools_3.9.2      DescTools_0.99.45
[13] forcats_0.5.1     stringr_1.4.0     dplyr_1.0.8       purrr_0.3.4      
[17] readr_2.1.2       tidyr_1.2.0       tibble_3.1.6      ggplot2_3.3.5    
[21] tidyverse_1.3.1   workflowr_1.7.0  

loaded via a namespace (and not attached):
 [1] minqa_1.2.4       colorspace_2.0-3  ellipsis_0.3.2    class_7.3-20     
 [5] rprojroot_2.0.3   fs_1.5.2          gld_2.6.5         rstudioapi_0.13  
 [9] proxy_0.4-27      farver_2.1.0      fansi_1.0.3       mvtnorm_1.1-3    
[13] lubridate_1.8.0   xml2_1.3.3        splines_4.1.3     extrafont_0.18   
[17] rootSolve_1.8.2.3 jsonlite_1.8.0    nloptr_2.0.0      broom_0.8.0      
[21] Rttf2pt1_1.3.10   dbplyr_2.1.1      compiler_4.1.3    httr_1.4.2       
[25] backports_1.4.1   assertthat_0.2.1  fastmap_1.1.0     cli_3.2.0        
[29] later_1.3.0       htmltools_0.5.2   tools_4.1.3       gtable_0.3.0     
[33] glue_1.6.2        lmom_2.9          Rcpp_1.0.8.3      carData_3.0-5    
[37] cellranger_1.1.0  jquerylib_0.1.4   vctrs_0.4.1       nlme_3.1-155     
[41] extrafontdb_1.0   xfun_0.30         ps_1.6.0          rvest_1.0.2      
[45] lifecycle_1.0.1   getPass_0.2-2     MASS_7.3-55       scales_1.2.0     
[49] hms_1.1.1         promises_1.2.0.1  expm_0.999-6      yaml_2.3.5       
[53] Exact_3.1         gdtools_0.2.4     sass_0.4.1        stringi_1.7.6    
[57] highr_0.9         e1071_1.7-11      boot_1.3-28       rlang_1.0.2      
[61] pkgconfig_2.0.3   systemfonts_1.0.4 evaluate_0.15     lattice_0.20-45  
[65] labeling_0.4.2    processx_3.5.3    tidyselect_1.1.2  plyr_1.8.7       
[69] magrittr_2.0.3    R6_2.5.1          generics_0.1.2    DBI_1.1.2        
[73] pillar_1.7.0      haven_2.5.0       whisker_0.4       withr_2.5.0      
[77] abind_1.4-5       modelr_0.1.8      crayon_1.5.1      car_3.0-12       
[81] utf8_1.2.2        tzdb_0.3.0        rmarkdown_2.13    readxl_1.4.0     
[85] data.table_1.14.2 callr_3.7.0       git2r_0.30.1      reprex_2.0.1     
[89] digest_0.6.29     httpuv_1.6.5      munsell_0.5.0     bslib_0.3.1

Experiment 3a: Dynamic RDK Direction Judgement, 4 Tasks, 500 msec Error Penalty

knowlabUnimelb

2021-04-27