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

Daniel R. Little¹, Ami Eidels², and Deborah J. Lin¹

¹ The University of Melbourne, ² The University of Newcastle

Method

Participants

We tested 49 participants (37 F, 9 M, 3 Undeclared). Participants were recruited through the Melbourne School of Psychological Sciences Research Experience Pool (Mean age = 20.44, range = 18 - 40). Participants were reimbursed with credit toward completion of a first year psychology subject. Datasets from 0 subjects were excluded for completing the experiment twice; i.e., only the first of the datasets for these subjects was retained.

Forty-nine were assigned to the Fixed Difficulty condition. In this condition, the location of the difficulty of the random dot kinematograms (RDK’s) was held constant across trials.

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 eight RDKs labelled from Very Easy to Very Hard. The labels corresponded to the coherence of the RDK; that is, the proportion of dots moving in a coherent direction was set to 80% to 10% in steps of 10%. From the set of eight RDKs, participants selected and completed one RDK at a time in any order. 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 resampled direction (so the direction may have changed), 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 eight RDK’s. Next, participants completed 30 trials with a 10 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 = 3) 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 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?

Average number of correctly completed tasks in each condition
condition	phase	mean
fixed	untimed	7.675381
fixed	deadline	7.210870

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?

RTs became shorter and more accurate as the difficulty of the RDK became easier. As expected, the RTs were shorter under a deadline than without a deadline. We visualised the response times in two ways: First, we simply took the average of each attempt on each RDK.

Second, we computed the time to complete an RDK as the cumulative sum across multiple attempts within a trial (termed Cumulative RT or cRT). That is, if an error is made and the RDK needs to be repeated, then the total RT is the sum of both attempts.

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	1	727	0.91	0.01	770.93	38.17	845.93	41.08
fixed	untimed	2	606	0.89	0.01	744.43	23.25	839.78	29.95
fixed	untimed	3	569	0.89	0.01	724.05	20.27	817.84	26.21
fixed	untimed	4	502	0.87	0.01	799.16	43.34	913.07	50.46
fixed	untimed	5	505	0.88	0.01	801.39	25.69	912.24	35.54
fixed	untimed	6	497	0.78	0.02	923.72	44.77	1186.45	61.30
fixed	untimed	7	502	0.73	0.02	970.36	41.66	1336.45	68.13
fixed	untimed	8	485	0.60	0.02	1068.34	47.91	1785.47	95.16
fixed	deadline	1	1761	0.92	0.01	532.96	4.28	576.90	5.82
fixed	deadline	2	1628	0.91	0.01	525.25	3.79	574.23	5.93
fixed	deadline	3	1507	0.91	0.01	542.01	4.56	595.54	6.29
fixed	deadline	4	1406	0.90	0.01	549.03	4.84	605.80	7.67
fixed	deadline	5	1430	0.87	0.01	565.69	5.61	642.98	8.53
fixed	deadline	6	1402	0.81	0.01	595.45	7.09	727.78	12.22
fixed	deadline	7	1442	0.71	0.01	620.58	7.98	840.51	15.32
fixed	deadline	8	1418	0.67	0.01	651.11	8.99	939.07	18.15

Reward Rate

Optimality in each condition

Having now established that the RDK’s are ordered in accuracy, difficulty, and reward rate, it is clear that the task set presented to each subject has an optimal solution, ordered from easiest to most difficult. We now ask:

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

We first compute the marginal distribution of the ranks of each of the tasks; in other words, what are the proportions of the ranks of each task in each rank position. These matrices indicate the proportions of responses for each difficulty level which were chosen first, second, third, or fourth, respectively. The matrix from a dataset in which choice is always optimal would have ones on the diagonal and zeros on the off-diagonal.

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	254.11	3	0
fixed	deadline	1338.19	3	0

It is evident at a glance that the ordering of choices is more optimal when the locations are fixed; that is, the proportions on the diagonal are higher. When the locations are fixed, choice order becomes more optimal under a deadline. By contrast, when locations are random, responding becomes less optimal under a deadline. This likely reflects the additional costs of having to search for the appropriate task to complete. This search is minimised in the fixed location condition.

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

Pearson’s chi-squared test
	Comparison	chi2	df	p
X-squared	Fixed: Untimed vs Deadline	154.23	63	0

How optimal were responses?

The next analysis computed the distance between the selected order and the optimal order (easiest to very hard for that trial), which ranges between 0 (perfect match) and 6 (maximally distant), for 4 options.

What we want is the distance of the selected options from the optimal solutions, which is the edit distance (or number of discordant pairs) between orders. However, because a participant may run out of time, there may be missing values. To handle these values, for each trial, we find the orders which partially match the selected order and compute three the average distance of those possible orders and the optimal solution (avg_distance).

The following figure compares the avg_distance between the fixed difficulty and random difficulty conditions as a function of deadline condition and phase. For each of these measures, lower values reflect respones which are closer to optimal.

# 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.

Selection Choice RTs

Mean RTs for each scheduling selection
condition	phase	mrt_sel1	mrt_sel2	mrt_sel3	mrt_sel4	mrt_sel5	mrt_sel6	mrt_sel7	mrt_sel8
fixed	untimed	1531	905	798	769	737	689	682	730
fixed	deadline	671	470	478	500	515	503	497	490

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 4: RDK Direction Judgement, 8 Tasks, 500 msec Error Penalty

knowlabUnimelb

2020-10-29