Experiment 3a: Dynamic RDK Direction Judgement, 4 Tasks, 500 msec Error Penalty
knowlabUnimelb
2021-04-27
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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
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.
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 | 58.657 | 1 | 58.657 | 55.725 | 0.000 | 0.385 |
| [“Phase”,“condition”] | Phase:condition | 1.869 | 1 | 1.869 | 1.775 | 0.186 | 0.020 |
| [“Phase”,“condition”,“.RES”] | Residual | 93.682 | 89 | 1.053 | NA | NA | NA |
| “Difficulty” | Difficulty | 127.041 | 3 | 42.347 | 35.433 | 0.000 | 0.285 |
| [“Difficulty”,“condition”] | Difficulty:condition | 8.578 | 3 | 2.859 | 2.392 | 0.069 | 0.026 |
| [“Difficulty”,“condition”,“.RES”] | Residual | 319.098 | 267 | 1.195 | NA | NA | NA |
| [“Phase”,“Difficulty”] | Phase:Difficulty | 2.717 | 3 | 0.906 | 1.860 | 0.137 | 0.020 |
| [“Phase”,“Difficulty”,“condition”] | Phase:Difficulty:condition | 1.267 | 3 | 0.422 | 0.868 | 0.458 | 0.010 |
| [“Phase”,“Difficulty”,“condition”,“.RES”] | Residual | 130.016 | 267 | 0.487 | NA | NA | NA |
| name | ss | df | ms | F | p | partEta | |
|---|---|---|---|---|---|---|---|
| “condition” | condition | 17.934 | 1 | 17.934 | 5.66 | 0.019 | 0.06 |
| “Residual” | Residual | 281.987 | 89 | 3.168 | 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).
| condition | phase | chi2 | df | p |
|---|---|---|---|---|
| fixed_moving_delay | untimed | 6.86 | 3 | 0.03 |
| fixed_moving_delay | deadline | 34.72 | 3 | 0.00 |
| random_moving_delay | untimed | 0.97 | 3 | 0.24 |
| random_moving_delay | deadline | 28.15 | 3 | 0.00 |
We compared the location conditions and phases using chi-2 analysis.
- How optimal were responses?
Stability of selections
Selection Choice RTs
| condition | phase | mrt_sel1 | mrt_sel2 | mrt_sel3 | mrt_sel4 |
|---|---|---|---|---|---|
| fixed_moving_delay | deadline | 769.29 | 773.83 | 736.46 | 641.89 |
| fixed_moving_delay | untimed | 1573.27 | 1446.25 | 1349.13 | 1277.06 |
| random_moving_delay | deadline | 789.28 | 781.54 | 683.74 | 571.73 |
| random_moving_delay | untimed | 1442.40 | 1267.05 | 1257.45 | 1207.59 |
REPEATED MEASURES ANOVA
Within Subjects Effects
───────────────────────────────────────────────────────────────────────────────────────────────────────────────
Sum of Squares df Mean Square F p η²-p
───────────────────────────────────────────────────────────────────────────────────────────────────────────────
Phase 6.601957e+7 1 6.601957e+7 262.1775758 < .0000001
0.7508431
Phase:condition 41878.49 1 41878.49 0.1663083 0.6844157 0.0019079
Residual 2.190768e+7 87 251812.41
Selection 3776174.89 3 1258724.96 17.0959933 < .0000001
0.1642330
Selection:condition 174865.57 3 58288.52 0.7916743 0.4994688
0.0090176
Residual 1.921662e+7 261 73626.90
Phase:Selection 584662.78 3 194887.59 3.1791846 0.0245836
0.0352541
Phase:Selection:condition 139754.53 3 46584.84 0.7599346 0.5174733
0.0086592
Residual 1.599959e+7 261 61301.13
───────────────────────────────────────────────────────────────────────────────────────────────────────────────
Note. Type 3 Sums of Squares
Between Subjects Effects
───────────────────────────────────────────────────────────────────────────────────────────
Sum of Squares df Mean Square F p η²-p
───────────────────────────────────────────────────────────────────────────────────────────
condition 313541.1 1 313541.1 0.7246950 0.3969453 0.0082610
Residual 3.764077e+7 87 432652.6
───────────────────────────────────────────────────────────────────────────────────────────
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 0.5904253 < .0000001 0.7717982 0.7941416
Phase:Selection 0.7021994 0.0000129 0.8113917 0.8364585
───────────────────────────────────────────────────────────────────────────────────────────
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
2.778678761 1 87 0.0991232
rt2_untimed 0.762235497 1 87 0.3850332
rt3_untimed 0.764902744 1 87 0.3842066
rt4_untimed 0.093919061 1 87 0.7599850
rt1_deadline 6.964069540 1 87 0.0098533
rt2_deadline 1.304233640 1 87 0.2565748
rt3_deadline 1.922184888 1 87 0.1691587
rt4_deadline 0.006713374 1 87 0.9348865
──────────────────────────────────────────────────────────
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.
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