## Diffusion modelling

### Drift rate

### Boundary separation

### Non-decision time

## Diffusion modelling of \(n-2\) task repetition costs

## Interim summary

## Episodic retrieval contributions to measures of inhibition

## The current study

## General method

### Participants

### Apparatus and stimuli

### Procedure

#### Design

### Individual experiment details

#### Mayr replication

#### Working memory

#### Healthy aging

#### New

## Results

### Data preparation

### Behavioural analysis

DV | Source | df | F | p | \(\eta _G^2\) |
---|---|---|---|---|---|

Response time | Sequence (S) | (1, 185) | 133.55 | < 0.001 | 0.02 |

Response (R) | (1, 185) | 44.91 | < 0.001 | < 0.01 | |

Experiment (E) | (3, 185) | 6.30 | < 0.001 | 0.09 | |

S \(\times \) R | (1, 185) | 83.31 | < 0.001 | 0.01 | |

S \(\times \) E | (3, 185) | 0.80 | 0.50 | < 0.01 | |

R \(\times \) E | (3, 185) | 1.82 | 0.15 | < 0.01 | |

S \(\times \) R \(\times \) E | (3, 185) | 2.42 | 0.07 | < 0.01 | |

Error | Sequence (S) | (1, 185) | 14.38 | < 0.001 | 0.01 |

Response (R) | (1, 185) | 7.71 | < 0.01 | < 0.01 | |

Experiment (E) | (3, 185) | 9.97 | < 0.001 | 0.09 | |

S \(\times \) R | (1, 185) | 61.84 | < 0.001 | 0.03 | |

S \(\times \) E | (3, 185) | 1.62 | 0.19 | < 0.01 | |

R \(\times \) E | (3, 185) | 0.58 | 0.63 | < 0.01 | |

S \(\times \) R \(\times \) E | (3, 185) | 4.52 | < 0.01 | < 0.01 |

#### Overview of Bayesian multilevel modelling

^{1}For the RT analysis, a linear Gaussian model was used for the distribution of the dependent variable, whereas a beta response distribution (i.e. a beta regression) was used for the error analysis (because proportion error falls within the range [0, 1]).

#### Response times

Dependent variable | Model | WAIC | SE | dWAIC | Weight |
---|---|---|---|---|---|

Response time | Interaction (S \(\times \) R) | − 1730 | 54 | 0 | 1 |

Main effects (S + R) | − 1606 | 54 | 124 | 0 | |

Sequence (S) | − 1573 | 53 | 157 | 0 | |

Response (R) | − 1528 | 45 | 202 | 0 | |

Error | Interaction (S \(\times \) R) | − 4073 | 51 | 0 | 1 |

Main effects (S + R) | − 4011 | 52 | 62 | 0 | |

Response (R) | − 4001 | 51 | 72 | 0 | |

Sequence (S) | − 3973 | 50 | 100 | 0 |

Dependent variable | Source | Estimate | Error | L-95% CI | U-95% CI |
---|---|---|---|---|---|

Response time | Intercept | 1.11 | 0.02 | 1.07 | 1.14 |

Sequence (CBA) | − 0.02 | 0.01 | − 0.03 | 0.00 | |

Response repetition (switch) | 0.07 | 0.01 | 0.06 | 0.09 | |

Interaction | − 0.09 | 0.01 | − 0.11 | − 0.07 | |

Error rates | Intercept | − 3.82 | 0.08 | − 3.98 | − 3.67 |

Sequence (CBA) | 0.12 | 0.07 | − 0.02 | 0.26 | |

Response repetition (switch) | 0.60 | 0.07 | 0.46 | 0.74 | |

Interaction | − 0.56 | 0.09 | − 0.74 | − 0.39 |

#### Error rates

### Diffusion modelling

#### Model parameterisation and fit procedure

Study | ABA repetition | ABA switch | CBA repetition | CBA switch |
---|---|---|---|---|

Aging | 53 | 160 | 54 | 157 |

Mayr | 55 | 167 | 55 | 168 |

WM | 58 | 168 | 56 | 165 |

New | 113 | 333 | 112 | 328 |

#### Goodness of fit assessment

#### Inferential analysis

Parameter | Source | df | F | p | \(\eta _G^2\) |
---|---|---|---|---|---|

Drift | Sequence (S) | (1, 185) | 39.39 | < 0.001 | 0.02 |

Response (R) | (1, 185) | 27.96 | < 0.001 | 0.01 | |

Experiment (E) | (3, 185) | 0.82 | 0.49 | 0.01 | |

S \(\times \) R | (1, 185) | 50.68 | < 0.001 | 0.01 | |

S \(\times \) E | (3, 185) | 1.90 | 0.13 | < 0.01 | |

R \(\times \) E | (3, 185) | 0.61 | 0.61 | < 0.01 | |

S \(\times \) R \(\times \) E | (3, 185) | 1.98 | 0.12 | < 0.01 | |

Boundary | Sequence (S) | (1, 185) | 25.70 | < 0.001 | 0.01 |

Response (R) | (1, 185) | 2.21 | 0.14 | < 0.01 | |

Experiment (E) | (3, 185) | 14.93 | < 0.001 | 0.15 | |

S \(\times \) R | (1, 185) | 0.42 | 0.52 | < 0.01 | |

S \(\times \) E | (3, 185) | 0.37 | 0.77 | < 0.01 | |

R \(\times \) E | (3, 185) | 0.77 | 0.51 | < 0.01 | |

S \(\times \) R \(\times \) E | (3, 185) | 1.00 | 0.39 | < 0.01 | |

Non-decision | Sequence (S) | (1, 185) | 0.00 | 0.96 | < 0.01 |

Response (R) | (1, 185) | 8.82 | < 0.01 | < 0.01 | |

Experiment (E) | (3, 185) | 5.51 | < 0.01 | 0.06 | |

S \(\times \) R | (1, 185) | 18.00 | < 0.01 | < 0.01 | |

S \(\times \) E | (3, 185) | 0.28 | 0.84 | < 0.01 | |

R \(\times \) E | (3, 185) | 1.45 | 0.23 | < 0.01 | |

S \(\times \) R \(\times \) E | (3, 185) | 1.81 | 0.15 | < 0.01 |

Parameter | Model | WAIC | SE | dWAIC | Weight |
---|---|---|---|---|---|

Drift | Interaction (S \(\times \) R) | − 75.24 | 52.29 | 0.00 | 1.00 |

Main effects (S + R) | 1.54 | 53.13 | 76.78 | 0.00 | |

Response (R) | 22.29 | 48.41 | 97.53 | 0.00 | |

Sequence (S) | 28.66 | 55.74 | 103.9 | 0.00 | |

Boundary | Interaction (S \(\times \) R) | 369.50 | 103.58 | 0.00 | 0.43 |

Main effects (S + R) | 369.82 | 104.53 | 0.32 | 0.37 | |

Sequence (S) | 370.99 | 106.32 | 1.49 | 0.20 | |

Response (R) | 389.73 | 102.66 | 20.23 | 0.00 | |

Non-decision | Interaction (S \(\times \) R) | − 2262.32 | 65.57 | 0.00 | 1.00 |

Response (R) | − 2225.62 | 66.17 | 36.70 | 0.00 | |

Main effects (S + R) | − 2224.20 | 66.00 | 38.12 | 0.00 | |

Sequence (S) | − 2213.69 | 67.55 | 48.63 | 0.00 |

Diffusion model parameter | Source | Estimate | Error | L-95% CI | U-95% CI |
---|---|---|---|---|---|

Drift rate | Intercept | 1.64 | 0.03 | 1.58 | 1.69 |

Sequence | 0.00 | 0.02 | − 0.04 | 0.05 | |

Response repetition | − 0.17 | 0.02 | − 0.21 | − 0.13 | |

Interaction | 0.20 | 0.03 | 0.15 | 0.26 | |

Boundary separation | Intercept | 2.36 | 0.04 | 2.28 | 2.44 |

Sequence | − 0.14 | 0.03 | − 0.20 | − 0.08 | |

Response repetition | − 0.06 | 0.03 | − 0.11 | 0.00 | |

Interaction | 0.04 | 0.04 | − 0.03 | 0.12 | |

Non-decision time | Intercept | 0.38 | 0.01 | 0.37 | 0.39 |

Sequence | 0.02 | 0.01 | 0.01 | 0.03 | |

Response repetition | 0.03 | 0.00 | 0.02 | 0.04 | |

Interaction | − 0.04 | 0.01 | − 0.05 | − 0.02 |

## General discussion

### Summary of results

#### Drift rate

#### Boundary separation

#### Non-decision time

### Limitations

^{2}One potential concern is that the diffusion model assumes that the decision component (e.g. response selection) and the non-decision component (e.g. motoric responding) are discrete stages which do not occur in parallel. The concern raised by a reviewer was that this independence might be violated in our paradigm given its spatial-responding. Given the task requires the participant to move their finger to the spatially congruent response key, there is the possibility that the participant can lift their finger and move it (i.e. engage in motoric aspects) before completion of response selection. This would violate the assumptions of the diffusion model, and could explain why we find that the exclusive effect of \(n-2\) task repetitions on drift rate found by Schuch (2016) and Schuch and Konrad (2017) spills over into other parameters in our study. Whilst this is certainly a possibility, we do not think that this issue is unique to our paradigm, so is unlikely to explain our results. For example, in typical task switching paradigms where just two response keys are used (as opposed to four in our study), it is typical in our experience to observe participants sometimes moving their fingers before a final response is executed. Thus, some degree of motoric action is very likely to occur in most choice response time tasks. Whilst this remains an interesting topic for diffusion model theorists to explore, we do not believe that the issue can uniquely explain our results of \(n-2\) task repetition effects in other non-drift parameters.