Topics in Drift Diffusion Models (DDM)

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### Statistical Modeling in Experimental Psychology
# W14 Drift Diffusion Models
## Decision Dynamics With RT
#### Han Hao @ Tarleton State University

---

## From SDT to DDM

- In many cognitive tasks, we sometimes observe:
  - A response choice
  - And **.red[a response time]**
- In those paradigms, RTs are used to reflect information so accuracy alone does not tell the whole story
  - How we process information and make decisions
  - **.red[Why is a person slower]**? Hard task/trial, cautious person, or they are just slow to process information and respond?
- Mean RT still does not tell the whole story, and due to the distribution of RT, it's **.red[less robust to assume normality]**
- We want a model that explains both together

---

# A familiar task: Stroop

.pull-left[
In a Stroop task, a participant may see a word like:

- **.red[RED]** shown in red ink  
- **.blue[RED]** shown in blue ink

The task is to name the **color** of the word, not read the word.
]

.pull-right[
#### Congruent
- **.blue[BLUE]**
- **.purple[PURPLE]**
#### Incongruent
- **.green[RED]**
- **.yellow[BLACK]**
]

---

## Basic idea of DDM

.pull-left[
For each trial, we usually record:
- whether the response was correct (Accuracy)
- how long the response took (RT)

The Drift Diffusion Model assumes that:
- Noisy evidence builds up over time
- The evidence "drifts" up or down
- A response is made when the evidence hits a boundary
]

.pull-right[
![:scale 100%](data:image/png;base64,#image/stroop_ddm_single_trial1.gif)
]

---

## What makes up the observed RT?

The 4 main parts constitute observed RT in a trial:

- **Non-Decision Time (NDT)**: time spent on things outside of decision

- **Drift rate**: how strong or clear the evidence is accumulating (plays a role similar to d' in SDT)

- **Boundaries**: how much evidence is needed before responding

- **Starting point**: any bias toward one response at start

---

## Non-decision time

.pull-left[
Non-Decision Time (NDT,$t_0$) reflects time that is **not** the decision itself.

Examples:
- General lags in "mental preparation"
- Sensing and perceiving the stimulus
- Pressing the response key
]

.pull-right[
![:scale 100%](data:image/png;base64,#image/stroop_ddm_single_trial1.gif)
]

---

.pull-left[
## Drift rate

Drift rate (`$\nu$`) reflects the **.red[quality of the information]** going into the decision ("the speed of information accumulation").
- Think about why I say this plays a role in DDM similar to d' in SDT

Higher drift rate usually means (in general):

- clearer evidence
- faster decisions
- more accurate decisions
]

.pull-right[
![:scale 90%](data:image/png;base64,#image/stroop_ddm_single_trial1.gif)
![:scale 90%](data:image/png;base64,#image/stroop_ddm_single_trial2.gif)
]

---

.pull-left[
## Boundaries

Boundary separation (`$\alpha$`) reflects **the amount of evidence needed** between the two possible choices ("the total distance it takes", a measure of caution?)

Larger boundary separation means:

- The person needs more evidence to make the choice
- Potentially slower responses
- But maybe higher accuracy
]

.pull-right[
## Bias

Bias (`$\beta$`, `$omega$` or `$z$`?) reflects the **preference or tendency** before the evidence begins to accumulate ("the start point")

A shifted starting point means:

- One option is favored early
- Less evidence is needed for that side
- Condition differences and/or individual differences
]

---

## A plain-language expectation for Stroop Task

.pull-left[
### Congruent trials

We would expect:

- easier evidence
- faster responses
- higher accuracy
]

.pull-right[
### Incongruent trials

We would expect:

- weaker or noisier evidence
- slower responses
- lower accuracy
]

---
class: inverse

# A small demo in R

## Simulated Stroop-style data

---

## The demo dataset

For the demo, I created a small simulated Stroop-style dataset.

- multiple participants
- congruent and incongruent trials
- trial-level RTs
- trial-level responses

In the hidden setup code, the dataset is also saved as:

`stroop_ddm_demo.csv`

``` r
stroop_dat <- stroop_dat |>
  mutate(response = case_when(correct == 0 ~ 1, correct == 1 ~ 2))
```

---

## Look at one subject

``` r
onep <- stroop_dat |>
  filter(id == "P01")
```

|           | vars| mean|   sd|  skew| kurtosis|   se|
|:----------|----:|----:|----:|-----:|--------:|----:|
|rt         |    1| 0.69| 0.32|  1.69|     2.80| 0.02|
|condition* |    2| 1.50| 0.50|  0.00|    -2.01| 0.04|
|correct    |    3| 0.88| 0.32| -2.40|     3.76| 0.02|
|response   |    4| 1.89| 0.32| -2.40|     3.76| 0.02|

![](data:image/png;base64,#PSYC5318W14Slides_files/figure-html/unnamed-chunk-4-1.png)

---

## Subject 1 SDT Info

``` r
knitr::kable(table(Condition = onep$condition,
             Correct = factor(onep$correct,
                              levels = c(0, 1),
                              labels = c("Incorrect", "Correct"))))
```

|            | Incorrect| Correct|
|:-----------|---------:|-------:|
|Congruent   |         0|     100|
|Incongruent |        23|      77|

---

## Subject 1 Accuracy and RT

![](data:image/png;base64,#PSYC5318W14Slides_files/figure-html/unnamed-chunk-6-1.png)

???

This helps students see that the model is trying to explain a familiar pattern, not replace it.  
The DDM is a way of explaining why these two bars differ, not a substitute for understanding the data.

---

# A simple DDM fit with "fddm" package

``` r
library(fddm)

model1 <- ddm(rt + response ~ 0 + condition, boundary = ~ condition, data = onep)

summary(model1)
```

```
## 
## Call:
## ddm(drift = rt + response ~ 0 + condition, boundary = ~condition, data = onep)
## 
## DDM fit with 3 estimated and 2 fixed distributional parameters.
## Fixed: bias = 0.5, sv = 0 
## 
## drift coefficients (identity link):
##                      Estimate Std. Error z value Pr(>|z|)    
## conditionCongruent     3.0317     0.1976  15.340  < 2e-16 ***
## conditionIncongruent   0.7898     0.1384   5.705 1.16e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## boundary coefficients (identity link):
##                      Estimate Std. Error
## (Intercept)           1.55236      0.069
## conditionIncongruent -0.02614      0.092
## 
## ndt coefficients (identity link):
##             Estimate Std. Error
## (Intercept)   0.2995      0.006
```

``` r
logLik(model1)
```

```
## 'log Lik.' -13.53722 (df=5)
```

---

# What this model is doing

The model uses both RT and response (accuracy recoded for fitting)
- The predictor is **condition**: In this simple example, we are mainly asking whether evidence quality (`$\nu$` and `$\alpha$`) differs across congruent and incongruent trials
  - With fixed bias and inter-trial variability in the drift rate
- This package uses C++ approximations to fit classical Diffusion Decision Model (5 parameters) using **.red[maximum likelihood estimation]**.
- FYI, for Subject 1, its responses were simulated based on the following set of parameters:
  - `$t_0$` = 0.305
  - `$\alpha$` = 1.57, `$\omega$` (relative bias) = 0.54
  - `$v_c$` = 2.55, `$v_i$` = 1.00
  
---

## Let's just run it across all subjects

**This will not mean much in real-world practice**, because for DDMs we usually want to have individual-level models and parameters.

**.red[Here I am only doing it for simple demo purpose (and it is a simulated dataset with pre-specified parameters anyway).]**

FYI, in the simulation, the average values for the key parameters are:
- `$\overline{t_0}$` = 0.35
- `$\overline{\alpha}$` = 1.69, `$\overline{\omega}$` = 0.50
- `$\overline{v_c}$` = 2.32, `$\overline{v_i}$` = 1.10

Let's see how well this model performs.

``` r
model_all <- ddm(rt + response ~ 0 + condition, boundary = ~ condition, data = stroop_dat)

summary(model_all)
```

```
## 
## Call:
## ddm(drift = rt + response ~ 0 + condition, boundary = ~condition, data = stroop_dat)
## 
## DDM fit with 3 estimated and 2 fixed distributional parameters.
## Fixed: bias = 0.5, sv = 0 
## 
## drift coefficients (identity link):
##                      Estimate Std. Error z value Pr(>|z|)    
## conditionCongruent    2.14192    0.03238   66.15   <2e-16 ***
## conditionIncongruent  1.02174    0.02604   39.24   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## boundary coefficients (identity link):
##                      Estimate Std. Error
## (Intercept)           1.79353      0.017
## conditionIncongruent -0.01005      0.023
## 
## ndt coefficients (identity link):
##             Estimate Std. Error
## (Intercept)   0.3112      0.002
```

``` r
logLik(model_all)
```

```
## 'log Lik.' -1677.184 (df=5)
```

---

## Takeaways

- Mean RT and accuracy are useful, but limited
  - They tell us what happened on average and whether one condition looks easier

- DDM adds a process explanation
  - Was the evidence weaker?
  - Was the person more cautious?
  - Was there extra time outside the decision?
  - ...

- The estimated parameters from DDM tell different cognitive stories and can be used for further correlational and/or computational modeling

---

## Final Notes

- Next week
- Final project (Presentation and Write-up)