Modelling Categorisation Learning Rate

library(tidyverse)
library(patchwork)

The Problem

Below is some (fake!) data representing the average performance across all subjects in our experiment. For both the rule-based (RB) and information-integration (II) conditions, the proportion accuracy increases as the number of blocks increases. This reflects that learning of the category structures is taking place, and as such these lines can be considered so-called learning curves.

Figure 1. Schematic (i.e., fake!) data from our Experiment.

Our current dependent variables for each participant are:

Overall proportion accuracy: The mean accuracy averaged across blocks.
Blocks to learn: The number of blocks it took the paricipant to achieve 100% accuracy. If this wasn’t achieved, then the number 6 is returned (the maximum number of blocks).

Whilst used in many categorisation studies, these DVs are a little crude for us if we want to know what the rate of learning was for our participants, and—more importantly—whether this rate of learning correlates with levels of depression symptomatology.

We need therefore to find a way to model these learning curves to get an estimate of the rate of learning for each participant.

One way we discussed in our last meeting would be to use linear regression, which is what geom_smooth() does. Linear regression returns two parameters: (1) Intercept, which is where the line should start from the y-axis, and (2) the Slope of the line. So, we could use the slope parameter to estimate the rate of learning: The steeper the slope, the more rapid the learning.

But note, linear regression fits a straight line, and our data ain’t linear!

Instead, we need something similar to a regression that is capable of taking into account the shape of our data.

The Model

We can model the learning curve (i.e., the growth of accuracy as block numbers increase) via the van Bertalanffy growth function. Similar to linear regression, the van Bertalanffy growth function has parameters that change the shape of the learning curve in different ways.

The equation of the van Bertalanffy growth function is

\[\begin{align} Pc_b = \varphi + (\alpha \textminus \varphi)(1 \textminus e^{-r(b\textminus1)}) \end{align}\]

where we have the following parameters:

\(Pc_b\) is the model’s estimate for the proportion accuracy at block b
\(\varphi\) is the first block performance (similar to the intercept in regression)
\(\alpha\) is the asymptotic performance (i.e., an estimate of the participant’s maximum accuracy)
\(r\) is the learning rate (similar to the slope in regresssion)
\(e\) is the mathematical exponential Euler’s number (\(e \sim 2.718\))

These parameters can be thought of as “tuning knobs” that alter particular aspects of the learning curve. In the plot below, I show the effect of changing just a single parameter across a range of values to show the effect that it has on the learning curve:

Figure 3. Estimated proportion accuracy per block. In each plot, a single parameter of the von Bertalanffy growth curve is altered whilst all other parameters remain constant.

As you can see:

Changing \(\varphi\) (first block performance) selectively alters where the learning curve starts from, but it doesn’t change the maximum accuracy achieved, and it doesn’t change the overall rate of learning across blocks
Changing \(\alpha\) (maximum accuracy achieved) selectively alters where the learning curve flattens out to signal learning is complete, but it doesn’t change the first block accuracy, and it doesn’t change the overall rate of learning across blocks
Changing \(r\) (the rate of learning) selectively alters how quickly the maximum accuracy is achieved, but it doesn’t change the first block accuracy, and it doesn’t change the maximum accuracy achieved

Applied to your data

We can fit the van Bertalanffy growth function to each participant in your experiment. This will provide us—for each participant and each condition—estimates of \(\varphi\), \(\alpha\), and—most importantly—\(r\). We can then analyse these in the same way as our previous dependent variables.

However, really we are only interested in r as this is the estimate of the learning rate. So, we would do the following analyses in a separate section at the end of your results section:

Perform t-tests for each of the three van Bertalanffy growth function model parameters, comparing whether the parameter is different in rule-based and information-integration category conditions
- One t-test comparing the \(r\) parameter across RB and II conditions
- (Note you could do additional t-tests for the other model parameters, but this is not essential.)
Correlate each parameter in each condition with the depression scores
- Correlate \(r\) in the RB condition with the BDI-II
- Correlate \(r\) in the RB condition with the CES
- Correlate \(r\) in the II condition with the BDI-II
- Correlate \(r\) in the II condition with the CES
- (Note you could do additional correlations for the other model parameters, but this is not essential.)

The van Bertalanffy data

You can download the van Bertalanffy data from our Teams channel (“vbf_data.csv”). This data is a result of fitting the van Bertalanffy to each participant in each condition. Read in the data and have a look at it:

vbf_data <- read_csv("vbf_data.csv") 
vbf_data

# A tibble: 348 × 7
        id condition   phi alpha      r   ces   bdi
     <dbl> <chr>     <dbl> <dbl>  <dbl> <dbl> <dbl>
 1 8470936 ii        0.500 0.783 2.05       7     2
 2 8470936 rb        0.889 0     0.0369     7     2
 3 8471027 ii        0.716 1     0.399     15    19
 4 8471027 rb        0.750 1     8.79      15    19
 5 8471122 ii        0.371 0.745 1.61      32    31
 6 8471122 rb        0.485 0.788 0.298     32    31
 7 8475499 ii        0.631 1     0.0984    20    17
 8 8475499 rb        0.750 1     8.79      20    17
 9 8476026 ii        0.630 1     0.864     33    34
10 8476026 rb        0.491 0.933 1.34      33    34
# ℹ 338 more rows

As you can see, for each participant and each condition we have the parameter estimates for phi (\(\varphi\)), alpha (\(\alpha\)), and r. We also have the participant’s CES and BDI scores.