Which function to use for fitting learning curves?
Sigmoids are more general and probably better for modeling learning than a simple exponential. Your exponential fits decently, but it probably doesn't describe the actual underlying process very well.
Importantly, learning is often not fastest at the start, there is often some level of &quot;aha!&quot; moment later on. Sigmoids can capture this.
There are a variety of sigmoidal functions, but they are all differentiable and have a central peak in the first derivative. The time to peak and slope at peak are both parameters you can use to measure learning rate, though they measure somewhat different things. Which one is more applicable will depend on how learning in your paradigm varies across subjects and conditions. Looking at the data you posted, the peak slope (peak of the first derivative) is probably a decent choice, although I'm not sure you really have enough data points in the first month to get a good measure.
Exponential fits are likely to be more unstable, and in the data you've posted it looks like your fit is overfitting the beginning and end of the learning process, so you're missing the most important part for determining the actual &quot;speed&quot;, which is between your first 3 data points; none of these are fit well. With an exponential fit, you are supposing that the most learning occurs at the point of the first trial. In most tasks without substantial prior instruction, you need at least a couple of trials to even figure out what is going on, and can't start to improve until then. If that first &quot;trial&quot; is a whole block then maybe this is not a terrible assumption, but that's mainly because you've already passed the peak slope of the sigmoidal function early on; if you used the entire data then a sigmoid would still fit better.
See also https://en.wikipedia.org/wiki/Learning_curve and:
Culler, E., &amp; Girden, E. (1951). The learning curve in relation to other psychometric functions. The American journal of psychology, 64(3), 327-349.
Gallistel, C. R., Fairhurst, S., &amp; Balsam, P. (2004). The learning curve: implications of a quantitative analysis. Proceedings of the National Academy of Sciences, 101(36), 13124-13131.
Daller, M., Turlik, J., &amp; Weir, I. (2013). Vocabulary acquisition and the learning curve. Vocabulary Knowledge: Human ratings and automated measures, 47, 185.