Which function to use for fitting learning curves? Background
I am fitting learning curves for cochlear implant (CI) users. These folks get a CI, and then need to start learning to understand speech again. They keep on improving for years. Along the way, speech recognition is tested frequently. The outcome measure of speech tests is a simple word or phoneme correct score. The %correct score runs from a minimum of 0%, to a maximum of 100%. No rocket science thus far.
Typically, these learning curves start at a value around 25% and end at 80% (barred the rock stars, CI users never reach 100%). In the beginning they learn quick, and this levels off as to what seems a maximum (Fig. 1).
Question premise
Now I seek to fit a function to these data, because I want to know how fast they learn. I did this using regression analysis using the least-squares method in MATLAB. The function I used I found in another software package called GraphPad Prism. While the latter package was not used to analyze the data, I did use it as an inspiration to find a proper model. The function I fitted through the data (red line in Fig. 1) GraphPad calls a 'one-phase association function'
Y = Y0 + (Plateau-Y0).(1-exp(-Kx)), which translates to Y = Y0 + (Plateau-Y0).(1-e(-Kx)) if I am not mistaken (I don't understand the fancy formula make up, hope this'll do...) (Fig. 2). The link explains the backgrounds, but in a nut shell, it is based on a biochemical process, namely the interaction between an enzyme and a substrate. My purpose is a totally different one, namely a psychophysical one to model a learning process.
I chose the 'one-phase association' function, because the main parameters in the model follow my needs very well: it has a start value Y0 that reflects the baseline score when people get their CI, and there is a maximum (Plateau). In addition, it has the K value, which in (psycho-)physics would be called tau, which describes the slope. And this slope is what we need, as we want to know how fast these people learn.
Specific questions

Is the 'one-phase association' a proper way to fit a learning curve? It is a function I only know from Biochemistry;
A colleague over in the USA used a 'logit' function for this purpose. Afaik, a logit function does not have a slope parameter and hence, from my pov, it would be less fitting for my purposes (pun not intended). Is a logit function the best way to fit a learning curve?*

* EDIT: after a literature search I found that the logit function is a sigmoid function, and, hence, it does have a slope parameter

Fig. 1. Learning curve obtained in quiet at 65 dB speech level. Blue crosses: raw experimental data. Red line: fitted logarithmic function. Diamonds: derived fitting features irrelevant to the question proper. source: own work

Fig. 2. One-phase association Y=Y0 + (Plateau-Y0)(1-exp(-Kx)). source: GraphPad
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.