# Prediction using linear regression: Why are the predicted values and actual values not equivalent in this example

Hope we are all safe, well and happy

I have a question that on the surface likely seems very trivial, but the empirical observations are not matching what should (in my head) theoretically happen. I suspect a very strong reason for this is the fact i'm very much a newbie in the stats space.

Background

Lets say I wish to use a linear regression equation to predict the maximum load (in kg/lbs) that an individual can lift using the velocity measured at i different submaximal loads (also recorded in kg/lbs). At present, one of the proposed practices is that you use only 2 loads, which are spaced as far apart on the sample space as possible (e.g., (1) 20% of the individuals maximum and (2) 80% of the individuals maximum. Once we have our regression we then extrapolate to a velocity value thought to correspond to an individual's maximum load. This can be done using either a pooled value (i.e., the mean velocity recorded during a maximal attempt across lots of individuals) or an individualised value (the actual value measured during a maximal lift). When using the individualised value, the assumption is made that this value is unique for each individual, but that it remains relatively stable. It's important to stress here that this regression is built for each person individually, and the goal is not to generalise. It is not for inferential purposes and it is not for hypothesis testing. It is simply to try and predict, for that specific individual, what load represents their maximum on a given day.

So, we get someone to come in, we load up the barbell incrementally until we reach their maximum. We then select the observations that represent 20% and 80% of that individuals maximum and create a linear model with these. Clearly, we get a regression that fits the observed data perfectly with R^2 =1 and RMSE=0. I have attached a very rudimentary illustration of what I mean below. Although these are observed values and do not represent that individuals true "performance", they are the "true" values for that given day, so to speak. Theoretically then, the biggest influence on our prediction should be our choice of the 'x' or the 'velocity' value used to represent that individuals maximum.

Question

If I use the group value and this =/= that velocity that corresponds to that individual during a maximal attempt, then the prediction will clearly be off. However, if that individuals actual velocity value for a maximum load on a given day were to be used (for example, if we had conducted the test above and retrospectively fit the model to the observed data, we would know the "truth") should the generated prediction and actual value not be exactly equal? I know this would not work prospectively though (when we do not know the "true" value.

To spin this another way: if we were to create a model with >2 loads, then we would have error due to regression (i.e., the line of best fit does not fit observed values perfectly) as well as error due to extrapolation. But with only 2 loads, this is not a factor. Again, note that the point is to predict the maximum load that a given individual can lift in a given training session or instance.

Hopefully that makes sense, but let me know if you need any clarification. This is just a generic question based on my very limited understanding. PS: I'm also not advocating that these practices are a good idea.... It's just what's done. Take from that what you will. Thanks in advance for any help!