Lets say we take some measurements at 3 equally spaced intervals of our independent variable. The same number of measurements are taken at each level. We then fit a nonlinear model with three parameters to the individual data and get one set of estimates for the parameters. We also fit the model to the mean at each level and got the same set of estimates as with the individual data. I have a feeling this seems to makes sense but I am not sure.
Our client wants us to fit the model to the individual values though our reference said to use the mean. We are also performing bootstrapping with this model and it looks like using the individual values is 30-50 times slower (which if the estimates are the same then it would makes sense that our reference would say to use the mean).
Ok I convinced myself that this makes sense. Within each of the three levels the point that would minimize the sums of squares would be the mean. Since there are three parameters the line can pass through all three means which would make the sums of squares as small as it could be. There are occasional small differences that could be from choice of starting value but all of the parameters from the bootstrapping match to at least 5 significant digits.
The parameters should be the same, because you are using the same data. However, the variance will be different, as some of the variability has been removed by aggregating the data.
As long as you restrict your inferences to your unit of analysis, (i.e., don't make inferences about individuals from your analysis of means, and vice-versa) you should be okay. The term "ecological fallacy" refers to making inferences about individuals from groups, and "individualistic fallacy" refers to making inferences about groups from individuals, and often results in inferential error.