How to get the best out of series of data with a single fit

In physical experiments it occurs often, that you make several measurements on the same experimental arrangement. Usually you fit each data set separately, due to experimental circumstances, which disallow the use all data points in one fit. In my PhD-thesis I described a way to fit these data in a single run, thus having more data points for the estimation of fit parameters of interest.

In my PhD-thesis I worked on light scattering at binary liquid mixtures with a miscibility gap. I used this method to fit the experimental data from turbidity measurments. The fit method is decribed in one of the appendices of my thesis. I wrote a special fit program to perform the fit, but I pointed out, that it should be possible with a standard (well not too standard, BMDP could do it) data processing program. The results between the standard program and my program were nearly the same.

• the standard situation
• and the trick I used.

The standard situation

You have a nice formula which describes the relation between two physical properties. You make your measurements in several separated runs, but when you plot your data, you will see, that each series has slight variations to the other, while each serie gives comparable good results on the parameters in your formula. The error on the parameters depends on the number of data points in each serie. If you could combine all series together you have more data points and the signal noise ratio would improve. But, the design of the experiment leads to these damn differences between each serie, so that a fit with all data gives results with a worse quality. There are some parameters which depend on the experiment condition rather than the underlying physical relationship.

The trick I used

is to separate the parameters in the formula which are related to the physical concept (p-parameters) from those parameters which are related to the condition of the experiment (e-parameters). In my fit of all data points in a single run the p-parameters are estimated only once, thus using all data points, while the e-parameters are estimated for each serie, thus using only the data points in the distinct serie. The signal noise ratio of the p-parameters is obviously better than the one for the e-parameters. But only the p-parameters are of interest.