Suppose you have a set of N₁ measurements obtained with one detector, and another set of N₂ measurements obtained with a second detector. And let’s say you wanted something as simple as an estimate of the mean of the quantity (say the intensity) being measured. Let us further stipulate that the measurement errors of each of the points is similar in magnitude and neither instrument displays any odd behavior. How does one combine the two datasets without appealing to subjective biases about the reliability or otherwise of the two instruments?

We’ve mentioned this problem before, but I don’t think there’s been a satisfactory answer.

The simplest thing to do would be to simply pool all the measurements into one dataset with N=N₁+N₂ measurements and compute the mean that way. But if the number of points in each dataset is very different, the simple combined sample mean is actually a statement of bias in favor of the dataset with more measurements.

In a Bayesian context, there seems to be at least a well-defined prescription: define a model, compute the posterior probability density for the model parameters using dataset 1 using some non-informative prior, use this posterior density as the prior density in the next step, where a new posterior density is computed from dataset 2.

What does one do in the frequentist universe?

[Update 9/30] After considerable discussion, it seems clear that there is no way to do this without making some assumption about the reliability of the detectors. In other words, disinterested objectivity is a mirage.

A couple of weeks ago Vinay suggested using the LOESS algorithm to create smooth curves (separately) through the SSD and FPC points. LOESS has been succeeded by LOWESS and, finally LOCFIT, which is the 800lb gorilla of local regression fitting.

The LOCFIT algorithm uses local regression (i.e. fits over samples of the data) to generate smooth curves. There is an enormous body of literature on this, much of it summarized in the book

Local Regression and Likelikhood, by C. Loader
ISBN 0-387-98775-4

which also serves as documentation for the LOCFIT software. The techniques seem well established and accepted by the statistical community.

LOCFIT looks to be a very elegant approach, but, unfortunately, I have still not been able to glean any information as to how one introduces experimental errors into the regressions. The voluminous research in this field certainly deals with experimental data, so I’m not quite sure what to make of this.

One way around this might be to take a Monte-Carlo approach: resample the data using the experimental errors, generate a new smoothing function, and generate a measure of the distribution of the fit functions.

For those interested, I have a copy of the above book on loan.
It’s fascinating reading.

More about the actual code is available at this web site:
http://locfit.herine.net/

In addition, Ping Zhao asks: (paraphrasing) if you combine two separate sets of observations with vastly different numbers of data points in each, how do you weight them during a combined loess/lowess/locfit fit?

Comments and suggestions from statisticians are much appreciated!