In a perfect world, a gauge should make measurements that are exact, meaning the measurement is exactly the value of the item's heat, for example. Unfortunately, we live in a not-so-perfect world and the measurements we receive are not exactly the true value of the heat. This error could come from changes in the rooms environment where the measurement was made, different operators, the gauge makes measurements that are too high, etc. It is the job of a statistician to somehow quantify all these sources of error and assess the gauge or the item being measured.
Anyway, back to the data set. The picture below is an example of heat standards being measured on a gauge (calorimeters if you are curious). The different shades of gray correspond to different heat standards (in watts). The y-axis is the measured value on the gauge divided by the claimed true value of the item (the ratio helps since these are assumed to be multiplicative error models, i.e., the error increases with the increasing heat standard) and the x-axis is time (in days). On the y-axis, I then subtracted one from all the values to help illustrate the bias more.
The blue line is a smoothing spline to show the trend in the data with associated confidence intervals (I think pointwise) about this line (gray ribbon). If this gauge was unbiased (meaning it doesn't make measurements too high or too low), the points would be centered around zero (or one if I had the raw data). This is not what we see, however. Instead, the bias seems to drift with time. If things were simple, we would expect to see a constant bias with time, e.g., all the values are centered about .001.
Now I know what you are thinking, on average it looks like there is not a bias, and this may be true, but this doesn't help when we are interested in the bias at a specified point in time; we would like to be able to account for that bias.
The researchers were excited to see this plot because their experience told them that there must be some sort of bias that is changing with time. They just hadn't gotten around to creating a plot like this. For those of you still curious why there is a drifting bias, the scientist think that there is a heat source correlation present. What I mean is that these units are put into a water bath. When they remove the item and put in a new one, there may still be some sort of residual heat or something left from the old item. Since these items are of different heat amounts, this residual heat could be large or small. Either way, the researchers think this residual heat is still present and affecting the current measurement.
Unfortunately I have not had the time to solve this problem, though I hope to one day and even have an interested colleague to collaborate with. I would love to hear your ideas.
LA-UR 11-02586
Update: Professor Ivan Ramler recently was given some funding for us to work on this! Stay tuned for what we discover.
