Ways of Expressing Variability in a Normally-non-linear Declining Slope

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    Cormac Raftery

    I’m new to six sigma and I’m sure there’s a standard way to handle what what I’m stuck at. Basically there is a process that produces a graph similar to the crude depiction that I attached to this post. It’s a naturally declining (but not linear) trend over time that’s perfectly normal. What I want to track is the variability between runs that produce this same behavior. I’m led to believe that over time the decline will become more pronounced which has implications. However I lack the six sigma ability to characterise this behavior with any particular technique. Any ideas?


    Robert Butler

    What you are trying to do is produce some kind of control chart on an entire line as opposed to a specific point measure. I did some checking last night and I wasn’t able to find anything. However, there are some possibilities. At the core what you have is a repeated measures problem. If you were to take a group of these “in control” lines you could use repeated measures regression methods and build a model which would describe the path of the grand average of these “in control” lines. This would give you your center line and the issue then becomes that of defining control limits for that average line.

    To that end the question is this: what would constitute an “out of control” line? Would a line have to be completely outside of some kind of envelope around the grand average line or would it just have to be outside of the envelope at any point or in some small region? If it was a small region would it have to be a single continuous region or could it be multiple regions or would there be just one specific region where a violation of the limits would signal out of control?

    Your statement “I’m led to believe that over time the decline will become more pronounced” would suggest the issue might be one violation of control in some region. In any case, if it makes sense to put control limits around an entire line such as this you could build the repeat measures regression average and then slice vertically through the cloud of lines at various points and identify the range of values at these slices. If the ranges are “similar” you could take their average and use the two point range method to estimate a standard deviation and use that to construct the +-3 standard deviation control limits around the repeated measures regression line.

    The other possibility is buried in your comment about a decline becoming more pronounced. This would suggest a change in slope of some part of the curve and it would suggest you have some idea as to which part of the curve that might be. If this is the case you could take that collection of “in control” curves and compute a grand average for the slope of the critical region and build control limits around that slope value in the usual fashion.



    @crafman – to augment the excellent advice from Robert, is there a specific time period (or series of time periods) which are critical for this process? If single (or even if multiple) you could plot the value a that point in time (or multiple plots for multiple time values) as a control chart. That would help to identify the outliers. Just realize that the further away, the more scatter you are going to get, and thus the larger the control limits are going to be. If your process is relatively stable except for the deviants of which you are concerned, you may be able to see them quite a ways out. And early deviations from the normal path will provide an early indicator – unless, as Robert queries, that the curve actually changes over the trajectory, in which case you can only measure at various time periods and plot the values.

    Is there another surrogate measurement that might be useful? Heat, pressure, angle, etc. that is directly tied to the curve but separate from the curve? That might be another way to measure a highly correlated value for a surrogate.

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