# Control charts where the center line is a trend or slope?

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#### Graeme

Thank you!

A large "Thank You!" to everyone who responded!

A couple of you refreshed my memory of prediction interval and confidence interval, and clarified the difference. To be honest, it's been so long since I used any of that I had forgotten the terms and definitions. I see I am going to have to dig for my old engineering statistics textbooks to see how to figure these things. Fortunately I have the four-day Thanksgiving weekend coming up, so I can squeeze that into the already crowded schedule.

A few comments on various responses:
• The drift of a standard resistor is determined by the physical properties of the resistance element alloy and the mechanical construction of the device. It is a combination (as I understand it) of ongoing strain relaxation and molecular changes in the alloys. The drift rate and direction can vary from unit to unit, and over the life of the unit. Generally the first several years have the most rapid change -- but not always!
• Charting the residuals may be a workable idea for me -- I doubt if most of the techs in the lab would understand it. I'll consider trying it, though.
• Several people mentioned using the Range. What are the pro's and con's of using that method instead of calculating the actual standard deviation of the data?
• The graph that Tim Folkerts attached does look like ones I recall seeing years ago -- Thank you! I don't have access to any statistics software here at the client's office, but I do have something at home. Now that I know what to try, I'll see if that can do it.
• I think I need to get a copy of the AIAG SPC manual ... it appears to cover some things that other ones I have do not.
Again, thanks to all!

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#### Rob Nix

I'm too late again. Doh!

Darius summed up what I've done in the past: used the residuals. It is not particularly elegant or "statistically correct", but it works very well. And I see you noted that Graeme in bullet two.

#### Statistical Steven

##### Statistician
Super Moderator
Just a caution when using regression methods to check the assumptions. If the errors (residuals) are not normally distributed, or when plotted versus the x-values does not give you a nice random pattern (see Daniel and Wood for residual analysis), plotting residuals on an SPC chart might be fruitless. Just a heads up.

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#### Camper

See Rolf Schumacher's paper on "Statistical Control in a Standards Laboratory" for some guidance on this issue...

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#### Camper

After I posted the above, I tried to find a reference to Shumacher's paper via Google and it was a lot harder that I thought... anyway here's the complete reference for those who are interested...

Schumacher, R., (1969), "Statistical Control in a Standards Laboratory", Measurements and Data, Vol. 3, No. 15, pp. 58-64.

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#### chalapathi

Trend control chart

Graeme said:
After some thought I have decided to go ahead and post this question, because I feel it is sufficiently different from others to be ... different. Besides, I want to use Excel to solve this, not rulers and pencils.

(Fair Notice: I have a long-winded discussion of the problem.)

Question – how are control limits for an SPC chart calculated when the center line is a trend?

What?” you say? Heresy! A trend is a “very bad thing” and a sure sign that things are out of control! Get rid of that nasty trend and all will be well in the world!

But wait, I say – maybe I’m talking about a special application – an application where a trend is expected, normal, and you can’t do anything to control it or get rid of it. Let me give a bit of background before re-stating the question.

Standard Shewhart statistical process control (SPC) charts are designed to monitor a process which produces a large number of measurements of the same "identical" feature over a relatively short period of time. Ideally the process holds a constant level. Random variation is easily visible, and that is good. The chart also gives visibility to variation that may not be random, such as points above or below the control limits, or an upward or downward trend. These things are considered bad.

SPC charts are also used in calibration labs to monitor measurement processes and measurement standards. Processes are typically monitored by use of a check standard – a device that is similar to the workload and measured at the same time as normal work. Recording and charting the values is similar to production SPC charts, except that there is much less data. Instead of producing hundreds of parts (measurements) per hour, the lab might generate 200 measurements a year – or only 20, depending on the workload. X and mR charts are much more common than X-bar and R charts. Still, things work about the same as the usual type.

The bigger problems come when trying to use SPC chart methods to monitor the variation of measurement standards. First, there is a data problem: if a standard is calibrated every 12 months, it naturally takes five years to get five measurement values. (In the calibration business, patience really is a virtue.) Calibrating more frequently than the usual interval will collect data more quickly, but that costs a lot of money and time. So we usually live with it. A bigger problem is that some things change in value over time – they drift. This is known and expected. Examples include shrinkage of gage blocks, drift of standard resistor values, change in the output of a DC volt standard, and many more examples. The drift is a natural phenomenon that can be compensated for but not controlled or adjusted. This means that values plotted on a graph will show a trend. On a standard SPC chart a trend may be a bad thing but for many measurement standards it’s just the way they are. The trend can be determined, often with linear regression, so I have a rate and direction of change over time. That can be used to determine an “assigned value” for right now, and to predict a value at times in the near future (up to one calibration interval). If control limits or – even better, 95% confidence limits – could be plotted around that trend, they would give an estimate of the standard deviation which is very useful in computing measurement uncertainty.

So the question before the group is this – how does one calculate control limits or confidence limits when the central line is a trend? I am looking for a method using Microsoft Excel, not a paper printout, ruler and pencil. Besides, I think confidence limits for this will be curves instead of straight lines.

Consider the following data set:
Code:
``````[FONT=Courier New][SIZE=3]Date (X)    Value (Y)[/SIZE][/FONT]
[FONT=Courier New][SIZE=3]1996-05-01  1.000 017[/SIZE][/FONT]
[FONT=Courier New][SIZE=3]1997-05-13  1.000 024 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]1998-06-05  1.000 022 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]1999-07-07  1.000 028 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2000-08-02  1.000 024 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2001-08-28  1.000 031 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2002-09-24  1.000 036 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2003-10-22  1.000 032 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2004-05-05  1.000 035 [/SIZE][/FONT]
[FONT=Courier New][SIZE=3]2005-05-11  1.000 036[/SIZE][/FONT]``````
When plotted, this series has an upward trend of about +5.4 x 10-9 per day or about +0.000 002 per year. From the regression line, I can be fairly confident in giving the device an assigned value of 1.000 037, and I can be fairly confident that in May 2006 the value will be near 1.000 039. My problem is that I don’t know (yet) how to quantify that confidence – the uncertainty – and without being able to do that these values are meaningless.

If the line was horizontal (slope = 0) then computing a value for the uncertainty of the device would be trivial. (Compute the standard deviation and multiply by an appropriate value of Student’s t for 95% confidence limits.) But the line has a non-zero slope … so,

How do I get the equivalent result when the center line is a trend?

(By the way, the device is a 1 Megohm standard resistor. But, it doesn’t really matter what it is – it could just as easily be the 1 Volt output of a DC volt standard, or a 1 Ohm standard resistor, or any of a number of other things. Also, precision of one part in 106 is fine for this application.)

TIA,

I am attaching trend control chart I developed in Excel. The sample data I have taken from Wheeler's book. You need to change this template for your data.

#### Attachments

• TrendControlChart.xls
55.5 KB · Views: 467

#### bobdoering

##### Stop X-bar/R Madness!!
Trusted Information Resource
What you are describing are not control limits but the confidence interval for the regression line. The reason the limits curve away is the confidence interval in the estimate of the slope of the line.

What I am describing is covered in the AIAG SPC manual under special types of control charts. The calculation of the control limits is the standard method based on the subgroup range. The only difference is that these limits are parallel to the Regression line at the same distance as the Central line of a standard control chart.

This special control chart is specifically designed for processes that are subject to a predictable and acceptable rate of tool wear. The intent is to accept the tool wear, but react to special causes, or accelerated rates of wear.

I have found that this old methodology for control charting tool wear is actually not the best. If you were to prepare the Total Variance Equation for the variances that participate in the machining process, you would find you would be ignoring the major variance (the tool wear itself) to focus on lesser order variances. The typical implementation of this technique generally tracks the effect of the measurement error that plagues it by not using enough within-part data.

A much better approach for trend data from tool wear is the X hi/lo-R charting methodology. Whether it is appropriate for other trending variances is still to be determined.

#### Statistical Steven

##### Statistician
Super Moderator
I have found that this old methodology for control charting tool wear is actually not the best. If you were to prepare the Total Variance Equation for the variances that participate in the machining process, you would find you would be ignoring the major variance (the tool wear itself) to focus on lesser order variances. The typical implementation of this technique generally tracks the effect of the measurement error that plagues it by not using enough within-part data.

A much better approach for trend data from tool wear is the X hi/lo-R charting methodology. Whether it is appropriate for other trending variances is still to be determined.

Bob

I know you will disagree, but using regression is trending the tool wear itself. There are actually other equally qualified methods besides the X hi/lo -R Chart methods.

#### bobdoering

##### Stop X-bar/R Madness!!
Trusted Information Resource
Bob

I know you will disagree, but using regression is trending the tool wear itself. There are actually other equally qualified methods besides the X hi/lo -R Chart methods.

Actually, it is focusing on the variation about the trend, not the trend itself. The control limits around the trend charts are calculated based on the variation that exists in spite of the trend itself. What variation would that be? The most typical - as I mentioned - the the measurement error from the range calculated by measuring one diameter per part. The error is primarily from within the part roundness (or parallelism if a length) - not variation over time that is key to SPC. It focuses on second-order variation or less. If you have a problem that you need to focus on a second order variation, then you might want to use the trend chart as an analytical tool - but it offers little for process control.

I agree, there are equally qualified methods besides the X hi/lo -R Chart methods, just not for precision machining. I have illustrated the errors of each, and why they are not as effective. I have also illustrated why the X hi/lo-R charting methodology provides far more meaningful information for making decisions on the process (in a relatively easy to implement technique) - again, for precision machining.

If you have electronic drift or other such trending variances, then one might want to consider the trend chart tool - of course verifying it is actually applicable.

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#### Statistical Steven

##### Statistician
Super Moderator
I agree, there are equally qualified methods besides the X hi/lo -R Chart methods, just not for precision machining. I have illustrated the errors of each, and why they are not as effective. I have also illustrated why the X hi/lo-R charting methodology provides far more meaningful information for making decisions on the process (in a relatively easy to implement technique) - again, for precision machining.
We can agree to disagree. If measurement error is an issue, then multiple readings of the same part can minimize that effect. The MSE from regression incorporates the inherent variability at any given time point. The slope and intercept should give me the overall trend.