Statistics and Action (2)*

[David Kerridge <dfkerridge@mac.com>]

Let us look at some practical choices that we might have to make, in
using a control chart. Suppose, to keep things simple, that we have
no choice in collecting the data. We get individual values at equal
intervals of time.

We might be concerned with a process that is subject to sudden
changes of mean, with nothing else (eg variation) changing.
This is rare, as gradual drift is more usual, but I saw it once,
in medical research.

This is selected for purposes of illustration because it is the
kind of situation in which the advocates of CUSUM charts say
that their favourite technique is superior to the Shewhart chart.

For those who are unfamiliar with CUSUM (Cumulative Sum)
charts, the idea is very simple. At each time you plot the total
of all observations up to that time, instead of the actual value
at that time. So if the values are varying about a constant mean,
the cumulative sum chart gives (approximately) a straight line,
at an inclination to the axis. If the mean of the values changes
suddenly, you get a sudden change in the slope of the graph.

As a visual display, this can be quite effective, because the
eye picks up a change in slope more easily than it does a
small change in the mean of a Shewhart chart.

But the claim made by those who favour CUSUM charts is
that you detect a change in mean sooner using rules based
on cumulative sums of values that by rules based on individual
values. As Paul Hollingworth mentioned, Don Wheeler disputes
this claim - but for the sake of argument, let us accept it.

If the change is large, it will be obvious, regardless of how
we plot or calculate. So let us assume the change is small.
And let us assume that the cause of the change is unknown
or action should have been taken to remove it long ago.

The question now is, what action should we take?

There are two obvious types of action.

1   To improve the process, by tracing and eliminating the
      cause of the changes of mean.

2   To reset the mean, so that the process is centred on the
      preferred value.

For neither of these purposes is it desirable to take immediate
action. To trace the cause of a change, it is most useful to have
plenty of observations both before and after the event, to
determine accurately when it took place.

Otherwise you may chase after apparent causes that are merely
coincidence.

To reset the mean, you need a sufficiently long series of
observations after the change to determine how large an
adjustment to make. If not, you will very likely make
things worse.

You also need accurate estimates of the response of the
process to adjustment, so as to be able to predict the effect
of future changes. It is easy to overcompensate - so easy that
it is almost the rule, if you do not follow the advice above.

Overall, there seems little point in elaborate rules to discover
that a change has taken place, when

1   If there is a catastrophic change, no rules are needed.

2   If the change is small, it pays to wait before acting.


The confusion has arisen, I believe, in the minds of those
who treat control charts as if they are significance tests.
The only purpose, from this point of view, is to "prove" that
a change has taken place - and as soon as possible.

 From Shewhart's operational point of view, the purpose is
to decide on economically worthwhile action - and the problem
dissolves at once.

Cumulative sum charts can be a useful graphical technique.

But I cannot think of any circumstance in which it would
be my main, let alone only, way to study a process.