COMPARISON OF PRODUCTION MEASUREMENTS AND ASSAY STANDARD
Figure 7. A CUSUM plot for production and assay reference data. The lack of correspondence between the plots gives reason
to eliminate the reference assay as the cause of the problem.

The CUSUM chart for production data shows a definite change on August 28 or 29 and possible changes at a few other dates.
The problemsolving process uses this information to confirm or eliminate potential causes. Figure 7 shows the CUSUM plots
for production data and for the assay reference standard, which may be a cause.
The two plots have a similar shape but the key change date for the production measurements does not have a complementary one
for the assay reference. An obvious change point for the reference standard on August 28 (or on a slightly earlier date)
would have confirmed that it was worthwhile to investigate this potential cause in more depth. The reference does rise (the
slope is upwards) starting on August 19 but similar short periods of high values in late June and late July and very high
values in October do not have a consistent effect on the production measurements. The differences between the two plots give
reason to eliminate the reference assay as the cause of the problem.
The best interpretation of the shape of the CUSUM plot for the reference assay is that it was on target up to May, was low
during June, and then it was slightly high. The other twists and turns in the line are probably just random variation and
serve to illustrate that it is possible to read a lot into a chart when nothing has really happened. If you think you might
be falling into this trap, then plot the raw data behind the CUSUM (as in Figure 5) because it is harder to imagine a pattern
or shape in the conventional trend chart.
STATISTICAL SIGNIFICANCE TESTS
The tests described here can be used to support the interpretation of a CUSUM chart, though it should be remembered that the
key purpose of the chart is to identify retrospectively when a change occurred. Judgment, knowledge of the process, and other
events around the process are key to interpretation. The results of statistical tests in these situations are not a core requirement
as in a clinical trial; they are an aid to problem solving. It is often enough to trust one's eyes, especially if a CUSUM
plot looks like two straight lines as has happened with these data.
T test to Compare Two Periods of Data
Figure 8. A t test for difference in averages

The t test is used to check for a difference in averages between two sets of data. Figure 8 shows how this is done in Excel using
the TTEST function:
 Array 1 contains the cells with data for one part of the chart, in this case March 9 to August 28. The numbers to the right
are the first data points in the selected range.
 Array 2 contains the cells with data for a second part of the chart (in this case, August 29 to October 21)
 Tails refers to a onetailed (is it bigger?) or a twotailed (are they different?) test. We have seen a difference and want
to check its significance, so choose a onetailed test
 Type 2 is a test that allows a different number of points in each array and assumes that the standard deviations in the two
sets are the same. If you have reason to think that the standard deviation and the average have changed, then put in Type
3.
The t test estimates the random, background variation from the spread of results in each array. It then compares the difference
between the average of the first and second array with this variation and calculates the likelihood that it is just random.
The answer given by the test is the probability that a difference this big would have occurred by chance. Values greater than
0.1 are generally considered statistically insignificant. If you get an answer of 0.05 and you believe that the difference
is real, then you carry a one in 20 risk that you are mistaken, an equivalent statement is that you are 95% confident that
the difference is real. Small values of 0.01 or less suggest that the difference is real or that you have been unlucky.
The increase in average seen by comparing data up to August 28 with data from August 29 onwards is almost certainly real because
the chance that it would have occurred through random variation is very low (the result is 7.76 x 10^{–18)}.
There is a major risk in applying the t test to a historical analysis, such as the one here, in which we have chosen a change point that already looks important,
and are then attempting to show that it is statistically significant. The answer in such a situation is often one which confirms
our subjective judgment. If we select a period when the average is high and compare it with a period when the average is low
then a statistical test will almost always say that the difference is real, particularly if the number of samples in each
average is high. It would be a serious mistake to perform t tests on many groupings of data until one is found to be significant; a significance level of one in 20 will be found in
about one in 20 random data sets. As stated above, the visual inspection of the CUSUM plot is primary and the statistical
test is secondary. A negative result (probability or pvalue of more than about 0.1) does, however, give a strong indication
that the observed, potential difference is just a random variation.
Sequential Tests
An alternative approach is to continually apply a test to each new data point and check if it is significantly different from
its predecessors. This is the principle behind statistical process control (SPC). The advantage of SPC over the t test is that the data indicate when there has been a significant change and the issue of false positive results does not
arise.
Figure 9. Statistical process control limits for individual values

The chart for individual values in Figure 9 has control limits that have been calculated using standard methods from SPC that
are based on the overall average and the moving range or difference between each sequential pair of values. The point for
August 29 and many of those that follow are above the upper control limit, confirming that the average has risen.
