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140 Statistics for Environmental Science and Management, Second Edition

With 108 observations altogether, it is expected that about five points will plot outside the warning limits. In fact, there are ten points outside these limits, and one point outside the lower action limit. It appears, therefore, that the mean pH level in the South Island of New Zealand was changing to some extent during the monitored period, with the overall plot suggesting that the pH level was high for about the first two years, and was lower from then on.

The range chart has seven points outside the warning limits, and one point above the upper action limit. Here again there is evidence that the process variation was not constant, with occasional spikes of high variability, particularly in the early part of the monitoring period.

As this example demonstrates, control charts are a relatively simple tool for getting some insight into the nature of the process being monitored, through the use of the action and control limits, which indicate the level of variation expected if the process is stable. See a book on industrial process control such as the one by Montgomery (2005) for more details about the use of x and range charts, as well as a number of other types of charts that are intended to detect changes in the average level and the variation in processes.

5.8  Detection of Changes Using CUSUM Charts

An alternative to the x chart that is sometimes used for monitoring the mean of a process is a CUSUM chart, as proposed by Page (1961). With this chart, instead of plotting the means of successive samples directly, the differences between the sample means and the desired target mean are calculated and summed. Thus if the means of successive samples are x1, x2, x3, and so on,

then S1 = (x1 − μ) is plotted against sample number 1, S2 = (x1 − μ) + (x2 − μ) is plotted against sample number 2, S3 = (x1 − μ) + (x2 − μ) + (x3 − μ) is plot-

ted against sample number 3, and so on, where μ is the target mean of the process. The idea is that because deviations from the target mean are accumulated, this type of chart will show small deviations from the target mean more quickly than the traditional x chart. See MacNally and Hart (1997) for an environmental application.

Setting control limits for CUSUM charts is more complicated than it is for an x chart (Montgomery 2005), and the details will not be considered here. There is, however, a variation on the usual CUSUM approach that will be discussed because of its potential value for monitoring under circumstances where repeated measurements are made on the same sites at different times. That is to say, the data are like the pH values for a fixed set of Norwegian lakes shown in Table 5.3, rather than the pH values for random samples of New Zealand rivers shown in Table 5.6.

This method proceeds as follows (Manly 1994; Manly and MacKenzie 2000). Suppose that there are n sample units measured at m different times. Let xij be the measurement on sample unit i at time tj, and let xi be the mean of all the measurements on the unit. Assume that the units are numbered in order of their values for xi, to make x1 the smallest mean and xn the largest mean. Then it is possible to construct m CUSUM charts by calculating

for j from 1 to m and i from 1 to n, and plotting the Sij values against i, for each of the m sample times.

The CUSUM chart for time tj indicates the manner in which the observations made on sites at that time differ from the average values for all sample times. For example, a positive slope for the CUSUM shows that the values for that time period are higher than the average values for all periods. Furthermore, a CUSUM slope of D for a series of sites (the rise divided by the number of observations) indicates that the observations on those sites are, on average, D higher than the corresponding means for the sites over all sample times. Thus a constant difference between the values on a sample unit at one time and the mean for all times is indicated by a constant slope of the CUSUM going either up or down from left to right. On the other hand, a positive slope on the left-hand side of the graph followed by a negative slope on the righthand side indicates that the values at the time being considered were high for sites with a low mean but low for sites with a high mean. Figure 5.7 illustrates some possible patterns that might be obtained, and their interpretation.

Randomization methods can be used to decide whether the CUSUM plot for time tj indicates systematic differences between the data for this time and the average for all times. In brief, four approaches based on the null hypothesis that the values for each sample unit are in a random order with respect to time are:

1.A large number of randomized CUSUM plots can be constructed where, for each one of these, the observations on each sample unit are randomly permuted. Then, for each value of i, the maximum

and minimum values obtained for Sij can be plotted on the CUSUM chart. This gives an envelope within which any CUSUM plot for real data can be expected to lie, as shown in Figure 5.8. If the real data plot goes outside the envelope, then there is clear evidence that the null hypothesis is not true.

2.Using the randomizations, it is possible to determine whether Sij is significantly different from zero for any particular values of i and j.

Thus if there are R − 1 randomizations, then the observed value of •Sij• is significantly different from zero at the 100α% level if it is among

the largest 100α% of the R values of •Sij• consisting of the observed value and the R − 1 randomizations.

3.To obtain a statistic that measures the overall extent to which a CUSUM plot differs from what is expected on the basis of the null hypothesis, the standardized deviation of Sij from zero,

Zij = Sij/√Var(Sij)

is calculated for each value of i. A measure of the overall deviation of the jth CUSUM from zero is then

Zmax,j = Max(•Z1j•, •Z2j•, …, •Znj•)

The value of Zmax,j for the observed data is significantly large at the 100α% level if it is among the largest 100α% of the R values consist-

ing of itself plus R − 1 other values obtained by randomization.

4.To measure the extent to which the CUSUM plots for all times differ from what is expected from the null hypothesis, the statistic

Zmax,T = Zmax,1 + Zmax,2 + … + Zmax,m

can be used. Again, the Zmax,T for a set of observed data is significantly large at the 100α% level if it is among the largest 100α% of the

R values consisting of itself plus R − 1 comparable values obtained from randomized data.

See Manly (1994) and Manly and MacKenzie (2000) for more details about how these tests are made.

Missing values can be handled easily. Again, see Manly (1994) and Manly and MacKenzie (2000) for more details. For the randomization tests, only the observations that are present are randomly permuted on a sample unit.

The way that the CUSUM values are calculated using equation (5.3) means that any average differences between sites are allowed for, so that there is no requirement for the mean values at different sites to be similar. However, for the randomization tests, there is an implicit assumption that there is negligible serial correlation between the successive observations at one sampling site, i.e., there is no tendency for these observations to be similar because they are relatively close in time. If this is not the case, as may well happen for many monitored variables, then it is possible to modify the randomization tests to allow for this. A method to do this is described by Manly and MacKenzie (2000), who also compare the power of the CUSUM procedure with the power of other tests for systematic changes in the distribution of the variable being studied. This modification will not be considered here.