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146 Statistics for Environmental Science and Management, Second Edition

Figure 5.11

Test for a change of shape in a distribution by counting points around a shifted line.

with 1 degree of freedom (df). This is significantly large at the 1% level, giving clear evidence of a shift in the general level of observations. Because most plotted points are above the 45 degree line, the observations tend to be higher at the second sample time than at the first sample time.

For the second test, the line at 45 degrees is shifted upward or downward so that an equal number of points are above and below it. Counts are then made of the number of points in four quadrats, as shown in Figure 5.11. The counts then form a 2 × 2 contingency table, and the contingency table chisquared test (Appendix 1, Section A1.4) can be used to see whether the counts above and below the line are significantly different. A significant result indicates a change in shape of the distribution from one time period to the next. In Figure 5.11 the observed counts are

10  10 10  10

These are exactly equal to the expected counts on the assumption that the probability of a point plotting above the line is the same for high and low observations, leading to a chi-squared value of zero with 1 df. There is, therefore, no evidence of a change in shape of the distribution from this test.

Finally, the third test involves dividing the points into quartiles, as shown in Figure 5.12. The counts in the different parts of the plot now make a 4 × 2 contingency table for which a significant chi-squared statistic again indicates a change in distribution. The contingency table from Figure 5.12 is shown in

Figure 5.12

Extension of the test for a change in the shape of a distribution with more division of points.

Table 5.7

Counts of Points above and below the Shifted 45 Degree Line Shown in Figure 5.12

Table 5.7. The chi-squared statistic is 0.80 with 3 df, again giving no evidence of a change in the shape of the distribution.

In carrying out these tests, it will usually be simplest to count the points based on drawing graphs like those shown in Figures 5.10 to 5.12 very accurately, with points on lines omitted from the counts.

Example 5.4:  The pH for Norwegian Lakes in 1976 and 1978

Figure 5.13 shows the 1978 pH values plotted against the 1976 values for the 44 lakes shown in Table 5.3 for which observations are available for both years. For the first of Stehman and Overton’s (1994) tests, it is found that there are 13 points that plot above, and 31 that plot below the 45 degree line. The chi-square statistic is therefore

(13 − 22)²/22 + (31 − 22)²/22 = 7.36

on whether the distribution of tree-ring widths appears to have been constant over the sampled period. The control chart for the mean should have 1 in 20 warning limits at the mean ±1.96 SÊ(x) and 1 in 500 action limits at the mean ±3.09 SÊ(x). The control chart for the range should be set up using the factors in Table 5.5.

Exercise 5.2

Use Stehman and Overton’s chi-squared tests as described in Section 5.9 to see whether the distribution of SO4 levels of Norwegian lakes seems to have changed from 1978 to 1981 and, if so, describe the nature of the apparent change. The data are in Table 1.1. It is simplest to plot the data, draw on the lines by hand, and count the points for this example.

6

Impact Assessment

6.1  Introduction

The before–after-control-impact (BACI) sampling design is often used to assess the effects of an environmental change made at a known point in time, and was called the optimal impact study design by Green (1979). The basic idea is that one or more potentially impacted sites are sampled both before and after the time of the impact, and one or more control sites that cannot receive any impact are sampled at the same time. The assumption is that any naturally occurring changes will be about the same at the two types of sites, so that any extreme changes at the potentially impacted sites can be attributed to the impact. An example of this type of study is given in Example 1.4, where the chlorophyll concentration and other variables were measured on two lakes on a number of occasions from June 1984 to August 1986, with one of the lakes receiving a large experimental manipulation in the piscivore and planktivore composition in May 1985.

Figure 6.1illustratesasituationwheretherearethreeobservationtimesbefore the impact and four observation times after the impact. Evidence for an impact is provided by a statistically significant change in the difference between the control and impact sites before and after the impact time. On the other hand, if the time plots for the two types of sites remain approximately parallel, then there is no evidence that the impact had an effect. Confidence in the existence of a lasting effect is also gained if the time plots are approximately parallel before the impact time, and then approximately parallel after the impact time, but with the difference between them either increased or decreased.

It is possible, of course, for an impact to have an effect that increases or decreases with time. Figure 6.2 illustrates the latter situation, where the impacted site apparently returns to its usual state by about two time periods after the impact.

As emphasized by Underwood (1994), it is desirable to have more than one control site to compare with the potentially impacted site, and where possible, these should be randomly selected from a population of sites that are physically similar to the impact site. It is also important to compare control sites

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Table 6.1

The Form of Results from a BACI Experiment with Three Observations before and Four Observations after the Impact Time

Note: Observations are indicated by X.

form of the data would be as shown in Table 6.1, which is for the case of three control and three impact sites, three samples before the impact, and four samples after the impact. For a repeated-measures analysis of variance, the two groups of sites give a single treatment factor at two levels (control and impact), and one within-site factor, which is the time relative to the impact, again at two levels (before and after). The repeated measurements are the observations at different times within the levels before and after, for one site. Interest is in the interaction between treatment factor and the time relative to the impact factor, because an impact will change the observations at the impact sites but not the control sites.

There are other analyses that can be carried out on data of the form shown in Table 6.1 that make different assumptions and may be more appropriate, depending on the circumstances (Von Ende 1993). Sometimes the data are obviously not normally distributed, or for some other reason a generalized- linear-model approach as discussed in Section 3.6 is needed rather than an analysis of variance. This is likely to be the case, for example, if the observations are counts or proportions. There are so many possibilities that it is not possible to cover them all here, and expert advice should be sought unless the appropriate analysis is very clear.

Various analyses have been proposed for the situation where there is only one control and one impact site (Manly 1992, chap. 6; Rasmussen et al. 1993). In the next section, a relatively straightforward approach is described that may properly allow for serial correlation in the observations from one site.

6.2  The Simple Difference Analysis with BACI Designs

Hurlbert (1984) highlighted the potential problem of pseudoreplication with BACI designs due to the use of repeated observations from sites. To