Multiple Sampling Locations with Multiple Observations

The preceding methods involve a single sampling location (station). However,

environmental data often consist of sets of data collected at several sampling locations (see Box 4-11). For example, data are often systematically collected at several fixed sites on a lake or river, or within a region or basin. The data collection plan (or experimental design) must be systematic in the sense that approximately the same sampling times should be used at all locations.

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In this situation, it is desirable to express the results by an overall regional summary statement across all sampling locations. However, there must be consistency in behavioral characteristics across sites over time in order for a single summary statement to be valid across all sampling locations. A useful plot to assess the consistency requirement is a single time plot (Section 2.3.8.1) of the measurements from all stations where a different symbol is used to represent each station.

Box 4-11: Data for Multiple Times and Multiple Stations

Let i = 1, 2, ..., n represent time, k = 1, 2, ..., K represent sampling locations, and Xik represent the measurement at time i for location k. This data can be summarized in matrix form, as shown below.

Stations 1 2 . . . K Time 1 2 . . . n X11 X21 . . . Xn1 X12 X22 . . . Xn2 . . . . . . . . . . . . . . . . . . X1K X2K . . . XnK S1 V(S1) Z1 S2 V(S2) Z2 . . . . . . . . . SK V(SK) ZK where Sk = Mann-Kendall statistic for station k (see STEP 3, Box 4-7),

V(Sk) = variance for S statistic for station k (see STEP 2, Box 4-9), and Zk = S_k/ VAR(S_k)

If the stations exhibit approximately steady trends in the same direction (upward or downward), with comparable slopes, then a single summary statement across stations is valid and this implies two relevant sets of hypotheses should be investigated:

Comparability of stations. H₀: Similar dynamics affect all K stations vs. H_A: At least two stations exhibit different dynamics.

Testing for overall monotonic trend. H₀*: Contaminant levels do not change over time vs. H_A': There is an increasing (or decreasing) trend consistently exhibited across all stations.

Therefore, the analyst must first test for homogeneity of stations, and then, if homogeneity is confirmed, test for an overall monotonic trend.

Ideally, the stations in Box 4-11 should have equal numbers. However, the numbers of observations at the stations can differ slightly, because of isolated missing values, but the overall time periods spanned must be similar. This guidance recommends that for less than 3 time periods, an equal

number of observations (a balanced design) is required. For 4 or more time periods, up to 1 missing value per sampling location may be tolerated.

a. One Observation per Time Period. When only one measurement is taken for each time period for each station, a generalization of the Mann-Kendall statistic can be used to test the above hypotheses. This procedure is described in Box 4-12.

Box 4-12: Testing for Comparability of Stations and an Overall Monotonic Trend Let i = 1, 2, ..., n represent time, k = 1, 2, ..., K represent sampling locations, and Xik represent the measurement at time i for location k. Let " represent the significance level for testing homogeneity and "* represent the significance level for testing for an overall trend.

STEP 1: Calculate the Mann-Kendall statistic Sk and its variance V(Sk) for each of the K stations using the methods of Section 4.3.4.1, Box 4-9.

STEP 2: For each of the K stations, calculate Z_k ' _S

k/ V(Sk).

STEP 3: Calculate the average Z¯ ' _j

K

k'1

Z_k/K.

STEP 4: Calculate the homogeneity chi-square statistic P2_h ' j K

k'1

Z_k2 & _{K Z}¯2 .

STEP 5: Using a chi-squared table (Table A-8 of Appendix A), find the critical value for P2 _{with (K-1) degrees} of freedom at an " significance level. For example, for a significance level of 5% and 5 degrees of freedom, P2

( 5 ) = 11.07, i.e., 11.07 is the cut point which puts 5% of the probability in the upper tail of a chi-square variable with 5 degrees of freedom.

STEP 6: If P2h#P2( K - 1 ), there are comparable dynamics across stations at significance level ". Go to Step 7. If P2h > P2( K - 1 ), the stations are not homogeneous (i.e., different dynamics at different stations) at the significance level ". Therefore, individual "*-level Mann-Kendall tests should be conducted at each station using the methods presented in Section 4.3.4.1.

STEP 7: Using a chi-squared table (Table A-8 of Appendix A), find the critical value for P2 _{with 1 degree of} freedom at an " significance level. If

K Z¯2 > O2_{(1 )},

then reject H0* and conclude that there is a significant (upward or downward) monotonic trend across all stations at significance level "*. The signs of the Sk indicate whether increasing or decreasing trends are present. If

K Z¯2 #O2₍₁₎,

there is not significant evidence at the "' level of a monotonic trend across all stations. That is, the stations appear approximately stable over time.

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b. Multiple Observations per Time Period. If multiple measurements are taken at some times and station, then the previous approaches are still applicable. However, the variance of the statistic S_k must be calculated using the equation for calculating V(S) given in Section 4.3.4.2. Note that S_k is computed for each station, so n, w_p, g, h, and u_q are all station-specific.