Community surveys through space and time: testing the space-time interaction in the absence of replication

Transcription

1 Community surveys through space and time: testing the space-time interaction in the absence of replication Pierre Legendre Département de sciences biologiques Université de Montréal Pierre Legendre 207

2 Outline of the presentation. The general problem 2. Two-way anova for space (S) and time (T) crossed factors by RDA 3. How can we test STI without replication? 4. Anova models 5. Simulation study 6. Examples 7. R software 8. Reference Space-time interaction (STI)

3 . The general problem Ecologists sample portions of the environment repeatedly across time for ecological monitoring, or to test hypotheses about changes in the environment induced by man, including climatic change. Problem: space-time studies are usually done without replication to maximize the study extent and minimize effort. Example: 0 Time coordinates (T) Spatial coordinates along transect (S)

4 This talk will describe a method for testing a space time interaction in repeated ecological survey data, when there is no replication at the level of individual sampling units (sites). This methodological development is important for the analysis of long-term monitoring data, including systems under anthropogenic influence. In these systems, an interaction may indicate - that the spatial structure of the community composition has changed in the course of time - or that the temporal evolution is not the same at all sites. => Classical statistics tells us that one cannot test an interaction without replication. Space-time interaction (STI)

5 2. Two-way anova for space (S) and time (T) crossed factors by RDA Analyze Y against space and time without replication First method: write matrices coding for space and time using dummy variables. Example: Space-time interaction (STI)

6 Y = Species presence-absence or abundance (columns) X = Sampling times W = Covariables: dummy variables coding for sites Sites Sites s s Time Time etc etc. Sites s Time etc. Sites s Time etc. Sites s Time etc.

7 Why can t we test the space-time interaction?

8 We would still like to test the space-time interaction because a significant interaction would indicate that the temporal structures differ from site to site, or that the spatial structures differ from time to time. When the interaction is significant, we have to carry out separate analyses of the temporal variance for the different points in space, or separate analyses of the spatial variance for the different times. The absence of a significant interaction would indicate either that the differences among times can be modelled in the same way at all points in space, and conversely; or that there were not enough data to obtain a significant result for the test of the interaction (n too small, lack of power; type II error).

9 3. How can we test STI without replication? Using dummy variables to code for space and time, we did not have enough degrees of freedom, in the no-replication case, to test the S-T interaction. We can solve that problem by using a more parsimonious way of coding for space and time. We will use Moran s eigenvector maps (MEM). The type of MEM that we will use in this method were formerly called Principal Coordinates of Neighbour Matrices (PCNM),2. Dray, S., P. Legendre and P. R. Peres-Neto Spatial modelling: a comprehensive framework for principal coordinate analysis of neighbour matrices (PCNM). Ecological Modelling 96: Borcard, D. and P. Legendre All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling 53: 5-68.

10 MEM eigenfunctions represent a spectral decomposition of the spatial (or temporal) relationships among sampling sites (or times). They are orthogonal to one another, and fewer in number than dummy variables coding for the same sites (or times). To model the Space and Time variation, we will use s/2 or t/2 MEM functions. These MEM are those that are modelling positive spatial (or temporal) correlation. We leave out the MEM modelling negative spatial (or temporal) correlation. For example, 0 equispaced sampling times are modelled by the following 5 MEM functions: Borcard, D. and P. Legendre All-scale spatial analysis of ecological data by means of principal coordinates of neighbour matrices. Ecological Modelling 53: Borcard, D., P. Legendre, C. Avois-Jacquet and H. Tuomisto Dissecting the spatial structure of ecological data at multiple scales. Ecology 85:

11 Testing the interaction in cross-designs without replication While it takes (s ) dummy variables to represent s sites, fewer MEM variables are necessary to analyze the spatial variation; likewise for time.

12 Simulated univariate data. Data with an S-T interaction Transect of s = 25 points, t = 5 sampling campaigns. Spatially autocorrelated data were generated in a 00 x 00 pixel field using the program SimSSD. A transect of 25 equidistant points (spacing = 4 units) was sampled in the middle of the field. A transect variable was the sum of two simulated vectors, with spatial ranges of 0 and 30 units respectively. 5 independent time realizations of the transect were created. Legendre, Dale, Fortin, Gurevitch, Hohn and Myers Ecography 25: Legendre, Dale, Fortin, Casgrain and Gurevitch Ecology 85: Legendre, Borcard and Peres-Neto Ecological Monographs 75:

13 5 Time Time Time Time Time Coordinates along the transect There is an S-T interaction because the 5 time realizations were created independently of one another.

14 3 S-MEM functions were created to model the spatial variation along the 25 points of the transect. 3 T-MEM functions were created to model the temporal variation across the 5 sampling times. To model the interaction, 39 ST functions were obtained by multiplying each S-MEM by each T-MEM. Canonical RDA was used to test the interaction in the presence of the main factors S and T. Results The S-T interaction was significant: p = (after 9999 perm.) ü Correct answer The spatial structures differed from time to time. One would have to test the spatial structure of each sampling time separately.

15 Simulated univariate data 2. Data without an S-T interaction Transect of s = 25 points, t = 5 sampling campaigns. We took one of the transects (Time 5) from the previous data set and created 5 new time replicates by adding N(0, 0.3) error to all data values. Space-time interaction (STI)

16 5 Time Time Time Time Time Coordinates along the transect There is no S-T interaction because the 5 time realizations were all constructed from the same, common spatial structure.

17 Results The S-T interaction was not significant: p =.0000 (9999 perm.) The main factor Time was not significant: p = (9999 perm.) The main factor Space was significant: p = (9999 perm.) ü Correct answer Space-time interaction (STI)

18 A paper was published in 200 describing the method, reporting extensive simulation results in Appendices A and B, as well as two examples involving real ecological survey data: Legendre, P., M. De Cáceres and D. Borcard Community surveys through space and time: testing the space-time interaction in the absence of replication. Ecology 9:

19 4. Anova models ANOVA models Model Space Time Interaction SS(X s ) SS(X t ) SS(X Int ) Residuals SS Res Model 2 SS(X s ) SS(X t ) SS Res Model 3 SS(X u ) SS(X t ) SS(X Int3 ) SS Res Model 4 SS(X u ) SS(X v ) SS(X Int4 ) SS Res Model 5 SS(X s ) SS(X t ) SS(X Int4 ) SS Res Sum of squares partitioning in five Anova models for space-time analysis (Legendre et al. 200, Fig. ).

20 5. Simulation study Comparison of Anova models for testing the space-time interaction Detailed results in Appendices A and B of the paper. Type I error of Model 5 was always correct for univariate response data and asymptotically correct for multivariate data, while type I error rates of Models 3 and 4 were too low. Hence the power of Model 5 was always equal to or higher than those of Models 3 and 4. Space-time interaction (STI)

21 Recommendations. Perform first a test of the S-T interaction using Anova Model Then proceed as follows: If the hypothesis of no interaction is not rejected at a high significance level (subject to the possibility of type II error), i.e. the presence of an interaction is not demonstrated: => Use Model 2 to test for space and time effects without replication. If the hypothesis of no interaction is rejected: => Model the spatial structure of each time period in a separate way; model the temporal structure of each sampling site separately.

22 6. Examples Example Trichoptera captured in 22 emergence traps sampled during the summer of 984 in the outflow stream of lac Cromwell, Québec. Space-time interaction (STI)

23 Lake Cromwell Outflow stream Google Maps

24 22 emergence traps visited daily during 00 days Data: 56 Trichoptera species abundances The insect counts were pooled in 0-day periods (many zeros) Data transformation: y' = log(y + ) Analysis: analysis of space-time interaction by RDA => The interaction was highly significant (R 2 = 0.205, p = after 9999 permutations) How can we visualize an interaction in a figure? Analysis: K-means partioning => 5 groups of observations.

25 0-day time periods Current direction Traps

26 Example 2 Barro Colorado Island permanent forest plot in Panama. Fully censused tropical forest plot, 50 ha. We used the stem-based plot data covering four censuses (982-83, 985, 990, and 995) and counted the trees with cm diameter at breast height (dbh) or more, by species, in grid cells of m. 250 grid cells 35 tree species A map of the Center for Tropical Forest Science (CTFS) forest plots in Africa and Asia, Europe, Latin America and North America, and details about each plot, are available on the Web page

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28 Data transformation: y' = log(y + ). Test of S-T interaction on the multivariate data (35 species): significant result (p = 0.00 after 999 permutations). 2. Test of S-T interaction for each species separately: about 43% of the tree species had significantly changed their spatial structures across the four censuses. Space-time interaction (STI)

29 a) Poulsenia armata 0 Meters b) Beilschmiedia pendula Fig. 4a Changes in numbers of individuals for two species associated with slopes that significantly changed their distributions between the and 995 censuses. Light gray squares: loss; dark gray squares: gain of individuals. 500

30 b) Beilschmiedia pendula 0 Meters Fig. 4b Changes in numbers of individuals for two species associated with slopes that significantly changed their distributions between the and 995 censuses. Light gray squares: loss; dark gray squares: gain of individuals.

31 7. R software An R package, called STI, is available on Ecological Archives E09-09-S of the Ecological Society of America ( year 200), and on the page => In 207, the stimodels() and quicksti() functions have been included in the adespatial package available on CRAN. The new functions are much faster than the old ones because the permutation test has been rewritten in C. Space-time interaction (STI)

32 8. Reference Legendre, P., M. De Cáceres & D. Borcard Community surveys through space and time: testing the space-time interaction in the absence of replication. Ecology 9: Space-time interaction (STI)

33 End of the presentation Space-time interaction (STI)