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, Miquel De Cáceres & Daniel Borcard Département de sciences biologiques, Université de Montréal College of the Environment and Ecology, Xiamen University, Xiamen, Fujian, China, May 7, 206

2 . 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 ecological studies are usually done without replication to maximize the study extent and minimize effort. Example: 0 Time coordinates (T) Spatial coordinates along transect (S)

3 . The general problem 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.

4 . The general problem 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.

5 2. Two-way anova for space (S) and time (T) crossed factors by canonical analysis (RDA)

6 Canonical redundancy analysis (RDA) The most common application of RDA in ecology is to test the relationship between a response Y and explanatory variables X: Simple RDA Response table Explanatory table Y Community composition data X Example: Environmental variables Partial RDA Response table Explanatory table Covariables Y Community composition data X Example: Environmental variables W Example: Spatial base functions

7 RDA as multivariate Anova RDA can be used to test the relationship between a response data matrix Y containing, for example, community composition data, and one or several experimental factors (crossed balanced designs) in matrix X. It is then a form of multivariate anova. Legendre, P. and M. J. Anderson Distance-based redundancy analysis: testing multispecies responses in multifactorial ecological experiments. Ecological Monographs 69: -24.

8 The anova factors can be coded in two different ways: () Coding for a single factor A with 3 levels: binary dummy variables Matrix X Factor A (3 levels) Factor A (3 levels) y = y y 2 y 3 y 4 y 2 y 22 y 23 y 24 y 3 y 32 y 33 y

9 The anova factors can be coded in two different ways: (2) Helmert contrasts Factor with 2 levels: variable Factor with 3 levels: 2 variables Factor with 4 levels: 3 variables Factor with 5 levels: 4 variables

10 The anova factors can be coded in two different ways: () Coding for a single (2) Coding for two crossed factors factor A with 3 levels: and their interaction: dummy variables Helmert contrasts A y 0 y 2 0 y 3 0 y 4 0 y 2 0 y = y 22 y 23 X = 0 0 y 24 0 y y y y

11 Tests of significance Test of a single factor Response table Y Community composition data Explanatory table X Factor to be tested Two crossed factors. Example: test of the interaction Response table Explanatory table Covariables Y Community composition data X Interaction to be tested W Main factors and other interactions

12 Analyze Y against space and time without replication First method: write matrices coding for space and time using dummy variables. Example:

13 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.

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

15 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).

16 3. How can we test the space-time interaction in analyses of Y against space (S) and time (T) without replication?

17 Analyze Y against space and time 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.

18 Analyze Y against space and time 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.

19 MEM eigenfunctions represent a spectral decomposition of the spatial (or temporal) relationships among sites (or times). Example dbmem along a transect: what do they look like? dbmem dbmem dbmem dbmem dbmem dbmem

20 dbmem dbmem 2 Example of a regular grid with 8 2 = 96 points. Maps showing ten of the 48 dbmem eigenfunctions that display positive spatial correlation. Shades of grey: values in each eigenvector, from white (largest negative value) to black (largest positive value). dbmem 3 dbmem 4 dbmem 5 dbmem 6 dbmem 0 dbmem 20 dbmem 30 dbmem 40

21 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). 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:

22 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:

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24 Sampling each point in space (S) during each sampling campaign (T) creates an orthogonal design. For that reason, the MEM eigenfunctions, which are orthogonal within each set (S, T), are also orthogonal between sets. The S-T interaction can be modelled by creating variables that are the products of each S-MEM by each T-MEM. The S-T interaction variables are orthogonal to the S-MEMs and the T-MEMs.

25 Sampling each point in space (S) during each sampling campaign (T) creates an orthogonal design. For that reason, the MEM eigenfunctions, which are orthogonal within each set (S, T), are also orthogonal between sets. The S-T interaction can be modelled by creating variables that are the products of each S-MEM by each T-MEM. The S-T interaction variables are orthogonal to the S-MEMs and the T-MEMs. Orthogonality of factors = happiness for statisticians Each portion of the model (main factor, interaction) accounts for a linearly independent fraction of variation, which is easier to interpret.

26 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.

27 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:

28 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.

29 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.)

30 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.

31 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.

32 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.

33 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

34 A paper was published in 200 describing the method, reporting extensive simulation results (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:

35 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. ).

36 Comparison of Anova models for testing the space-time interaction 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. 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.

37 We can test the significance of the temporal structures that were independently generated at the 30 sites along the transect, as follows: Times Times t t Y Simulated response variable Site Site 2 T-PCNM X Model of independent temporal structures at the various sites T-PCNM Times t Site 3 0 T-PCNM Times t Site s T-PCNM Result: p = (block permutations in Canoco, 9999 perm.) The independent temporal structures, at the 30 sites, are significant.

38 Likewise, we can test the significance of the spatial structures that were independently generated at the 0 sampling times, as follows: Sites Sites Sites s s s Y Simulated response variable Time Time 2 Time 3 S-PCNM X Model of independent spatial structures at the various times S-PCNM S-PCNM Sites s Time t S-PCNM Result: p = (block permutations in Canoco, 9999 perm.) The independent spatial structures, at the 0 times, are significant.

39 Example Trichoptera captured in 22 emergence traps sampled during summer 984 in outflow stream of lac Cromwell, Québec Google Maps

40 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) Analysis: K-means partioning to visualize the interaction => 5 groups of observations.

41 0-day time periods Current direction Traps

42 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, and details about each plot, are available on the Web page

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44 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 0 mm diameter at breast height (dbh) or more, by species, in grid cells of m. 250 grid cells 35 tree species 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. A map of the Center for Tropical Forest Science (CTFS) forest plots, and details about each plot, are available on the Web page

45 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

46 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.

47 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

48 Avez-vous des questions? Do you have questions?