Size Overlap

Topics

  1. Introduction to Size Overlap
  2. Data Format
  3. Defaults
  4. Options
  5. Output
  6. Caveats
  7. Size Overlap Tutorial
  8. Literature Cited


1. Introduction to Size Overlap

A seminal paper in the history of ecology is Hutchinson (1959): "Homage to Santa Rosalia, or why are there so many kinds of animals?" In this paper, Hutchinson described a visit to the shrine of Santa Rosalia in Palermo, Sicily. In the pool at the shrine, Hutchinson found three species of co-existing corixiid water beetles. He noticed that when the species were ordered from largest to smallest, the ratio of the body size of each species to the next smallest was about 1.3. He speculated that a body size ratio of 1.3 might represent a minimum size difference between animal species that was necessary to ensure coexistence. Species that are "too similar" in body size might not be able to coexist because they overlap too much in the use of shared resources. This modest suggestion spawned a vast amount of ecological research in which ecologists searched for patterns in the body sizes of coexisting species, often without an appropriate statistical analysis (Simberloff and Boecklen 1981; Wiens 1982).

This module of EcoSim allows you to test for unusual patterns in the body sizes of coexisting species, and to compare those patterns to what might be expected in a random assemblage that was not structured by competition.

This module can also be used to test a related hypothesis that has been important in the study of flowering plant communities: peak flowering times in a guild of coexisting plant species may be segregated in time (Stiles 1977). If competition for shared pollinators is important, each species will do best if flowering times are staggered through the season. Segregation of flowering times may also be favored if it reduces the chances of infertile hybridization between species (Levin 1971).

On the other hand, there may be other forces that promote the convergence of animal body sizes or the aggregation of plant flowering times. If certain resource states are very abundant, similarity in body size may be selected for because there is little competition for those resources. In plant communities, convergence in flowering times could result if flowering species act as "pollinator magnets" (Schemske 1981), or if species flower at certain times to avoid seed predators. EcoSim will allow you to evaluate whether patterns of body sizes or flowering times are aggregated, segregated, or random.

2. Data Format

The input for size overlap analyses is a matrix of body size measurements or flowering times. Each row is a different species and each column is a different site. In the size overlap analyses, each entry represents the body size of a particular species in a particular site. In phenology analyses, each entry represents the peak flowering time of a particular species in a particular site.

The entries for this module must be positive real numbers. It is not necessary to arrange the data in order from smallest to largest. EcoSim will sort the data before it performs its analyses. For phenology analyses, the data will frequently be in integer form (the week or day of peak flowering) and EcoSim has a special analysis option for handling integer data. Blanks (no entry) are also acceptable. EcoSim treats blanks as species that did not occur in particular sites, so they are ignored in the analysis.

As in all EcoSim modules, the first column is reserved for species names, and the first row is reserved for site names. See Importing Data for restrictions on species and site names.

Although EcoSim accepts an entire data matrix as input, it only analyzes a single column of data, which is chosen by the user for a particular run.

3. Defaults

EcoSim generates 1000 random matrices as the default. The default size overlap metric is the variance in segment length. This measures the overall tendency for body sizes to be evenly spaced on a number line. The default Body size distribution is a Parametric distribution, which is Log uniform and Data defined. This samples body sizes within a log-uniform range. For a set of n species, EcoSim uses the largest and smallest sizes as boundaries on a random uniform number line. It then simulates the placement of the remaining n - 2 species along the number line to create each random assemblage. The default transformation is Log, so that the null model tests for size ratio differences between adjacent species.

4. Options

Column to Analyze

You must specify which column of data you want to analyze in the size overlap module. One column of data is analyzed at a time. The default is column 2, the first column of data in the matrix (remember that the first column in your data set always contains the species labels).

Metrics

EcoSim offers you two choices for the size overlap metric. The default option is the variance in segment length. This metric measures the overall tendency for the observations to be evenly spaced, which would result in a variance that is significantly smaller than expected by chance.

The other metric you may select is the minimum segment length. With this metric, EcoSim scans the data matrix and finds the pair of adjacent species with the smallest size difference.

The variance in segment length is the most appropriate metric for analyses of size overlap in animal communities, although minimum size ratios may also be analyzed this way. The minimum size difference is the most appropriate metric for analyses of phenological overlap in plant (or animal) communities.

Body size distribution

EcoSim provides 4 possible parametric distributions from which body sizes can be drawn:

Uniform

Normal

Log uniform

Log normal

Both of the uniform distributions are defined by a minimum and a maximum value. Both of the normal distributions are defined by a mean and a standard deviation. If the distribution is data-defined and uniform, EcoSim uses the largest and smallest species in the data set to set the limits. If the distribution is data-defined and normal, EcoSim calculates the mean and standard deviation of the observed data. If these functions are chosen with user-defined, an edit box opens up for you to insert the minimum and maximum (for the uniform distribution) or the mean and standard deviation (for the normal). If you are using the log uniform or log normal distributions, be sure that the values you provide are also given on a logarithmic scale. EcoSim then samples from the specified distribution to create the body sizes of a null community. If the normal distribution is selected, negative values are discarded because these are not possible for body size measurements. However, negative values are retained for the log uniform or log normal distributions because these would correspond to small positive values (< 1.0) on an untransformed scale.

The logarithmic transformation leads to an analysis of size ratios, because of the relationship:

log(A/B) = log(A) - log(B)

Thus, analyzing differences on a log-scale is equivalent to analyzing ratios. EcoSim uses a base 10 logarithmic transformation, although the results would be identical with other logarithm bases.

Rounding

If the default option of no rounding is used, the simulated data are treated as continuous measures. This is appropriate for measures of body size. Treating the data as continuous means there is virtually no chance of a tie, which would generate a segment of length zero.

If the rounding option is chosen (only available for the Uniform or Normal body size distributions), the simulation still generates continuous observations, but these are then rounded to the nearest whole integer before the segment lengths are calculated. If the rounding option is chosen, there is a chance that two adjacent species in the null community will be the same size.

The rounding option is most important for phenological data, in which the day or week of peak flowering is recorded. You may need to rescale your data to an appropriate interval to use the integer option. For example, if you are measuring flowering times at intervals of 0.5 months, with a season from 1.0 to 3.0 months (1.0, 1.5, 2.0, 2.5, 3.0), you should rescale your data as integers from 1 to 5 (1, 2, 3, 4, 5) before analyzing them with EcoSim.

Data-defined and User-defined

For each of the 4 Parametric distributions of body size (Uniform, Normal, Log uniform, Log normal), EcoSim allows you to either estimate the parameters of the distribution from the data themselves (Data defined), or you can supply your own values (User defined).

1) Data-defined (Uniform or Log uniform) This is the simplest option, and the one that EcoSim uses for a default. For the uniform distributions, EcoSim uses the smallest and largest of the n species in the data to set the minimum and maximum boundaries for the simulation. It then randomly and uniformly selects a point greater than or equal to the smallest boundary and less than or equal to the largest boundary. It continues selecting points until it has accumulated n - 2 interior points. This randomized assemblage is then used to calculate segment lengths and other metrics.

2) User-defined (Uniform or Log uniform) When this option is chosen, a small edit box appears, and the user enters a minimum and a maximum body size, on either the untransformed or the log-10 transformed scale. The maximum body size must be greater than the minimum body size. If the data are untransformed, the minimum and maximum body sizes must also be greater than zero.

If the Log uniform has been chosen, EcoSim will accept negative numbers for the minimum and maximum. With the log-transform, be very careful to recognize that you are sampling from a log-uniform distribution, and the limits you choose should be appropriate for this scale.

EcoSim provides a set of "default" values: the default minimum is 10% less than the observed minimum, and the default maximum is 10% more than the maximum. If you have chosen the log transformation, the same rules hold for the log-transformed minimum and maximum. These are by no means "appropriate" default values, they are just examples of the sort of limits that might be expected for these data. You will need to use phylogenetic information (e.g., the largest and smallest species in a genus) and/or ecological information (e.g., the largest and smallest species in a biologically realistic source pool) to decide what these limits should be.

Once the user has set the limits, EcoSim will then simulate the placement of n species within these boundaries. Note the important difference between this and the "uniform option, in which EcoSim takes the observed minimum and maximum as the endpoints and then simulates the placement of the remaining n - 2 species.

3) Data-defined (Normal and Log normal) For these options, EcoSim will measure the mean and variance of the observed data (on either the untransformed or log-transformed scale) to define the normal distribution.

If the analysis is based on the (untransformed) Normal, EcoSim will discard any negative values that it draws. If the analysis is based on the Log normal, EcoSim will retain negative values, because these represent body sizes less than 1.0 (remember that the log10 of a positive number that is less than 1.0 is negative).

4) User-defined (Normal and Log normal) For this option, a dialog box opens and the user is asked to supply the mean and the standard deviation of a normal distribution that is used to create body weights for each null assemblage. The body weight of each of the n species is drawn from this normal distribution.

As before, you will need to be careful that the mean and standard deviation that you supply to EcoSim are appropriate for the transformation you have selected (none or log-transformed). The default value that EcoSim provides is the observed mean and standard deviation of the original data.

5) User-defined (Histogram) This option lets you construct a source pool with colonization weights and body sizes that are tailored to a particular assemblage. When this option is checked, a new screen of options appears. At the top, the number of species in the source pool is listed. The default value is the number of species in the data matrix. Clicking the change button allows you to add or remove rows. Obviously, the minimum number of species in the source pool must be at least as large as the number of species on the island. The file pull down menu allows you to save your source pool data to disk, or to load a previously created source pool data file.

EcoSim then lists each of the species in the source pool, their observed body sizes, and their colonization weights. All three of these entries can be changed by the user. The colonization weights must be positive real numbers that represent the relative chances that a particular species is drawn. The default value for each species is 1.0, meaning that all species have an equal chance of being drawn.

Once these parameters are set, EcoSim constructs each random assemblage by sampling randomly without replacement n species from the source pool list. The probability that a species is selected is proportional to its colonization weight. Once a species is chosen in a null assemblage, it cannot be drawn a second time.

The rounding option is not available when a user-defined source pool has been created.

5. Output

Input Column Tab

The Input tab shows you the original column of data with species names and body size or phenology records. You cannot edit the data in this window, but you can refer back to the original data set as you study the simulation results.

Simulation Tab

The simulation tab shows the ordered body sizes from one of the simulated data sets. If you used the default option of data-defined (Log uniform), the first and last rows give the logarithms of the observed body sizes for the smallest and largest species respectively, with random draws for the body sizes of the interior n-2 species.

Simulation Segments Tab

This tab shows the calculation of the segment lengths from the data in the Simulation Tab. Each entry is the segment length, calculated as the difference in body sizes of two adjacent species.

Size Histogram Tab

This histogram illustrates the distribution of body sizes that has been created by EcoSim according to the options that you have selected. This allows you to confirm that the body size distribution is appropriate for the null hypothesis that you have envisioned. To create this histogram, EcoSim randomly selects one of the n species that was created from each random assemblage and constructs the histogram of body sizes.

Note that the statistical tests of pattern in your data set are not compared to this histogram, but to the histogram of minimum segments or variances in segment length.

Size Overlap Tab

This tab gives the actual probability test in which the observed size overlap metric (minimum segment length or variance in segment length) is compared to the metric in the simulated communities. In the left-hand column of this tab, you will see an observed size overlap index.

The next three columns form the histogram window, which summarizes the distribution of the size overlap indices for the simulated communities. The first two columns give the low and high boundaries of 12 evenly spaced histogram bins. In the right-hand column, the number of simulations tells you how many of the simulated indices were in each bin. These integers sum up to the total number of iterations that were specified for the run.

The placement of the observed index shows you, graphically, where the observed size overlap index fell in the histogram distribution. You can use these data to plot the histogram and the observed value if you want to illustrate your results with a graph.

Summary Tab

The summary tab gives the simulation conditions, including the name of the input file, the randomization constraints, transformation, size overlap index, number of iterations, and random number seed. Next, it presents the information that was contained in the size overlap tab.

The summary window also supplies you with the standardized effect size, which is calculated as:

observed index - mean(simulated indices)/standard deviation(simulated indices)

This metric is analogous to the standardized effect size that is used in meta-analyses (Gurevitch et al. 1992). It scales the results in units of standard deviations, which allows for meaningful comparisons among different tests. Roughly speaking a standardized effect size that is greater than 2 or less than -2 is statistically significant with a tail probability of less than 0.05. However, this is only an approximation, and it assumes that the data are normally distributed, which is often not the case for null model tests. For any individual study, you should always report the actual tail probability, which is calculated directly from the simulation, and does not require any assumptions about normality of the data.

Finally, the summary tab shows the original column of body size data, with labels, that was analyzed.

All of these data can be edited, deleted, or annotated. The output can then be saved (Save to File) or discarded (Close). There is also a small time clock in the lower right-hand corner so you can tell how long your simulation took.

6. Caveats

Much of the early controversy over statistical analysis of size overlap data emphasized the shape of the underlying body size distribution from which random assemblages were constructed. Critics argued that the log-uniform distribution that was used by Simberloff and Boecklen (1981) was prone to Type II error and led to an incorrect acceptance of the null hypothesis. However, subsequent studies showed that the results are robust to the shape of the body size distribution in the null assemblage (Boecklen and NeSmith 1985, Tonkyn and Cole 1986). In fact, most data sets yield higher p values for log-normal versus log-uniform null distributions, suggesting that the uniform distribution is not prone to Type II errors.

A more important problem is deciding what the appropriate boundaries are for minimum and maximum body sizes. Using the observed data is the easiest choice, but it sacrifices two data points to fix the boundaries, and thereby reduces the power of the test, which is then based on the randomization of only n - 2 species. Phylogenetic or ecological limits may be very different from observed minimum and maximum body sizes in an assemblages, which will affect the simulation results.

More generally, the problem is that species are rarely independent of one another. This issue arises in all null model analyses of community structure, but it is especially important in the analysis of body size, because body sizes of closely related species are often very similar to one another. There are a variety of opinions on how, if at all, phylogenetic information should be incorporated into null model analyses (e.g, Harvey and Pagel 1991, Brown 1995, Kochmer and Handel 1986). The user-defined histogram offers the greatest potential for constructing realistic source pool distributions, but it may require a great deal of historical and phylogenetic information (e.g. Schluter and Grant 1984, Losos 1990).

7. Size Overlap Tutorial

Desert rodents

Launch EcoSim and you will see the familiar opening 5 x 5 matrix of species and sites. Use the file menu to open the file called "Desert rodents.txt". This data file gives the body size in grams of coexisting rodent species in the Sonoran and Great Basin deserts. These data come from Brown (1975). Brown and his colleagues have studied competitive interactions among desert rodents (and among rodents and ants) for many years at these sites (see Brown 1998 for a summary of this work). Experimental studies have established that, in some locations, rodents compete for seed resources (Brown et al. 1986). We can now use EcoSim to see if competition is manifest in the body sizes of coexisting species.

Each row is a different rodent species, and each column is a different site (Great Basin and Sonoran deserts). Each entry in the matrix is the average body size of a particular species in a particular site. A blank indicates that a species does not occur in a particular site. EcoSim ignores these blanks. In this module, it also ignores zeros or negative numbers.

As in most analyses of body size variation, these data ignore small scale among and within populations due to factors such as clinal variation, age and size structure of populations, and sexual size dimorphism.The tests in this module only analyze the pattern of spacing of body sizes or peak flowering time. If you have quantitative data on resource use or flowering times, you should use one of Ecosim's other modules for the analysis of niche overlap.

Understanding segment lengths

Before we can start analyzing the Brown data set, we need to understand how EcoSim uses size data to calculate patterns. For the Sonoran data set, the body sizes are:

7.2, 11.4, 17.1, 24.3, 45.3, and 120.

These are in order from smallest to largest. You do not have to enter your data this way, because EcoSim will sort them in order before it starts working. The first thing that EcoSim does it to create a set of segments from the set of body sizes. Each segment represents the difference in size between two consecutive species. Thus, for the Sonoran data set, the segments are:

4.2, 5.7, 7.2, 21.0, 74.7

The first segment is calculated as 11.4 - 7.2 = 4.2, and the last segment is calculated as 120 - 45.3 = 74.7. Because the body size data have been ordered from smallest to largest, the segments will always be non-negative numbers, but they need not increase in length. Whether the segments are large or small depends on whether two consecutive species are very similar in size (small segment) or very different in size (large segment). Also, notice that if there are n species in the community, there will be only n - 1 segment lengths created.

It is essential that you grasp the distinction between the original body sizes and these newly created segments. EcoSim will use a variety of randomization methods to create null communities in which body sizes are randomly chosen. However, the calculation of the pattern in body sizes is based on the segments, as we will explain.

Understanding size metrics

EcoSim gives you two choices for analyzing size patterns, both of which are based on segment lengths calculated between adjacent species.

Minimum segment length is the smallest segment for the assemblage. This measure describes the minimum spacing between adjacent species. Minimum segment length corresponds to Hutchinson's hypothesis that a minimum spacing between species is necessary for coexistence. Minimum segment length is also an appropriate for measuring spacing patterns in flowering phenology. If species are separated by a critical minimum, then the observed minimum segment length would be significantly larger than that predicted by the null model. On the other hand, if there is convergence in body size (perhaps because of common environmental or foraging constraints), the observed minimum might be smaller than expected by chance.

Whereas minimum segment length tests the hypothesis that there is a critical minimum separation necessary for coexistence, the variance in segment length tests the hypothesis that species sizes (or flowering phenologies) are evenly spaced, even if there is no particular minimum separation (Poole and Rathche 1979).

The more regular the spacing of species, the more similar the segments are to one another in size. Therefore, the more regular the spacing of species, the smaller the variance in segment length. For example, suppose we measure the peak flowering time for four coexisting plant species and find that the number of days into the season that the peak occurs is:

15, 30, 45, 60

The calculated segment lengths for this assemblage will be:

15, 15, and 15.

These numbers have a variance of zero, indicating perfectly even spacing of all species in the assemblage. Thus, if competition (or other forces) have led to unusual spacing of species, the observed variance should be significantly less than expected by chance. Conversely, if the variance is unusually large, it means that some species are very similar in body size (small segment length) and other adjacent species are very different in body size (large segment length).

Your first size overlap analysis

Now select size overlap from the analysis menu. Immediately switch to the "general" tab and set the random number seed to 10. Normally, you should use the default seed of 0, which instructs EcoSim to get a fresh random number seed from the system clock each time a new analysis is requested. In this case, by choosing a particular random number seed, your results will match up exactly with those in this tutorial. As always, EcoSim will remember your settings until you change them or restart the program.

Return to the Preferences tab and examine the choices that you have. The first choice is which column to analyze. In this module, each column (= site) is analyzed separately. That is, the analysis is not based on an archipelago of sites, as in the co-occurrence module. The second choice is the metric that you are going to use. You can select either the minimum segment length, which measures the spacing of the closest pair of species, or the variance of segment lengths, which measures the overall tendency for even spacing of all of the species. The next choice is the transformation. If the data are untransformed, you are testing for patterns in size differences between species. If the data are log-transformed, you are testing for patterns in the size ratios between species.

The next choice is the algorithm that creates the body size distribution. The default choice is Log uniform (data-defined). In this simulation, EcoSim uses the logarithm of the largest and smallest body sizes in your data set as fixed endpoints, and then chooses random uniform numbers between those limits to generate the logarithm of the body sizes of the other species. The default choice for the parameters of the log uniform distribution is Data defined, which is appropriate unless you have additional data or information on the underlying body size distribution limits.

After setting the random number seed to 10, keep all of the other default values and run the simulation on the Sonoran data set.

Output from size overlap

The first tab in the output window is labelled "Input column", and just shows you the original data set that you analyzed. The next tab, "Simulation", shows you one of the null assemblages with randomly chosen body sizes. At first glance these numbers appear very different, but remember that the default uses a log transformation of the data. EcoSim always uses a base 10 for this transformation, but the results would be identical with any other log base.

Notice that the smallest body size is 0.85733 and the largest body size is 2.07918. If you take the anti-logs of these numbers, you get 7.2 and 120, which were the largest and smallest species in the Sonoran data set. In this simulation model, the largest and smallest species always form the fixed endpoints of the distribution.

The next tab, "Simulation Segments", shows the corresponding segments for the simulated data. These segments are calculated as the difference between the sizes (log-transformed) of consecutive species. Note that with 6 species in the Sonoran data set, 5 segments are created.

The "Size Histogram" tab shows you the distribution of simulated body sizes, which are again displayed on a log scale. The window below the histogram gives the mean (1.45155) and variance (0.12999) of these values. For each of the 1000 iterations of this model, one of the body sizes that was randomly generated was chosen and used to construct this histogram.

Because the default null model specified a (log) uniform distribution of body sizes, there are approximately equal numbers of species in each of the bins of the size histogram. It is important to appreciate that although the simulation creates this distribution of body sizes, the statistical test is based on the properties of the segment lengths, calculated from all pairs of adjacent species in each simulation.

This statistical test is illustrated in the "Size Overlap" tab. This histogram shows the distribution of the variances of segment lengths for each of the 1000 simulated communities. The variance in segment length for the Sonoran data is 0.01193, shown in the first panel. The second two panels give the low and high cut points for 12 evenly spaced bins. The final column shows the frequency of simulations in each of the 12 bins. These integer values add up to 1000, the number of iterations specified.

The observed variance of 0.01193 was smaller than 944 of the simulated variances, generating a tail probability of 0.056. The observed variance in segment lengths for the Sonoran size data is suspiciously small, suggesting a pattern of constant body size ratios in the Sonoran rodents.

As always, a complete paper trail of your analysis can be found in the "Summary" tab, which can be annotated as a text window. Now that you have seen how EcoSim analyzes body size data, we will move on to a more detailed consideration of the simulation options and talk about how to interpret your output.

Understanding transformations

The choice of the transformation in size overlap is important because it dictates the type of pattern EcoSim will detect. If you do not transform the data, the simulations will reveal patterns in the absolute spacing of body sizes or flowering times. However, if you choose the log transformation, you are testing for patterns in the ratios of adjacent species.

Consider two species with average body sizes of A and B. If you use the untransformed analysis, each segment corresponds to the size difference:

A - B

However, with the log transformation, the segment length is:

log(A) - log(B)

A basic property of logarithms is:

log(A/B) = log(A) - log(B)

Therefore, the log transformation analyzes patterns in the size ratios of adjacent species.

The untransformed analysis would be most appropriate for the analysis of temporal flowering patterns because the absolute difference in the timing of flowering is important, not the ratio of flowering times. However, for Hutchinsonian analyses of body size, we might expect the ratio analysis to be more relevant.

These transformations can make quite a difference. For example, in the Brown Sonoran data set, the segment lengths for the untransformed data are:

4.2, 5.1, 7.2, 21.0, 74.7

These are quite heterogenous. If you repeat the analysis without the log transformation, you find a variance of 894.57, compared to an expected variance of 689.72. The tail probability for this pattern is 0.255, which does not suggest the distribution is unusual.

However, the sequence of segment lengths is suspicious because the larger the body sizes, the larger the segments between adjacent body sizes. This is exactly the sort of pattern we would expect if size ratios, rather than size differences, were evenly spaced. Using the log transformation, the segment lengths for the Sonoran data set are:

0.20, 0.18, 0.15, 0.27, 0.42

In contrast to the untransformed data, these segment lengths are fairly similar to one another. When you analyzed the transformed data, you found a variance of 0.01193, compared to an expected variance of 0.075. The tail probability for this pattern was 0.04806, suggesting that the variance in segment length on the log scale is somewhat smaller than expected by chance. This pattern is consistent with the hypothesis that body sizes of coexisting rodent species are evenly spaced, with a constant ratio of adjacent body sizes.

Understanding rounding

If you do not select the log transform, another option appears on the Size Overlap preferences screen for Rounding. If no rounding is chosen, then EcoSim draws sizes from a continuous number line. Therefore, there is essentially no chance that two species would have an identical value.

This is an appropriate choice when the data represent continuous body size measurements. However, if the data represent peak flowering times or other measures taken on an integer scale, the rounding option would be appropriate. In the simulated communities, fractional values are rounded to the nearest whole number. As a consequence, there is a chance of ties between species, particularly if the range of flowering times in the data is limited.

This option is not available for the Log normal and Log uniform, because the hypothesis of size ratios presumes that you have measured your data on a continuous scale. For size differences, and especially for flowering time differences, the rounding option is appropriate if you think that tied observations are likely. Of course, if there are ties in your continuous data, these observations weigh against the hypothesis of even or minimum spacing. Ties will increase the variance in segment length, and cause the minimum segment length be smaller than in the simulated communities.

Understanding Body size distributions

An important determinant of the outcome of your simulation is the shape of the body-size distribution, and the maximum and minimum sizes possible in the null model simulation. EcoSim gives you four options for setting distributions: Uniform, Normal, Log normal, and Log uniform. We have chosen the Log uniform as the default value, but the other distributions are also valid candidates. The normal distributions usually give larger p values than the uniform distributions applied to the same data set. For the Sonoran rodents, we can test the null hypothesis that the observed variance in segment length is no different than that expected from the null distribution. The alternative hypothesis is that the observed variance is unusually small, indicating constant size ratios in the assemblage. The tail values for this test are:

Body Size DistributionP value
Log uniform0.056
Log normal0.180
Uniform0.024
Normal0.317

The two normal distributions give the largest p values. The Normal distribution should be used cautiously when the mean is relative small and the variance is large (as in these data). Because negative values have to be tossed out, the mean and variance of the simulated distribution may not match the mean and variance of the observed data.

Results are more consistent for the test for a minimum size ratio. To conduct this test, select "Log" for the transformation, and "Minimum segment length" for the metric. In this case, we are testing whether the observed minimum size difference is unusually large compared to the null hypothesis:

Body Size DistributionP value
Log uniform0.018
Log normal0.030
Uniform0.007
Normal0.002

Moving beyond the data-defined choices allows the user to set the endpoints of the uniform distribution or the means and variances of the normal distribution. These values could be based on other phylogenetic or biogeographic information about the assemblage.

What are the effects of user-defined endpoints or means and variances on the null model test? If the new boundaries are very far removed from the real data, the observed variance will be large because the first and last segments will be very large compared to all the rest. On the other hand, if the endpoints are very close to the largest and smallest species in the assemblage, the observed variance will also increase because the first and last segment will be very small compared to all the rest. The variance will be minimized when the distances from the terminal species to the endpoints are most similar to the average distance between any pair of consecutive species.

Thus, adding user-defined endpoints may cause a data set to show patterns that are more extreme or less extreme than the data-defined endpoints. In general, if you can supply user-defined endpoints that are biologically realistic, your test will be more powerful because you won't have to sacrifice two of your data points to establish the endpoints; the test will be based on the randomization of n species rather than n - 2.

The final body size option (user-defined (histogram)) let's you "roll your own" and create a source pool of possible colonizing species. When you check this option, an edit window will open in which each row is a species. There are two columns, one for the size, and one for the colonization weight. Initially, EcoSim will show you only the observed species, each with a weight of 1.0, but you will certainly want to add additional species to the hypothetical source pool. To add species, click the "change" button in the upper right hand corner and increase the number. You can also load in a file from disk that you may have created previously. For each species in the source pool, assign a size and a colonization weight. Each weight must be a positive number. The weight describes the colonization probability of each species relative to all others in the source pool. Thus, the default values of 1.0 mean that each species has an equally likely chance of being selected. If you increase the weight for one of the first species to 3.0, it has 3 times as great a chance of being represented in the null community as any other species.

Once the species list is completed and the weights are assigned, EcoSim uses this information to construct each random community. n species are drawn at random from the source pool. The probability that a species is drawn is directly proportional to its assigned weight. Once a species has been drawn, it will not be drawn again for that particular null assemblage.

Once an assemblage is created in this way, EcoSim will then calculate segment lengths, variances and other statistics as you have specified in your other preferences.

This module lets you create a customized source pool from which null communities are assembled and compared to the real data. Some possibilities here would be to create a source pool based on species that occur within a certain radius of the site (Graves and Gotelli 1983). Using data on seed availability and size, Schluter and Grant (1984) constructed null models that specified the probability that a certain sized species would occur in an assemblage that was not structured by competition. The weighting functions could be used to create similar models for other taxa. Such tests are potentially very powerful, but they require considerably more data than a list of the body sizes of coexisting species.

Alpine plants

As an illustration of the analysis of flowering peaks, open the data set "RMBL plant data.txt". This is an idealized data set that gives the approximate flowering peak of 13 common alpine wildflowers near the Rocky Mountain Biological Laboratory, Crested Butte, CO. We thank Alison Brody for constructing this data set for us. Each entry gives the common name of the species and the approximate week in the summer flowering season when the species peaks in its flowering. Week 1 is defined as the start of the growing season, approximately the first week of June.

For the analysis of these data, use size differences, not size ratios, so there will be no transformation. In addition, you should use the rounding option because the data are discrete, not continuous.

First, try analyzing minimum segment length. You will see that all the simulations generate a minimum segment length of 0.0, due to ties in the peak flowering times of species in both the simulated and the observed data set.

Now analyze the pattern for the variance in segment length. Using the random number seed of 10, you will see that the observed variance (0.35897) is less than expected (0.81081), but not significantly so (p = 0.0910). For this idealized data set, there is no evidence of unusually even spacing in flowering times.

8. Literature Cited

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Brown, J. H. 1998. The desert granivory experiments at Portal. Pages 71-95 in W. J. Resetarits, Jr. and J. Bernardo, eds. Experimental ecology: issues and perspectives. Oxford University Press, New York.

Brown, J. H., D. W. Davidson, J. C. Munger, and R. S. Inouye. 1986. Experimental community ecology: the desert granivore system. Pages 41-61 in J. Diamond and T. J. Case, eds. Community ecology. Harper and Row, New York.

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Levin, D. A. 1971. The origin of reproductive isolating mechanisms in flowering plants. Taxon 20: 91-113.

Losos, J. B. 1990. A phylogenetic analysis of character displacement in Caribbean Anolis lizards. Evolution 44: 558-569.

Schemske, D. W. 1981. Floral convergence and pollinatory sharing in two bee-pollinated tropical herbs. Ecology 62: 946-954.

Schluter, D., and P. R. Grant. 1984. Determinants of morphological patterns in communities of Darwin's finches. The American Naturalist 123: 175-196.

Simberloff, D., and W. Boecklen. 1981. Santa Rosalia reconsidered: size ratios and competition. Evolution 35: 1206-1228.

Stiles, F. G. 1977. Coadapted competitors: the flowering seasons of hummingbird-pollinated plants in a tropical forest. Science 198: 1177-1178.

Tonkyn, D. W., and B. J. Cole. 1986. The statistical analysis of size ratios. The American Naturalist 128: 66-81.

Wiens, J. A. 1982. On size ratios and sequences in ecological communities: Are there no rules? Annales Zoologici Fennici 19: 297-308.


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