Sunday, March 29, 2009

How to analyze limitation experiments

Figure 4.2 from RSWP

Resource limitation in plants can only truly be assessed with resource addition experiments. Many other correlates with resource limitation have been proposed, but to tell for sure, you have to add the resource in question. Although resource addition can persist in some cases even after the addition of the resource, this is the best approach. 

In RSWP, I discuss how the patterns of responses to resource addition can be used to help constrain the mechanisms by which the resources limit plants. The case in point was “co-limitation”, generally defined where more than one resource limits plant production. Yet, there can be many patterns to co-limitation (see above figure). For example, plants can be co-limited by two resources if the two resources are supplied in the exact ratio they are demanded by plants. Alternatively, plants can be co-limited by two resources if there are trade-offs in allocation to acquire the two resources. For example, plants might need to acquire both water deep in the soil and N shallow in the soil, but the same unit of biomass cannot be allocated to acquire both. Serial limitation exists when resource A limits biomass first and after limitation by A is relieved, resource B limits biomass.

There is a lot to learn from past and future resource addition experiments by beginning to analyze the patterns of limitation. Yet, one thing I hadn’t developed was how one does this statistically.

Most resource addition experiments are analyzed with a two-way ANOVA, which compares the marginal changes in biomass with resource addition relative to the biomass of plants without resources added. If one of these marginal changes is significant, then the limitation is considered “single-resource” limitation. If both are significant, there is co-limitation. If only the interaction term is significant (no response to either resource A or B, but A and B together), then again there is co-limitation.

The problem with this approach is when you have a significant effect of one resource and a positive interaction term. Addition of resource A increases biomass by a certain amount, addition of both A and B increase biomass by even more. A significant main effect of one resource and a positive interaction term could signify either serial limitation or co-limitation by supply. If the main effect of resource A is small, then the patterns imply co-limitation by supply. If the main effect of resource A is large relative to the interaction, then there is serial limitation.

To differentiate these two cases with an ANOVA, one would have to set an arbitrary cut-off for what a “small” significant main effect. For example, if biomass increases less than 15% with resource A addition then the statistically significant main effect is not biologically significant and we can assume co-limitation by supply.

This approach is tenable, but arbitrary. A better way to analyze resource addition experiments is by comparing levels. Not ANOVA’s, but Tukey’s tests that determine with categories are significantly different. This approach is functionally the same as ANOVA in many ways, but allows one to separate the two types of co-limitation. All one has to do is calculate the average of biomass with no additional resources added and biomass with both resources added. Then one compares the amount of biomass with one resource added to the average of control and 2-resources. If it’s less, co-limitation by supply. If it’s more, serial limitation. Determining whether the levels are significantly more or less than the average requires combining errors, which is simple to do, although it remains to be seen how many experiments have enough power to separate the cases.

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