Letters

# Questions about floral (dis)integration

Article first published online: 21 MAY 2009

DOI: 10.1111/j.1469-8137.2009.02881.x

© The Author (2009). Journal compilation © New Phytologist (2009)

Additional Information

#### How to Cite

Harder, L. D. (2009), Questions about floral (dis)integration. New Phytologist, 183: 247–248. doi: 10.1111/j.1469-8137.2009.02881.x

#### Publication History

- Issue published online: 25 JUN 2009
- Article first published online: 21 MAY 2009

- Abstract
- Article
- References
- Cited By

### Keywords:

- eigenvalue variance;
- floral integration;
- floral morphology;
- null distribution;
- pollination

Successful pollen dispersal requires complex interactions for pollen to be loaded from flowers onto a pollen vector and then be deposited on stigmas as the vector subsequently encounters other flowers. These interactions involve many floral and inflorescence traits, so that coordination of their functions should promote pollen dispersal, leading to the hypothesis that angiosperm flowers are functionally integrated organs (reviewed by Armbruster *et al*., 2004). Several specialized pollination systems, such as heterostyly, secondary pollen presentation, and the fusion of anther(s) and stigma(s) into the orchid column and asclepiad gynostegium, provide obvious examples of floral integration, but what is the evidence for general integration? This question has been examined for almost 50 years (Berg, 1960), but with inconsistent results (reviewed by Armbruster *et al*., 2004; see Ashman & Majetic, 2006, for results based on genetic correlations), and variation in the extent of floral integration within angiosperms remains poorly understood. In this context, the recent article by Ordano *et al*. (2008) in *New Phytologist*, reporting a survey of 55 studies of floral integration, is of particular interest. Based on this survey, they concluded that ‘flowering plants have lower floral integration ... than expected by a randomly generated distribution’ (page 1189), a finding clearly at odds with the integration hypothesis. However, certain aspects of Ordano *et al*.'s analysis seem inappropriate and I demonstrate here that correction of this problem leads instead to strong evidence that flowers are usually integrated.

The studies surveyed by Ordano *et al*. (2008) used the population variance of eigenvalues, *V*(λ), from principal-components analyses of floral traits as an index of floral integration. In such an analysis, all eigenvalues will equal one when the *m* traits are uncorrelated (not integrated), resulting in a minimum variance of zero; whereas, if all traits are perfectly, positively correlated (highly integrated), the first eigenvalue would equal *m*, the number of variables, and the remaining eigenvalues would equal zero, resulting in the maximum possible variance (which coincidentally equals *m* numerically). To generate their random ‘null’ distribution, Ordano *et al*. calculated the integration index for simulated samples based on trait correlations ‘randomly chosen from a uniform probability distribution’ (page 1186) and found that the average ‘expected’ integration index was 32.7% (SD = 8.5%) of the maximum possible. Use of a uniform distribution proposes that when traits are uncorrelated strong correlations occur as often as weak correlations, owing solely to sampling error. For product–moment correlations, which are used in most studies of floral integration, such a uniform distribution is appropriate only for samples of four observations (Stuart & Ord, 1987). By contrast, the null distribution of product–moment correlations approaches a normal distribution as the sample size increases (Stuart & Ord, 1987), so strong correlations should occur rarely for studies with reasonable samples when the null hypothesis is true. As a result of their use of a uniform distribution of correlations, Ordano *et al*.'s simulations should generally have overestimated the expected integration index for randomly associated traits.

I illustrate this problem with the results from two sets of 1000 simulations for *n* = 30 observations of *m* = 5 traits: one using a uniform distribution of correlation coefficients; and the other based on the null distribution of product–moment correlation coefficients. The mean integration index for simulations based on uniform correlations is 26.7% of the maximum possible (Fig. 1a, white histogram) and of the magnitude of the results of Ordano *et al*., which were based on varying numbers of traits and observations. By contrast, the mean integration index based on the null distribution of product–moment correlations was an order of magnitude smaller, at 2.76% of the maximum (Fig. 1a, grey histogram). This outcome is consistent with the prediction of 100(*m* – 1)/*mn* = 2.67%, based on the expected variance of eigenvalues for product–moment correlations when the null hypothesis is true (Wagner, 1984; also see Chevrud *et al*., 1989: note that this expectation approaches an asymptotic maximum of 100/*n* as the number of traits considered increases).

In light of these results, the average integration index of 21.5% (SD = 15.4%), observed for 36 plant species in 16 families in the survey of Ordano *et al*. (Fig. 1b), indicates, in contrast to their conclusion, that flowers are much more highly integrated than expected based on random trait correlations. Indeed, over 80% of the observed integration indices exceed the maximum null simulation based on product–moment correlations (compare grey histograms in Fig. 1a,b), indicating that floral integration is the rule, rather than the exception. Thus, rather than questioning whether flowers are integrated, attention should now focus on the causes of the extensive variation in integration (Fig. 1b) and its reproductive consequences.

### References

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