Brodiaea santarosae
The Santa Rosa Basalt Brodiaea

Tom Chester, Wayne Armstrong and Kay Madore

Principal Components Analysis of Four Brodiaea Species

The measurements of seven floral parameters (perianth lobe, perianth tube, filament, anther, ovary, style, and staminode) were analyzed for principal components, with the results separated by geographic populations in Fig. 3. These parameters, along with a number of other parameters, are explicitly analyzed separately as well.

Fig. 3. Principal Components Analysis of floral parameters not scaled by standard deviation. The mean values with their one-standard deviation error bars are shown for the eleven geographically-distinct populations given in Table 1. Rectangles delineate individual taxa with more than one population; circles denote the mean for each taxon. To convert to PCA values from floral parameters scaled by standard deviation, multiply PCA1 by 0.389 mm-1 and PCA2 by 0.425 mm-1. Zoomable PDF version

We removed the mean from each floral parameter and then computed the principal components in two ways, once dividing each parameter by its standard deviation and once without such division. Since the resulting plots were essentially identical except for scale (r2=0.95 for PCA1 and 0.96 for PCA2), we present here the version without such scaling which allows the coefficients to be more easily interpreted. The scale factors that will closely give the other version are given in the caption for Fig. 3.

The coefficients of the first two principal components (PC1 and PC2) are given in Table 1, along with the variation in PC2 contributed by each variable. These two components accounted for 89% of the total variance in the data.

Principal Component 1 is a length parameter, essentially a scale factor times the perianth length plus the portion of the length of the floral parts that is correlated to the perianth length. Principal Component 2 is essentially a scale factor times the lengths of the filament plus the style minus the perianth tube minus the perianth lobe minus the staminode, where the staminode contributes roughly twice the variation of the other parameters. The variable staminodes of B. santarosae account for most of the spread in values of PC2 for that taxon. Table 1 also shows the variation in PC2 contributed by each parameter.

The error bars in Fig. 3 were computed by dividing the population standard deviation for each species by (n-1)1/2, where n is the number of individuals in each population. Note that some populations, such as B. filifolia and B. filifolia X B. orcuttii from San Marcos, are represented by only three specimens, which accounts for some of the larger error bars.

Fig. 3 shows four distinct taxa that are nearly equally distant from each other, the three distinct clusters denoted by labeled rectangles and the point corresponding to B. terrestris ssp. kernensis. The B. filifolia X B. orcuttii specimens plot almost exactly intermediately between the parent species, as expected for F1 hybrids. Since these specimens are found only in the single location where the parent species occur together, the evidence for their hybrid origin is strong.

The separation of the taxa in Fig. 3 demonstrates that B. santarosae is a taxon as distinct as any of the other three taxa, with characteristics not intermediate to any of those three taxa. In particular, these specimens are not direct hybrids of B. filifolia and B. orcuttii, nor are they part of any hybrid swarm between those species. Our discovery of clear hybrids between B. filifolia and B. orcuttii, from the only location where the populations overlap, confirms these conclusions.

The clusters denoted by the labeled rectangles in Fig. 3 are robustly defined. Note in particular the very small error bars on the PC2 values for all taxa except B. santarosae; within each of those clusters, all populations show extremely good concordance for PC2 in Fig. 3, reflecting the uniform staminodes of each taxon. The populations of B. santarosae also show good concordance, with each having significant spreads in the values of PC2, reflecting the staminode variability of that species being consistent between populations.

For both PC1 and PC2, the mean of each population is consistent with the corresponding species mean (Fig. 3), except for the B. orcuttii population with the highest value of PC1. That population (San Marcos) is 3.3 standard deviations away from the species mean value. Since PC1 is essentially the perianth length, this simply means that the San Marcos flowers in our sample were significantly larger than the other two populations from Miramar and Cuyamaca Lake.

Although the San Marcos specimens possessed somewhat-larger flowers, they are otherwise consistent with the other two B. orcuttii populations in all respects. In particular, they have the distinctive properties of B. orcuttii described below, including the complete absence of staminodes. Herbarium specimens show a much smaller separation of San Marcos flowers from the other two populations in other years, indicating that the apparent difference in our samples was either sampling variation or perhaps related to the unusual rainfall pattern in 2006.

Table 1. Coefficients for Principal Components PC1 and PC2, and Variation in PC2, By Parameter.

ParameterPC1PC2PC2 Variation (mm)

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Copyright © 2007 by Tom Chester, Wayne Armstrong and Kay Madore.
Comments and feedback: Tom Chester
Last update: 16 October 2007