Pedro R. Peres-Neto, Donald A. Jackson, and Keith M. Somers. 2003. Giving meaningful interpretation to ordination axes: assessing loading significance in principal component analysis. Ecology 84:2347–2363.

Appendix C. An examination of real ecological data using these methods.

We selected five data sets representing a gradient from weak to strong correlation structures in order to observe the performance of methods in real ecological situations. These data sets were chosen because in each case the raw data were available in the original publication, with the exception of the stream data set used to construct Table 1. Data set Morph1 represents seven morphological variables for 38 bird species from Burgundy (France) and California (USA) studied by Blondel et al. (1984; raw data are presented in their Table 1). Data set Morph2 comprises six morphological variables for 13 species of West Indian Anolis lizards (Losos 1990). Because of missing data in the performance measurements, two species were excluded from the analysis. Data set Behavior represents six behavioral variables for the same 13 lizard species in Morph2. Raw data for Morph2 and Behavior data sets were presented in Losos (1990: Table 1) and they were both log10 transformed. Finally, data set Lake represents eight environmental variables for 42 lakes sampled by Robinson and Tonn (1989: Appendix 1) in the Athabasca River basin. Three lakes presenting missing data were deleted from the analysis. All variables with the exception of pH were log10 transformed.

Assessments of the ecological data sets were based only on the bootstrapped eigenvector and bootstrapped broken-stick due to their superior performance. Overall, both methods tended to agree in terms of the magnitude of their probabilities (Table C1). In only three situations did the two tests provide contrasting results indicating that the loading was significant or not for an alpha at 0.05. As expected, the magnitude of loading is not as important as the associated eigenvalue. For instance, a loading of 0.741 associated with an eigenvalue representing 53.1% of the total variation presented larger probabilities of rejection for both methods than when compared to a loading of 0.733 for Morph1 where PC-1 retained 70.0% of the total variance. However, three variables (two in Behavior and one in Lake) were marginally significant according to the bootstrapped broken-stick but not to the bootstrapped eigenvector. In cases of non-rejection, the bootstrapped broken-stick tended to provide greater probabilities of rejection than the bootstrapped eigenvector. This result is indicative of  several variables being associated with more than one component.

Table C1. Loadings for the ecological data sets in the first two principal components and the associated probabilities for the bootstrapped eigenvector (Bt-eigv) and the bootstrapped broken-stick (Bt-bs). Percentages of variation retained by the axes are presented below the data set being assessed. Note that both methods largely agreed, indicating that perhaps in real situations, where there is a mix of both unique and complex variables, the two methods may be similarly efficient.

Data set		Loading (PC-1)	Bt-eigv	Bt-bs		Loading (PC-2)	Bt-eigv	Bt-bs
Stream (53.1%/24%)		-0.458	0.161	0.606		0.788	0.258	0.146
		0.741	0.186	0.038		0.559	0.258	0.440
		-0.786	0.010	0.055		-0.003	0.703	1.000
		0.931	0.000	0.000		0.252	0.262	0.999
		0.645	0.185	0.343		-0.448	0.297	0.549
Morph1 (70.0%/11.9%)		0.882	0.000	0.000		-0.295	0.169	1.000
		0.862	0.000	0.000		-0.250	0.155	0.990
		0.688	0.002	0.019		-0.451	0.171	0.736
		0.733	0.000	0.000		0.598	0.147	0.259
		0.878	0.000	0.000		0.276	0.163	0.996
		0.851	0.000	0.000		0.202	0.221	0.987
		0.935	0.000	0.000		-0.070	0.235	1.000
Morph2 (71.3%/11.9%)		0.972	0.001	0.000		-0.020	0.491	1.000
		0.983	0.001	0.000		-0.018	0.369	1.000
		0.840	0.001	0.000		-0.321	0.253	0.993
		0.953	0.000	0.000		0.173	0.369	1.000
		0.734	0.062	0.015		0.626	0.337	0.513
		0.734	0.017	0.043		-0.433	0.373	0.785
Behavior (42.5%/28.1%)		0.768	0.027	0.077		-0.166	0.517	0.994
		-0.807	0.024	0.009		-0.456	0.367	0.723
		0.946	0.025	0.001		0.181	0.397	0.989
		-0.130	0.438	0.998		0.880	0.380	0.064
		0.304	0.353	0.966		-0.798	0.382	0.060
		0.550	0.116	0.418		-0.100	0.529	0.999
Lake (25.4%/20.1%)		0.560	0.086	0.196		0.428	0.309	0.686
		-0.536	0.126	0.490		0.635	0.315	0.076
		0.586	0.119	0.121		-0.492	0.362	0.681
		0.139	0.325	0.960		-0.442	0.403	0.450
		0.756	0.081	0.064		0.245	0.316	0.900
		0.204	0.331	0.936		0.314	0.376	0.723
		-0.050	0.426	0.971		0.560	0.309	0.206
		0.671	0.101	0.162		0.341	0.357	0.682