Estimate Selection

Estimating selection on shape is challenging, both in terms of the data collection required and in terms of the methods to be used for the analysis. Perhaps this explains why few such studies have been published, even though the methods have long been published (Lande and Arnold 1983; Phillips and Arnold 1989). A rare example in the context of geometric morphometrics is a study of selection on flower shape in a cruciferous plant (Gómez et al. 2006).

Background

The fitness measure

Estimating selection is done by regressing relative fitness on shape. Relative fitness is computed from a measure of fitness, which must be provided by the user as a covariate. Relative fitness for each individual in the analysis is computed by dividing the individual's fitness measure by the average fitness of all individuals. Therefore, it does not matter in which units the fitness measure is provided (e.g. number of offspring, survival probability), because the division by the average will produce a meaningful scaling. However, the fitness measure must not be standardized or centred (re-scaled to a zero mean) because that would make it impossible to compute the relative fitness.

Methods

Dimensionality. There is an inherent trade-off in estimating selection for shape and other high-dimensonal features. On the one hand, the analysis should consider as much as possible of the dimensionality of the phenotype, and on the other hand, but on the other hand, the robustness of the quadratic regression analysis of fitness on shape is reduced if the sample size is not large relative to the number of dimensions. The number of independent variables is roughly proportional to the square of the number of dimensions.

MorphoJ uses a principal component analysis to reduce dimensionality. The number of principal components (PCs) of shape that are included is either the number of PCs whose variances are greater than zero (actually, a threshold of 10-14 is used to cope with rounding errors) or the square root of the sample size, whichever is less. Large sample size is a crucial factor for obtaining reliable estimates of selection; investigators should make every possible effort to include as many individuals in the sample as is possible. For the presentation of results, the data are rotated back to the coordinate system of the landmark coordinates after the Procrustes fit.

Linear selection. Linear or directional selection is estimated by multiple regression of relative fitness on shape. The resulting selection gradient is not ashape change (in short, it is in the wrong units; see Klingenberg and Monteiro 2005), but is can be considered to be an auxiliary variable perfectly correlated with a shape variable, and that shape variable can then be visulaized.

Nonlinear selection. MorphoJ can estimate nonlinear selection as quadratic selection, by fitting a quadratic response surface to the data (Phillips and Arnold 1989).

Standard errors for selection coefficients. The standard errors for the coefficcients of the selection gradients are estimated using the usual methods for multiple linear regression.

The standard errors for the eigenvalues of the matrix of nonlinear selection are computed using the method of double linear regression (Bisgaard and Ankenman 1996).

Requesting the analysis

 

References

Bisgaard, S., and B. Ankenman. 1996. Standard errors for the eigenvalues in second-order response surface models. Technometrics 38:238–246.

Gómez, J. M., F. Perfectti, and J. P. M. Camacho. 2006. Natural selection on Erysimum mediohispanicum flower shape: Insights into the evolution of zygomorphy. American Naturalist 168:531–545.

Klingenberg, C. P., and L. R. Monteiro. 2005. Distances and directions in multidimensional shape spaces: implications for morphometric applications. Systematic Biology 54:678–688.

Lande, R., and S. J. Arnold. 1983. The measurement of selection on correlated characters. Evolution 37:1210–1226.

Phillips, P. C., and S. J. Arnold. 1989. Visualizing multivariate selection. Evolution 43:1209–1222.