scaled_variance_of_eigenvalues: Compute scaled variance of eigenvalues

In fruciano/GeometricMorphometricsMix: Miscellaneous functions useful for geometric morphometrics

Compute scaled variance of eigenvalues

Description

Compute estimates of the scaled variance of eigenvalues using only the positive eigenvalues

Usage

scaled_variance_of_eigenvalues(
  data_matrix,
  boot = 999,
  rarefy = FALSE,
  shrinkage = FALSE
)

Arguments

data_matrix	Matrix or data frame containing the original data (observations in rows, variables in columns).
boot	number of bootstrap resamples (no bootstrap if 0)
rarefy	either a logical to determine whether rarefaction will be performed or a number indicating the sample size at which the samples will be rarefied
shrinkage	logical on whether the analysis should be based on a covariance matrix obtained through linear shrinkage

Details

The function allows computing the scaled variance of eigenvalues (Pavlicev et al. 2009) under a variety of settings. The scaled variance of eigenvalues is a commonly used index of morphological integration. Its value is comprised between 0 and 1, with larger values suggesting stronger integration.

Only positive eigenvalues are used in the computations used in this function.

The function employes two possible strategies to obtain eigenvalues:

• a singular value decomposition of the data matrix (default)

• an eigenvalue decomposition of the covariance matrix estimated using linear shrinkage (option shrinkage=TRUE; Ledoit & Wolf 2004)

Further, the function allows obtaining bootstrapped estimates by setting boot to the number of bootstrap replicates (resampling with replacement)

It is also possible to obtain rarefied estimates by setting rarefy to the desired sample size. This is useful when comparing the scaled variance of eigenvalues across multiple groups with different sample sizes. In this case, the suggestion is to use either the smallest sample size or less

Using a bootstrap estimate with the singular value decomposition approach represents a good compromise between computation time and accuracy. This should be complemented by rarefaction to the smallest sample size (or lower) in case one wants to compare the value obtained across different groups.

Value

If boot=0 the function outputs a vector containing the scaled variance of eigenvalues and the number of dimensions used in the computations. If, instead, boot>0 (recommended) the function outputs a list containing

• the mean scaled variance of eigenvalues across all bootstrap samples

• the median number of dimensions used across all bootstrap samples

• the 95

• the scaled variance of eigenvalues and dimensions used for each of the bootstrap replicates

Notice

When boot>0 the rarefied estimates are based on sampling with replacement (bootstrap). However, if boot=0, then a single rarefied estimate is obtained by sampling without replacement. In this case, the user should repeat the same operation multiple times (e.g., 100) and take the average of the scaled variance of eigenvalues obtained.

Also notice that using the shrinkage-based estimation of the covariance matrix requires longer computational time and memory. This option requires the package nlshrink

Computational details

For the computation, this function uses only positive eigenvalues (which are also used to identify data dimensionality). The eigenvalues are first scaled by dividing them by their sum (Young 2006), then their variance is computed as population variance (rather than sample variance; see Haber 2011). Finally, this value is standardized to a scale between 0 and 1 by dividing it by its maximum theoretical value of (p-1)/p^2 (where p is the number of dimensions) - this is the same scaling used in the software MorphoJ (Klingenberg 2011).

References

Ledoit O, Wolf M. 2004. A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis 88:365-411.

Young NM. 2006. Function, ontogeny and canalization of shape variance in the primate scapula. Journal of Anatomy 209:623-636.

Pavlicev M, Cheverud JM, Wagner GP. Measuring Morphological Integration Using Eigenvalue Variance. Evolutionary Biology 36(1):157–170.

Haber A. 2011. A Comparative Analysis of Integration Indices. Evolutionary Biology 38:476-488.

Klingenberg CP. 2011. MorphoJ: an integrated software package for geometric morphometrics. Molecolar Ecology Resources 11:353-357.

fruciano/GeometricMorphometricsMix documentation built on Jan. 31, 2024, 6:24 a.m.