a-score: Compute and optimize a-score for Discriminant Analysis of Principal Components (DAPC)

Description

These functions are under development. Please email the author before using them for published results.

Usage

a.score(x, n.sim=10, ...)
optim.a.score(x, n.pca=1:ncol(x$tab), smart=TRUE, n=10, plot=TRUE,
              n.sim=10, n.da=length(levels(x$grp)), ...)

Arguments

a dapc object.

n.pca

a vector of integers indicating the number of axes retained in the Principal Component Analysis (PCA) steps of DAPC. nsim DAPC will be run for each value in n.pca, unless the smart approach is used (see details

smart

a logical indicating whether a smart, less computer-intensive approach should be used (TRUE, default) or not (FALSE). See details section.

an integer indicating the numbers of values spanning the range of n.pca to be used in the smart approach.

plot

a logical indicating whether the results should be displayed graphically (TRUE, default) or not (FALSE).

n.sim

an integer indicating the number of simulations to be performed for each number of retained PC.

n.da

an integer indicating the number of axes retained in the Discriminant Analysis step.

...

further arguments passed to other methods; currently unused..

Value

=== a.score === a.score returns a list with the following components:
taba matrix of a-scores with groups in columns and simulations in row.
pop.scorea vector giving the mean a-score for each population.
meanthe overall mean a-score.
=== optim.a.score === optima.score returns a list with the following components:
pop.scorea list giving the mean a-score of the populations for each number of retained PC (each element of the list corresponds to a number of retained PCs).
meana vector giving the overall mean a-score for each number of retained PCs.
pred(only when smart is TRUE) the predictions of the spline, given in x and y coordinates.
bestthe optimal number of PCs to be retained.

encoding

UTF-8

Details

The Discriminant Analysis of Principal Components seeks a reduced space inside which observations are best discriminated into pre-defined groups. One way to assess the quality of the discrimination is looking at re-assignment of individuals to their prior group, successful re-assignment being a sign of strong discrimination.

However, when the original space is very large, ad hoc solutions can be found, which discriminate very well the sampled individuals but would perform poorly on new samples. In such a case, DAPC re-assignment would be high even for randomly chosen clusters. The a-score measures this bias. It is computed as (Pt-Pr), where Pt is the reassignment probability using the true cluster, and Pr is the reassignment probability for randomly permuted clusters. A a-score close to one is a sign that the DAPC solution is both strongly discriminating and stable, while low values (toward 0 or lower) indicate either weak discrimination or instability of the results.

The a-score can serve as a criterion for choosing the optimal number of PCs in the PCA step of DAPC, i.e. the number of PC maximizing the a-score. Two procedures are implemented in optim.a.score. The smart procedure selects evenly distributed number of PCs in a pre-defined range, compute the a-score for each, and then interpolate the results using splines, predicting an approximate optimal number of PCs. The other procedure (when smart is FALSE) performs the computations for all number of PCs request by the user. The 'optimal' number is then the one giving the highest mean a-score (computed over the groups).

References

Jombart T, Devillard S and Balloux F (2010) Discriminant analysis of principal components: a new method for the analysis of genetically structured populations. BMC Genetics11:94. doi:10.1186/1471-2156-11-94