Dataset formats
The input dataset is in gd_ped format. The output dataset is html with links to a pdf for a graphical output and text files. (Dataset missing?)
What it does
The user selects a gd_ped dataset generated by the Prepare Input tool. The PCA tool runs a Principal Components Analysis on the input genotype data and constructs a plot of the top two principal components. It also reports the following estimates of the statistical significance of the analysis.
1. Average divergence between each pair of populations. Specifically, from the covariance matrix X whose eigenvectors were computed, we can compute a "distance", d, for each pair of individuals (i,j): d(i,j) = X(i,i) + X(j,j) - 2X(i,j). For each pair of populations (a,b) now define an average distance: D(a,b) = sum d(i,j) (in pop a, in pop b) / (|pop a| * |pop b|). We then normalize D so that the diagonal has mean 1 and report it.
2. Anova statistics for population differences along each eigenvector. For each eigenvector, a P-value for statistical significance of differences between each pair of populations along that eigenvector is printed. +++ is used to highlight P-values less than 1e-06. *** is used to highlight P-values between 1e-06 and 1e-03. If there are more than 2 populations, then an overall P-value is also printed for that eigenvector, as are the populations with minimum (minv) and maximum (maxv) eigenvector coordinate. [If there is only 1 population, no Anova statistics are printed.]
3. Statistical significance of differences between populations. For each pair of populations, the above Anova statistics are summed across eigenvectors. The result is approximately chisq with d.o.f. equal to the number of eigenvectors. The chisq statistic and its p-value are printed. [If there is only 1 population, no statistics are printed.]
We post-process the output of the PCA tool to estimate "admixture fractions". For this, we take three populations at a time and determine each one's average point in the PCA plot (by separately averaging first and second coordinates). For each combination of two center points, modeling two ancestral populations, we try to model the third central point as having a certain fraction, r, of its SNP genotypes from the second ancestral population and the remainder from the first ancestral population, where we estimate r. The output file "coordinates.txt" then contains pairs of lines like
where the number (in this case 0.1245) is the estimation of r. Computations with simulated data suggests that the true r is systematically underestimated, perhaps giving roughly 0.6 times r.
Acknowledgments
We use the programs "smartpca" and "ploteig" downloaded from
http://genepath.med.harvard.edu/~reich/Software.htm
and described in the paper "Population structure and eigenanalysis" by Nick Patterson, Alkes L. Price, and David Reich, PLoS Genetics, 2 (2006), e190.