## Results: 10

Figure 1. The Tracy–Widom Density. From: Population Structure and Eigenanalysis.

*P*= 0.05,

*x*= .9794;

*P*= 0.01,

*x*= 2.0236;

*P*= 0.001,

*x*= 3.2730.

Figure 7. Simulation of an Admixed Population. From: Population Structure and Eigenanalysis.

*A,B,D*trifurcated 100 generations ago, while population

*C*is a recent admixture of

*A*and

*B*. Admixture weights for the proportion of population

*A*in population

*C*are Beta-distributed with parameters (3.5,1.5). Effective population sizes are 10,000.

Figure 9. LD Correction with no LD Present. From: Population Structure and Eigenanalysis.

*k*) of our LD correction. We first show this (A) for

*m*= 500,

*n*= 5,000, and then (B) for

*m*= 200,

*n*= 50,000. In both cases the LD correction makes little difference to the fit.

Figure 3. Testing the Fit of the Second Eigenvalue. From: Population Structure and Eigenanalysis.

*F*= .01,

_{ST}*m*= 100,

*n*= 5,000) with two equal-sized subpopulations. We show P–P plots for the TW statistic computed from the

*second*eigenvalue. The fit at the high end is excellent.

Figure 4. Three African Populations. From: Population Structure and Eigenanalysis.

Figure 6. The BBP Phase Change. From: Population Structure and Eigenanalysis.

*m*and number of markers

*n*but keeping the product at

*mn*= 2

^{20}. Thus the predicted phase change threshold is

*F*= 2

_{ST}^{−10}. We vary

*F*and plot the log

_{S}*p*-value of the Tracy–Widom statistic. (We clipped −log

_{10}

*p*at 20.) Note that below the threshold there is no statistical significance, while above threshold, we tend to get enormous significance.

Figure 5. Three East Asian Populations. From: Population Structure and Eigenanalysis.

Figure 10. LD Correction with Strong LD. From: Population Structure and Eigenanalysis.

*m*= 100,

*n*= 5,000) with large blocks of complete LD. Uncorrected, the TW statistic is hopelessly poor, but after correction the fit is again good. Here, we show 1,000 runs with the same data size parameters as in Figure 2A,

*m*= 500,

*n*= 5,000, varying

*k,*the number of columns used to “correct” for LD. The fit is adequate for any nonzero value of

*k*.

(B) Shows a similar analysis with

*m*= 200,

*n*= 50,000.

Figure 2. Testing the Fit of the TW Distribution. From: Population Structure and Eigenanalysis.

*m*= 100 and

*n*= 5,000 unlinked markers. We give a P–P plot of the TW statistic against the theoretical distribution; this shows the empirical cumulative distribution against the theoretical cumulative distribution for a given quantile. If the fit is good, we expect the plot will lie along the line

*y*=

*x*. Interest is primarily at the top right, corresponding to low

*p*-values.

(B) P–P plot corresponding to a sample size of

*m*= 200 and

*n*= 50,000 markers. The fit is again excellent, demonstrating the appropriateness of the Johnstone normalization.