The GPS (Geographic Population Structure) method is claimed to be a method that can predict the location of an individual's ancestors 1000 years ago, at least according to the University press release that announced it. I'm interested in ancient migrations and statistics in archaeology, and I follow the application of genetics in archaeology. So it sounds like just my thing!

The GPS was first published by Elhaik et al. (2014) and, despite the authors’ disclaimer of financial interests, immediately formed the basis of a commercial offering at Prosapia Genetics. Recently competing financial interests were acknowledged in a corrigendum to the original paper published two and a half years after the original despite the conflict being obvious within a few days of publication. Since publication, the method has been the subject of some online criticism and has recently been offered by another company, GPS Origins. Critical overviews of these offerings have been given by Debbie Kennett in her blog and Pavel Flegontov on Facebook. GPS has formed the basis for some subsequent publications from the same group. One on the origins of the Jews (Das et al. 2016) has been notably controversial (Flegontov et al.2016, Aptroot et al. 2016, and see the set of links provided by Pavel Flegontov on Facebook). Recently Das et al. published a preprint of a lengthy response to their critics (Das et al 2016).

Prompted by this I have gone back to look at the original paper to understand its methods a bit better.

The method is simple. Elhaik et al. take genetic data from a set of individuals of known population origin (I shall call this the reference dataset), and compute ADMIXTURE proportions of putative ancestral populations for those individuals. Comparison of the differences in the proportions to the geographical distance between the populations reveals a linear relationship. This relationship is then exploited to locate genetically characterised individuals of unknown geographical origin (I shall call this the test sample) with respect to the reference populations.

However, the devil is in the detail. Joe Pickrell has blogged about the conceptual problems of relating genetic variation to spatial distance in this way. I’m not going to comment on the genetics, but I do know about prediction of spatial locations from other data. The rest of this post covers some mathematical and technical aspects of the paper. For those who wish to cut to the chase, I give a summary and conclusions here.

__The regression model__treats composition data as if it did not have to sum to 100% which is clearly untrue. The results in the paper cannot be replicated from the supplementary material, using either the method described in the paper or the one actually implemented in the code. The paper describes removing points from the dataset but unless the removed points are improperly replaced with zeroes, the correlation is too weak to be useful for prediction. Consequently, the regression equation calculated is probably invalid as it is methodologically unsound. The supplementary code does not do what is described in the paper.

__The predictive model__implemented in the code does not use the equations in the paper. In addition, in both the paper and the code if there were two populations of the same genetic distance from the test sample, but different geographical distance from the population that best matches the test sample, the geographically more distant population will contribute more to determining the direction of the GPS arrow. Usually more distant populations should contribute less to a prediction not more.

__Test calculations__: the (previously available) online tool and downloadable code differ by hundreds of miles in their predicted location for the same data.

__Overall conclusion__: the mathematical methods described are incoherent, the supplementary data is not that used to create the figures or equations in the paper, and the supplementary code does not implement the methods described.

*The paper is methodologically unsound and not reproducible.*Now here's the detail ...

### The regression model

Elhaik et al. (2014) calculated the genetic distances between individuals as “Euclidean distances between … admixture proportions for the analysed individuals”. This is a mistake. Euclidean distances, are designed to apply to unconstrained variables which can take any value between minus infinity and plus infinity. With compositional data, including admixture proportions, the values are not unconstrained as they must sum to 1 or 100%. It has been known since at least 1897 that applying Euclidean methods to compositional data leads to errors, particularly overestimation of correlation coefficients (Pearson 1897, Aitchison 1986, Aitchison 2005).

Elhaik et al. (2014) give a confusing account of how they calculated geographical distances. They state “we calculated the Euclidean distances between all the populations within the GEN and GEO data sets” but then “distances Δ

_{GEO}(X, Y) between two individuals with known latitude and longitude were calculated using the Haversine formula”. The Haversine formula is not Euclidean; it is a method for calculating great circle distances on the earth’s surface (as explained on Wikipedia). This is much better than a Euclidean distance. However, the publically released R program code reads latitude and longitude data and then applies the default method of R's dist function. That default method is Euclidean distance! Given that the distance represented by a degree of longitude varies with latitude, this measure is not one that is even approximately linearly related to distance in miles. It is impossible that Figure 9 of Elhaik et al. could have been produced using the code that has been released to accompany the paper.

The third step is to correlate the (incoherently calculated) genetic distances and geographical distances. Each individual in the reference dataset is compared to all the others and the points are analysed. But there is a big problem here. This approach overstates the number of independent data points. If I know that A is close to both B and C, then I know a lot about the possible values of distance between B and C before I measure it. The distance B-C is not completely independent of the distances A-B and A-C. Regression methods assume independence of all the data points, and if the points are not independent the reliability of the regression is over-estimated.

The paper notes that the relationship is linear only up to about 5000 miles geographical distance so they remove datapoints above 4000 miles distance. The published code does something very strange at this point. Having calculated the distances, it replaces any point with geographical distance over 70 or genetic distance over 0.8 with a point (0,0) and then runs the regression. 70 degrees of latitude is approximately 4800 miles, so this may be an approximation to the 4000 mile cut-off described in the paper. But removing genetic distances over 0.8 is not mentioned in the paper. These points are not removed from the regression in the manner described in the paper; they are

*moved*to lie at the origin. As these are 67% of the data points in the supplementary data files (on http://chcb.saban-chla.usc.edu/gps/downloads.php as of 16 August 2016), this very strongly weights the regression to pass close to the origin. As coded, the regression line obtained is Δ

_{GEO}= 0.94(±0.09) + 68.2(±0.4) Δ

_{GEN}, with r

^{2}= 0.78; leaving out these points it is Δ

_{GEO}= 9.50(±0.49) + 51.8(±1.1) Δ

_{GEN}with r

^{2}= 0.41. This latter equation is rather awkward: it would predict that even at zero genetic distance an average geographical distance of 9.5 “degrees” is expected.

If the data provided with the code is converted to miles using the Haversine formula it is clearly a different dataset to that in the paper (see figures below). As the first is stated to be of individuals and the second of reference populations this is not very surprising.

figure 9 from Elhaik et al., described as plotting individuals, |

reference population data provided with the code, converted to miles using the Haversine formula. |

The paper reports a regression equation of Δ

_{GEO}= 38.7 + 2523 Δ

_{GEN}. Running the regression on the supplementary dataset converted to miles but limited to geographical distances less than 4000 miles gives: Δ

_{GEO}= 747(±22) + 2422(±35) Δ

_{GEN}(r

^{2}= 0.50) which is not very similar to the equation reported in the paper. Removing the points with Δ

_{GEN}>0.8 gives Δ

_{GEO}= 504(±27) + 3095(±58) Δ

_{GEN}(r

^{2}= 0.46). Replacing the removed points (67% of the data) with (0,0) gives Δ

_{GEO}= 50(±5) + 3969(±19) Δ

_{GEN}(r

^{2}= 0.81).

Summary: the results cannot be replicated from the supplementary material, using either the method described in the paper or the one actually implemented in the code. Unless the removed points are improperly replaced with (0,0), the r

^{2}value is unacceptably low for prediction (below 50%).

Conclusion: the regression equation calculated is probably invalid as it is methodologically unsound The supplementary code does not do what is described in the paper.

### The predictive model

Having established their regression equation, Elhaik et al. propose a method to predict the location of a new genetic sample. They calculate the genetic distance to the closest population and use the regression equation to calibrate it to a geographical distance. The sample is assumed to lie at that distance from the closest population and the next nine nearest populations are used to determine the direction of that distance by calculating a weighted average of their vectors from the best match. How the admixture proportions attributed to populations are calculated from the data for individuals is not stated. Presumably they are means. Why a regression equation calculated from distances between pairs of individuals can be applied to the distance between an individual and a population mean is not explained.

In the code the Δ

_{GEO}

^{min}value is calculated not with the full regression equation but only by multiplying the Δ

_{GEO}

^{min}by the gradient; the intercept is omitted!

The weights for the second to tenth distances given in the paper are

but in the code these are given by

W <- (minE[1]/minE)^4;

W = W/(sum(W));

As the minE of the code are the Δ

_{GEN}of the equations in the paper, the ratio in the first line makes sense, but somehow a fourth power has appeared and then a normalisation so that weights sum to one. Bizarrely the way the code and the paper are constructed, if there were two populations of the same genetic distance but different geographical distance, the geographically more distant population will contribute more to determining the direction of the result because its vector will be longer.

The equations in the paper are constructed so that the result is a distance Δ

_{GEO}

^{min}from the best matching population. The code does not do this. The code places the result at that distance or closer if the weighted average vector is shorter.

Conclusion: the code does not implement the predictive equations of the paper, and in any case the method using inappropriate weightings..

### Test calculations

The download site also provides an online calculator and partial results for the test data. Entering the first line of test data in the online calculator gives a predictions of Latitude: 34.4237, Longitude: 52.7614 (as of 16 August 2016: http://chcb.saban-chla.usc.edu/gps/record.php?jobID=GPS-16082016-013652-4819). The partial results file has 39.5307070554391, 46.885015601252 (yes the values are reported to a precision better than the size of an atom), and is the same as running the supplied code. The instructions on the download page (http://chcb.saban-chla.usc.edu/gps/downloads.php) state that the online tool gives Latitude: 38.2188 Longitude: 47.2863 for that first line. [Note: as of 25 November 2016 this site seems to have been hacked and now redirects to a Chinese site.]

Conclusion: the online tool and downloadable code differ by hundreds of miles in the predicted location.

###
Overall conclusion

The mathematical methods described are incoherent, the supplementary data is not that used to create the figures and equations in the paper, and the supplementary code does not implement the methods described.

**The paper is methodologically unsound and not reproducible.**### References

Aitchison J. 1986. The statistical analysis of compositional data. London: Chapman and Hall

Aitchison J. 2005. A concise guide to compositional data analysis. University of Glasgow, http://ima.udg.edu/activitats/codawork05/A_concise_guide_to_compositional_data_analysis.pdf

Aptroot M. 2016. Yiddish Language and Ashkenazic Jews: A Perspective from Culture, Language, and Literature. Genome Biology and Evolution 8:1948-1949. http://gbe.oxfordjournals.org/citmgr?gca=gbe%3B8%2F6%2F1948

Das R, Wexler P, Pirooznia M, Elhaik E. 2016. Localizing Ashkenazic Jews to Primeval Villages in the Ancient Iranian Lands of Ashkenaz. Genome Biology and Evolution 8:1132-1149. http://gbe.oxfordjournals.org/content/8/4/1132.abstract

Das R, Wexler P, Pirooznia M, Elhaik E. 2016. Responding to an enquiry concerning the geographic population structure (GPS) approach and the origin of Ashkenazic Jews - a reply to Flegontov et al. arXiv 1608.02038. https://arxiv.org/abs/1608.02038

Elhaik E, Tatarinova T, Chebotarev D, Piras IS, Maria Calò C, De Montis A, Atzori M, Marini M, Tofanelli S, Francalacci P, Pagani L, Tyler-Smith C, Xue Y, Cucca F, Schurr TG, Gaieski JB, Melendez C, Vilar MG, Owings AC, Gómez R, Fujita R, Santos FR, Comas D, Balanovsky O, Balanovska E, Zalloua P, Soodyall H, Pitchappan R, GaneshPrasad A, Hammer M, Matisoo-Smith L, Wells RS. 2014. Geographic population structure analysis of worldwide human populations infers their biogeographical origins. Nature Communications 5:3513. http://www.nature.com/articles/ncomms4513

Flegontov P, Kassian A, Thomas MG, Fedchenko V, Changmai P, Starostin G. 2016. Pitfalls of the Geographic Population Structure (GPS) Approach Applied to Human Genetic History: A Case Study of Ashkenazi Jews. Genome Biology and Evolution 8:2259-2265. http://gbe.oxfordjournals.org/citmgr?gca=gbe%3B8%2F7%2F2259

Pearson K. 1897. Mathematical Contributions to the Theory of Evolution - On a Form of Spurious Correlation Which May Arise When Indices Are Used in the Measurement of Organs. Proceedings of the Royal Society of London 60:489-498. http://rspl.royalsocietypublishing.org/content/60/359-367/489.full.pdf+html