Scratching my head! - Pt 2 - Major statistical analysis gap in Narasimhan paper undermines steppe->Brahmin theory

In a previous post, I had critiqued Narasimhan et al 2019 for the inference that steppe ancestry is significantly related to modern Indian Brahmin groups, and hence provides evidence that this steppe ancestry is causal to the Indo Aryan languages.

Nevertheless, the fact that traditional custodians of liturgy in Sanskrit (Brahmins) tend to have more Steppe ancestry than is predicted by a simple ASI-ANI mixture model provides an an independent line of evidence—beyond the distinctive ancestry profile shared between South Asia and Bronze Eastern Europe mirroring the shared features of Indo-Iranian and Balto- Slavic languages (58)—for a Bronze Age Steppe the origin for South Asia’s Indo-European languages.

I gave 3 main reasons why this reasoning and analysis was faulty.

1. Correlation is not causation, Brahmin steppe can be enriched by way of intermarriage with steppe rich brides. It may not have anything to do with the steppe causing the formation of brahmin jAtis.

2. There are other brahmin jAtis in the same dataset from states other than UP and Bihar which do not show high Z scores as per the table given in the supplement. What explains that?

3. The main reason was that Steppe/Indus periphery ratio is a useless indicator in a 3 way model with Onge, steppe & Indus periphery. It would mean something only in a 2 way model between Steppe & Indus Periphery. I explain this with an example in my previous post, please read that if my reasoning is unclear.

Since then, I have studied this Z-Score table in the paper, and have found serious inconsistencies.

What is Z-Score?

The Z-score for each item in an array of numbers is simply the number of standard deviations that number is away from the mean of the array of numbers. Z>3 or Z<-3 (for eg. some may use 2, some 4) is indicative of that particular item being more than 3 standard deviations away from mean, at the extreme tail of the distribution, right and left end respectively. Therefore it is to be considered an outlier.

Z>3 means 3 sigma standard deviation away from the mean, in the tail

Z-Scores from the Narasimhan Paper

There are a couple of places where the paper defines the Z - Scores. 

Our analysis of Steppe ancestry also identified six groupswith a highly elevated ratio of Central_ Steppe_MLBA– to Indus_Periphery_West–related ancestry compared with the expectation for the model at the Z < −4.5 level (Fig. 4). The strongest two signals were in Brahmin_Tiwari (Z = −7.9) and Bhumihar_Bihar (Z = −7.0). More generally, there is a notable enrichment in groups that consider themselves to be of traditionally priestly

status: five of the six groups with Z < −4.5 were Brahmins or Bhumihars.

Read this - Distribution of Z Scores for the ratio of steppe pastoralist ancestry to indus pool related ancestry shows upto 8 standard deviation from the mean for brahmin groups as per the paper. THIS IS AN EXTRAORDINARY CLAIM.

I have been trying to reproduce these numbers for past 2 days, but just cannot arrive at such large numbers. 

Z Scores have the property of having 0 mean and standard deviation as 1. But the Narasimhan paper has a standard deviation of 140 Z-Scores as 2.79 (indus periphery pool) and 2.06 (indus periphery west). There is clearly something seriously wrong in this part of the paper and its conclusion.

Methodology:

1. I copied the qpAdm coefficients, aposteriori hierarchical model coefficients and Z scores from Narasimhan et al 2019 excel supplement for all 140 populations into 2 excel sheets, one for Indus Periphery West and 1 for Indus periphery Pool models. 

Indus Periphery West is sample I8726 who has the least AASI of all the Indus periphery samples from ShahrSokhta and Gonur. Indus Periphery Pool includes all 11 samples from both these locations, it has a higher AASI than I8726.

2. I made a column for Indus/steppe ratio in each sheet for all populations, and calculated the Z Scores for each population. Steppe/Inpe ratio will give same Z scores with sign reversed. I also used correlation on my calculated Z Score and the paper's Z-score using function '=correl(array1, array2)'.

3. Formula used for Z-Score: standardize(), which uses average() and stdev.p() as inputs for mean and standard deviation of the array of 140 ratios. ie '=standardize(x, mean, stdev.p)' for each of 140 x's. 

The correlation of my Z Scores to the paper's Z Score for InPeWest is just 0.49, and 0.60 for InPePool.

Clearly this ratio is not what the paper used.

4. Next i calculated 140 Z Scores for the value 'Indus periphery MINUS steppe', using same formulae in step 3. 

The correlation between this Z-Score and the paper's Z-Score was 0.9975 for InPeWest and 0.992 for InPe Pool. This seems to be what the paper used to calculate Z Scores - InPe MINUS Steppe ancestry %.

5. In none of these 2 Z Score arrays i calculated is the Z-Score for Brahmins >3 or <-3. They are all square in the meat of the normal curve.

Distribution of ZScores according to Indus/steppe ratio does not give <3 Z for Brahmins

Distribution of ZScores according to Indus-steppe does not give <3 Z for Brahmins

As you can see, none of the Z-scores for Brahmins cross -3. So how did the paper get values of -7 and -8? I have tried various other combinations of the data, but none gave me high correlation of 99.8% with the paper's Z scores.

The paper's Z-scores look to be simply multipled 2-2.8 times of the actual Z-scores. I cannot fathom what calculation Narasimhan et al would have used to reach Z-scores of 7-8. It seems virtually impossible to me.

CONCLUSIONS:

1. The Z-scores based on which Narasimhan et al computed looks seriously flawed, and cannot be reproduced. The Z-scores do not have a standard deviation of 1, rather it's between 2-2.8. This means what the paper uses as ZScore isn't even Z-Score.

2. The Z-scores look artificially high, and cannot be reproduced.

3. The Z-scores seem to be calculated with Indus Periphery MINUS steppe ancestry as array input rather than ratios, which is a seemingly useless indicator.

4. The Z scores calculated based on either the ratio or the difference in indus/steppe ancestries do not cross the >3 threshold and are therefore not significant. It undermines one of the core arguments in the paper.

If any of you see some logic by which the paper's Z scores are correct, do let me know.

My workbook is below.Its downloadable.

References

Scratching my head! - Criticism of Narsimhan VM et al 2019 - Pt 1

Narasimhan VM, Patterson N, Moorjani P, et al. The formation of human populations in South and Central Asia. Science. 2019;365(6457):eaat7487. doi:10.1126/science.aat7487

Excel supplement from the paper