# Detecting Rounding of Results

Hans Erren brought to my notice a study he did on artifacts in the Quelccaya ice core record, first observed by Steve McIntyre (of climateaudit.org). I thought I would run this through benford() to see if it could pick up the problem. The study is here at Hans’ home site and the dataset is here at the NOAA data archive.

Here the original figure with the artifact shown as fanned out lines.

According to Hans’ study the artifact is caused by rounding up of the accumulation results to whole centimeters. This form of ‘result management’ reduces the degrees of freedom of results, reducing essentially real numbers to a more limited integer domain, clearly seen in Hans’ figure.

I created a little space between some abutting numbers, trimmed off the header and footer of the file and ran the benford() time series analysis from the Audit-0.1 package on the 6th and 7th column, the O18 and the accumulation results. Below is the code in R:

``` d< -read.table("q83cor1.txt")
benford(d[,6]+10,plot=TRUE,n=50)
benford(d[,7],plot=TRUE,n=50)
```

Here is the accumulation, with the x axis indicating greater depth (time) of core:

Here is the O18, with the x axis indicating greater depth (time) of core:

Results management is indicated by the red line falling below the green
line along the series. This red line is the probability of the second digit frequency being consistent with Benford’s Law. The accumulation figures show a clear deviation with increasing depth, consistent with the figure produced by Hans Erren above. The O18 also show signs of results management at greater depths, though to a lesser degree than the accumulation.

The package used to perform these analyses is freely available here

## 0 thoughts on “Detecting Rounding of Results”

1. Almost certainly there would be rounding in data of the final figure. I should say that rounding does not necessarily lead to errors in mean values. It is just one of the possible explanation for deviations from expected digit frequencies.

2. Almost certainly there would be rounding in data of the final figure. I should say that rounding does not necessarily lead to errors in mean values. It is just one of the possible explanation for deviations from expected digit frequencies.

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