# Statistical Tests

Here is another example of the book I mentioned in the previous blog post. The nurse Lucia de Berk was convicted of murdering patients on the basis of statistical data. For details, refer to the book. The main block of data was the following.

• She did 134 of 1021 shifts.
• All 8 incidents of resuscitation happened during her shifts.

This leads to a table of data as follows.

 Shifts Incidents Without Lucia 887 0 887 With Lucia 134 8 142 1021 8 1029

There are tests to verify the „independence“ of the rows from the columns of the table. One of them is the Fisher test. It is based on the hyper-geometrical distribution and as such is quite exact. The book does not explain the simple argument behind the test. In fact, my feeling is that this case is an abuse of the test.

I tend to think of the problem in the following way: The incidents are distributed randomly over the shifts, which are set by the hospital for each nurse. Then the probability of all 8 incidents to occur during the shifts of Lucia is a simple application of the product rule. We get

$$(134/1021)^8 \approx 10^{-7}$$

That is one in 10 millions. Certainly this is small. But it could easily happen to some nurse at some time. If there is no other evidence, the case should be closed. Of course, digging in the dirt the prosecution found other hints. Read the book for details.

Later, when more thorough investigation was done,  it turned out that two incidents had been left out and Lucia had not been present at two of the incidents. So the table changed much in favor of  Lucia.

 Shifts Incidents Without Lucia 883 4 887 With Lucia 136 6 142 1021 10 1029

This means, we have only 6 of the 10 incidents happening during her shift. To compute how likely 6 or more incidents are, we can use the binomial distribution.

$$\sum\limits_{k=6}^{10} \dbinom{10}{k} p^k (1-p)^{10-k} \approx 0.0007$$

This happens much more often, in fact I estimate it to happen once every 2 years to some nurse in the Netherlands. Moreover, the additional evidence was more or less destroyed on closer examination. By that time, Lucia had already spent a life in jail and had been suffering a stroke.

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