Download Gregor Mendel and the chi-squared distribution File

Gregor Mendel and the left tail of the chi-squared distribution
We usually use the right tail of the chi-squared distribution, but we can use the left tail as well, for example
when the fit between data and prediction looks too good to be true. An interesting example is the results of
Mendel’s famous experiments on peas, which laid the foundations of the modern science of genetics. Mendel
published his work in 1866, just 6 years after the publication of Charles Darwin’s Origin of the Species,
although it was the turn of the century before anyone seems to have noticed it and later still before the two
theories were brought together. It probably didn’t help that his paper was entitled “Experiments in plant
hybridization”, and was published in the proceedings of the Natural History Society of Brünn (now Brno).
It was also in German, but in those days that wouldn’t have been so much of a disadvantage as English had
not yet become as dominant in science as it is today.
In Mendel’s time, people believed in blending inheritance. The idea was that when a child is conceived,
the characters from the two parents are mixed rather like what happens when you mix two tins of paint.
That’s why if one parent is light skinned and the other is dark, the offspring are generally somewhere in
between.
Mendel’s hypothesis was that inheritance is particulate, which means that what is passed on are discrete
particles and that these influence what happens. This has the advantage that it explains why in things like
eye-colour, children often resemble one parent entirely: if one parent has brown eyes and the other blue, the
children are more likely to be brown-eyed and less likely to be blue-eyed, but they’ll almost always be one
or the other, not some mixture.
The story is really very interesting. For one thing, Mendel was not an ignorant monk pottering around in
the monastery garden and noticing something peculiar about the flowers. He was an educated man who had
studied both physics and biology at the University of Vienna. The experiments were deliberately designed
because landowners in the area had discovered a great deal about how to breed horses to produce the features
they wanted and were interested to find out why their methods worked and how they could be improved. It
is also known that the abbott of the monastery had been in correspondence with Gauss, but the letters have
been lost, so we do not know what they were communicating about. It is just possible that Gauss was the
first person to work out the principles of population genetics, like so much that was discovered in pure and
applied mathematics in the nineteenth century.
I’ve never been able to understand why no one else seems to have thought of particulate inheritance,
especially since blending inheritance is such a problem for Darwin’s theory of natural selection. The problem
is that any mutation (as we would now say) that occurs would be diluted out before it had a chance to be
selected. It would be like putting one drop of coloured paint into a tin of white. And the idea of particulate
inheritance had been around for a very long time. In 55BC, the year Caesar invaded Britain, Lucretius wrote
of it in his De Rerum Natura. He argued that inheritance had to be by “seeds”, because otherwise you can’t
explain how features can skip generations, as they often do. By the way, he explained why children often
resemble one parent more than the other by the intriguing idea that it depends on which parent showed the
most passion at the moment of conception.
I once asked a professor of classics why educated people in the eighteenth and nineteenth centuries
wouldn’t have known about this, as so many of them spent a good deal of their time at school reading
ancient works in Latin and Greek. She said it was probably because Lucretius was considered too racy and
subversive to be on the curriculum. He also, by the way, wrote of natural selection, although he didn’t call
it that and he didn’t have the idea of evolution. He imagined unfit – or less fit – species becoming extinct,
not gradually changing into something else.
Perhaps it was Mendel’s studies led him to the idea of particles. Statistical mechanics as we know it had
not yet been developed when he went to Vienna in 1851; Maxwell’s work was done in 1859 and Boltzmann
was even later, but the idea that gases are composed of particles dates from Daniel Bernoulli in 1738, so
Mendel may well have been influenced by it.
Mendel crossed round yellow (RY) pea plants with wrinkled green (WG) ones. According to his theory,
there should have been progeny of all four possible combinations of the characters and they should have
occurred in the ratio RY:RG:WY:WG::9:3:3:1.
He had 556 plants, so the expected numbers were 312.75, 104.25, 104.25, 34.75. The frequencies he
reported were 315, 108, 101, 32.
We can easily do a chi-squared test on the data. Mendel couldn’t because statistics hadn’t been invented
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yet. The appropriate sum is
2.252
3.752
(−3.25)2
(−2.75)2
+
+
+
= 0.47
312.75 104.25
104.25
34.75
There are no parameters estimated from the data because the probabilities are derived from the theory.
So we compare 0.47 with χ295% which for 3df is 7.81. This is clearly not a significant deviation from the
prediction, so there is no reason to doubt Mendel’s hypothesis.
On the other hand, that is an awfully small chi-squared value. So let’s just look at the left side of the
distribution. On 3 df, χ25% = 0.352. That tells us that the deviation he got was really very small. In fact,
the probability of getting a χ2 value of 0.47 or less is about 0.075. In other words, the fit looks too good.
And things look even worse if you include his other experiments as well.
There’s been a lot of debate about what we’re to make of all this. One suspects that he (some people
say, his assistant) simply kept only the best results out of several, or counted until they got the proportions
they expected and then stopped counting, or whatever.
Actually, it turns out that inheritance in peas is more complicated than Mendel or most other people
realised, so he was really a bit lucky that the experiment worked out at all. On the other hand, he was
basically right, so maybe he deserved a bit of luck.
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