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From: Nathan Myhrvold <nathanm@MICROSOFT.com>
> Given the example discussed in this thread
> N = 72 X (number of dinos) = 12
> M = 54 (75% extinction rate) then the probability of killing all the
> dinos is 2.2%.
This allows one to reject the null hypothesis (no selectivity and 75%
extinction) at the 5% level. In other words, if this model is a
reasonable one, then either the effective extinction rate was higher
or there were more non-dinosaur species or there was some selective
factor against the dinosaurs.
> M = 61 (84.7% extinction rate) the probability of killing all dinos is
> M = 65 (90.3% extinction rate) it rises to 26.2%
These two, however, would fail to reject the null hypothesis.
In short they are consistant with the dinosaur extinction being
[With correlated extinctions]:
> 75% extinction M = 54, probability of 11 or more species going
> extinct = 13.5%
> 84.7% extinction M = 61, probability of 11 or more species going extinct
> = 41.2%
> 90.3% extinction M = 65, probability of 11 or more species going extinct
> = 67%
Now, none of them reject the null hypothesis.
I suppose this is about a good a place as any to put in my data from
Archibald on the Lancian fauna. (Though keep in mind, the smaller
terrestrial species will be undersampled relative to the larger species,
and the aquatic species will be more completely sampled than any
The groups with the highest extinction rates were:
Sharks & Rays (5 spp) 100% extinction
(Archibald suggests that this is a
Marsupials (11 species) 91% extinction
Lizards (10 species) 70% extinction
Ornithischians (10) 100% extinction
Saurischians (9) 100% extinction
Totalled this gives 45 species and an 91% extinction rate,
or, excluding the sharks and rays, 40 species and a 90% extinction
rate. This would be 40-45 marbles in three to five colors.
Applying the hypergeometric with N = 40, M = 36, X = 19
what do we we get? (I do not have a calculator with either factorial
or hypergeometric available here).
The remaining groups with extinction above the background rate are:
Bony fish: (15 species) 40% extinction
Multituberculates (10) 50% extinction
Combined with the above high rate groups (and still excluding
the sharks and rays) we get 65 species, and 74% extinction.
[N = 65, M = 47, X = 19]. This is close enough to example #1
that we can reject the basic null hypothesis here. Thus, at this
level either there is a systematic difference between the high and
intermediate rate groups, or the multi's are *way* undersampled,
or there is some correlation in extinctions.
The low (background) extinction groups are:
Amphibians (8 species) 0% extinction
Placentals (6 species) 0% extinction
Turtles (17 species) 12% extinction
Crocodiles (5 species) 20% extinction
(this last is included because the 20% is only one species,
which is the minimum possible non-zero value)
Among these the extinction rate is 8% out of 36 species.
For the whole Lancian fauna the numbers are:
101 species (excluding sharks), 50% extinction
or 106 species (with sharks), and 52% extinction
In the hypergemoteric, this would be N=101, M=50, X=19.
My guess is that this, also, will reject the null hypothesis, but
I need to get the numbers to be sure. (I will try to do this at home
tonight, as I think my calculator there has factorial).
The peace of God be with you.