Populations III:
evidence, uncertainty, and
decisions
Bio 415/615
Questions
1. What types of uncertainty are involved in
conservation decisions?
2. How would you ask a conservation question for
an event that involves chance?
3. Why do our estimates of uncertainty usually
involve the normal distribution (bell curve)?
4. In statistics, what is a P value?
5. What is one way that ‘frequentist’ statistical
methods differ from Bayesian methods?
Four threats to small populations:
1. Loss of genetic variability
2. Demographic variation
3. Environmental variation
4. Natural (rare) catastrophes
Variation in populations
• We tend to first ask about central
tendencies:
– what is the mean growth rate?
• But variation about the mean—variance—
can often be more important in making
conservations decisions than central
tendencies.
• Stochasticity is unpredictable variation,
and requires us to deal with
probabilities.
Probabilistic outcomes
• In a deterministic world, we can ask: will
the population go extinct if a road is
built?
• In a stochastic world, we have to ask:
what is the probability the population
will go extinct if a road is built (and
over what time period?)
Recall minimum viable population:
Estimate of the # of individuals needed
to perpetuate a population:
1) For a given length of time, e.g. for 100
or 1000 years
2) With a specified level of (un)certainty,
e.g. 95% or 99%
Comparing outcomes (risk)
• Probabilities are often most useful when
comparing the outcomes of two or more
conservation decisions.
– What is the probability Silene regia will
decline to fewer than 10 individuals after
50 years of burning vs. non-burning?
– How will the probability of extinction in 100
years of the Florida panther change if we
translocate individuals from Texas?
Types of uncertainty
• Measurement uncertainty: variation in a
parameter estimate due to precision and
accuracy of measurement (sampling
error)
• Model uncertainty: do we know how
factors relate to one another, scale,
etc? Do we know all the factors to
include?
• Process uncertainty: can we know
everything about nature? Climate?
Random events and the bell curve
Karl Gauss (17771855)
German
mathematician
Formulated the
normal (Gaussian)
distribution
http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html
Random events and the bell curve
Normal
distribution is
pervasive in
stochastic
models because
it represents
the expected
error of
random,
independent
events
Dealing with uncertainty usually means figuring out
the SHAPE (mean, variance) of a normal distribution.
Probability density function
Area under the curve = 1
Tells probability of events between two limits
Frequentist methods (classical
hypothesis testing)
Involves comparing a null hypothesis (Ho)
to an alternative hypothesis (Ha).
• Ho: The population is not declining.
• Ha: The population is declining.
Frequentist methods involve significance
tests involving the null hypothesis. What
is the probability of the data, if the null
hypothesis is true? = P value
P values
If the P value is smaller than the
significance level (e.g., α = 5%), then the
null hypothesis is rejected.
This is NOT the probability that the null
hypothesis is true! In fact frequentist
statistics cannot evaluate the
probability of hypothesis (Ho OR Ha),
only the probability of the data.
Coin flips: is it a fair coin?
Ho: The coin is fair.
Ha: The coin is not fair.
Data: 14 heads out of 20 flips.
If each flip is 50% chance of heads, this
is a binomial trial. Under a normal
distribution of 20 samples with a mean
of 0.5, the probability of 14 heads is
about 12%. So the coin is fair.
BUT 15 heads is not fair! (P=0.04)
Why alpha of 5%?
• We will accept Ha even though it is
false (Type I error, or false positive) 1
time out of 20. Why not more stringent
criterion?
Why alpha of 5%?
• We will accept Ha even though it is
false (Type I error, or false positive) 1
time out of 20. Why not more stringent
criterion?
• If alpha 0.1%, we would probably ignore
many Ha that are in fact correct (false
negative, or Type II error)
• So why 5%? It’s an arbitrary
compromise.
Why frequentist methods are
losing ground to other methods
• Null hypotheses are usually not
interesting.
• Decisions based on rejecting null
hypotheses are often insensitive.
• Probabilities are usually misinterpreted as
commenting on hypotheses, when actually
they comment on the data.
• There is a greater risk of using
inappropriate uncertainty estimates (e.g.,
often normality assumed).
Why frequentist methods are
losing ground to other methods
• No straightforward way to combine
different types of data, or pre-existing
expert opinion.
• Often difficult to evaluate strength of
different models (hypotheses).
• Poor integration with ‘the next step’—
decision theory (again because
probabilities are not associated with
models, but data).
Bayesian methods
Reverend
Thomas Bayes
(1702-1761)
Bayes’
theorem
Describes both a
method of statistical
inference (quantifying
probabilities of events)
AND a statistical
philosophy
Bayesian methods
Reverend
Thomas Bayes
(1702-1761)
Bayes’
theorem
Philosophy is based on
putting probabilities on
partial belief versus
those established by
frequencies.
New information does
not replace old
information… it adjusts
old information.
Diversion: baseball averages
What can the first month of at bats tell you
about the final batting average of a hitter?
Bayesian methods
The statistical method
is based on calculating
inverse probabilities.
Bayes’ theorem
Example: Forward probability is
establishing probability of
tossing heads once you know
something about the coin.
Inverse probability is rather,
what can you tell about the coin,
given data on heads and tails?
Bayesian methods
Normalizing
constant
Posterior:
Likelihood:
Prior:
Probability of a
hypothesis, given
new data
Probability of the
data, given a
hypothesis of
what uncertainty
should look like
(eg, bell curve)
Our initial
hypothesis
before we got
new data
We make decisions
based on this
Medical example
• http://yudkowsky.net/bayes/bayes.html
• 1% of women at age forty who participate in
routine screening have breast cancer. 80% of
women with breast cancer will get positive
mammographies. 9.6% of women without
breast cancer will also get positive
mammographies. A woman in this age group
had a positive mammography in a routine
screening. What is the probability that she
actually has breast cancer?
Medical example
• http://yudkowsky.net/bayes/bayes.html
• 1% of women
at doctors
age forty
who participate
in
Most
say about
70-80%.
routine screening
have breast cancer. 80% of
Actual answer is 7.8%
women with breast cancer will get positive
Why is our
intuition
wrong?
Because it
mammographies.
9.6%
of women
without
ignores the fact that very few women in
breast cancer
will
also
get
positive
the screened group actually have breast
mammographies.
in doesn’t
this age
group
cancer, A
andwoman
the test
change
that.
had a positive
a routine
Thatmammography
is, the prior is ain
very
low
screening. probability.
What is the probability that she
actually has breast cancer?
Wade 2000 example
1. Priors and
posteriors are
probability density
functions.
Wade 2000 example
2. When we have no
initial expectation,
we use a
noniformative prior
(usually a uniform
distribution, or flat
line).
Wade 2000 example
3. With a flat prior,
our posterior
distribution has
the same
properties as the
likelihood, because
it is based only on
the current data.
Here: mean of 3000.
(Variance?)
Wade 2000 example
4. With more data
(this time N=2000),
our new estimate of
population size
somewhere in the
middle, because we
had prior information
(data-based prior).
Why does statistical method
matter in conservation?
• We are rarely concerned with yes/no
answers
• Wade 2000:
Compare Ho: population is not declining to
Ha: population is declining.
If there is much variation, Ho is difficult
to reject. But we’re not just interested
in Ho!
Wade 2000
Which one is declining?
Which one is declining
fast?
Two populations: which
to care about?
Final comment: the precautionary
principle
• “Where there are threats of serious or
irreversible environmental damage, lack
of full scientific certainty should not be
used as a reason for postponing
measures to prevent environmental
degradation.” (West German Env
Legislation, late 1960s)
• Burden of proof on developers?
• Can you ‘prove’ lack of effect?