Populations III: evidence, uncertainty, and

decisionsBio 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

http://www.ms.uky.edu/~mai/java/stat/GaltonMachine.html

Karl Gauss (1777-1855)

German mathematician

Formulated the normal (Gaussian) distribution

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

Bayes’ theorem

The statistical method is based on calculating inverse probabilities. 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

Posterior:

Probability of a hypothesis, given new data

We make decisions based on this

Likelihood:

Probability of the data, given a hypothesis of what uncertainty should look like (eg, bell curve)

Prior:

Our initial hypothesis before we got new data

Normalizing constant

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 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?

Most doctors say about 70-80%.

Actual answer is 7.8%

Why is our intuition wrong? Because it ignores the fact that very few women in the screened group actually have breast cancer, and the test doesn’t change that. That is, the prior is a very low probability.

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 toHa: population is declining.If there is much variation, Ho is

difficult to reject. But we’re not just interested in Ho!

Wade 2000

Two populations: which to care about?

Which one is declining?

Which one is declining fast?

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?