1. Probability distributions and likelihood

2. Readings Ecological detective: Wikipedia (seriously!)
Chapter 3 Probability distributions
Wikipedia (seriously!)
e.g. Beta distribution, lognormal distribution, etc.

3. Overview (three lectures)
Probability vs. likelihood
Probability distributions: binomial, poisson, normal, lognormal, negative binomial, beta, gamma, multinomial
Likelihood profile
The concept of support
Model selection, likelihood ratio, AIC
Robustness
Contradictory data

4. Probability Likelihood
If I flip a fair coin 10 times, what is the probability of it landing heads up every time? Given the fixed parameter (p = 0.5), what is the probability of different outcomes? Probabilities add up to 1.
I flipped a coin 10 times and obtained 10 heads. What is the likelihood that the coin is fair?
Given the fixed outcomes (data), what is the likelihood of different parameter values?
Likelihoods do not add up to 1.
Hypotheses (parameter values) are compared using likelihood values (higher = better).

5. Probability Likelihood
Area under curve between 5 and 10
Height of curve at x = 10
Height of curve at x = 14
What is the probability that 5 ≤ x ≤ 10 given a normal distribution with µ = 13 and σ = 4? Answer: What is the probability that –1000 ≤ x ≤ 1000 given a normal distribution with µ = 13 and σ = 4? Answer: 1.000
What is the likelihood that µ = 13 and σ = 4 if you observed a value of
x = 10 (answer: the likelihood is 0.075)
x = 14 (answer: the likelihood is 0.097)
Conclusion: if the observed value was 14, it is more likely that the parameters are µ = 13 and σ = 4, because is higher than

6. We use the same (in this case normal) probability distribution function for both probability and likelihood!
Likelihood is the height, probability is the area

7. Common probability distributions
Discrete: binomial, Poisson, negative binomial, multinomial
Continuous: normal, lognormal, beta, gamma, (negative binomial)
For all of these the Excel sheet gives the likelihood at each value of x, given the parameters of the distribution

8. 7 Statistical distributions
7 Statistical distributions.xlsx Many distributions defined here, including Excel functions and functions defined directly in the spreadsheet

9. SD and CV (all distributions)

10. Binomial probability distribution
Number of trials
Number of successes
Probability of success [0,1]
The “factorial term”
How many ways are there of selecting k objects from among N objects
Example: probability of getting k = 5 heads when flipping a coin N = 10 times, if the coin is fair (p = 0.5). Note: known number of trials.
7 Statistical distributions.xlsx, sheet Binomial

11. Poisson probability distribution
Expected number of events
Number of events
Example: On average there are λ = 9.4 fatal traffic accidents in Washington State every week. What is the probability that there would be k = 0 in a week? (Note: rare event out of large number of possible events.)
7 Statistical distributions.xlsx, sheet Poisson

12. Limitations of Poisson
Has only one parameter, which is both the mean and the variance
We often have discrete count data, but in real-life data the variance is often larger than predicted by the Poisson

13. Thus we often use the negative binomial
Closely related to the Poisson and binomial
One extra parameter related to the variance
VERY useful

14. Standard negative binomial
Number of failures
Probability of a success
Number of successes
Squint a lot and this looks
kind of like a binomial
Example: a factory makes widgets successfully with probability p. How many successful widgets have been made when r = 3 failed widgets have been made? The distribution predicts the probability of k = 0, 1, 2, … successful widgets being made.
7 Statistical distributions.xlsx, sheet Neg binomial (std)

15. Ecological usefulness?
Great for factories, but almost no ecological problems can be thought of as successes or failures in this way
Great for factory production problems
But we want a function with parameters for
Mean
Overdispersion (increased variance = increased chance of extreme events)
Integer events are rare in nature, we want to deal with real numbers

16. Practitioner’s negative binomial
Gamma function (factorial that accepts non-integers, see later)
Overdispersion parameter
Predicted mean
As θ increases, variance increases, hence “overdispersion”
As θ → , var(Z) → 
As θ → 0, var(Z) = λ, just like a Poisson!
Example: our data contain observations k, with mean λ and variance greater than λ. Find the value of overdispersion θ that best accounts for this increased variance.
7 Statistical distributions.xlsx, sheet Neg binomial (practitioner)

17. Weird facts about the practitioner’s negative binomial
When θ → 0 this doesn’t just smell like a Poisson, and act like a Poisson, it is the Poisson
By replacing the factorials with gamma functions, the r and k can be real numbers not just integers
What on earth is a gamma function???

18. Gamma function Γ()
For background information only: this is a generalized factorial function that accepts real numbers not just integers
Excel: does not have a gamma function but has a ln of gamma function (GAMMALN)

19. Multinomial probability distribution
Observed number
in category k
Total number observed
Model-predicted proportion
in category k
Example: fitting a model to proportions at age (or proportions at length) data. Model produces predicted proportions pi and data gives observed numbers xi in each category. Total numbers sampled = n = x1 + x2+ … + xk
7 Statistical distributions.xlsx, sheet Multinomial

20. 7 Statistical distributions.xlsx, sheet Multinomial

21. Unrealism of multinomial
(and other distributions too!)
Assumes every sampling event is completely independent
But there is much correlation in reality
Same trawl, area, time of day, day of year, gender, etc.
Real data never ever fit a multinomial this well
Later lectures will introduce the concept of “effective sample size” neff, which will be smaller than reported sample size n.

22. Normal distribution
7 Statistical distributions.xlsx, sheet Normal

23. Lognormal distribution
7 Statistical distributions.xlsx, sheet Lognormal

24. Lognormal: key notes 0 < x <  Mean(x) is not µ
If we want the mean to be µ, then replace the model parameter with:
Used widely for abundance and biomass

25. Beta distribution
0.5,0.5
1,1
1.3,1.3
4,4
0.5,2
2,6
50,50
7 Statistical distributions.xlsx, sheet Beta

26. Beta: key notes Values confined to be 0 < x < 1
Can mimic almost any shape within those bounds
Although bounded, can change the bounds by multiplying / dividing x values
E.g. survival parameters

27. Gamma distribution
4, 1
4, 2
1.1, 0.5
0.9,
60, 5
7 Statistical distributions.xlsx, sheet Gamma

28. Gamma: key notes 0 ≤ x < 
Somewhat like an exponential, lognormal, or normal
Flexibility without being bounded like the beta distribution
E.g. salmon arrival numbers plotted over time
Excel function beta.dist() assumes parameters α* = α and β* =1/β