1. Probability Distributions and Dataset Properties Lecture 2 Likelihood Methods in Forest Ecology October 9 th – 20 th, 2006

2. Statistical Inference Data Scientific Model (Scientific hypothesis) Probability Model (Statistical hypothesis) Inference

3. Parametric perspective on inference Scientific Model (Hypothesis test) Often with linear models Probability Model (Normal typically) Inference

4. Likelihood perspective on inference Data Scientific Model (hypothesis) Probability Model Inference

5. An example... The Data: x i = measurements of DBH on 50 trees y i = measurements of crown radius on those trees The Scientific Model: y i =  x i +  (linear relationship, with 2 parameters (  and an error term (  ) (the residuals)) The Probability Model:  is normally distributed, with E[  ] and variance estimated from the observed variance of the residuals...

6. The triangle of statistical inference: Model Models clarify our understanding of nature. Help us understand the importance (or unimportance) of individuals processes and mechanisms. Since they are not hypotheses, they can never be “correct”. We don’t “reject” models; we assess their validity. Establish what’s “true” by establishing which model the data support.

7. The triangle of statistical inference: Probability distributions Data are never “clean”. Most models are deterministic, they describe the average behavior of a system but not the noise or variability. To compare models with data, we need a statistical model which describes the variability. We must understand the the processes giving rise to variability to select the correct probability density function (error structure) that gives rise to the variability or noise.

8. The Data: x i = measurements of DBH on 50 trees y i = measurements of crown radius on those trees The Scientific Model: y i =  DBH i +  The Probability Model:  is normally distributed. Data Scientific Probability Model Inference Data Scientific Model Probability Model Inference An example: Can we predict crown radius using tree diameter?

9. Why do we care about probability? Foundation of theory of statistics. Description of uncertainty (error). –Measurement error –Process error Needed to understand likelihood theory which is required for: Estimating model parameters. Model selection (What hypothesis do data support?).

10. Error (noise, variability) is your friend! Classical statistics are built around the assumption that the variability is normally distributed. But…normality is in fact rare in ecology. Non-normality is an opportunity to: Represent variability in a more realistic way. Gain insights into the process of interest.

11. The likelihood framework Ask biological question Collect data Probability Model Model noise Ecological Model Model signal Estimate parameters Estimate support regions Answer questions Model selection Bolker, Notes

12. Probability Concepts An experiment is an operation with uncertain outcome. A sample space is a set of all possible outcomes of an experiment. An event is a particular outcome of an experiment, a subset of the sample space.

13. Random Variables A random variable is a function that assigns a numeric value to every outcome of an experiment (event) or sample. For instance EventRandom variable Tree Growth = f (DBH, light, soil…)

14. Function: formula expressing a relationship between two variables. All pdf’s are functions BUT NOT all functions are PDF’s. Functions and probability density functions Functions = Scientific Model pdf’s Crown radius =  DBH WE WILL TALK ABOUT THIS LATER Used to model noise:  Y-(  DBH)

15. Probability Density Functions: properties A function that assigns probabilities to ALL the possible values of a random variable (x). x Probability density f(x)

16. Probability Density Functions: Expectations The expectation of a random variable x is the weighted value of the possible values that x can take, each value weighted by the probability that x assumes it. Analogous to “center of gravity”. First moment. -1 0 1 2 p(-1)=0.10 p(0)=0.25 p(1)=0.3 p(2)=0.35

17. Probability Density Functions: Variance The variance of a random variable reflects the spread of X values around the expected value. Second moment of a distribution.

18. Probability Distributions A function that assigns probabilities to the possible values of a random variable (X). They come in two flavors: DISCRETE: outcomes are a set of discrete possibilities such as integers (e.g, counting). CONTINUOUS: A probability distribution over a continuous range (real numbers or the non-negative real numbers).

19. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 01234567891011121314151617181920 Event (x) Probability Probability Mass Functions For a discrete random variable, X, the probability that x takes on a value x is a discrete density function, f(x) also known as probability mass or distribution function.

20. Probability density f(x) Probability Density Functions: Continuous variables A probability density function (f(x)) gives the probability that a random variable X takes on values within a range.        1 0 dx)x(f )x(f }bXa{P )x(f b a a b

21. Some rules of probability assuming independence AB

22. Real data: Histograms

23. Histograms and PDF’s Probability density functions approximate the distribution of finite data sets. VARIABLE -10-50510 0 50 100 150 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 n = 1000 Probability

24. Uses of Frequency Distributions Empirical (frequentist): Make predictions about the frequency of a particular event. Judge whether an observation belongs to a population. Theoretical: Predictions about the distribution of the data based on some basic assumptions about the nature of the forces acting on a particular biological system. Describe the randomness in the data.

25. Some useful distributions 1.Discrete  Binomial : Two possible outcomes.  Poisson: Counts.  Negative binomial: Counts.  Multinomial: Multiple categorical outcomes. 2.Continuous  Normal.  Lognormal.  Exponential  Gamma  Beta

26. An example: Seed predation x =no seeds taken 0 to N Assume each seed has equal probability (p) of being taken. Then: Normalization constant t1 t2 ( )

27. Zero-inflated binomial

28. Binomial distribution: Discrete events that can take one of two values 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 01234567891011121314151617181920 Event (x) Probability E[x] = np Variance =np(1-p) n = number of sites p = prob. of survival Example: Probability of survival derived from pop data n =20 p = 0.5

29. Binomial distribution

30. Poisson Distribution: Counts (or getting hit in the head by a horse) k = number of seedlings λ= arrival rate 5000.5 01234567 POISSON 0 100 200 300 400 Count 0.0 0.1 0.2 0.3 0.4 Proportion per Bar Number of Seedlings/quadrat **Alt param= λ=rt

31. Poisson distribution

32. Example: Number of seedlings in census quad. 0102030405060708090100 Number of seedlings/trap 0 10 20 30 40 50 60 Count 0.0 0.1 0.2 0.3 0.4 Proportion per Bar Alchornea latifolia (Data from LFDP, Puerto Rico)

33. Clustering in space or time Poisson process E[X]=Variance[X] Poisson process E[X]<Variance[X] Overdispersed  Clumped or patchy Negative binomial?

34. Negative binomial: Table 4.2 & 4.3 in H&M Bycatch Data E[X]=0.279 Variance[X]=1.56 Suggests temporal or spatial aggregation in the data!!

35. Negative Binomial: Counts 01020304050 NEGBIN 0 10 20 30 40 50 60 70 80 90 100 Count 0.0 0.1 0.2 Proportion per Bar Number of Seeds

36. Negative Binomial: Counts 01020304050 NEGBIN 0 10 20 30 40 50 60 70 80 90 100 Count 0.0 0.1 0.2 Proportion per Bar Number of Seeds

37. Negative binomial

38. Negative Binomial: Count data 0102030405060708090100 No seedlings/quad. 0 10 20 30 Count 0.0 0.1 0.2 Proportion per Bar Prestoea acuminata (Data from LFDP, Puerto Rico)

39. Normal PDF with mean = 0 Normal Distribution E[x] = m Variance = δ 2

40. Normal Distribution with increasing variance

41. Lognormal: One tail and no negative values 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 010203040506070 x is always positive f(x) x

42. Lognormal: Radial growth data (Data from Date Creek, British Columbia)

43. Exponential Variable Count 0123456 0 10 20 30 40 50 60 70 80 0.0 0.1 0.2 0.3 0.4 Proportion per Bar

44. Exponential: Growth data (negatives assumed 0) 012345678 Growth (mm/yr) 0 200 400 600 800 1000 1200 Count 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Proportion per Bar Beilschemedia pendula (Data from BCI, Panama)

45. Gamma: One tail and flexibility

46. Gamma: “raw” growth data 0123456789 Growth (mm/yr) 0 200 400 600 800 1000 Count Alseis blackiana (Data from BCI, Panama) 0102030 0 50 100 150 200 Cordia bicolor Growth (mm/yr)

47. Beta distribution

48. Beta: Light interception by crown trees (Data from Luquillo, PR)

49. Mixture models What do you do when your data don’t fit any known distribution? –Add covariates –Mixture models Discrete Continuous

50. Discrete mixtures

51. Discrete mixture: Zero-inflated binomial

52. Continuous (compounded) mixtures

53. The Method of Moments You can match up the sample values of the moments of the distributions and match them up with the theoretical moments. Recall that: The MOM is a good way to get a first (but biased) estimate of the parameters of a distribution. ML estimators are more reliable.

54. MOM: Negative binomial