Introduction to statistical estimation methods Finse Alpine Research Center, September 2010

OUTLINE TODAY: Mostly maximum likelihood Focus of the course: Introduce essential methods for statistical modelling in ecology Construction of biologically sound models Estimation of parameter values and associated uncertainties Interpretation of results Introduce concepts that are important for the course next week TOMORROW: Mostly Bayesian statistics SUNDAY: Day off / Glacier hike MONDAY TO FRIDAY: Occupancy modelling workshop (3 new lecturers – joining on the glacier hike) – Some lectures – Many Exercises – Tutoring – – Help each other– Ask questions – Be active! MATERIAL ON:

Quantification of …: … relationship between variables … differences between groups of individuals … the effect of experimental treatments … predictions for the future (effects of climate change) … of effect of management strategies and not the least: Quantification of uncertainty! Most studies in ecology require quantification in some way: Quantification of anything requires: … some sort of model … ways to estimate parameters / distributions of random variables

Claim: In ecology, the main question is seldom IF something has an effect The questions are more about HOW and HOW MUCH

Energy expenditure (Field metabolic rate) Body mass HABITAT Season Sex Reproductive state Temperature Weather Activity / Behaviour OTHER THINGS (biological things + measurement error) ? Example: How does habitat quality affect energy expenditure?  The question should not be IF these variables have an effect – from biological theory we can be almost certain that all these variables have an effect.  Relationships in ecology are almost infinitely complex (there is no true model)  “All models are wrong, but some are useful” (Box)

Energy expenditure (Field metabolic rate) Body mass HABITAT Season Sex Reproductive state Temperature Weather Activity / Behaviour OTHER THINGS (biological things + measurement error) ? “Typical approach”: 1.Put everything into a linear model (multiple regression) 2.Remove non-significant effects 3.Reporting p-values Without thinking about HOW the various predictor variables can affect the response variables Without thinking about what you are really interested in Without quantifying HOW MUCH the predictor variables affect the response variable, and without thinking about BIOLOGICAL SIGNIFICANCE

Effect size Small Large p<0.001 NSp<0.05NSp<0.05 Biologically significant Not biologically significant Could be important – more data needed Statistical significance vs. biological relevance 5 different confidence intervals: Null-hypothesis tests are often used erroneously to make a classification of “no effect” (not significant) and “significant effect” with no consideration of the potential biological significance (a somewhat thoughtless process). E.g. statements like “Predator density did not affect prey survival” with no further detail on effect size.

Number of papers questioning the utility of null hypothesis testing in scientific research Anderson et al Null-hypothesis testing in ecological science:  Yoccoz, N. G Use, overuse, and misuse of significance tests in evolutionary biology and ecology. Bulletin of the Ecological Society of America 72:  Anderson, D. R., K. P. Burnham, and W. L. Thompson Null hypothesis testing: Problems, prevalence, and an alternative. Journal of Wildlife Management 64: Web-page:  Null-hypothesis testing in ecological science:  Yoccoz, N. G Use, overuse, and misuse of significance tests in evolutionary biology and ecology. Bulletin of the Ecological Society of America 72:  Anderson, D. R., K. P. Burnham, and W. L. Thompson Null hypothesis testing: Problems, prevalence, and an alternative. Journal of Wildlife Management 64: Web-page: 

= deviance + 2 × no. parameters + small sample correction In a set of models, the model with the lowest AIC will, on average, be the model with the lowest K-L distance (i.e., give predictions closest to the truth). Using p-values for model selection is a different thing Bias 2 Variance (uncertainty) Number of parameters (K) Prediction Freq. Truth Too simple model Too complex model

Example: Influence of testosterone on size of home-range in voles. 60 sites … Testosterone treated male Control male Do testosterone treated males have larger home-ranges at high densities? What are the effects at low densities? Response variable:Predictor variables: Home range size measured by radio-telemetry ~ Treatment Density Body mass Think about HOW things are related

F Value Pr(F) Body mass 15.79<0.001 Treatment Density <0.001 Body mass × Treatment Density × Treatment Full model: F Value Pr(F) Body mass <0.001 Treatment Density <0.001 Density × Treatment Step 1: F Value Pr(F) Body mass <0.001 Treatment Density <0.001 Step 2: F Value Pr(F) Body mass <0.001 Density <0.001 Step 3: Conclusion: There is no significant effect of ‘Treatment’ (p = 0.32) or a ‘Density × Treatment’ interaction (p = 0.11). Conclusion: There is no significant effect of ‘Treatment’ (p = 0.32) or a ‘Density × Treatment’ interaction (p = 0.11). Response variable: home range size

Density Home range size log(Density) log(Home range size) D<c: y = constant D≥c: y = c*(D-c) b D<c: log(y) = constant D≥c: log(y) = log(c) + b*log(D-c) Body mass (M) Home range size log(Body mass) log(Home range size) y = aM b log(y) = a+ b*log(M)

log(Population density) log(Home range size) Treatment + Treatment × Density Treatment + DensityTreatment × Density Candidate models