1. Friday: Lab 3 & A3 due Mon Oct 1: Exam I  this room, 12 pm Please, no computers or smartphones Mon Oct 1: No grad seminar Next grad seminar: Wednesday, Oct 10 Type II error & Power

2. Today

3. Table 7.1 Generic recipe for decision making with statistics 1.State population, conditions for taking sample 2.State the model or measure of pattern……………………………ST 3.State null hypothesis about population……………………………H0 4.State alternative hypothesis……………………………………… HA 5.State tolerance for Type I error…………………………………… α 6.State frequency distribution that gives probability of outcomes when the Null Hypothesis is true. Choices: a)Permutations: distributions of all possible outcomes b)Empirical distribution obtained by random sampling of all possible outcomes when H0 is true c)Cumulative distribution function (cdf) that applies when H 0 is true State assumptions when using a cdf such as Normal, F, t or chisquare 7.Calculate the statistic. This is the observed outcome 8.Calculate p-value for observed outcome relative to distribution of outcomes when H0 is true 9.If p less than α then reject H0 in favour of HA If greater than α then not reject H0 10.Report statistic, p-value, sample size Declare decision

4. Table 7.2 Key for choosing a FD of a statistic Statistic of the population is a mean If data are normal or cluster around a central value If sample size is large(n>30)……....…………Normal distribution If sample size is small(n<30)……....…………t distribution If data are Poisson………………………………..Poisson distribution If data are Binomial………………………………Binomial distribution If data do not cluster around central value, examine residuals If residuals are normal or cluster around a central value If sample size is large(n>30)……....…………Normal distribution If sample size is small(n<30)……....…………t distribution If residuals are not normal………………………Empirical distribution Statistic of the population is a variance If data are normal or cluster around a central value……...Chi-square If data do not cluster around a central value If sample size is large(n>30)……....… …Chi-square distribution If sample size is small(n<30)……....…………Empirical

5. Table 7.2 Key for choosing a FD of a statistic - continued Statistic of the population: ratio of 2 variances (ANOVA tables) If data are normal or cluster around a central value…………….F dist If data do not cluster around central value, calculate residuals If residuals are normal or cluster around a central value……….F dist If residuals do not cluster around a central value If sample size is large(n>30)……....………………F distribution If sample size is small(n<30)……....………………..…Empirical Statistic is none of the above Search statistical literature for apropriate distribution or confer with a statistician If not in literature or can not be found…....………………..…Empirical

6. Example: jackal bones - revisited

7. 1. 2. 3. 4. 5.

8. Example: jackal bones - revisited 6. Key 7.

9. Example: jackal bones - revisited 8. Calculate p from t dist

10. Example: jackal bones - revisited 9. 10.

11. Example: jackal bones - revisited Is your data normal?

12. Example: jackal bones - revisited Is your data normal? It really does not matter! The assumption is that the residuals follow a normal distribution

13. Example: jackal bones - revisited Are your residuals normal?

14. Example: roach survival Data: Survival (T s ) in days of the roach Blatella vaga when kept without food or water Females n=10 mean(T s )=8.5 days var(T s )=3.6 days Males n=10 mean(T s )=4.8 days var(T s )=0.9 days Is the variation in survival time equal between male and female roaches? Data from Sokal & Rohlf 1995, p 189

15. Example: roach survival 1. 2. 3. 4. 5.

16. Example: roach survival 6. Key  7. 8.

17. Example: roach survival 9. 10.

18. Parameters Formal models (equations) consist of variable quantities and parameters Parameters have a fixed value in a particular situation Parameters are found in functional expressions of causal relations statistical or empirical functions theoretical frequency distributions Parameters are obtained from data by estimation

19. Parameters - examples 1.Functional relationship. Scallops density M scal =k 1 if R=5 or 6 M scal =k 2 if R not equal to 5 or 6 M scal = kg caught pr unit area of seafloor R = sediment roughness from 1 (sand) to 100 (cobble) k = mean scallop catch Red for params, blue for variables

20. Parameters - examples 2.Statistical relationship. Morphoedaphic equation M fish = 1.38 MEI 0.4661 M fish = kg ha -1 yr -1 fish caught per year from lake MEI = ppm m -1 dissolved organics/lake depth 0.4661 1.38 kg ha -1 ppm -0.4661 m 0.4661 Red for params, blue for variables

21. Parameters - examples 3.Frequency distribution. Normal distribution Red for params, blue for variables Y X μ = mean σ = standard deviation

22. Parameter estimates 1.Scallops density M scal = μ 1 if R=5 or 6 M scal = μ 2 if R not equal to 5 or 6 Theoretical model to calculate μ 1 and μ 2 ? Non-existent  estimate from data recorded in 28 tows M scal = μ 1 =mean(M R=5,6 )n=13 M scal = μ 2 =mean(M R<>5,6 )n=15

23. Parameter estimates 2.Ryder’s morphoedaphic equation pM = α MEI β ln(pM) = + population =+ln(MEI) sample

24. Statistical Inference Two categories: 1.Hypothesis testing Make decisions about an unknown population parameter 2. Estimation specific values of an unknown population parameter

25. Parameters Estimation: 1.Analytic formula e.g. slope, mean 2. Iterative methods criterion: maximize the likelihood of the parameter common ways to measure the likelihood: sums of squared deviations of data from model G-statistic (Poisson, binomial)

26. Parameters Uncertainty: Confidence limit: 2 values between which we have a specified level of confidence (e.g. 95%) that the population parameter lies