1. Today: Quizz 8 Friday: GLM review Monday: Exam 2

2. Part IV The General Linear Model Multiple Explanatory Variables Chapter 14 ANCOVA 1 categorical, 1 continuous

3. Analysis of covariance: 1 categorical 1 continuous 2 different analysis: 1. comparison of 2 regression slopes Ch 14.1 2. Statistical control for a continuous variable within an ANOVA design Ch 14.2 ANCOVA

4. Part IV The General Linear Model Multiple Explanatory Variables Chapter 14.1 ANCOVA Comparison of slopes

5. Heterozygosity (H) of fruit flies from Yosemite Park, Dobzhansky’s investigations H is a measure of genetic variability Altitude = harsh environment Does genetic variability decrease at higher altitudes, due to stronger selection in extreme environments? GLM | ANCOVA

6. 1. Construct Model Response variable: H (%) = inversion heterozigosity (%) Explanatory variables: 1. Altitude (km) Continuous 2. Species Drosophila pseudoobscura Drosophila persimilis

7. Verbal: Inversion heterozygosity changes with altitude, depending on species Graphical: 1. Construct Model

8. Formal:

9. 1. Construct Model

10. 2. Execute analysis Data in model format lm1 <- lm(H~Alt+Sp+Alt*Sp, data=dros)

11. 2. Execute analysis grand mean species means common slope deviations from common slope species slopes Regression equations per species

12. 3. Evaluate model a. Straight line □ Straight line model ok? b. Need to revise model? □ Errors homogeneous? c. Assumptions for computing p-values □ Errors normal? □ Errors independent?

13. a. Straight line □ Straight line model ok? b. Need to revise model? □ Errors homogeneous? c. Assumptions for computing p-values □ Errors normal? □ Errors independent? 3. Evaluate model

14. 3. Evaluate model a. Straight line □ Straight line model ok? b. Need to revise model? □ Errors homogeneous? c. Assumptions for computing p-values □ Errors normal? □ Errors independent?

15. 4. State the population and whether the sample is representative. Not enough information about how flies were collected All measurements that could have been obtained on this collection of flies, given the procedural statement

16. 5. Decide on mode of inference. Is hypothesis testing appropriate? 6. State H A / H o pair, test statistic, distribution, tolerance for Type I error. Interaction Term: Are the gradients in heterozigosity equal between species? Is there variance due to the interaction term?

17. State H A / H o pair, test statistic, distribution, tolerance for Type I error. Species Term: Does the mean heterozigosity for D. persimilis differ from that of D. pseudoobscura?

18. State H A / H o pair, test statistic, distribution, tolerance for Type I error. Altitude Term: Is the slope less than zero? More specific hypotheses?

19. 6. State H A / H o pair, test statistic, distribution, tolerance for Type I error. Test Statistic Distribution of test statitstic Tolerance for Type I error

20. 7. ANOVA From multiple regression lecture (Ch 12) Remember Type I SS: sequential sums of squares partitioning of SS is done in the order the terms are written in the model Type III SS: adjusted sums of squares SS allocated to each term when entered last into the model, i.e. controlled for the rest of the variables Minitab provides Type III R: use Anova{cars}, eg: library(cars); Anova(lm1,type=3)

21. 7. ANOVA Anova(lm1,type=3)

22. 8. Decide whether to recompute p-value Assumptions met, skip step

23. 9. Declare decision about terms Interaction term p=0.003< α =0.05 Reject H 0  The rate of decrease in heterozygosity with altitude differs between species F sp*alt =15.33 df=1,10 p=0.003 No sense in checking if common slope = 0 Appropriate to check if slope for each species = 0

24. 10. Report and interpret parameters of biological interest Let’s examine species separately D. persimilis H = 0.58 – 0.127 Alt D. pseudoobscura No Δ with Alt mean(H pseu ) = 68.6 %

25. Part IV The General Linear Model Multiple Explanatory Variables Chapter 14.2 ANCOVA Statistical control

26. Crawley 1993 Response variable: Fruit production (mg) Explanatory variables: 1. Plant size (root diameter) 2. Grazed? Yes OR No We are interested in the effect of grazing on fruit production, controlled for the effect of plant size GLM | ANCOVA

30. Quizz 8 Good luck! Clock