General Linear Model

1. General Linear Model Generalized Linear Model GeneralizedLinearMixed Model

2. General Linear Model Generalized Linear Model GeneralizedLinearMixed Model

3. GLMM LMM GeneralizedLinearMixed Model LMEM HLM MultilevelModel

5. Tagliamonte& Baayen(2012: 7 of preprint) Tagliamonte, S. A., & Baayen, R. H. (2012). Models, forests, and trees of York English: Was/were variation as a case study for statistical practice. Language Variation and Change, 24(02), 135-178.

6. The Beauty of Mixed Models • Account for clusters without averaging • Different distributions (generalized LMM) • Interpretation at the trial-level • Everything in one model • Excellent for individual differences studies (cf. Drager & Hay, 2012; Dan Mirman’s work) More Power!! (see e.g., Barr et al., 2013)

7. Problems of Mixed Models • Issues surrounding p-values • People misuse them … in a way that doesn’t improve Type I error rate(Schielzeth & Forstmeier, 2009; Barr et al., 2013) • Sometimes take A LOT of time • Some models don’t converge

8. The Linear Model response ~ intercept + slope * fixed effect + error structural partsystematic partdeterministic part probabilistic part stochastic partrandom part

9. The Linear Mixed Effects Model response ~ intercept + slope * fixed effect + error structural partsystematic partdeterministic part probabilistic part stochastic partrandom part

10. Important terminology assumed to be constantacross experiments StructuralPart StochasticPart “Fixed-effects factors are those in which the populations to which we wish to generalize are precisely the levels represented in our analysis.” Fixed effect Random effect - repeatable - non-repeatable - systematic influence - random influence - exhaust the population- sample the population - generally of interest - often not of interest - can be continuous - have to be categorical or categorical

11. Crawley (2013: 681)

12. Subjects as a fixed effect? NO… why: not repeatable  not systematic  often, not of interest  small subset of population

13. Repetitions as a fixed effect? Yes… why:  repeatable  systematic [ often, not of interest] “exhausts the population”

14. Common experimental data Subject Item... Item #1 Item... Rep 1 Rep 3 Rep 2

15. Finnish Norwegian Swedish English German Hungarian French Romanian Italian Spanish Turkish

16. In R: library(lme4) lmer(y ~ x + (1|subject), mydata)

18. Random intercepts versus Random slopes

19. Random intercepts RT (ms) Subjects

20. Random slopes

21. Randomintercepts RT (ms) Experiment time

22. Randominterceptsand slopes RT (ms) Experiment time

23. Random intercept vs. slope models Random intercept model = the fixed effect is evaluated against an error term that captures subject- or item-specific variability in the response Random slope model = the fixed effect is evaluated against an error term that captures subject- or item-specific variability in how the fixed effect affects the response In R: (1|subject) In R: (1+pred|subject)

24. http://anythingbutrbitrary.blogspot.com/2012/06/random-regression-coefficients-using.htmlhttp://anythingbutrbitrary.blogspot.com/2012/06/random-regression-coefficients-using.html

25. Random intercept examples • Some people are fast responders, some people are slow responders (their “intercepts” for response time are different) • Some people are very sensitive / accurate listeners, some are less sensitive (their “intercepts” for accuracy are different) • Some people have high or low voices with respect to their gender (their “intercepts” for pitch are different)

26. Random slope examples • Some people speed up during a long experiment,some slow down • Some people become more accurate during a long experiment, some less • Some people raise their pitch more for focus than others

27. An example RT ~

28. An example RT ~ Condition + (1|Subject)

29. An example RT ~ Condition + + (1+Condition|Subject)

30. An example RT ~ Condition + + (1+Condition|Subject) + (1|Item)

31. An example RT ~ Condition + + (1+Condition|Subject) + (1+Condition|Item)

32. An example RT ~ Condition + TrialOrder + + (1+Condition|Subject) + (1+Condition|Item)

33. An example RT ~ Condition + TrialOrder + + (1+Condition+TrialOrder|Subject) + (1+Condition|Item)

34. Model specificationfor random effects (1|subject) random intercept (0+fixedeffect|subject) random slope (1+fixedeffect|subject) … with correlation term

35. Assumptions Absence ofCollinearity No influentialdata points Normality of Errors Homoskedasticity of Errors Independence