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General Linear Model Generalized Linear Model Generalized Linear Mixed Model General Linear Model Generalized Linear Model Generalized Linear Mixed Model GLMM LMM LMEM HLM Generalized Linear Mixed Model Multilevel Model 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. 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) 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 The Linear Model response ~ intercept + slope * fixed effect structural part systematic part deterministic part + error probabilistic part stochastic part random part The Linear Mixed Effects Model response ~ intercept + slope * fixed effect structural part systematic part deterministic part + error probabilistic part stochastic part random part Important terminology assumed to be constant across experiments Structural Stochastic “Fixed-effects factors are those inPart which the Part 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 or categorical - have to be categorical Crawley (2013: 681) Subjects as a fixed effect? NO… why: not repeatable not systematic often, not of interest small subset of population Repetitions as a fixed effect? Yes… why: repeatable systematic [ often, not of interest] “exhausts the population” Common experimental data Subject Item #1 Rep 1 Rep 3 Rep 2 Item ... Item ... Finnish Norwegian Swedish English French Spanish Germa n Hungarian Romanian Italian Turkish In R: library(lme4) lmer(y ~ x + (1|subject), mydata) Random intercepts versus Random slopes 3000 2000 1000 RT (ms) 4000 Random intercepts 2 3 4 5 6 7 8 9 10 11 12 13 14 Subjects 15 16 17 18 19 20 21 22 23 Random slopes 0 50 100150200 Subject Subject Subject Subject 1400 1200 1000 800 600 400 Subject Subject Subject RT 1400 1200 1000 800 600 400 Subject Subject Subject Subject Subject Subject Subject Subject Subject Subject 1400 1200 1000 800 600 400 1400 1200 1000 800 600 400 Subject Subject Subject Subject Subject 1400 1200 1000 800 600 400 0 50 100150200 0 50 100150200 Block 0 50 100150200 RT (ms) Random intercepts Experiment time RT (ms) Random intercepts and slopes Experiment time 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 In R: (1|subject) In R: (1+pred|subject) 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 http://anythingbutrbitrary.blogspot.com/2012/06/ra ndom-regression-coefficients-using.html 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) 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 An example RT ~ An example RT ~ Condition + (1|Subject) An example RT ~ Condition + + (1+Condition|Subject) An example RT ~ Condition + + (1+Condition|Subject) + (1|Item) An example RT ~ Condition + + (1+Condition|Subject) + (1+Condition|Item) An example RT ~ Condition + TrialOrder + + (1+Condition|Subject) + (1+Condition|Item) An example RT ~ Condition + TrialOrder + + (1+Condition+ TrialOrder|Subject) + (1+Condition|Item) Model specification for random effects (1|subject) random intercept (0+fixedeffect|subject) random slope (1+fixedeffect|subject) … with correlation term Assumptions Absence of Collinearity No influential data points Normality of Errors Homoskedasticity of Errors Independence