General Linear Model General Linear Model Generalized Linear Model Generalized Linear Model Generalized Linear Mixed Model

General Linear Model General Linear Model Generalized 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),

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

response ~ intercept + slope * fixed effect + error The Linear Model structural part systematic part deterministic part probabilistic part stochastic part random part

response ~ intercept + slope * fixed effect + error structural part systematic part deterministic part probabilistic part stochastic part random part The Linear Mixed Effects Model

Important terminology - 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 Fixed effectRandom effect “Fixed-effects factors are those in which the populations to which we wish to generalize are precisely the levels represented in our analysis.” assumed to be constant across experiments Structural Part Stochastic Part

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”

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

Germa n French English Spanish Italian Swedish Norwegian Finnish Hungarian Turkish Romanian

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

Random intercepts versus Random slopes

RT (ms) Subjects Random intercepts

Random slopes

Experiment time RT (ms) Random intercepts

Experiment time RT (ms) Random intercepts and slopes

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)

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 Normality of Errors Homoskedasticity of Errors No influential data points Independence