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Genera l Linear Model Generalize d Linear Model Generali zed Linear Mixed Model

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Page 1: General Linear Model General Linear Model Generalized Linear Model Generalized Linear Model Generalized Linear Mixed Model

GeneralLinear Model

GeneralizedLinear Model

GeneralizedLinearMixed Model

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GeneralLinear Model

GeneralizedLinear Model

GeneralizedLinearMixed Model

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GLMM

LMM

LMEM

HLM

GeneralizedLinearMixed Model

MultilevelModel

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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.

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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)

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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

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response ~ intercept + slope * fixed effect + error

The Linear Model

structural partsystematic

partdeterministic

part

probabilistic part

stochastic partrandom part

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response ~ intercept + slope * fixed effect + error

structural partsystematic

partdeterministic

part

probabilistic part

stochastic part

random part

The Linear Mixed Effects Model

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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 constantacross experimentsStructural

PartStochastic

Part

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Crawley (2013: 681)

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Subjects as a fixed effect?

NO… why:

not repeatable not systematic often, not of interest small subset of population

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Repetitions as a fixed effect?

Yes… why:

repeatable systematic[ often, not of interest] “exhausts the population”

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Rep 1

Rep 2

Rep 3

Item #1

Subject

Common experimental data

Item...

Item...

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German

French

English

Spanish Italian

Swedish

NorwegianFinnish

Hungarian

Turkish

Romanian

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library(lme4)

lmer(y ~ x + (1|subject), mydata)

In R:

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Random intercepts

versus

Random slopes

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RT (m

s)

Subjects

Random intercepts

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Random slopes

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Experiment time

RT (m

s)

Randomintercepts

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Experiment time

RT (m

s)

Randominterceptsand slopes

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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)

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

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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)

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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

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An example

RT ~

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An example

RT ~ Condition+ (1|Subject)

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An example

RT ~ Condition ++ (1+Condition|Subject)

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An example

RT ~ Condition ++ (1+Condition|Subject)+ (1|Item)

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An example

RT ~ Condition ++ (1+Condition|Subject)+ (1+Condition|Item)

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An example

RT ~ Condition + TrialOrder ++ (1+Condition|Subject)+ (1+Condition|Item)

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An example

RT ~ Condition + TrialOrder ++ (1+Condition+

TrialOrder|Subject)+ (1+Condition|Item)

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Model specificationfor random effects

(1|subject)random intercept

(0+fixedeffect|subject) random slope

(1+fixedeffect|subject) … with correlation

term

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Assumptions

Absence ofCollinearity

Normality of Errors

Homoskedasticity of Errors

No influentialdata points

Independence

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